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Magnetorotational instability in protostellar discs
Raquel Salmeron

E R E· M E

N

E S· EAD

M

·MUT A

A thesis submitted for the degree of Doctor of Philosophy at the University of Sydney

August 2004

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Abstract
We investigate the linear growth and vertical structure of the magnetorotational instability (MRI) in weakly ionised, stratified accretion discs. The magnetic field is initially vertical and p erturbations have vertical wavevectors only. Solutions are obtained at representative radial lo cations from the central protostar for different choices of the initial magnetic field strength, sources of ionisation, disc structure and configuration of the conductivity tensor. The MRI is active over a wide range of magnetic field strengths and fluid conditions in low conductivity discs. For the minimum-mass solar nebula mo del, incorp orating cosmic ray and x-ray ionisation and assuming that charges are carried by ions and electrons only, p erturbations grow at 1 AU for B 8 G. For a significant subset of these strengths (200 mG B 5 G), the growth rate is of order the ideal MHD rate (0.75). Hall conductivity mo difies the structure and growth rate of global unstable mo des at 1 AU for all magnetic field strengths that supp ort MRI. As a result, at this radius, mo des obtained with a full conductivity tensor grow faster and are active over a more extended cross-section of the disc, than p erturbations in the ambip olar diffusion limit. For relatively strong fields (e.g. B 200 mG), ambip olar diffusion alters the envelop e shap es of the unstable mo des, which p eak at an intermediate height, instead of b eing mostly flat as mo des in the Hall limit are in this region of parameter space. Similarly, when cosmic rays are assumed to b e excluded from the disc by the winds emitted by the magnetically active protostar, unstable mo des grow at this radius for B 2 G. exist. For strong fields, p erturbations exhibit a kink at the height where x-ray ionisation b ecomes active. Finally, for R = 5 AU (10 AU), unstable mo des exist for B 800 mG (B 250 mG) and the maximum growth rate is close to the ideal-MHD rate for 20 mG B 500 mG (2 mG B 50 mG). Similarly, p erturbations incorp orating Hall conductivity have a higher wavenumb er and grow faster than solutions in the ambip olar diffusion limit for B 100 mG (B 10 mG). Unstable mo des grow even at the midplane for B 100 mG (B 1 mG), but for weaker fields, a small dead region exists. When a p opulation of 0.1µm grains is assumed to b e present, p erturbations grow at 10 AU for B 10 mG. We estimate that the figure for R = 1 AU would b e of order 400 mG. We conclude that, despite the low magnetic coupling, the magnetic field is dynamically imp ortant for a large range of fluid conditions and field strengths in protostellar discs. An example of such magnetic activity is the generation of MRI unstable mo des, which are supp orted at 1 AU for field strengths up to a few gauss. Hall diffusion largely determines the structure and growth rate of these p erturbations for all studied radii. At radii of order 1 AU, in particular, it is crucial to incorp orate the full conductivity tensor in the analysis of this instability, and more generally, in studies of the dynamics of astrophysical discs.


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To Andr´ es


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vii

Acknowledgements
I thank my sup ervisors, Mark Wardle and Anne Green for their constant supp ort and encouragement and for making my transition to astrophysics, from a career in aviation, most enjoyable. You have help ed me achieve a childho o d dream.

I am grateful to you, Mark, for teaching me so much and for b eing so nice to work with. Your enthusiasm and dedication have inspired me. To you, Anne, my gratitude for your care and genuine interest in my progress.

Finally, I sincerely thank my husband C´ esar and little son Andr´ for supp orting es me so much during my studies. Your love and care keep me going.


viii

Declaration of originality
This thesis describ es research carried out in the Scho ol of Physics at the University of Sydney. The content is my own work, except where acknowledged in the text. No material included here has b een presented for a degree at the University of Sydney or any other university.

Raquel Salmeron August 2004

Refereed publications
Part of the material in this thesis has b een published in the following articles: Salmeron, R. & Wardle, M. 2004. Magnetorotational instability in protoplanetary discs (submitted ). Salmeron, R. & Wardle, M. 2003. Magnetorotational instability in stratified, weakly ionised accretion discs. MNRAS 345: 992-1008. Salmeron, R. & Wardle, M. 2003. Magnetorotational instability in weakly ionised, stratified accretion discs. Magnetic fields and star formation: theory versus observations, eds. A. I. Gomez de Castro et al, in press (astro-ph/0307109).


Contents
1 Intro duction 1.1 1.2 1.3 1.4 The angular momentum transp ort problem . . . . . . . . . . . . . . . Magnetorotational instability: mechanism and prop erties . . . . . . . Metho dology of this study . . . . . . . . . . . . . . . . . . . . . . . . 1 3 5 8

Thesis outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 13

2 The instability in parameter space 2.1 2.2

Intro duction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.2.1 2.2.2 2.2.3 2.2.4 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . 16 The conductivity tensor . . . . . . . . . . . . . . . . . . . . 18 Comparison with the multifluid approach . . . . . . . . . . . . 20 Disc Mo del . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 Linearised Equations . . . . . . . . . . . . . . . . . . . . . . . 25 Equations in dimensionless form . . . . . . . . . . . . . . . . . 26 Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.3

Linearisation 2.3.1 2.3.2 2.3.3

2.4 2.5

Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 2.5.1 2.5.2 2.5.3 2.5.4 Test Mo dels . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 Comparison with lo cal analysis . . . . . . . . . . . . . . . . . 35 Structure of the Perturbations . . . . . . . . . . . . . . . . . . 36 The p erturbations in parameter space . . . . . . . . . . . . . . 47

2.6 2.7

Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 ix


x 3 The instability in protoplanetary discs

Contents 59

3.1 Intro duction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 3.2 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 3.2.1 3.2.2 3.2.3 3.3.1 3.3.2 3.3.3 3.4.1 3.4.2 3.4.3 3.4.4 3.5.1 3.5.2 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . 62 Disc Mo del . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 Ionisation balance . . . . . . . . . . . . . . . . . . . . . . . . . 68 Linearisation . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . 78 Test Mo dels . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 Ionisation Rates . . . . . . . . . . . . . . . . . . . . . . . . . . 81 Magnetic coupling and conductivity regimes as a function of height . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 Magnetic field strength . . . . . . . . . . . . . . . . . . . . . . 88 Structure of the p erturbations . . . . . . . . . . . . . . . . . . 89 Growth rate of the p erturbations . . . . . . . . . . . . . . . . 106 3.5 MAGNETOROTATIONAL INSTABILITY . . . . . . . . . . . . . . . 89

3.3 METHODOLOGY . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

3.4 DISC CONDUCTIVITY . . . . . . . . . . . . . . . . . . . . . . . . . 80

3.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 3.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 4 Impact of dust grains 121

4.1 Intro duction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 4.2 Disc mo del . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 4.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 4.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 5 Conclusions References 133 139


Chapter 1 Introduction
Magnetic fields play imp ortant roles in the dynamics of star formation and, in particular, in the evolution of astrophysical discs. They are thought to drive the acceleration and collimation of jets and outflows frequently observed in star forming systems (Blandford & Payne 1982, Wardle & Konigl 1993, Li 1996, Contop oulos & Saunty 2001, see also review by K¨ onigl & Pudritz 2000). They can also effectively regulate the evolution of the `accretion phase' of protostellar discs (e.g. Adams & Lin 1993), by providing mechanisms that remove angular momentum from the disc material, enabling most of it to b e accreted (e.g. Weintraub, Sandell & Duncan 1989; Adams, Emerson & Fuller 1990; Beckwith et al. 1990). In the weakly ionised environment of such discs, the ionisation fraction of the gas is not enough to pro duce go o d magnetic coupling over their entire vertical and radial extention. Consequently, the gas close to the midplane is likely to form a magnetically inactive `dead' zone (Gammie 1996, Wardle 1997), while the fluid near the surface is generally well coupled to the field, as a result of the ionisation contributed by cosmic rays and the x-rays emitted by the protostar. The ability of the magnetic field to effectively couple to the fluid and drive the magnetic pro cesses mentioned ab ove is, therefore, strongly dep endent on the conductivity of the gas and its spatial dep endency. Most studies of low conductivity astrophysical systems, including not only protostellar, but also quiescent ­and the outer regions of hot-state­ dwarf novae discs (Gammie & Menou 1998; Menou 2000; Stone et al. 2000), adopt either the ambip olar diffusion (`plasma drift') or resistive approximations. Ambip olar diffusion is a go o d approximation in relatively low density regions, where collisions are not very frequent and charged particles can follow the magnetic field and slip with it through 1


2

Chapter 1. Intro duction

the neutrals. This assumption has b een used extensively to mo del the dynamics of molecular clouds and star formation regions (e.g. Mestel & Spitzer 1956, Shu et al. 1993). Conversely, the resistive (`Ohmic') approximation is valid when the gas density is high enough for charged sp ecies to b e mainly tied to the neutrals, by collisions; rather than to the magnetic field, by magnetic stresses. This limit has b een used to mo del the evolution of the magnetic field in protostellar discs (e.g. Hayashi 1981). There is, however, an intermediate density range where the degree of magnetic coupling vary amongst charged sp ecies. In this regime, Hall currents are imp ortant. It has b een shown that this regime dominates over vast regions in weakly ionised discs (Wardle & Ng 1999; Sano & Stone 2002a, 2002b, 2003) and, therefore, should not b e ignored when mo delling magnetic pro cesses in these systems. Despite this, Hall conductivity has only recently b een included in such studies (Wardle 1999, Balbus & Terquem 2001, Sano & Stone 2002a,b; 2003, Salmeron & Wardle 2003 and Desch 2004). Results indicate that when Hall conductivity is taken into account, the extent of the inner magnetically dead zone is reduced and the magnetic field is dynamically imp ortant over a wider range of fluid conditions. In this thesis we explore the magnetic activity of low conductivity astrophysical discs, by mo delling one imp ortant example of such activity: the linear growth and vertical structure of the magnetorotational instability (MRI; Balbus & Hawley 1991, 1998; Hawley & Balbus 1991). This instability has b een identified as the most promising candidate to generate and sustain angular momentum transp ort radially outwards, so accretion can pro ceed. Our metho d includes the effects of the magnetic coupling, the conductivity regime of the fluid and the strength of the magnetic field, which is initially vertical. The conductivity is treated as a tensor and can b e a function of lo cation, so the effect of different conductivity regimes b eing dominant at different heights can b e examined. Although the adopted formulation restricts the unstable mo des that are mo delled, as well as the geometrical configuration of the assumed initial magnetic field (section 2.3); it can, nonetheless, illustrate the far reaching effects of a weak magnetic field in the dynamics of low conductivity discs. In the next sections we review in more detail the concepts asso ciated with the `angular momentum transp ort problem' in protostellar discs and how the MRI offers a simple, yet efficient, mechanism to solve this long-standing problem of accretion theories. We finalise this intro duction by summarising our metho dology and pre-


1.1. The angular momentum transp ort problem senting a brief outline of this work.

3

1.1

The angular momentum transport problem

The pro cess of star formation, triggered by the collapse of protostellar cores, leads to the development of a central ob ject ­a protostar­ which will eventually b ecome the new star. A sizable fraction of the collapsing mass forms a disc of material that surrounds this ob ject (e.g. Lo oney, Mundy & Welch 2003), and which has an The material in the disc is then slowly accreted towards the centre in the `accretion average mass of 0.04 times the mass of the central protostar (Natta et al. 2000).

phase' of the star formation pro cess. Observational evidence for the presence of discs around young stellar ob jects are comp elling. They comprise imaging in the near infrared and optical wavebands (e.g. see review by McCaughrean et. al. 2000) as well as interferometric studies that have resolved the velo city profile and structure in the inner regions of the discs, up to a few tens of AU from the centre (e.g. Wilner & Lay 2000). These `protostellar discs' are differentially rotating, with a typical angular momentum profile that increases with radius (dL/dR > 0). As a result, accretion can only pro ceed if angular momentum is carried away to larger radii by a small fraction of disc material, enabling most of the disc to b e accreted (e.g. Weintraub, Sandell & Duncan 1989; Adams, Emerson & Fuller 1990; Beckwith et al. 1990). Indeed, the efficiency and evolution of this phase of the star formation pro cess is regulated by the rate at which angular momentum can b e transferred outwards (e.g. Adams & Lin 1993). The mechanism(s) resp onsible for generating and sustaining this transp ort are not well understo o d. Different options have b een invoked by many authors over several decades, with various degrees of success. It is known, however, that the molecular viscosity of accretion discs is far to o low to explain observed accretion rates (Pringle 1981; see also Frank, King & Raine 2002). Viscous diffusion is able to propagate disturbances in timescales l 2 / , where l is the distance the disturbance travels as a result of the action of the (kinematic) viscosity . Taking l 10
10

cm

to explain the variability observed in some accreting systems. Consequently, some form of `turbulent' viscosity is thought to b e in action.

and = 105 cm2 s-1 , this timescale is 3 â 107 years (Balbus 2003), far to o long


4

Chapter 1. Intro duction Most current mo dels of accretion discs adopt, in some form, the -prescription

of Shakura & Sunyaev (1973). In this formulation, the radial-azimuthal comp onent of the stress tensor is assumed to scale with the gas pressure, w
r

= P . The

asso ciated turbulent viscosity is parameterised as t = cs H , where cs is the sound sp eed and H , the scaleheight of the disc. However useful to supp ort the mo delling of accretion discs, this metho dology do es not offer any explanation for the nature of the accretion torque, whose origin remains unsp ecified. In fact, using this formulation, all the unknowns and uncertainties of the accretion stress are lump ed into the single parameter . One of the most natural ways of generating turbulence is via hydro dynamical instabilities. Lab oratory shear flows, for example, readily break into turbulence at sufficiently high Reynolds numb ers. However, Keplerian discs satisfy the Rayleigh's hydro dynamical stability criterion (angular momentum increases with radius), so the generation and sustaining of hydro dynamic turbulence in these systems is, at b est, unproven (e.g. see review by Balbus & Hawley 1998). Convective turbulence has also b een considered as an option (Lin & Papaloizou 1980; Lin, Papaloizou & Kley 1993), but further studies app ear to indicate that this mechanism transp orts angular momentum towards the central ob ject (Cab ot & Pollack 1992, Ryu & Go o dman 1992, Cab ot 1996, Stone & Balbus 1996). More generally, the sign of the radial flux generated by convective turbulence may dep end on the ratio of the epicyclic frequency to wave frequency of the fluid (Balbus 2003): angular momentum is likely to b e transp orted inwards by long-p erio d, incompressible p erturbations; and outwards by high frequency, compressible disturbances. Still, other authors have p ointed out that hydro dynamic waves originated by the gravitational field of a companion star could p ossibly transp ort angular momentum (Vishniac & Diamond 1989, Rozyczka & Spruit 1993). This mechanism, however, requires an `external' source to excite, and maintain, such density waves; and can not explain accretion in stars that do not b elong to multiple systems. It is likely that outflows, commonly observed in star forming systems, transp ort angular momentum away from the central ob ject and thus, play also an imp ortant role in regulating accretion pro cesses (e.g. see review by K¨ onigl & Pudritz 2000). The apparent correlation b etween outflow and accretion signatures (e.g. Cabrit et. al. 1990; Cabrit & Andr´ 1991; Hartigan, Edwards & Ghandour 1995), lend e supp ort to this interpretation. This inflow-outflow mechanism is far from b eing


1.2. Magnetorotational instability: mechanism and prop erties

5

well understo o d, but it is thought that it may b e mediated by magnetic stresses (Blandford & Payne 1982, Wardle & Konigl 1993, Li 1996, Contop oulos & Saunty 2001, see also review by K¨ onigl & Pudritz 2000). This takes us to an imp ortant p oint: It is b elieved that protostellar discs are magnetised. Evidence of this comes from different sources. On the one hand, there is strong evidence for an enhanced magnetic activity in young stellar ob jects (e.g. see review by Glassgold, Feigelson & Montmerle 2000). As these authors p oint out, it is likely that the strong magnetic field near the stellar surface, extends out to the circumstellar disc. On the other hand, the remnant magnetisation of primitive meteorites suggest that magnetic fields were imp ortant in the solar nebula as well (Levy & Sonnett 1978). This is imp ortant for the present topic, as the presence of even a very weak magnetic field, changes dramatically the stability prop erties of differentially rotating systems (Chandrasekhar 1961). In fact, it is now well accepted that the turbulent viscosity required for accretion is most likely originated by hydromagnetic stresses (Balbus & Hawley 1991, Hawley & Balbus 1991). The study of this `MHD turbulence' in protostellar dics is further complicated by the low ionisation fraction of the disc, which makes essential to account for the departure from ideal-MHD fluid conditions. These topics are discussed in the next section.

1.2

Magnetorotational instability: mechanism and properties

The notion that magnetic stresses can play an imp ortant role in the generation of the `turbulent' viscosity required for accretion was first flagged by Lynden-Bell (1969). Indeed, the existence of a magnetohydro dynamical (MHD) turbulence in magnetised fluids had b een describ ed even earlier by Velikhov (1959) and Chandrasekhar (1960, 1961) through their analysis of the stability prop erties of Couette flows. These flows are stable when the angular velo city (not the angular momentum, as in the unmagnetised discs) increases with radius (Chandrasekhar 1961). In Keplerian flows, this condition is not satisfied. In fact, as Balbus & Hawley (1991) first p ointed out, in the presence of a magnetic field, differentially rotating discs can efficiently generate and sustain angular momentum transp ort away from the centre through the magnetorotational instability (MRI). This instability acts by converting the free energy


6

Chapter 1. Intro duction

source contributed by the differential rotation of the disc into turbulent p erturbations (e.g. Balbus 2003). This p ossibility has its origin in the additional degrees of freedom intro duced in the fluid by magnetic fields, which allow fluid elements to exchange angular momentum non-lo cally via the distortion of the magnetic field lines that connect them (Christo doulou, Contop oulos & Kazanas 1996). A numb er of numerical simulations conducted to date have confirmed that the generated MHD-turbulence transp orts angular momentum outwards (e.g. Brandenburg et. al. 1995; Hawley, Gammie & Balbus 1995, 1996; Matsumoto & Ta jima 1995). The mechanism by which the magnetorotational instability works is surprisingly simple: We b egin by imagining an initial, steady state disc configuration in which fluid elements are in orbital motion ab out the central ob ject while joined by weak magnetic field lines. For simplicity, these lines are assumed to b e vertical (fig. 1.1, top panel). In this circumstance, any small disturbance that displaces the elements to different (inner and outer) radii, will also generate magnetic tension, by stretching the magnetic field lines that connect them (see Fig. 1.1, b ottom panel). In this configuration, the element in the inner orbit is rotating more rapidly than the other, so the tension in the line transfers some of its angular momentum to the element in the outer orbit. As a result, the inner element drops even closer towards the star, as its new equilibrium p osition must lie in an orbit asso ciated with less angular momentum (i.e. at a smaller radius). Evidently, the opp osite is true for the element displaced away from the star: the tension in the line increases its angular momentum, so it moves outwards. As the elements move away from each other, the tension in the field line increases and the pro cess runs away. Perturbations, once initiated, will amplify. By this mechanism, some fluid elements lose angular momentum, and move inwards; while others gain it, and carry it away from the centre. Under ideal-MHD conditions, MRI unstable mo des exist in astrophysical discs that are differentially rotating, with the angular velo city profile decreasing outwards. Axisymmetric p erturbations grow when the magnetic field is `weak' (subthermal) and a p oloidal comp onent is present. These p erturbations have a characteristic length scale = vA /, where vA is the alfv´ sp eed and is the angular (Kepen lerian) frequency of the disc. Their maximum growth rate is
max

= q /2, where

q dln/dlnr = 1.5 for a Keplerian rotation profile (Balbus & Hawley 1992a). Imp ortantly, this growth rate is indep endent of the strength (or direction) of the


1.2. Magnetorotational instability: mechanism and prop erties

7

magnetic field, as long as the p oloidal comp onent exists. Moreover, when the magnetic field is purely toroidal, non-axisymetric mo des can still grow, although they are most unstable under the influence of a p oloidal field (Balbus & Hawley 1992b). It is exp ected that non-ideal MHD effects are imp ortant in protostellar discs, outside the innermost regions (R 0.1 AU), where thermal ionisation is not relevant (Hayashi 1981). The low ionisation fraction of the fluid, esp ecially close to the midplane, can p otentially affect the growth and structure of MRI unstable mo des. netic coupling (e.g. Balbus 2003), actual values may b e even smaller, particularly when chemistry o ccurring on grain surfaces is considered (Umebayashi & Nakano 1988). As a result, it is unlikely that the magnetic coupling of these discs is significant over their entire vertical and radial extents (Gammie 1996) and non-ideal MHD effects must b e taken into account. Several approximations have b een adopted in order to mo del the MRI in low conductivity accretion discs. In relatively low density regions, the ambip olar diffusion approximation is generally adequate. It is valid when the electron-ion drift is small in relation to the ion-neutral drift, so the magnetic field is frozen into the ionised comp onents of the fluid, with the charged sp ecies effectively b ehaving as a single fluid. The effect of ambip olar diffusion in the linear regime of the MRI has b een studied by Blaes & Balbus (1994), who found that discs are lo cally unstable when the ion-neutral collision frequency is shorter than the epicyclic frequency. The effectiveness of the MRI as an angular momentum transp ort mechanism must b e studied, however, in the non-linear regime. Such studies have b een conducted by Brandenburg et. al. (1995), MacLow et. al. (1995) and Hawley & Stone (1998). This last work followed a two-fluid (ion-neutral) evolution, in order to explore the effect of lowering the magnetic coupling in the prop erties of MRI mo des. Significant turbulence and angular momentum transp ort was found when the collision frequency is of the order of 100. For lower frequencies, the MRI prop erties app ear to b e determined primarily by the evolution of the ions alone. In relatively high density regions, all ionised sp ecies are more strongly tied to the neutrals, by collisions, than they are to the magnetic field by magnetic stresses. In these circumstances, the magnetic field lines can not b e assumed to b e frozen into any fluid comp onent and Ohmic diffusion dominates. Linear studies in this limit have b een conducted by Jin (1996), Balbus & Hawley (1998), Papaloizou & Terquem Although even a very small ionisation fraction ( 10
-13

) can generate sufficient mag-


8

Chapter 1. Intro duction

(1997), Sano & Miyama (1999) and Sano et. al. (2000). When recombination pro cesses on grain surfaces are taken into account, Ohmic dissipation app ears to central ob ject (Sano et. al. 2000). The extent of this region diminishes as grain size b e able to stabilise a minimum-mass solar nebula disc within 20 AU from the

increases and/or particles settle towards the midplane. The non-linear phase of the MRI in this regime has b een mo delled by Sano et. al. (1998, 2004), Fleming, Stone & Hawley (2000) and Stone & Fleming (2003). In this last study, the ionisation fraction is a function of height. The authors find that the MRI grows in the upp er regions of the disc, where the magnetic Reynolds numb er, Rem c2 / , exceeds a s critical value. Imp ortantly, significant mixing may o ccur b etween the inner `dead zone' and the active layers ab ove, so angular momentum transp ort may take place in the dead zone, via non-axisymmetric density waves driven by the active layers (Stone & Fleming 2003). Finally, there is an intermediate density range, in which some charged sp ecies (typically electrons) are tied to the magnetic field, while more massive particles (i.e. ions or grains) follow the neutrals. In these conditions, Hall currents are imp ortant. Hall diffusion has b een shown to b e relevant in the low ionisation environment typical of protostellar discs (Wardle & Ng 1999). MRI studies in this limit have b een conducted in the linear (Wardle 1999, Balbus & Terquem 2001 and Salmeron & Wardle 2003) and non-linear regimes (Sano & Stone 2002a, b; 2003). In the non-linear phase, the Hall effect was found to enhance the saturation level of the Maxwell stress, although the critical magnetic Reynolds numb er did not change by much (Sano & Stone 2002b).

1.3

Methodology of this study

In this thesis, we investigate the linear growth and vertical structure of MRI p erturbations in low-conductivity accretion discs. Different forms of diffusion of the magnetic field through the neutrals ­ambip olar, Hall or resistive­ are exp ected to dominate at different heights, as a result of the density stratification. Our formulation incorp orates all three limits by treating the conductivity as a tensor (Cowling 1957, Norman & Heyvaertz 1985, Nakano & Umebayashi 1986, Wardle & Ng 1999). In general, the comp onents of this tensor are the field-parallel conductivity ( ), the Hall conductivity (1 ) and the Pedersen conductivity (2 ). They are a function of


1.3. Metho dology of this study

9

Figure 1.1 The mechanism by which the magnetorotational instability transfer angular momentum outwards is conceptually simple. Initially, two fluid elements are in the same orbit, joined by a (vertical) magnetic field line (see top panel). The tension in the line is negligible. If the elements are displaced to different orbits (b ottom panel), the magnetic field line develops tension. This tension lowers the angular momentum of the element in the inner orbit, which is now rotating faster than the other. As a result, it drops even closer to the centre. The opp osite is true for the other element: The line tension increases its angular momentum and forces it to move outwards. As the elements separate, the tension in the line increases and the pro cess runs away.


10

Chapter 1. Intro duction

height, as a result of the z dep endence of the ionisation fraction. The present study is the first to incorp orate all three conductivity comp onents in a stratified disc. This is interesting b ecause the effect of different regimes b eing lo cally dominant at different heights can b e fully explored. The metho dology also includes the effects of the magnetic coupling and the strength of the magnetic field, which is initially vertical. Perturbations are restricted to have vertical wavevectors only (k = kz ). These are the fastest growing mo des, from an initially vertical magnetic field, in b oth the Hall and Ohmic conductivity regimes (Balbus & Hawley 1991, Sano & Miyama 1999). However, as has b een recently p ointed out by Kunz & Balbus (2004), this is not necessarily true in the ambip olar diffusion limit; where the most unstable mo des can have radial wavenumb ers as well. The appropriate governing equations of non-ideal MHD were written ab out a local Keplerian frame corotating with the disc at the angular frequency = GM /r 3 . These equations were linearised ab out an initial steady state where the fluid motion is exactly Keplerian. We note that , the comp onent of the conductivity tensor parallel to the field do es not app ear in the final, linearised system of ordinary differential equations (ODE), which signals that under the adopted approximations, the ambip olar diffusion and resistive limits b ehave identically. The obtained ODE system was integrated vertically from the midplane to the surface of the disc, using appropriate b oundary conditions formulated at b oth ends. This can b e treated as a two-p oint b oundary value problem for coupled ODE; which is solved by `sho oting' from the midplane to the surface of the disc, while adjusting the appropriate variables until the solution converges. The adjustment is done via a multidimensional, globally convergent Newton-Raphson metho d.

1.4

Thesis outline

This investigation is presented as follows: Firstly, in chapter 2 we conduct a parameter-space analysis of the prop erties of MRI p erturbations in low conductivity discs. For simplicity, we assume that the comp onents of the conductivity tensor are constant with height and examine how the structure and growth of the p erturbations are affected when different conductivity regimes dominate over the entire cross-section of the disc. The following


1.4. Thesis outline

11

configurations of the conductivity tensor are explored: the ambip olar diffusion (or resistive) limit, b oth Hall limits, and the cases where b oth effects are imp ortant. The effects of varying the strength of the field and the degree of coupling b etween ionised and neutral comp onents of the fluid are also studied. We are able to determine the regions of parameter space that supp ort MRI unstable mo des in weakly ionised discs, as well as the subset of these for which the Hall effect mo difies the structure and growth of unstable p erturbations. We find that for weak coupling, p erturbations obtained with the full conductivity tensor grow faster and act over a more extended cross-section of the disc than those obtained using the ambip olar diffusion approximation. Similarly, we explore the impact of the alignment of the magnetic field and the angular velo city vectors of the disc when Hall conductivity is imp ortant. Finally, in this chapter we derive an approximate criterion for when Hall diffusion drives the growth of the magnetorotational instability. This criterion is satisfied for a broad range of fluid conditions in protostellar discs. In a real disc, the conductivities vary with height as a result of the z -dep endency of the charged particle abundances and fluid density. In these conditions, different conductivity regimes are exp ected to dominate at different heights (Wardle 2003). Therefore, in chapter 3 we expand this study to include height-dep endent conductivity comp onents, calculated using a realistic ionisation profile. Dust grains are assumed to have settled towards the midplane of the disc, so electrons and ions are the sole charge carriers. We obtain solutions at representative radial lo cations from the central ob ject (R = 1, 5 and 10 AU), for different choices of the initial magnetic field strength and configurations of the conductivity tensor. As the ionisation fraction of the disc is exp ected to b e heavily dep endent on whether cosmic rays are able to p enetrate it, or are excluded by the winds emitted by the magnetically active protostar, we compare results obtained in b oth scenarios. In all cases, the ionisation rates contributed by x-rays and radioactive materials are included. Solutions are computed for the minimum-mass solar nebula mo del (Hayashi 1981), as well as for a more massive disc. Results indicate that the magnetic field is dynamically imp ortant for a large range of fluid conditions and field strengths in protostellar discs. One imp ortant instance of such activity is the generation of MRI unstable mo des, which exist at 1 AU (minimum solar nebula disc incorp orating cosmic ray ionisation) for B 8 G. Moreover, when 200 mG B 5 G, these mo des grow at ab out the ideal-MHD rate (0.75). Hall diffusion largely determines the prop erties


12 of the p erturbations for all studied radii.

Chapter 1. Intro duction

In the solutions presented in chapters 2 and 3, dust grains has not b een taken into account. This is a valid approximation in relatively late evolutionary stages of accretion, after dust grains have settled enough towards the midplane that they do not appreciably affect the dynamics of the gas at higher vertical lo cations. However, this settling is exp ected to b e affected by MHD turbulence, which could p otentially prevent dust particles from settling b elow a certain height (Dullemond & Dominik 2004 and references therein). On the other hand, dust grains lower the ionisation fraction of the gas, via recombination pro cesses that o ccur on their surfaces, so their presence is likely to mo dify the efficiency of MHD turbulence itself. As a result, it is exp ected that the prop erties of MRI unstable mo des will b e changed by the presence of dust particles. As an illustration of this effect, in chapter 4 we compare the solutions presented in the previous chapter for R = 10 AU (and no grains), with results obtained at the same radius and with the same magnetic field strengths, but assuming 0.1µm dust grains are present and fully mixed with the gas. We found that unstable mo des are supp orted in this case for B B 10 mG, down from 250 mG when grains were settled. Similarly, the central dead zone (which was 3. Assuming that the maximum field 400 mG.

practically nonexistent b efore, given the relatively high magnetic coupling at this radius) extends in this scenario to z /H strength that supp orts MRI p erturbations is the equipartition field at this height (Wardle in prep.), we estimate that unstable mo des exist at 1 AU for B The overall conclusions of this investigation are summarised in chapter 5. Broader implications for the study of astrophysical discs and directions for future work are discussed. Finally, we acknowledge that chapters 2 and 3 contain some rep etition, esp ecially in the intro duction and metho dology sections. This has b een unavoidable, as they are complete pap ers either published (chapter 2) or submitted for publication (chapter 3).


Chapter 2 The instability in parameter space
2.1 Introduction

The collapse of protostellar cores lead to the development of a central mass or protostar, surrounded by a disc of material, which is accreted towards the center. During this pro cess angular momentum is transferred to a small p ercentage of disc material at large radii, enabling the collapse of most of the disc towards the central star (e.g. Weintraub, Sandell & Duncan 1989; Adams, Emerson & Fuller 1990; Beckwith et al. 1990). The evolution of this `disc accretion' phase is dep endent up on the rate of angular momentum transp ort in the disc (e.g. Adams & Lin 1993). A variety of mechanisms have b een invoked to explain this transp ort. As the molecular viscosity of accretion discs is to o low to explain observed accretion rates (Pringle 1981), some form of turbulent viscosity must b e present. The origin and characteristics of this turbulence remains an imp ortant problem in star formation theories. Convective turbulence has b een considered as an option (Lin & Papaloizou 1980), but further studies suggest that this mechanism may transp ort angular momentum towards the central star instead of away from it (Cab ot & Pollack 1992; Ryu & Go o dman 1992; Stone & Balbus 1996). The gravitational field of a companion star may trigger hydro dynamic waves which can transp ort angular momentum (Vishniac & Diamond 1989; Rozyczka & Spruit 1993), but as a significant fraction of stars do not b elong to binary systems, this mechanism is not general enough to explain accretion pro cesses in all stars. Balbus and Hawley have p ointed out that the nature of this anomalous viscosity 13


14

Chapter 2. The instability in parameter space

can b e hydromagnetic (Balbus & Hawley 1991; Hawley & Balbus 1991; Stone et al. 1996). This `magnetorotational' instability (MRI) had b een describ ed initially by Velikov (1959) and Chandrasekhar (1961) through their analysis of magnetised Couette flows. It drives turbulent motions which transp ort angular momentum radially outwards, as fluid elements exchange angular momentum non-lo cally by means of the distortion of the magnetic field lines that connect them. Under ideal MHD conditions, MRI p erturbations grow in discs that are differentially rotating, with the angular velo city increasing outwards. Axisymmetric mo des need a magnetic field with a weak, p oloidal comp onent. In this context, `weak' means that the magnetic energy density of the field is less than the thermal energy density. Alfv´ sp eed and is the Keplerian angular frequency in the disc, and a maximum en These p erturbations have a characteristic length scale vA /, where vA is the

growth rate q /2, with q 1.5 for Keplerian discs (Balbus & Hawley 1992a). This growth rate do es not dep end on the strength or direction of the magnetic field most unstable under the influence of a p oloidal field, but also grow at a reduced rate if the field is purely toroidal (Balbus & Hawley 1992b). These p erturbations are of interest for the analysis of field amplification mechanisms, as dynamo amplification can not o ccur through axisymmetric p erturbations (Moffatt 1978). With no strong dissipation pro cesses, no other conditions are required. Because of its robustness and the general conditions under which it develops, the magnetorotational instability is a promising source of turbulent viscosity in accretion discs. Ideal MHD conditions are a go o d approximation to mo del astrophysical systems where the ionisation fraction of the gas is high enough to ensure neutral and ionised comp onents of the fluid are well coupled. Active dwarf novae (except p ossibly in the outer regions) and black hole accretion discs are examples of such systems. However, in dense, co ol environments such as those of protostellar discs, it is doubtful that magnetic coupling is significant over the entire radial and vertical dimensions of the discs (Gammie 1996; Wardle 1997). Similar conditions are thought to apply in quiescent and the outer regions of hot-state dwarf novae discs (Gammie & Menou 1998; Menou 2000; Stone et al. 2000). In these cases, low conductivity significantly affects the growth and structure of MRI p erturbations. Different approximations have b een adopted to account for the departure from ideal MHD in low conductivity astrophysical discs (see section 2.2.1). as long as a p oloidal comp onent is present. Non-axisymmetric p erturbations are


2.1. Intro duction

15

Most mo dels of the MRI in low conductivity discs have used the ambip olar diffusion (Blaes & Balbus 1994; MacLow et al. 1995; Hawley & Stone 1998) or resistive (Jin 1996; Balbus & Hawley 1998; Sano, Inutsuka & Miyama 1998) limits. Recently, it has b een recognised the imp ortance of the Hall conductivity terms in addition to resistivity for the analysis of low conductivity discs (Wardle 1999, hereafter W99, Balbus & Terquem 2001 and Sano & Stone 2002a,b; 2003). The huge variation of fluid variables over the vertical and radial extension of astrophysical discs is a further complication. Vertical stratification is particularly relevant, as these ob jects are generally thin and changes in the plane of the disc are much more gradual than those in the direction p erp endicular to it. Previous mo dels of the MRI have not included density stratification and Hall conductivity simultaneously. It is exp ected that solutions will b e strongly mo dified when b oth factors are present. This motivates the present study. This chapter examines the structure and linear growth of the magnetorotational instability in vertically stratified, non self-gravitating accretion discs. We assume the disc is isothermal and geometrically thin, so variations in the fluid variables in the radial direction can b e ignored. The initial magnetic field is vertical and the analysis is restricted to p erturbations with wavevector p erp endicular to the plane of the disc (k = kz ). These are the most unstable p erturbations with the adopted field geometry, as magnetic pressure strongly supresses displacements with kr = 0 (Balbus & Hawley 1991, Sano & Miyama 1999). The conductivity of the gas is treated as a tensor and assumed constant with height in this initial study, although the formulation is also valid for a z -dep endent conductivity. This makes the present metho d a p owerful to ol for the analysis of more realistic discs (see section 2.6 Discussion). Section 2.2 presents the governing equations for a weakly ionised, magnetised disc in near-Keplerian motion around the central star and details the adopted disc mo del. Section 2.3 summarises the linearisation of the equations and presents the final linear system in dimensionless form. It also describ es the three parameters that control the dynamics of the fluid. Section 2.4 discusses the b oundary conditions used to integrate the equations from the midplane to the surface of the disc and the integration metho d. Section 2.5 presents the test cases used to characterise the conductivity regimes relevant for this work and compares our results with a previous lo cal analysis. It also details key findings on the dep endency of the structure and


16

Chapter 2. The instability in parameter space

growth rate of the p erturbations with the conductivity regime, the strength of the magnetic field and its coupling with the neutral gas. These results are discussed in section 2.6. By way of example, this section also calculates the structure and growth rate of the MRI under fluid conditions where different conductivity regimes are dominant at different heights ab ove the midplane. This reflects (qualitatively) the conditions exp ected to b e found in real discs. Finally, the metho dology and key findings in this chapter are summarised in section 2.7.

2.2
2.2.1

Formulation
Governing Equations

The equations of non-ideal MHD are written ab out a lo cal Keplerian frame corotating with the disc at the angular frequency asso ciated with the particular radius of interest. Consequently, the velo city of the fluid can b e expressed as a departure dard lab oratory co ordinate system (r, , z ) anchored at the central mass M , and ^ vK = GM /r is the Keplerian velo city at the radius r . Similarly, if / t is the ratory frame can b e expressed as / t + / . We also assume that the fluid is weakly ionised, meaning that the abundances of charged sp ecies are so low that their inertia and thermal pressure, as well as the effect of ionisation and recombination pro cesses in the neutral gas, are negligible. These assumptions effectively restrict the range of frequencies that can b e studied with this formulation to b e smaller than the collision frequency of any of the charged sp ecies with the neutrals. Accordingly, separate equations of motion for the charged sp ecies are not required and their effect on the neutrals is contained in a conductivity tensor (see section 2.2.2). The governing equations are the continuity equation, + t the equation of motion, v + (v · t
2 2 ^ vr c )v - 2v^ + 2 vr - K ^ + s + r1 r

from exact Keplerian motion v = V - vK , where V is the velo city in the stan-

time derivative in the lo cal Keplerian frame, then the time derivative in the lab o-

· (v) = 0 ,

(2.1)

-

JâB = 0, c

(2.2)


2.2. Formulation and the induction equation, B = t mass, given by =- GM (r + z
2 1 2) 2

17

^ â(vâB) - c âE - 3 Br . 2

(2.3)

In the equation of motion (2.2), is the gravitational p otential due to the central

,

(2.4)

2 and vK /r is the centrip etal term generated by exact Keplerian motion. At the disc

midplane this term balances the radial comp onent of the gravitational p otential. ^ The terms 2v^ and 1 vr are the coriolis terms asso ciated with the use of a r 2 lo cal Keplerian frame, cs = meanings. MHD. E is the electric field in the frame comoving with the neutrals and the term 3 ^ Br accounts for the generation of toroidal field from the radial comp onent due 2 to the differential rotation of the disc. Additionally, the magnetic field must satisfy the constraint: ·B = 0, and the current density must satisfy Amp ere's law, J= and Ohm's law, J=·E . (2.7) c âB 4 (2.6) (2.5) In the induction equation (2.3), the term c âE contains the effects of non-ideal P / is the isothermal sound sp eed, = vK /r is the Keplerian frequency and c is the sp eed of light. Other symb ols have their usual

Note that the conductivity, which dep ends on the abundance and drifts of the charged sp ecies through the neutral gas is treated as a tensor , as detailed in the following section. This formulation is compared to the drift velo city approach in section 2.2.3.


18

Chapter 2. The instability in parameter space

2.2.2

The conductivity tensor

The electric conductivity is a tensor whenever the gyrofrequency of the charged carriers is larger than the frequency of momentum exchange by collisions with the bayashi 1986). On the contrary, when collisions with the neutrals are dominant the conductivity is a scalar, the ordinary ohmic resistivity. To obtain expressions for the comp onents of this tensor, we b egin by writing down the equations of motion of the ionised sp ecies. As inertia and thermal pressure are neglected, the motion of the charged particles is given by the balance of the Lorentz force and the drag force from collisions with the neutrals, Zj e E + vj â B - j mj vj = 0 , c (2.8) neutrals, or | | 1 (Cowling 1957, Norman & Hayvaerts 1985, Nakano & Ume-

where each charged sp ecies j is characterised by its numb er density nj , particle mass mj , charge Zj e and drift velo city vj . In the ab ove equation, j = < v >j , mj + m (2.9)

where m is the mean mass of the neutral particles and < v >j is the rate co efficient of momentum exchange by collisions with the neutrals. We will also make use of the Hall parameter, j = Zj eB 1 , mj c j (2.10)

given by the ratio of the gyrofrequency and the collision frequency of charged sp ecies j with the neutrals. It represents the relative imp ortance of the Lorentz and drag terms in equation (2.8). Following the treatment of Wardle & Ng (1999) and W99 we use the following expression for Ohm's law, ^ J = · E = E + 1 BâE + 2 E , ing J = e nj Zj vj together with the charge neutrality assumption (2.11)

obtained by inverting (2.8) to express vj as a function of E and B and then usj j

nj Zj = 0.

In equation (2.11), E and E are the comp onents of the electric field E , paral-


2.2. Formulation

19

lel and p erp endicular to the magnetic field, resp ectively. The comp onents of the conductivity tensor are the conductivity parallel to the magnetic field, = the Hall conductivity, 1 = and the Pedersen conductivity, 2 = ec B n j Z j j . 2 1 + j (2.14) ec B nj Z j , 2 1 + j (2.13) ec B n j Z j j ,
j

(2.12)

j

j

The relative values of the comp onents of the conductivity tensor differentiate three conductivity regimes: 1. The ambipolar diffusion regime o ccurs when
2

most charged sp ecies. This implies that most charged particles are strongly tied to the magnetic field by electromagnetic stresses. This regime is dominant at relatively low densities, where the magnetic field is frozen into the ionised comp onent of the fluid and drifts with it through the neutrals. The linear b ehaviour of the MRI in this regime has b een analysed by Blaes & Balbus (1994) and the non-linear growth by MacLow et al. (1995) and Hawley & Stone (1998). 2. The resistive (Ohmic) regime is obtained when most charged sp ecies are linked to the neutrals via collisions. This o ccurs when | |
2

|1 |, or | |

1 for

1. This regime is predominant closest to the midplane, where the

|1 |, implying

high density makes the collision frequency of the charged particles with the neutrals high enough to prevent the former from drifting. This case has b een

studied under a linear approximation by Jin (1996), Balbus & Hawley (1998), Papaloizou & Terquem (1997), Sano & Miyama (1999) and Sano et. al. (2000). The non-linear regime has b een studied by Sano et al. (1998); Fleming, Stone & Hawley (2000) and Stone & Fleming, (2003). This last work includes a z -dep endent resistivity.


20

Chapter 2. The instability in parameter space 3. Finally, the Hal l regime o ccurs when charged particles of one sign are tied to the magnetic field while those of the other sign follow the neutrals. In this case |1 | 2 < and | | 1. It is imp ortant at intermediate densities, b etween the ones asso ciated with the ambip olar and Ohmic difussion regimes. Recent Terquem 2001) and non-linear regimes (Sano & Stone 2002a,b; 2003).

studies have explored the MRI with Hall effects in the linear (W99, Balbus &

2.2.3

Comparison with the multifluid approach

Another commonly used form of the induction equation is obtained by assuming that ions and electrons are the main charge carriers and drift through the neutrals (e.g Balbus & Terquem 2001, Sano & Stone 2002). Using this approach, the induction equation is B = t =

â (ve âB) â vâB - 4 J JâB (JâB)âB - + c ene ci i (2.15)

where ve is the electron drift sp eed, = c2 /4 is the resistivity and subscripts e and i refer to electrons and ions, resp ectively. The four terms in the right hand side of equation (2.15) are, from left to right, the inductive, resistive, Hall and ambip olar diffusion terms. We now express equation (2.3) in terms of J, B and the comp onents of the conductivity tensor in order to show that, in the appropriate limit, it corresp onds to equation (2.15), as exp ected. We b egin by inverting equation (2.11) to find an expression for E , E= where = 1 JâB J +2 - B 2 1 - 2 (JâB)âB B2 (2.16)

2 2 1 + 2 is the total conductivity p erp endicular to the magnetic field.

Assuming that the only charged sp ecies are ions and electrons with Hall parameters i and e (< 0), resp ectively, and that charge neutrality is satisfied (ni = ne ), we obtain the following expressions for the comp onents of the conductivity tensor: = cene (i - e ) , B (2.17)


2.2. Formulation

21

1 =

cene (i + e )(e - i ) , and 2 B (1 + e )(1 + i2 ) cene (1 - i e )(i - e ) . 2 B (1 + e )(1 + i2 )

(2.18)

2 =

(2.19)

From (2.18) and (2.19), we find: = cene (i - e ) 2 B [(1 + e )(1 + i2 )] . (2.20)

1/2

Substituting these expressions into (2.16) gives E= e (JâB)âB e J e + i JâB - + i - e e - i cene e - i c2 i i (2.21)

where is the electrical conductivity due to electrons. Finally, simplifying (2.21) by using |e | i (as is the case) and substituting the resulting E in equation (2.3) (without the coriolis term), yields the standard result, shown in (2.15). Each of the last three terms in the right hand side of equation (2.15) dominate when the fluid is in a particular conductivity regime (section 2.2.2). These limits can also b e recovered with the appropriate assumptions, through equations (2.16) and (2.21). To get the resistive regime, for example, we substitute 1 = 0 and = 2 into equation (2.16). In this limit the conductivity is a scalar (the resistive approximation), so E = J/ and the induction equation reduces to the familiar form, B = t â vâB - c2 âB . 4 (2.22)

On the other hand, to mo del the Hall limit, we regard the charged sp ecies to b e either `ions', which are strongly tied to the neutrals through collisions ( 1) (see also discussion in W99). In this limit the Hall conductivity is |1 | = while the Pedersen conductivity is 2 = |1 | i - 1 e 1 . (2.24) cen B
e i

1), or

`electrons', for which the only imp ortant forces are electromagnetic stresses (|e |

(2.23)


22

Chapter 2. The instability in parameter space As the ions are effectively lo cked with the neutrals, the current density will b e

given by the drift of the electrons through the neutral gas. Collisions are unimp ortant in their equation of motion, so they drift p erp endicular to the plane of the electric and magnetic fields, in order to annul the Lorentz force acting up on them. In this limit [(JâB)âB]/B 2 = -J and E= JâB , cene (2.25)

consistent with the Hall term in equation (2.15). Finally, the ambip olar diffusion approximation is recovered by assuming 1 = 0 and |e |
i

2 ,

1. In this limit, the Pedersen conductivity is given by 2 = ce ni , and B i (JâB)âB , c2 i i (2.26)

E =-

(2.27)

which originates the ambip olar diffusion term of (2.15). Although the multifluid drift and conductivity tensor formulations are ultimately equivalent, which one is more convenient dep ends on the problem at hand. In particular, the presence of dust grains tends to make the treatment of different sp ecies esp ecially complex. In protostellar discs, dust grains can b e the more abundant charged sp ecies over extended regions. For example, assuming 0.1 µm grains, negatively charged grains dominate whenever n grains are the most abundant ions for n
H H 14

10 cm

11

cm-3 , while p ositive charged (Wardle & Ng 1999). Having

10

-3

separate equations of motion for different charged sp ecies would generally involve dividing the grain size distribution of interest into an appropriate numb er of discrete intervals and explicitly treating each one. Unless the numb er of such intervals is small it is easy to see that this metho d can b ecome very cumb ersome. In these circumstances, incorp orating the contribution of each charged sp ecies into a conductivity tensor can b e a valuable approach.

2.2.4

Disc Mo del

Our mo del incorp orates the vertical structure of the disc, but neglects fluid variations in the radial direction. This is appropriate as astrophysical accretion discs are


2.2. Formulation

23

generally thin and changes in the radial direction o ccur in a much bigger length scale than those in the vertical direction. Including the vertical structure means that p erturbations of spatial dimensions comparable to the scale height of the disc, which are asso ciated with a strong magnetic field (v can b e explored.
A

cs ), or low conductivity,

The balance b etween the vertical comp onent of the central gravitational force and the pressure gradient within the disc determines its equilibrium structure. The vertical density distribution of a (vertically isothermal) disc in hydrostatic equilibrium is given by (r, z ) z2 = exp - 2 o (r ) 2H (r ) . (2.28)

In the ab ove equation, o (r ) is the gas density at the midplane and H (r ) = cs / is the scaleheight of the disc. A self-consistent treatment of this problem, would involve adopting a particular dep endency of o and H with r using a suitable mo del, such as the minimum solar nebula (Hayashi, Nakazawa & Nakagawa 1985) and calculating (r, z ) by means of (2.28). This density, together with the adopted strength of the magnetic field B and the values of the conductivity tensor as a function of height would b e used to evaluate the parameters that govern the fluid evolution (see section 2.3.3) and solve the fluid equations. The realistic evaluation of the conductivity tensor is a complex undertaking, as it dep ends critically on the abundances of charged sp ecies (ions, electrons and charged dust grains) which, in turn, are a function of the ionisation balance in the disc. This balance is given by the equilibrium b etween ionisation pro cesses by cosmic rays, radioactive elements and X-rays from the central star (e.g. Hayashi 1981; Glassgold, Na jita & Igea 1997, Igea & Glassgold 1999; Fromang, Terquem & Balbus 2002) and recombinations taking place b oth in the gas phase and in grain surfaces (e.g. Nishi, Nakano & Umebayashi 1991). In low conductivity discs, the level of ionisation is insufficient to pro duce go o d coupling b etween the magnetic field and the neutral comp onent of the fluid over their entire vertical structure. In protostellar star, the coupling will b e significant only in the surface layers, where X-rays and cosmic rays can p enetrate and ionise the gas (Gammie 1996; Wardle 1997). In discs, for example, it is exp ected that in the region outside 0.1 AU from the central


24

Chapter 2. The instability in parameter space

these environments, the z -dep endent attenuation of the ionisation rate typical of interstellar space 10
-17

H

-1

s

-1

has to b e taken into account.

On the other hand, the contribution of dust grains to recombination pro cesses is particularly complex, b ecause they generally have a distribution of sizes and corresp onding collision cross sections (Mathis, Rumpl & Nordsieck 1977; Umebayashi & Nakano 1990). Moreover, the dynamics of grain particles dep ends on the activity of the disc. In quiescent environments, they tend to settle towards the midplane and b egin to agglomerate into bigger structures that could eventually b ecome planets (e.g. Weidenschilling & Cuzzi 1993). This removes grains from regions at relatively high z and causes the ionisation level to increase by reducing recombination pro cesses taking place in their surfaces. Simulations of the evolution of dust grain distributions in dense cores show that they grow icy mantles and coagulate efficiently (Ossenkopf 1993). By this pro cess, the smallest particles grow quickly while the upp er grain size limit is only slightly changed (Ossenkopf 1993). As a result, the surface area of dust grains can b e significantly mo dified by grain evolution, which ultimately affects the ionisation balance in the disc. In the present work, a simpler treatment has b een adopted. In order to study the MRI under different conductivity regimes, the values of the comp onents of the conductivity tensor have b een selected so that they satisfy the conditions outlined in section 2.2.2 for each regime. For simplicity, these values are assumed to b e constant, although the formulation allows them to b e a function of height. We found that the parameters that control the evolution of the fluid are the ratio of the comp onents of the conductivity tensor p erp endicular to the magnetic field (1 /2 ), the strength of the magnetic field and its degree of coupling with the neutral comp onent of the fluid (see section 2.3.3). The midplane values of these parameters have b een selected in order to simulate the fluid conditions we were interested in mo delling. This approach will b e discussed in section 2.5.1 where the chosen test cases are detailed. With this approach we are able to study the dep endency of the growth rate and structure (characterised by the height of maximum amplitude and the wavenumb er of unstable mo des) of the instability with the parameters of the fluid in a stratified disc. This is relevant, as the region where linear p erturbations p eak is also exp ected to b e the region where non-linear p erturbations grow fastest, until turbulence finally sets in and causes all wavenumb ers to interact so that, eventually, the longest wavelengths carry the greatest angular momentum transp ort (e.g. Hawley & Balbus


2.3. Linearisation 1995).

25

2.3

Linearisation

We linearised the system of equations (2.1) - (2.3), (2.6) and (2.11) ab out an initial steady state where J = v = E = 0 and B = B z . In the initial state b oth E and ^ J vanish, so the changes in the conductivity tensor due to the p erturbations do not app ear in the linearised equations. As a result, it is not necessary to explore how the p erturbations affect the conductivity and only the values in the initial steady state are required.

2.3.1

Linearised Equations

We assume the wavevector of the p erturbations is p erp endicular to the plane of the disc (k = kz ). Perturbations with a vertical wavenumb er, initiated from a vertically aligned equilibrium magnetic field, exhibit the fastest growth rate for a given set of parameters b ecause magnetic pressure strongly suppresses displacements with kr = 0 (Balbus & Hawley 1991; Sano & Miyama 1999). Taking p erturbations of the form q = q0 + q(z )e
i t

ab out the initial state, linearising and neglecting terms

of order H/r or smaller, we find that the equations decouple into two subsystems. One of them corresp onds to sound waves propagating in the vertical direction and the other describ es MHD p erturbations in the plane of the disc, with vanishing z comp onent. With these simplifying assumptions, the final linear system of equations that describ es the MHD p erturbations within the disc is, i vr - 2 v - i v + 1 vr + 2 B0 J = 0 , c B0 Jr = 0 , c (2.29)

(2.30)

i Br - c i B + c

E = 0, z

(2.31)

Er 3 + 2 Br = 0 , z

(2.32)


26

Chapter 2. The instability in parameter space

Jr = - J =

c B , 4 z

(2.33)

c Br , 4 z

(2.34)

Jr = 2 Er - 1 E , J = 1 Er + 2 E ,

(2.35)

(2.36)

where E and Er are the p erturbations of the electric field in the lab oratory frame, given by E = E + B0 vr , and c (2.37)

B0 v . (2.38) c We note that , the comp onent of the conductivity tensor parallel to the mag Er = Er - netic field, do es not app ear in the final linearised equations. Because the ambip olar diffusion and resistive conductivity regimes are differentiated by the value of (section 2.2.2), this means that during the linear stage of the MRI, under the adopted approximations, these regimes are identical.

2.3.2

Equations in dimensionless form

Equations (2.29) to (2.38) can b e expressed in dimensionless form, normalising the variables as follows: z = z H v cs = (r, z ) 0 (r ) c E cs B 0 B = B B0 c E cs B 0 0

v =

E =

E =

J =

c J cs B 0

=

0

=
0


2.3. Linearisation

27

Subscript `o' is used to denote variables at the midplane of the disc. Effecting these changes and dropping the asterisks to keep the notation simple, we finally express the dimensionless system of equations in matrix form as: B
r

0 -C1 A B 0 d = dz 3 Er - 2 - 0 E 0 0 J = C2 v = A A
2





0

0

C1 A

1

C1 A

2

2

C1 A3 0 0 E
1 2



B

r

B Er E

(2.39)

-A A -2
2

3

(2.40)

1

o 1 1 + 2
2





- -
1

J

(2.41)

where

1 E = 2


2

-

1

-

J

(2.42)

=

i , vA cs
-2

(2.43)

C1 = o o 1 C2 = 1 + 1+ A1 =



C2 ,
-1

(2.44)

2

5 1 2 o + 2 + 2



,

(2.45)

1 o 1 +2 , 1 + 2

(2.46)


28

Chapter 2. The instability in parameter space

A2 =

2 o 1 , and + 1 + 2 1 1 o 1 . + 2 1 + 2

(2.47)

A3 = In the ab ove expressions,

(2.48)

Bo 4 o is the Alfv´ sp eed at the midplane of the disc, and en vA = o =

(2.49)

2 1 Bo o co = (2.50) o c2 is a parameter that characterises the midplane coupling b etween the magnetic field

and the disc (see section 2.3.3). To understand the information contained in this parameter it is useful to recall that the effect of finite conductivity is different for p erturbations of different wavelengths. Finite conductivity is imp ortant when the term c âE in the induction equation (2.3) is of order when k v
A

scale L 1/k vA / and , it is found that these two terms are comparable wavemo des at, or ab ove, the critical frequency, c =

â(vâB). Adopting a length

(c2 /4 )k 2 / , in other words, non-ideal effects will strongly mo dify

collision frequency of the neutrals with any of the charged sp ecies. Generally, c is smaller than this value and much smaller than j , the collision frequency of charged sp ecies j with the neutrals. In dense clouds, c G G , which is the smallest j j , the collision frequency of neutrals with grains. As G 0.01, the treatment of this study, restricted to < j by neglecting the inertia of the charged sp ecies, remains valid for 100c. For p erturbations with lower frequencies (longer wavelength)

B 2 . c2 It can b e shown (W99) that in the limit |j | , c reduces to

(2.51)
j

j j , the

than c , ideal MHD (the flux-freezing approximation) is valid.

2.3.3
fluid:

Parameters

As these equations reveal, three imp ortant parameters control the evolution of the


2.3. Linearisation

29

1. vA /cs , the ratio of the Alfv´ sp eed to the isothermal sound sp eed of the gas en at the midplane. It is a measure of the strength of the magnetic field. In ideal MHD unstable mo des grow when the magnetic field is subthermal (vA /cs < 1). scaleheight of the disc and the growth rate decreases rapidly. When vA cs the minimum wavelength of the instability is of the order of the

2. o , a parameter that characterises the strength of the coupling b etween the magnetic field and the disc at the midplane (see equation 2.50). It is given by the ratio of the critical frequency ab ove which flux-freezing conditions break down and the dynamical frequency of the disc at the midplane. If o = co / < 1 the disc is p o orly coupled to the disc at the frequencies of interest for dynamical analysis. As the growth rate of the most unstable mo des are of order in ideal MHD conditions, these are also the interesting frequencies for the study of this instability. 3. 1 /2 , the ratio of the conductivity terms p erp endicular to the magnetic field. It is an indication of the conductivity regime of the fluid, as discussed in section 2.2.2. Note that the density of the disc decreases with z , so the lo cal values of and vA /cs increase with height. The parameters of the mo del are defined as the corresp onding values at the midplane. It is common practice to characterise the magnetic coupling of a weakly ionised fluid by its electron density ne . Before finishing this section, we discuss how relates to this fluid parameter. We b egin by writing the magnetic Reynolds numb er as (e.g. Balbus & Terquem 2001), B2 vA H ReM = = c2 then = In the resistive regime, | | ReM vA 2 )1/2 (1 + 2 ) cs (1 + e i . (2.53) 4 cs = , B vA /cs (2.52)

where we have used = c2 /4 . If ions and electrons are the only charged sp ecies,

1/2

1 for b oth charged sp ecies and (2.53) shows that

the criterion for non-Hall MRI p erturbations to grow (W99), > vA /cs , is equivalent to ReM > 1.


30

Chapter 2. The instability in parameter space We can now obtain an expression for in terms of the electron fraction xe =

ne /nH at the midplane, ene B i - e 1 B 2 = 2 2 )1/2 (1 + 2 )1/2 c c (1 + e i 2 ne < v > m e e , 2 (me + mn ) (1 + e )1/2
i

=

(2.54)

where we used expression (2.20) for and assumed < v > 1 â 10
-15

|e |. In the ab ove equation cm
-2

128k T 9 me

1/2

(2.55)

is the momentum-transfer rate co efficient for electron-neutral scattering. Note that the dep endence of on magnetic field strength now enters only through the electron Hall parameter e . Following Fromang, Terquem & Balbus (2003) we assume that grains have settled out and the electron (and ion) numb er density is determined by the recombination of metal ions given by, ne where 3 â 10 g cm
-2 -1 -11

nH

1/2

,

(2.56)

T

-1/2

cm3 s

-1

is the radiative recombination rate for metal ions.
-17

The ionisation rate is assumed to b e due to cosmic rays at a rate 10 s H
-1

where is the disc surface density. This dominates x-ray ionisation

exp(-/96)

for the column densities we shall consider here. Results are shown in Table 3.1 for a nominal 1 solar mass star and B = 10 mG. (up to ab out 200 mG) at this radius. On the other hand, e is greater than 1 at 5 and 10 AU, so will scale linearly with B (see equation 2.54). In particular for Wardle (2003). B = 100 mG at 1 AU, 0.00088, consistent with the detailed calculations in For this strength of the magnetic field, |e | 1 at 1 AU and scales as B
2

2.4

Boundary Conditions

To solve equations (2.39) to (2.42) it is necessary first to integrate the system of ordinary differential equations (ODE) in (2.39). This problem can b e treated as


2.4. Boundary Conditions
Comparison of the magnetic coupling parameter o and the ionisation fraction xe at the midplane for different radial

Table 2.1

p ositions ro , with B = 10 mG and assuming grains have settled out (Fromang, Terquem & Balbus 2002). Also shown are the assumed temp erature To and calculated values of nH , vA /cs , , e and i .

ro (AU) 1 5 10

To (K)

nH (cm-3 )
14 12 12

vA /c

s

(s-1 H-1 )
-22 -18 -18

280 6 â 10 130 7 â 10 90 1 â 10

0.00076 5.76 â 10 0.010 0.033

|e |



i -5 -3

x

e -13 -10 -9



o -6

4.81 â 10 9.76 â 10

0.035 7.68 â 10 4.44 37.31 0.046

6.58 â 10

7.32 â 10 1.76 â 10 5.11 â 10

8.9 â 10 1.9 19

31


32

Chapter 2. The instability in parameter space

a two-p oint b oundary value problem for coupled ODE. Five b oundary conditions must b e formulated, prescrib ed either at the midplane or at the surface of the disc. At the midplane: A set of b oundary conditions can b e arrived at by asuming fluid variables have either `o dd' or `even' symmetry ab out the midplane. `Odd' symmetry means the variable is an o dd function of z and vanishes at z = 0. Conversely, when `even' symmetry is applied, the variable is assumed to b e an even function of z and its gradient is zero at the midplane. In this study we applied the o dd - even symmetry criteria to the p erturbations in sp onds to even (o dd) symmetry conditions. This contrasts with Lovelace, the magnetic field, B(z ) = ± B(-z ), where the upp er (lower) sign correWang & Sulkanen (1987), who applied the symmetry criteria to the flux function (r, z ) = r A with A the toroidal comp onent of the vector p otential and obtained Br, (r, z ) = try of a particular fluid variable is assigned arbitrarily, sub ject to the constraint Br, (r, -z ) as their symmetry criteria. The symme-

that fluid equations are satisfied. This means that two sets of b oundary conditions are equally valid, obtained by reversing the assumed symmetry of the fluid variables. Perturbations obtained with a particular set of b oundary conditions are displaced a quarter of a wavelength from those found with the other one. The growth rates of these solutions lie at intermediate p oints of the curve vs k obtained from the lo cal analysis (W99), as exp ected. Evidently, no generality is lost by fo cusing in one of these two p ossible sets of solutions. We compare the growth rate versus numb er of no des of p erturbations obtained with `o dd' and `even' symmetry in section 2.5.2 (Comparison with lo cal analysis). For the rest of the analysis presented in this study we chose to assign o dd symmetry to Br and B , so they vanish at z = 0. This gives us two b oundary conditions at the midplane. As the equations are linear, their overall scaling is arbitrary, so a third b oundary condition can b e obtained by setting one of the fluid variables to any convenient value. To that effect, we assigned a value of 1 to Er . Summarising, three b oundary conditions are applied at the midplane:

Br = B = 0, and


2.4. Boundary Conditions

33

Er = 1 . At the surface: At sufficiently high z ab ove the midplane, ideal MHD conditions hold. This assumption is appropriate in this case b ecause the lo cal coupling parameter is inversely prop ortional to the density and so it is stronger at higher z regions where the density is smaller. When > 10 the growth rate and characteristic wavenumb er of unstable mo des differ little from the ideal limit (W99), so even though for simplicity we have assumed the conductivity tensor to b e spatially constant, we can assume that flux-freezing conditions hold at the surface and the lo cal disp ersion relation is k v Hawley 1991). As the Alfv´ sp eed increases with en
1/2 A

= (Balbus &

, the wavelengths of

magnetic field p erturbations increase with z and given the dep endency of in the plane of the disc of an infinitely stretched p erturbation should effectively vanish, so Br and B should b e zero at infinity. This gives us the remaining two b oundary conditions required to integrate the system of equations (2.39). Consistently with them, b oth E and J vanish as well. Interestingly, this solution is consistent with E and v b eing non-zero at infinity. The only requirement is that the gradient of the velo city in the vertical of the magnetic field. It may seem puzzling at first that v is non-vanishing at infinity. This can b e understo o d taking into account that these p erturbations travel to infinity in a finite time t


with z (see equation 2.28), must tend to infinity as z . The displacements

direction v / z b e zero when z , to prevent any horizontal stretching

given by,
0

t



=
0

dz H = vAl vA

exp -

z2 4

dz =

H , vA
Al

(2.57) is the

where z is the vertical co ordinate in units of the scaleheight H and v

lo cal value of the Alfv´ sp eed. Because of this finite travel time to infinity, the en fluid can retain a finite velo city when z . Furthermore, through equations seen from the lab oratory frame, are finite at infinity as well. These b oundary conditions are strictly valid at infinity, but will also hold at a b oundary lo cated sufficiently high ab ove the midplane. We chose to lo cate (2.37) and (2.38) it is clear that E, the p erturbations in the electric field as


34

Chapter 2. The instability in parameter space

Table 2.2

Comparison of maximum growth rate

max

and numb er of no des N of

the fastest growing mo des for all conductivity regimes and two different lo cations of the b oundary. In all cases vA /cs = 0.1 and o = 10.

z /H = 5 Conductivity Regime Ambip olar Diffusion Hall limit (1 Bz > 0) Hall limit (1 Bz < 0) Comparable conductivities Opp osite conductivities
max

z /H = 7 N 5 5 5 5 5
max

N 5 5 5 5 5

0.7303865 0.7498761 0.7461857 0.7345455 0.7340035

0.7303869 0.7498761 0.7461854 0.7345459 0.7340039

the b oundary at z /H = 5 after confirming that increasing this height do es not significantly affect either the structure or the growth rate of unstable mo des. This can b e appreciated in Table 2.2, which compares the maximum growth rate
max

and numb er of no des N (proxy for wavenumb er) of the

p erturbations in all conductivity regimes for two different lo cations of the b oundary. Summarising, the b oundary conditions adopted at the surface are:

Br = B = 0 , at z /H = 5. This system of equations is solved as a two-p oint b oundary value problem for coupled ODE by `sho oting' from the midplane to the surface of the disc and simultaneously adjusting the growth rate and E until the solution converges.

2.5
2.5.1

Results
Test Mo dels

We solved the system of equations (2.39) for different conductivity regimes, coupling b etween fluid comp onents and initial magnetic field strengths. As discussed in section 2.3.1, under the linear approximation and disc mo del adopted in this thesis, the ambip olar diffusion and resistive conductivity regimes are identical, so even though throughout this work we have lab elled the case when 1 = 0 as the `ambip olar


2.5. Results

35

diffusion' limit, it should b e kept in mind that this condition describ es the resistive regime as well. Two different Hall limits exist, as the growth rate of the MRI dep ends on the orientation of the initial magnetic field with resp ect to the disc angular velo city vector (W99). The case when Bo is parallel (antiparallel) to is characterised by 1 Bz > 0 (1 Bz < 0). We calculated the growth rate and vertical structure of all unstable p erturbations for different conductivity regimes with vA /cs = 0.1. The degree of coupling b etween the magnetic field and the neutral comp onent of the fluid was characterised by either o = 10 (go o d coupling) or o = 2 or 1 (p o or coupling). The choice of o for the low coupling analysis is dep endent on the conductivity regime of the fluid. We to ok o = 1 for all regimes, except the Hall (1 Bz < 0) limit, where o = 2 was adopted as our co de fails to converge for < 2. In this regime, the lo cal analysis shows that when 0.5 < < 2 all wavenumb ers grow (W99). We b elieve that this complex structure of the p erturbations at ever increasing k prevents our co de from converging when o < 2. We also examined the dep endency of the structure of the fastest growing mo des, their growth rate and the height of maximum amplitude, with the coupling o and the strength of the magnetic field for all conductivity regimes. To study the effect of the magnetic coupling, the value of vA /cs was fixed at 0.1. The impact of the strength of the field was explored for go o d and p o or coupling conditions. Finally, we studied the dep endency of the growth rate of the most unstable p erturbations with the coupling o for vA /cs = 0.1 and 0.01 and with the magnetic field strength for o = 10, 2 and 0.1. The relative values of 1 and 2 used to characterise each conductivity regime, together with the values of vA /cs and o used to explore the structure of the p erturbations are summarised in Table 2.3.

2.5.2

Comparison with lo cal analysis

The linear growth of the MRI as a function of wavenumb er in a lo cal analysis shows that vs k generally takes the form of inverted quadratics (W99). The lo cal wavenumb er of the p erturbations change with z , so we use the numb er of no des of Br over the entire thickness of the disc, from z = -5 to z = +5, as a proxy for the wavenumb er k to compare our results with those of W99. Results are shown in


36

Chapter 2. The instability in parameter space

Table 2.3

Relative values of the comp onents of the conductivity tensor p erp endicuo

lar to the magnetic field 1 and 2 and fiducial values of the coupling parameters p erturbations for all conductivity regimes.

(go o d coupling, p o or coupling limits) and v A /cs adopted to explore the structure of the

Conductivity Regime Ambip olar Diffusion Hall Limit 1 Bz > 0 Hall Limit 1 Bz < 0 Comparable conductivities Opp osite conductivities

1 = 0



o

vA /c

s

10, 1 0.1

2 = 0, 1 > 0 10, 1 0.1 2 = 0, 1 < 0 10, 2 0.1 1 =
2 2

10, 1 0.1 10, 1 0.1

1 = -

Fig. 2.1 for the ambip olar diffusion and Hall (1 Bz > 0) limits for go o d and p o or coupling. In b oth regimes the reduction of the wavenumb er of the fastest growing p erturbation with o is obtained, as exp ected from the lo cal analysis. Reducing but remains unchanged for the Hall regime, as exp ected from the lo cal results. These results confirm our exp ectation that applying b oundary conditions and integrating the fluid equations in the vertical direction would restrict the unstable frequencies from the continuous curve vs k obtained in the lo cal analysis to a discrete subset of global unstable mo des supp orted by the fluid. Also shown as crosses in Fig. 2.1 are the growth rates obtained with `even' b oundary conditions applied to Br and B at the midplane (see section 2.4), for the ambip olar diffusion limit with o = 10. As exp ected, the p erturbations are displaced a quarter of a wavelength (one no de) from those obtained with `o dd' b oundary conditions.
o

also diminishes the growth rate of the instability in the ambip olar diffusion limit,

2.5.3

Structure of the Perturbations

Fig. 2.2 shows the p erturbations in all fluid variables as a function of height, from the midplane (z /H = 0) to the surface of the disc (z /H = 5), for the ambip olar diffusion regime and go o d coupling (o = 10). They are obtained through equations (2.40) - (2.42) once the ODE system (2.39) has b een integrated. Note the non-zero values of v and E at the surface, as discussed in section 2.4. From this p oint onwards, discussion will b e fo cussed in the p erturbations of the magnetic field only.


2.5. Results

37

Figure 2.1

Growth rate versus numb er of no des (proxy for wavenumb er) of the MRI

for different conductivity regimes and coupling at the midplane o . Circles show the ambip olar diffusion limit (1 = 0) and triangles the Hall limit (2 = 0, 1 Bz > 0). Filled symb ols corresp ond to the go o d coupling case o = 10 and op en ones to the p o or coupling case o = 1. Crosses show the ambip olar diffusion limit with `even' b oundary conditions applied to Br and B . Note that results in this case are displaced a quarter of a wavelength (one no de) from those obtained with `o dd' b oundary conditions. In all cases vA /cs = 0.1.

Unless otherwise stated, the radial comp onent of the field Br is plotted with a solid line and the azimuthal comp onent B with a dashed line. As the overall scale of our linear equations is arbitrary, plots depicting the structure of the p erturbations either do not show the scale of the vertical axis (corresp onding to the amplitude of the p erturbations) or show a conveniently normalised scale, for reference purp oses. Effect of the Conductivity Regime Fig. 2.3 compares the structure of all unstable p erturbations for the ambip olar diffusion and Hall (1 Bz > 0) regimes under go o d coupling (o = 10). The mo des are ordered by the numb er of no des. We find that at small , the structure of the p erturbations in b oth cases is very similar, but significant differences arise when the growth rate is close to maximum. Then, ambip olar diffusion p erturbations p eak


38

Chapter 2. The instability in parameter space

Figure 2.2

Structure of the p erturbations in all fluid variables as a function of height

for the most unstable mo de in the ambip olar diffusion limit, for go o d coupling ( o = 10) and vA /cs = 0.1. The growth rate is = 0.7304. Note the non-zero values of v and E at the surface, due to the finite travel time to infinity of the p erturbations.

at the no de closest to the surface, while Hall (1 Bz > 0) ones p eak closest to the midplane. This b ehaviour is linked to the change in the lo cal coupling with z and its effect in the structure of the p erturbations for different conductivity regimes. It will b e discussed further in the next section. At this go o d coupling level, there are no appreciable differences in the structure or growth rate of p erturbations b etween b oth Hall limits results, as exp ected from the lo cal analysis (W99). When b oth bations is similar to the ambip olar diffusion limit. This prop erty is also dep endent found 9 to 10 unstable p erturbations in all cases. Under low coupling conditions, o = 2 or 1 dep ending on the conductivity regime (see Table 2.3), fewer unstable mo des grow for b oth the ambip olar diffusion, Hall (1 Bz > 0) and the comparable conductivities (1 = 2 ) regimes. Six to eight unstable p erturbations are found in these cases. As exp ected from the lo cal analysis (W99), the range of wavenumb ers for which unstable mo des exist is reduced as conductivity comp onents are present (1 = ±2 cases), the structure of the p ertur-

on the value of the lo cal coupling and will b e analysed in the next section. We


2.5. Results

39

Figure 2.3 Structure and growth rate of all unstable mo des of the MRI for the ambip olar diffusion (1 = 0) and Hall (2 = 0, 1 Bz > 0) cases. In all plots o = 10 and vA /cs = 0.1. For this go o d coupling, there are no differences b etween b oth Hall limits. When b oth conductivity comp onents are present ( 1 = ±2 cases), results resemble the ambip olar diffusion limit shown.


40

Chapter 2. The instability in parameter space

compared with the go o d coupling cases.

Results are quite different for the two remaining conductivity regimes (Hall 1 Bz < 0 and 1 = -2 ). More unstable mo des are found in these cases; 12 in the opp osite conductivities case (1 = -2 ) and a total of 27 for the Hall (1 Bz < 0)

limit. In this last case in particular, despite the low coupling of ionised and neutral comp onents of the fluid, unstable mo des have a very complex structure (high wavenumb er). Fig. 2.4 shows the structure of two such mo des at the low growth rate, high wavenumb er region of the - k space. Note that unstable mo des are so closely spaced that increasing the numb er of no des by a few only maginally changes their wavenumb er and has little effect on their growth rate. This complexity is exp ected from the form of the disp ersion relation at low coupling from the lo cal analysis (W99). Non-linear simulations (Sano & Stone 2002a, b) confirm that the many growing mo des in this regime strongly interact with each other and the instability develops into MHD turbulence. This turbulence is a transient phase that eventually dies away in two-dimensional simulations (Sano & Stone 2002a) but it is sustained in full three-dimensional mo dels (Sano & Stone 2002b). In b oth cases the non-emergence of the typical two-channel flow obtained in other regimes is noted by the authors.

Finally, some of the p erturbations show a structure resembling an interference pattern (see Fig. 2.5). They were obtained sp ecifically in regimes where ambip olar diffusion is present, for go o d and p o or coupling, but not in any of the Hall limits. This pattern can b e explained recalling that lo cal results show that two unstable mo des exist with the same growth rate and different wavenumb er. Despite this, just one global mo de is found for each in this analysis. Again, the application of b oundary conditions and integration along the vertical direction restricts global unstable mo des from those p ossible under a lo cal analysis. The interference pattern suggests that global mo des are a sup erp osition of two WKB mo des with (nearly) the same growth rate and which are not global solutions themselves. The interference of Fig. 2.5 was successfully replicated through the sup erp osition of two lo cal mo des with = 0.7004 using the analytical expresions in W99 for the ambip olar diffusion limit.


2.5. Results

41

Figure 2.4

Structure of two unstable mo des in the Hall ( 1 Bz < 0) limit for p o or

coupling (o = 2) and vA /cs = 0.1. Note the complex structure of the p erturbations (high wavenumb er). Unstable mo des are so closely spaced that increasing the numb er of no des only marginally changes their wavenumb er and growth rate.


42

Chapter 2. The instability in parameter space

Figure 2.5

Structure of an unstable mo de in the ambip olar diffusion limit ( 1 = 0)

for go o d coupling (o = 10) and vA /cs = 0.01. Note the interference pattern of the p erturbation, which suggests that this global mo de is a sup erp osition of two WKB mo des of similar growth rates.

Effect of the coupling parameter

o

Fig. 2.6 compares the structure and growth rate of the most unstable mo des of the MRI for all conductivity regimes as a function of the coupling parameter o . In all cases vA /cs = 0.1. We notice that reducing the coupling o causes the wavenumb er (i.e. the numb er of no des) of unstable mo des to diminish in all conductivity regimes except the Hall 1 Bz < 0 limit (rightmost column of Fig. 2.6), for which this dep endency is inverted. The growth rate is also reduced at a rate that dep ends on the conductivity regime of the fluid. These results are exp ected from the findings of the lo cal analysis (W99). the fluid is close to ideal MHD conditions and results obtained in all conductivity It is evident from Fig. 2.6 that at very high magnetic coupling (o 100),

regimes are alike. When the coupling is reduced to o 10, we b egin to appreciate differences b etween them. In particular, the amplitude of the p erturbations when ambip olar diffusion is present p eaks close to the surface while in b oth Hall limits the


2.5. Results

43

maximum amplitude is closest to the midplane. This is more clearly appreciated in Fig. 2.7, which plots the height of maximum amplitude of the fastest growing mo des as a function of the coupling parameter o and the conductivity regime of the fluid for vA /cs = 0.1. This figure shows that pure Hall regimes (1 Bz > 0 and 1 Bz < 0) p eak closer to the midplane, for all o studied, than the cases when ambip olar diffusion is present. This b ehaviour can b e explained by the dep endency of the lo cal growth of the instability with for different conductivity regimes. The maximum growth rate of ambip olar diffusion p erturbations increases with the lo cal (W99), which in turn is a function of height. As a result, at higher z , the lo cal growth of the instability increases, driving the amplitude of global p erturbations to increase. Hall (1 Bz > 0) p erturbations, on the contrary, have the same
max

for all , so

the instability is not driven from any particular vertical lo cation, which explains the flatness of their envelop e. This also explains why in Fig. 2.3 the differences b etween these regimes are apparent at close to maximum growth: The increment in the lo cal growth rate with the coupling is less marked for slow growing p erturbations. This is also appreciated in the lo cal analysis by the form of the disp ersion relation for different in the ambip olar diffusion regime (W99). It is also clear from Fig. 2.7 that p erturbations in the ambip olar diffusion and Hall (1 Bz > 0) limits p eak at higher z when o is reduced. This dep endency is driven by the reduction of the wavenumb er of most unstable p erturbations with the coupling in these regimes (W99, see also Fig. 2.6). On the contrary, in the Hall (1 Bz < 0) limit, the wavenumb er of the fastest growing mo de increases as o is reduced and the p erturbations p eak closer to the midplane with weaker o . Finally, lo oking at the first three columns of Fig. 2.6 it is evident that the structure of unstable mo des in the 1 = ±2 conductivity regimes are remarkably for
o

similar to the ambip olar diffusion limit shown in the leftmost column of the figure vA /cs = 0.1. When the coupling is weaker than this value, the structure

of unstable mo des in these regimes is no longer alike. To explain this we recall that the minimum degree of coupling for unstable mo des to grow, determined by the requirement that a wavelength fit in the disc scaleheight, is given by the ambip olar diffusion limit, and vA 2 /c
s 2

vA /cs in
o

in the Hall case (W99). For

0.1

then, the growth rate of ambip olar diffusion (1 = 0) p erturbations is exp ected to drop markedly and the envelop e of the p erturbations will b e mainly determined by the Hall effect. This transition in b oth 1 = ±2 cases, is clearly seen in Fig. 2.7.


44 In b oth cases, for
o

Chapter 2. The instability in parameter space 0.1 the p erturbations resemble the ambip olar diffusion limit

(1 = 0). In the 1 = 2 regime, when o < vA /cs the p erturbations resemble those in the Hall (1 Bz > 0) limit (compare lowest panels in the second and fourth columns of Fig. 2.6), consistent with the notion that ambip olar diffusion effects are no longer imp ortant in this region of parameter space. This implies a constant contrary, in the 1 = -2 case,
max

for weaker o , so the p erturbations tend to p eak closer to the midplane. On the
max

continues to diminish with once < vA /c

s

(W99). In this case, the range of growing mo des is always finite (as opp osed to the Hall 1 Bz < 0 limit) and there is a fastest growing mo de for every . As a result, the instability is driven at intermediate z (as in the ambip olar diffusion limit) and accordingly, the p erturbations p eak at higher z with decreasing o . The height of maximum amplitude increases faster as o is reduced in this regime than in the ambip olar diffusion limit. This o ccurs b ecause further.
max

for a given lo cal is greater

in this case (W99, see also section 2.5.4) so global p erturbations are amplified even

Effect of the Magnetic Field Strength In ideal MHD, the weaker the magnetic field the higher the minimum wavenumb er of the p erturbations (Balbus & Hawley 1991). The results of this study are consistent with this finding. With vA /cs 0.005 the p erturbations grow with very high wavenumb ers in all conductivity regimes. This can b e appreciated in Fig. 2.8 and 2.9 for the ambip olar diffusion and Hall (1 Bz > 0) limits, resp ectively, for go o d coupling (o = 10). At this o , solutions for b oth Hall limits are similar. Also, 1 = ±2 regimes are similar to the ambip olar diffusion limit, as exp ected. Note the interference pattern of p erturbations in lower panels of Fig. 2.8. We also studied the dep endency of the height of maximum amplitude with the strength of the magnetic field (Fig. 2.10) for go o d (o = 10) and p o or coupling (o = 2). In the former case, when ambip olar diffusion is present, with and without the Hall effect and regardless of the sign of 1 Bz , the lo cation of maximum amplitude of the fastest growing p erturbations as a function of vA /cs is similar, which is exp ected as o > vA /cs (see Fig. 2.10, panels a, b and c). We obtained unstable mo des for vA /cs up to 1. As vA /cs is reduced from this value, the p erturbations p eak at higher z until vA /cs 0.04. However, the lo cation of maximum amplitude b egins to


2.5. Results

45

Figure 2.6

Structure and growth rate of the most unstable mo des of the MRI for all

conductivity regimes and different values of o . In all cases vA /cs = 0.1. The value of the coupling parameter o is indicated at the top right corner of each panel. The growth rate ( ) of the p erturbations is shown in the lower right corner. Note that the Hall 1 Bz < 0 regime is explored for o 2 as in this region of parameter space the range of wavenumb ers for which lo cal unstable mo des exist b ecomes infinite (W99).


46

Chapter 2. The instability in parameter space

Figure 2.7

Height of maximum amplitude of the most unstable p erturbations as a

function of o for all conductivity regimes. In all cases v A /cs = 0.1

diminish as the field is further reduced, which could b e caused by the interference pattern, and very high wavenumb er, of the p erturbations (see also Fig. 2.8). The height of maximum amplitude p eaks at z 2.8. In b oth Hall limits the height of maximum amplitude increases with the strength

of the magnetic field until vA /cs 0.3 and then remains unaffected with further increments of vA /cs (panels d and e of Fig. 2.10). This o ccurs b ecause as the strength of the magnetic field increases, the wavenumb er of the p erturbations diminish, which pushes the maximum amplitude to higher z . For vA /cs 0.3 the p erturbations have only one no de, so any further increase in the magnetic field strength have little effect in the lo cation of the maximum amplitude. In the low coupling case results are very similar to the o = 10 cases. We note that in regimes where ambip olar diffusion is present (left hand side panels of Fig. 2.10), p erturbations tend to p eak at a lower z /H than in the go o d coupling cases when vA /c
s

0.1. In the Hall (1 Bz < 0) regime, there are unstable mo des for

`suprathermal' field strengths (vA /cs up to 2.9). This will b e further analysed in section 2.5.4, dealing with the dep endency of the growth rate of this instability with the strength of the magnetic field. In this case, the height of maximum amplitude p eaks at vA /cs 0.5 and then gradually diminishes as the field is incremented


2.5. Results

47

Figure 2.8

Structure as a function of height and growth rate of the MRI for different

choices of vA /cs under the ambip olar diffusion limit ( 1 = 0) and go o d coupling o = 10. The value of vA /cs is indicated at the top right corner of each panel. The growth rate is shown in the lower right corner.

b eyond this value.

2.5.4

The p erturbations in parameter space

Effect of the coupling Fig. 2.11 shows the growth rate of the most unstable mo des as a function of the coupling o for all conductivity regimes with vA /cs = 0.1 (top panel) and 0.01 (b ottom panel). The Hall (1 Bz < 0) limit could not b e mo delled for o < 2 b ecause in this region of parameter space the range of wavenumb ers for which unstable mo des grow b ecomes infinite (see section 2.5.1). We find that at the go o d coupling limit the instability grows at a rate similar to its ideal value of 0.75 for all conductivity regimes. As the coupling diminishes, the growth rate is reduced at a rate that dep ends on the conductivity regime of the fluid. In the Hall (1 Bz > 0) case the growth rate remains unaffected until o 0.01, and then diminishes drastically to 0.1 when o 0.005. The ambip olar diffusion limit has a much more gradual


48

Chapter 2. The instability in parameter space

Figure 2.9

As for Fig. 2.8, but for the Hall limit 1 Bz > 0. The Hall limit when

1 Bz < 0 (not shown) exhibits the same dep endency with v A /cs for this o .

3

2

1

0 3

2

1

0 3 -2 -1 0 1

2

1

0 -2 -1 0 1

Figure 2.10 Height of maximum amplitude of the p erturbations as a function of v A /cs , for all conductivity regimes. Triangles corresp ond to go o d coupling ( o = 10) and circles to p o or coupling (o = 2).


2.5. Results reduction of

49
max

with o . In this case, the growth rate departs significantly from the vA /c
s

ideal value for o 0.1 and then drops rapidly, reaching 0.007 for o 0.008. This is in agreement with findings by W99 that unstable mo des grow when in the ambip olar diffusion limit, and vA /c
2 s 2

in the Hall case (see also section

2.5.3). When is less than these values, p erturbations are strongly damp ed. p erturbations is constant at ab out the ideal rate 0.75 until o 10-4 . Below this, it plummets to zero as exp ected. On the contrary, in the ambip olar diffusion
max

When vA /cs 0.01 (b ottom panel of Fig. 2.11), the growth of Hall (1 Bz > 0)

and 1 = ±2 regimes,

b egins to diminish much so oner. It is also noticed

that, in the ambip olar diffusion and 1 = 2 cases, the growth rate increases again of the p erturbations due to the weakness of the magnetic field. In these conditions minishes with the coupling as p er lo cal results (W99). As the wavenumb ers of the low enough for global effects to b e imp ortant again and mo dify the growth rate of unstable p erturbations. With the lo cal increasing with height ab ove the midplane, stratification will tend to increase the growth of global mo des at low o . p erturbations decrease when o is reduced, for sufficiently low o ( 0.05), k is

after reaching a minimum for o 0.05. This is caused by the high wavenumb er

global effects (stratification) are less imp ortant and the maximum growth rate di-

Effect of the magnetic field strength The dep endency of the maximum growth rate with the strength of the magnetic field for all conductivity regimes is shown in Fig. 2.12 for o = 10, 2 and 0.1. In the go o d coupling case (top panel), increasing the strength of the magnetic field has little effect in the growth rate of the most unstable mo des of all conductivity regimes until vA /cs 1, where it drastically drops to zero. These results are similar to the ideal MHD case, which predicts that at this strength of the magnetic field the

the p erturbations are strongly damp ed.

wavelength of most unstable mo des b ecome H, the scaleheight of the disc, and In the o = 2 case shown in the middle panel, we found unstable mo des in the

Hall limit (1 Bz < 0) for vA /cs up to 2.9. We know from the lo cal analysis (W99) that once the lo cal 2, unstable mo des exist for every k vA / in this conductivity mo des with k H 1 growing within the disc. regime. As a result, even for suprathermal fields (vA /cs > 1), there are still unstable


50

Chapter 2. The instability in parameter space

0

-0.5

-1

-1.5

-2

-2 0 -0.2 -0.4 -0.6 -0.8 -1 -1.2 -4

-1

0

1

-2

0

Figure 2.11

Growth rate of the fastest growing mo des of the MRI as a function of

o

for different conductivity regimes. v A /cs = 0.1 (top panel) and 0.01 (b ottom panel).


2.6. Discussion Results for o = 0.1 (b ottom panel) show clearly how vA /c
s

51
max

plummets when

(ambip olar diffusion limit) or vA /c

2

s

2

(Hall limit), as exp ected.

Finally, turning our attention to the dep endency of the growth rate of the instability with the field strength at low vA /cs , we appreciate in all panels of Fig. 2.12 that
max

initially increases as vA /cs is reduced, until it reaches a maximum.

Further reductions in vA /cs cause the growth rate to diminish monotonically. Comparing these results with the findings of the lo cal analysis (W99), we can show that the growth rates of global unstable mo des at weak magnetic field strengths tend to the lo cal values for the relevant coupling o . This can b e explained simply by the increase in wavenumb er of unstable mo des as vA /cs is reduced, which causes global effects (driven by stratification) to b e less imp ortant. As a result, the growth of global mo des do es not differ significantly from the lo cal values at the same coupling.

2.6

Discussion

The solutions presented in the previous sections illustrate the growth and structure of the MRI when different comp onents of the conductivity tensor are dominant throughout the entire cross-section of the disc. Density stratification causes the local growth of unstable mo des, and the amplitude of global p erturbations, to b e a function of height. The envelop es of short-wavelength solutions are shap ed by this comp etition b etween different growth rates acting at different vertical lo cations. Moreover, when is weak at the midplane, long wavelength p erturbations are imp ortant and vertical stratification is crucial in determining the growth of global MRI solutions. The results presented in this contribution confirm these exp ectations. When the Hall and Pedersen comp onents are comparable, the Hall effect alters the structure and growth of unstable p erturbations for o < vA /cs . In this region of parameter space, ambip olar diffusion p erturbations have negligible growth, but unstable mo des that include Hall conductivity still grow at = 0.2 - 0.3. Hall weak magnetic field (vA /c
s

1 Bz > 0 p erturbations grow faster than 1 Bz < 0 ones. Furthermore, under a 0.01), the Hall effect significantly increases the growth

rate of unstable mo des at low coupling. When it dominates, unstable mo des grow at close to the ideal rate for o 10-4 . The height ab ove the midplane where the most unstable p erturbations p eak is de-

p endent on the conductivity regime of the fluid. Consequently, the vertical lo cation


52

Chapter 2. The instability in parameter space

Figure 2.12 Growth rate of the fastest growing mo des of the MRI as a function of v A /c for different conductivity regimes. From top to b ottom the coupling o is 10, 2 and 0.1.

s


2.6. Discussion

53

of the active zones within the disc, in which the MRI pro duces angular momentum transp ort and disc material is b eing accreted, is dep endent on the configuration of the conductivity tensor. Perturbations including ambip olar diffusion p eak consisp eak at different heights, signalling that the Hall effect is dep endent on the orientation of the magnetic field with resp ect to the disc angular velo city vector . In this region of parameter space, when 1 Bz < 0, mo des p eak at a higher z than when 1 Bz > 0. The Hall effect do es not significantly mo dify the dep endency of the height of maximum amplitude of unstable mo des with the strength of the magnetic field for go o d coupling. In this region of parameter space ambip olar diffusion dominates and causes the p erturbations to p eak at higher z when vA /cs is reduced. When 1 Bz < 0, Hall p erturbations can have a very complex structure (high wavenumb er), even at low coupling, and many mo des are found to grow. In the non-linear stage, the interaction b etween these mo des causes the MRI to develop into MHD turbulence with non-emergence of the typical two-channel flow obtained in other conductivity regimes (Sano & Stone 2002a, 2002b). When do es Hall diffusion determine the b ehaviour of the instability? Naively b etween the magnetic field and the gas. For example if one might prop ose |1 | 2 as a criterion, but this ignores the level of coupling 10, ideal MHD almost tently higher than those in b oth Hall limits. Also, for o < vA /cs , 1 = ±2 mo des

holds and there is little dep endence of growth rate and structure on the diffusion regime (see Fig. 2.3). A useful criterion can b e derived using the results of the lo cal analysis in W99 and comparing the maximum growth rates with and without Hall diffusion in the weak-coupling limit. In the absence of Hall diffusion (i.e. 1 = 0), the maximum growth rate for 1 is 3/4. When Hall diffusion is present and
3 |1 |/ , the maximum growth rate is either 4 |1 |/( + 2 ) if 1 Bz > 0 or the

instability is suppressed if 1 Bz < 0. In either case, Hall diffusion dominates the b ehaviour of the instability when |1 | .

Thus even when |1 |



crit

=

(2.58)

2 , the structure and growth rate of the magnetorotational

for example, for the nominal conditions at the disc midplane 1 AU from the central

instability are dominated by Hall diffusion if < |1 |/ . This is easily satisfied, protostar (see Table 3.1), where |1 |/ |e | = 0.035 (using eqs. 2.18 and 2.20


54 with |e |

Chapter 2. The instability in parameter space 1) and = 9 â 10-6 . At 5 AU, 2 and
crit

1

2 , so Hall diffusion

dominates here also. Although these results dep end on the assumed magnetic field strength, the conditions under which < a large range of radii. Despite this, Hall diffusion has generally b een neglected in studies of accretion discs in favour of the ambip olar diffusion or resistive limits. Here we illustrate the severity of this approximation by comparing the structure and growth rate of the unstable p erturbations for a mo del with 1 = 2 to those for `simplified' pure ambip olar/resistive diffusion and Hall (1 Bz > 0) mo dels obtained by setting
2

are so broad that we can conclude

that Hall diffusion determines the growth rate and structure of the instability over

or 1 to zero resp ectively and reducing the coupling parameter o by a factor of 2 to reflect its dep endence on (1 2 + 2 2 )1/2 (see eq. 2.50). The full mo del has vA /cs = 0.01 and o = 0.01414, whereas the the corresp onding ambip olar diffusion and Hall limits have o = vA /cs = 0.01 and the appropriate values of 1 and 2 as p er Table 2.3. The comparison is presented in Fig. 2.13. The structure and growth rate of pure ambip olar diffusion and Hall (1 Bz > 0) p erturbations are as exp ected for envelop e of the pure-Hall solutions b eing fairly constant. In b oth the ambip olar diffusion and the 1 = 2 cases we obtain a magnetic `dead zone' near the midplane where no p erturbations grow (Gammie 1996, Wardle 1997). Comparing the top panel of Fig. 2.13 with the middle and b ottom ones it is clear that the Hall effect mo difies b oth the structure and growth of unstable mo des. In particular, the extent of the dead zone is reduced and the growth rate is increased. According to these results, b oth the depth of the active zones within the disc and the rate of angular momentum transp ort by unstable mo des can b e significantly mo dified by the Hall effect. Although our solutions incorp orate the effect of density stratification on the coupling parameter and the Alfven sp eed, we have assumed that the comp onents of the conductivity tensor do not vary with height. While this simplification p ermitted us to compare the b ehaviour of the instability in different regimes, the conductivities in a real disc will reflect the height-dep endence of charged particle abundances and different heights (Wardle 2003). their Hall parameters (see §2.2.1). Different regimes are exp ected to dominate at o vA /cs , with the ambip olar diffusion decaying towards the midplane and the


2.7. Summary

55

Nonetheless, it is clear from the simplified comparison presented here that Hall diffusion is an essential part of accretion in low-conductivity discs, and that it determines the extent of the magnetically-inactive `dead zone' (Gammie 1996, Wardle 1997). Further, Hall diffusion will mo dify any angular momentum transp ort within the dead zone that o ccurs via non-axisymmetric density waves driven by the active surface layers (Stone & Fleming 2003) b ecause it will dominate the marginally magnetically-active regions of the disc just ab ove the dead zone. Hall diffusion may therefore affect the ability of dust grains to settle towards the midplane and b egin to assemble into planetesimals (e.g. Weidenschilling & Cuzzi 1993).

2.7

Summary

In this chapter we have examined the structure and linear growth of the magnetorotational instability (MRI) in weakly ionised, stratified accretion discs, assuming an initially vertical magnetic field. This work is relevant for the study of lowconductivity accretion systems, such as protostellar and quiescent dwarf novae discs, where non-ideal MHD effects are imp ortant (Gammie & Menou 1998; Menou 2000; Stone et al. 2000). The formulation allows for a height-dep endent conductivity but in this initial study we assumed the comp onents of the conductivity tensor were constant with height. The analysis was restricted to p erturbations with a vertical wavevector (k = kz ), which are the most unstable mo des when initiated from a vertically aligned magnetic field (Balbus & Hawley 1991; Sano & Miyama 1999). In this case, the field-parallel comp onent of the conductivity tensor plays no role and the ambip olar diffusion and resistive limits are identical. The linearised system of ODE was integrated from the midplane to the surface of the disc under appropriate b oundary conditions and global unstable mo des were obtained. The parameters that control the evolution of the fluid are: (i ) The coupling b etween ionised and neutral comp onents of the fluid evaluated at the midplane (o ), which relates the frequency at which non-ideal effects are imp ortant with the dynamical (Keplerian) frequency of the disc; (ii ) the magnetic field strength characterised by the ratio vA /cs at the midplane; and (iii ) the ratio of the comp onents of the conductivity tensor p erp endicular to the magnetic field 1 /2 . In order to explore the growth and structure of unstable mo des when different conductivity regimes dominate over the entire cross-section of the disc, we exam-


56

Chapter 2. The instability in parameter space

Figure 2.13

Comparison of the structure and growth rate of the MRI for different

configurations of the conductivity tensor for v A /cs = 0.01. Top panel shows the case where b oth ambip olar diffusion and Hall ( 1 Bz > 0) conductivity terms are imp ortant. In this configuration the Hall regime is dominant close to the midplane and ambip olar diffusion dominates near the surface. Middle and b ottom panels show the instability under the ambip olar diffusion and Hall approximations, resp ectively.


2.7. Summary

57

ined the following configurations of the conductivity tensor: 1 = 0 (the ambip olar diffusion or resistive limits), 2 = 0 (b oth Hall limits 1 Bz > 0 and 1 Bz < 0), and the cases where b oth effects are imp ortant (1 = ±2 ). The main results of this study are highlighted b elow:

1. Global mo des are a discrete subset from the continuous curve of p ossible vs. k combinations obtained with a lo cal analysis (W99). These unstable mo des can b e expressed as a sup erp osition of two WKB mo des with similar growth rate, which explains the interference patterns found in some of the p erturbations. 2. Ambip olar diffusion p erturbations p eak consistently higher ab ove the midplane than solutions where Hall conductivity dominates. 3. For go o d coupling (o > vA /cs ), the structure and growth of the p erturbations are mainly determined by ambip olar diffusion. For a weaker coupling, Hall conductivity significantly mo difies unstable mo des. In this case, 1 = while 1 = -2 mo des have their maximum amplitude closer to the surface. 4. Hall limit (1 B
z 2

p erturbations resemble the Hall 1 Bz > 0 limit and p eak closer to the midplane

< 0) p erturbations can have a complex structure (high

wavenumb er) even for p o or coupling (o = 2). There are also many unstable mo des, which supp orts findings that in this case the MRI evolves into MHD turbulence with non-emergence of the two-channel flow obtained in other regimes (Sano & Stone 2002a, 2002b). 5. As the coupling parameter o is reduced, departure from ideal growth 0.75 o ccurs at a rate that dep ends on the conductivity regime. Hall limit p erturbations grow at close to the ideal limit for o > vA 2 /cs 2 . In the ambip olar diffusion approximation the growth rate decreases when results are in agreement with predictions from W99. 6. The weaker the magnetic field the higher the p erturbations p eak in all regimes where ambip olar diffusion is present. On the contrary, b oth Hall limits p eak closer to the surface with weaker vA /cs . 7. Unstable mo des grow when vA /cs is increased until a critical value vA /c go o d coupling vA /c
s cr it s cr it o

vA /cs . These

is reached. At the critical vA /cs the growth rate abruptly drops to zero. At 1 for all conductivity regimes. At the p o or coupling


58

Chapter 2. The instability in parameter space limit (o = 2), results are different only for the Hall regime (1 Bz < 0). In this case we obtain unstable mo des for vA /cs 2.9. 8. At very weak magnetic fields (vA /c
s

0.01), global effects are less imp ortant,

due to the high wavenumb er of the p erturbations. In this region of parameter space the growth rates of MRI p erturbations tend to the corresp onding lo cal values for the relevant fluid parameters. 9. Hall difussion determines the growth of the MRI when |1 |/ . This

condition is satisfied over a large range of radii in protostellar discs.

10. When the Hall regime dominates near the midplane and ambip olar diffusion is dominant closer to the surface, a larger section of the disc is unstable to MRI p erturbations and unstable mo des grow faster than those obtained using the ambip olar diffusion approximation


Chapter 3 The instability in protoplanetary discs
3.1 Introduction

The magnetorotational instability (MRI; Balbus & Hawley 1991, 1998; Hawley & Balbus 1991) generates and sustains angular momentum transp ort in differentially rotating astrophysical discs. It do es so by converting the free energy source contributed by differential rotation into turbulent motions (e.g. Balbus 2003), which transp ort angular momentum via Maxwell stresses. Most MRI mo dels in non-ideal MHD conditions adopt either the ambip olar diffusion (Blaes & Balbus 1994, MacLow et al. 1995 and Hawley & Stone 1998) or resistive approximations (Jin 1996, Balbus & Hawley 1998, Papaloizou & Terquem 1997, Sano & Miyama 1999; Sano, Inutsuka & Miyama 1998, Sano et. al. 2000, Fleming, Stone & Hawley 2000 and Stone & Fleming 2003). The inclusion of Hall diffusion is a relatively recent development (Wardle 1999 (W99 hereafter), Balbus & Terquem 2001, Sano & Stone 2002a,b; 2003, Salmeron & Wardle 2003 (SW03 hereafter) and Desch 2004). When the Hall effect dominates over ambip olar diffusion, fluid dynamics is dep endent on the alignment of the magnetic field with the angular velo city vector of the disc (Wardle & Ng 1999), and wave mo des supp orted by the fluid are intrinsically mo dified. For example, left and right-circularly p olarised Alfven waves travel at different sp eeds and damp at different rates in this regime (Pilipp et al. 1987, Wardle & Ng 1999). Both the structure and growth rate 59


60

Chapter 3. The instability in protoplanetary discs

of MRI p erturbations can b e substantially mo dified by Hall conductivity, esp ecially when the coupling b etween ionised and neutral comp onents of the fluid is low (W99, SW03). In a previous pap er we presented a linear analysis of the vertical structure and growth of the MRI in weakly ionised, stratified accretion discs (SW03). In that study the comp onents of the conductivity tensor were assumed to b e constant with height. The obtained solutions illustrate the prop erties of the MRI when different conductivity regimes are dominant over the entire cross-section of the disc. We found that when the magnetic coupling is weak, mo des computed with a non-zero Hall conductivity grow faster and act over a more extended cross-section than those obtained using the ambip olar diffusion approximation. The height ab ove the midplane where the fastest growing mo des p eak dep ends on the conductivity regime of the fluid. When ambip olar diffusion is imp ortant, p erturbations p eak at a higher z than when the fluid is in the Hall limit. Furthermore, when the coupling is weak, p erturbations computed with a full conductivity tensor p eak at different heights dep ending on the orientation of the magnetic field with resp ect to the angular velo city vector of the disc. This is a consequence of the dep endency of the Hall effect on the sign of (k · )(k · B) (Balbus & Terquem 2001), which reduces to · B for vertical bations can have a very complex structure (high wavenumb er), particularly when the magnetic field is weak. In these conditions, many mo des were found to grow, even with a very weak magnetic coupling. These results suggest that significant accretion can o ccur in regions closer to the midplane of astrophysical discs, despite the low magnetic coupling, due to the large column density of the fluid. This idea contrasts with the commonly accepted view that accretion is imp ortant primarily in the surface regions, where the coupling b etween ionised and neutral comp onents of the fluid is much stronger, but the column density is significantly smaller. In a real disc, the comp onents of the conductivity tensor vary with height (e.g. Wardle 2003) as a result of changes in the abundances of charged particles, fluid density and magnetic field strength. This, in turn, is a consequence of changes in the ionisation balance within the disc, which reflects the equilibrium b etween ionisation and recombination pro cesses. The former are primarily non-thermal, triggered by cosmic rays, x-rays emitted by the central protostar and radioactive materials. The later can, in general, o ccur b oth in the gas phase and on grain surfaces (e.g. Opp enfields and wavenumb ers (W99). Finally, we showed that in the Hall regime p ertur-


3.1. Intro duction

61

heimer & Dalgarno 1974, Spitzer 1978, Umebayashi & Nakano 1980, Nishi, Nakano & Umebayashi 1991, Sano et. al. 2000). As a result, different conductivity regimes are exp ected to dominate at different heights (Wardle 2003). In this pap er we revisit the linear growth and structure of MRI p erturbations using a height-dep endent conductivity tensor. We assume that the disc is thin and isothermal and adopt a fiducial disc mo del based in the minimum-mass solar nebula (Hayashi 1981, Hayashi, Nakazawa & Nakagawa 1985). Further, we assume that ions and electrons are the main charge carriers, which is a valid approximation in late stages of accretion, after dust grains have settled towards the midplane of the disc. As an indication of the timescales for this settling to o ccur, Nakagawa, Nakazawa & Hayashi (1981) show settled), drops from 10 that the mass fraction of 1 - 10 µm grains well mixed with the gas phase, (i.e. not
-1

to 10

-4

although the timescale for dust settling all the way to the equator may exceed the midplane in a much shorter timescale (Dullemond & Dominik 2004).

in t 2 â 103 to 1 â 105 years. Furthermore,

lifetime of the disc, grains can settle within a few pressure scaleheights ab out the

This pap er is structured as follows: Section 3.2 summarises the formulation, including the governing equations, fiducial disc mo del and ionisation balance. Section 3.3 describ es the linearisation, parameters of the problem and b oundary conditions. Results are presented in sections 3.4 and 3.5. We summarise in section 3.4 the test mo dels used in this study and present the ionisation rates as a function of z at representative radial lo cations from the central protostar (R = 1, 5 and 10 AU). The imp ortance of different conductivity regimes at different heights is also describ ed for a range of magnetic field strengths. Then, in section 3.5, the structure and growth rate of MRI p erturbations at the radii of interest are analysed, including a comparison with results obtained using different configurations of the conductivity tensor, sources of ionisation and disc structure. We find that the MRI is active over a wide range of fluid conditions and magnetic field strengths. For example, for the fiducial mo del at R = 1 AU, and including cosmic ray ionisation, unstable mo des are found for B 8 G. When 200 mG B 5 G, these p erturbations grow at ab out the ideal-MHD rate 0.75 times the dynamical (Keplerian) frequency of the disc. Resummarised in section 3.7.

sults are discussed in section 3.6 and the pap er's formulation and key findings are


62

Chapter 3. The instability in protoplanetary discs

3.2
3.2.1

Formulation
Governing Equations

Following W99 and references therein, we formulate the conservation equations ab out a lo cal Keplerian frame corotating with the disc at the Keplerian frequency = GM /r 3 . Time derivatives in this frame, / t, corresp ond to / t + / in the standard lab oratory system (r, , z ) anchored at the central mass M. The fluid velo city is expressed as a departure from Keplerian motion v = V - vK , where V is ^ the velo city in the lab oratory frame and vK = GM /r is the Keplerian velo city at the radius r . The fluid is assumed to b e weakly ionised, so the effect of ionisation and recombination pro cesses on the neutral gas, as well as the ionised sp ecies' inertia and thermal pressure, are negligible. Under this approximation, separate equations of motion for the ionised sp ecies are not required and their effect on the neutrals is treated via a conductivity tensor (W99 and references therein), which is a function of lo cation (r ,z ). The governing equations are the continuity equation, + t the equation of motion, · (v) = 0 , (3.1)

v + (v · t

2 2 ^ vr c )v - 2v^ + 2 vr - K ^ + s + r1 r JâB - = 0, (3.2) c

and the induction equation, B = t given by =- GM (r + z
2 1 2) 2

^ â(vâB) - c âE - 3 Br . 2

(3.3)

In the equation of motion (3.2), is the gravitational p otential of the central ob ject,

,

(3.4)


3.2. Formulation

63

2 and vK /r is the centrip etal term generated by exact Keplerian motion. Similarly, ^ 2v^ and 1 vr are the coriolis terms asso ciated with the use of a lo cal Kepler 2

rian frame, cs =

P / is the isothermal sound sp eed, = vK /r is the Keplerian

frequency and c is the sp eed of light. Other symb ols have their usual meanings. In the induction equation (3.3), E is the electric field in the frame comoving 3 ^ with the neutrals and the term 2 Br accounts for the generation of toroidal field from the radial comp onent due to the differential rotation of the disc. Additionally, the magnetic field must satisfy the constraint: ·B = 0, and the current density must satisfy Amp ere's law, J= and Ohm's law, ^ J = · E = E + 1 BâE + 2 E . (3.7) c âB 4 (3.6) (3.5)

Note that we have intro duced the conductivity tensor in equation (3.7). We refer the reader to W99 and references therein for derivations and additional details of this formulation. Assuming that the only charged sp ecies are ions and electrons, and that charge neutrality is satisfied (ni = ne ), the comp onents of the conductivity tensor can b e expressed as (SW03), the conductivity parallel to the magnetic field, = the Hall conductivity, 1 = and the Pedersen conductivity, 2 = From (3.9) and (3.10), we find: = (i - e ) cene 2 B [(1 + e )(1 + i2 )] . (3.11) cene (1 - i e )(i - e ) . 2 B (1 + e )(1 + i2 ) (3.10) cene (i + e )(e - i ) , 2 B (1 + e )(1 + i2 ) (3.9) cene (i - e ) , B (3.8)

1/2


64 where =

Chapter 3. The instability in protoplanetary discs
2 2 1 + 2 is the total conductivity p erp endicular to the magnetic field.

In equations (3.8) to (3.11), j = Zj eB 1 mj c j (3.12)

is the Hall parameter, given by the ratio of the gyrofrequency and the collision frequency of charged sp ecies j with the neutrals. It represents the relative imp ortance of the Lorentz and drag terms in the charged sp ecies' motion. In equation (3.12), j = < v >j , mj + m (3.13)

where m is the mean mass of the neutral particles and < v >j is the rate co efficient of momentum exchange by collisions with the neutrals. The ion-neutral momentum rate co efficient is given by, < v >i = 1.6 â 10
-9

cm3 s

-1

,

(3.14)

an expression that ignores differences in the values of elastic cross-sections of H, H2 and He (Draine, Rob erge & Dalgarno 1983). Similarly, the rate co efficient of momentum exchange of electrons with the neutrals is (Draine et al. 1983): < v >e 1 â 10 and the mean ion mass mi = 30mH . The relative magnitudes of the comp onents of the conductivity tensor differentiate three conductivity regimes: When
2 -15

cm

2

128k T 9 me

1/2

,

(3.15)

ambip olar diffusion dominates and the magnetic field is effectively frozen into the

|1 | for most charged sp ecies,

ionised comp onents of the fluid. Electron-ion drift is small compared with ionneutral drift and the ionised sp ecies act as a single fluid. MRI studies in this regime include Blaes & Balbus (1994), MacLow et al. (1995) and Hawley & Stone (1998). Conversely, when
2

familiar ohmic diffusion, and the magnetic field is no longer frozen into any fluid Hawley (1998), Papaloizou & Terquem (1997), Sano & Miyama (1999), Sano et. al.

|1 |, the conductivity is a scalar, giving rise to the

comp onent. Examples of studies of the MRI in this regime are Jin (1996), Balbus & (1998, 2000, 2004), Fleming, Stone & Hawley (2000) and Stone & Fleming, (2003). Ambip olar diffusion dominates in low density regions, where magnetic stresses are


3.2. Formulation

65

more imp ortant than collisions with the neutrals and the ionised sp ecies are mainly tied to the magnetic field rather than to the neutral particles. On the contrary, in relatively high density environments, the ionised sp ecies are primarily linked to the neutrals via collisions and Ohmic diffusion dominates. There is, however, an intermediate density range characterised by a varying degree of coupling amongst charged sp ecies. In these circumstances, there is a comp onent of the conductivity tensor that is p erp endicular b oth to the electric and magnetic fields. It is this term that gives rise to Hall currents. It has b een shown that this regime can b e imp ortant in the weakly ionised environment of accretion discs. For example, using an MRN grain mo del (Mathis, Rumpl & Nordsieck 1977) with a p ower law distribution of grain sizes b etween 50 and 2500 Angstrom, Wardle & Ng (1999) showed that Hall conductivity is imp ortant for 10
7

n

H

1011 cm-3 . MRI studies including Hall

diffusion have b een conducted by W99, Balbus & Terquem (2001), Sano & Stone (2002a,b; 2003), SW03 and Desch (2004). The ionisation balance within the disc (section 3.2.3) determines the abundances of charged sp ecies (ions and electrons). These, in turn, determine the configuration of the conductivity tensor. In protostellar discs, outside of the central 0.1 AU, ionisation sources are non-thermal (Hayashi 1981) and the ionisation fraction is not enough to pro duce go o d magnetic coupling over the entire cross-section of the disc. In these conditions, the region around the midplane is likely to b e a magnetically "dead zone" (Gammie 1996).

3.2.2

Disc Mo del

Our mo del incorp orates the vertical structure of the disc, but neglects fluid gradients in the radial direction. This is an appropriate approximation, as astrophysical accretion discs are generally thin and changes in the radial direction o ccur on a much bigger length scale than those in the vertical direction. Including the vertical structure means that p erturbations of spatial dimensions comparable to the scale height of the disc, which are asso ciated with either a strong magnetic field (vA cs ) or low conductivity, can b e explored. We adopted, as our fiducial mo del, a disc based in the minimum-mass solar nebula (Hayashi 1981, Hayashi et. al. 1985). In this mo del, the surface density (r ), sound sp eed cs (r ), midplane density o (r ), scaleheight H (r ) and temp erature


66 T (r ) are:

Chapter 3. The instability in protoplanetary discs

(r ) = r 1AU
-
11 4

o

r 1AU
1 4

-

3 2

, 2.34 µ
1 2 1 2

(3.16)

cs (r ) = c

-

so

L L
-
1 8

1 8

, 2.34 µ
1 2

(3.17)

0 (r ) =

o

r 1AU

L L
0

M M
1.25

,

(3.18)

H (r ) = H and, T (r ) = T In the previous expressions,

r 1AU

,

(3.19)

o

r 1AU

-1/2

L L

1/4

.

(3.20)

o = 1.7 â 103 g cm

-2

,

(3.21)

cso = 9.9 â 104 cm s o = 1.4 â 10
-9

-1

,

(3.22)

g cm

-3

,

(3.23)

H0 = 5.0 â 1011 cm , and, To = 280 K .

(3.24)

(3.25)

In equations (3.16) to (3.20), M and L are the stellar mass and luminosity, resp ectively, and µ is the mean molecular mass of the gas. This mo del describ es the minimum mass distribution of the solar nebula, estimated assuming an efficient planet formation with no significant migration. With these assumptions, the current mass distribution and comp osition of the planets is a go o d indication of that of dust in the original nebula. This mo del has b een used extensively, but there are


3.2. Formulation

67

theoretical grounds to exp ect that a typical protostellar disc may b e more massive, with a different large scale structure, shap ed ultimately by the action of MHD turbulence (e.g. Balbus & Papaloizou 1999). Furthermore, disc masses of up to 0.1M , asso ciated with surface density distributions following a p ower-law of index p b etween 0 and 1, have b een derived for T Tauri stars in Taurus (Kitamura et. al. 2002). It can b e shown that the disc surface density may b e roughly up to an order of magnitude higher than the one sp ecified by the minimum-mass solar nebula mo del b efore self gravity b ecomes imp ortant. To account for the p ossibility of discs b eing more massive than the minimum-mass solar nebula mo del, we also studied the prop erties of MRI unstable mo des using a disc structure with an increased surface density o = 10o and mass density o = 10o . The To omre parameter (To omre valid. For simplicity, we assume that the temp erature To is unchanged, so Ho and c
so

1964) Q 6 in this case, so the assumption of a non self-gravitating disc is still are the same as in the fiducial mo del.

As the disc is gravitationally stable, the gravitational force in the vertical direction comes from the central protostar. Under these conditions, the balance b etween the vertical comp onent of the central gravitational force and the pressure gradient within the disc determines its equilibrium structure. The vertical density distribution in hydrostatic equilibrium is given by (r, z ) z2 = exp - 2 . o (r ) 2H (r ) n = 0.2nH2 , the neutral gas mass and numb er densities are, resp ectively, = ni mi = 1.4mH n and n = 1.2n which gives, nH (r, z ) = (r, z ) 1.4mH (3.29) , (3.28) (3.27) (3.26)

Assuming a neutral gas comp osed of molecular hydrogen and helium such that
He

H

H

2

and µ = /n = 2.34. For simplicity, we take L /L = M /M = 1 in all our mo dels.


68

Chapter 3. The instability in protoplanetary discs

3.2.3

Ionisation balance

The ionisation balance within the disc is given by the equilibrium b etween ionisation and recombination pro cesses. In general, recombination can take place b oth in the gas phase and on grain surfaces (e.g. Opp enheimer & Dalgarno 1974, Spitzer 1978, Umebayashi & Nakano 1980, Nishi, Nakano & Umebayashi 1991, Sano et. al. 2000), but here we have assumed that grains have settled, so we are including only gasphase recombination rates. Except in the innermost sections of the disc (R 0.1 AU), where thermal effects are imp ortant (Hayashi 1981), ionisation pro cesses in protostellar discs are mainly non-thermal. Ionising agents are typically cosmic rays, x-rays emitted by the magnetically active protostar, and the decay of radioactive materials present within the disc. Some authors (e.g. Fromang, Terquem & Balbus 2002) have argued that the low energy particles imp ortant for cosmic-ray ionisation are likely to b e excluded by the protostar's winds. Given the uncertainties involved, and the exp ectation that cosmic rays (if present), may b e more imp ortant than xrays near the midplane (esp ecially for R 1 AU, where the surface density is larger than the attenuation length of x-rays), we explore b oth options in this study. The treatment of ionisation and recombination pro cesses is detailed b elow. Cosmic ray ionisation The cosmic ray ionisation rate,
CR

(r, z ), is given by,



CR

(r, z ) =

(r, z ) 2 C R (r ) - (r, z ) exp - , C R
CR

exp -

+ (3.30)

where

CR

= 10

-17 -1

s

is the ionisation rate due to cosmic rays in the interstellar
-2

medium and

CR

= 96 gr cm

is the attenuation length, a measure of how deep

cosmic rays can p enetrate the disc (Umebayashi & Nakano 1981). When the surface density is larger than 2
CR

, most cosmic rays do not reach the midplane. Also,


(r, z ) =
z

(r, z )dz ,

(3.31)

is the vertical column density from the lo cation of interest to infinity. The two terms in brackets in equation (3.30) measure the ionisation rate at the p osition (r , z ) by


3.2. Formulation cosmic rays p enetrating the disc from ab ove, and b elow, resp ectively.

69

X-ray ionisation There is strong evidence for an enhanced magnetic activity in young stellar ob jects (i.e. see review by Glassgold, Feigelson & Montmerle 2000). Typical soft x-ray luminosities (0.2 - 2 kev) of these ob jects are in the range of 10
28

- 1030 erg s-1 , ab out

102 - 103 times more energetic than solar levels (Glassgold et. al. 2000). Mo dels of the p enetration of stellar x-rays into a protostellar disc by Igea & Glassgold (1999), show that even discounting the low energy photons that are attenuated by stellar winds, the ionisation rate due to hard x-rays close to the central ob ject is many orders authors investigated the ionisation rate by x-rays from a central protostar mo delled as a x-ray source of total luminosity Lx 1029 erg s
-1

of magnitude larger than that of cosmic rays, esp ecially ab ove z /H 2. These

and temp erature k Tx in the

range of 3 - 8 kev. The transp ort of x-rays through the disc was followed using material. The incorp oration of scattering is imp ortant, as x-rays can b e scattered not only out of the disc, but also towards the midplane, enhancing the ionisation level deep er within the disc. Results indicate that, at each radius, the x-ray ionisation rate is a function of the vertical column density into the disc N (r, z ), irresp ective of its structural details. To calculate the x-ray ionisation rate X (r, z ) in the upp er half of the disc we added the contribution of x-rays arriving from b oth sides. We used the values of X (r, z ) ( s-1 ) as a function of the vertical column density, N (cm-2 ), plotted in Fig. 3 of Igea & Glassgold (1999), for R = 1, 5 and 10 AU and k Tx = 5 Kev. The vertical column density, appropriate for x-rays arriving from the top is


a Monte Carlo pro cedure which included b oth absorption and scattering by disc

N (r, z ) =
z

nH (r, z )dz ,

(3.32)

with n

H

given by equation (3.29). Similarly, the ionisation contributed by x-rays

arriving at (r, z > 0) from the other hemisphere of the disc is obtained substituting N ab ove by 2N (r, 0) - N (r, z ), although this contribution is usually negligible.


70 Radioactivity

Chapter 3. The instability in protoplanetary discs

The ionisation rate R contributed by the decay of radioactive materials (primarily
40

K ) have also b een considered in previous work (i.e. Consolmagno & Jokipii 1978,

Sano et al. 2000). This rate can b e calculated as (Consolmagno & Jokipii 1978): nr E ,

R =

(3.33)

where ( s-1 ) is the decay rate of the radioactive isotop es, nr their numb er density relative to hydrogen, and E (eV) the energy of the pro duced radiation. Similarly, = 37 eV is the average energy for the pro duction of an ion pair in H2 gas (Consolmagno & Jokippi 1978 and references therein, Shull & Van Steenburg 1985, Voit 1991). In the present study we have adopted,

R = 6.9 â 10

-23

[2 + (1 - 2 )fg ] s

-1

(3.34)

atoms in the gas phase, estimated via measurements of interstellar absorption lines

(Umebayashi & Nakano 1981, 1990). Here, 2 0.02 is the fraction of heavy metal in diffuse clouds (Morton 1974). It is, of course, p ossible that 2 in discs could b e even smaller than this value. On the other hand, fg is a parameter that takes into

account the degree of sedimentation of dust grains in protostellar discs with resp ect to interstellar values (Hayashi 1981, Sano et. al. 2000). Although the ionisation effect contributed by this agent is very small compared with that of x-rays and cosmic rays, we included it b ecause it may well b e the only mechanism active in regions close to the midplane (particularly for R 5 AU) in the scenario where cosmic rays are excluded from the disc. In this case, it is imp ortant to explore the sensitivity of MRI p erturbations to changes in the level of depletion of dust grains (fg ). We have assumed here that the ionised comp onent of the fluid is made of ions and electrons only, a case that corresp onds to late evolutionary stages of protostellar discs, after dust grains have settled towards the midplane. Accordingly, we assume fg = 0 in all our mo dels, except when sp ecifically exploring the role of radioactivity in a disc where cosmic rays are assumed to b e excluded from it.


3.2. Formulation Recombination rate

71

Gas-phase recombination o ccurs through disso ciative recombination of electrons with molecular ions and, at a slower rate, via radiative recombination with metal ions. It has b een p ointed out by previous authors that the ionisation balance may b e esp ecially sensitive to the presence of metal atoms within the disc. For some disc configurations, a numb er density of metals as small as 10
-7

times the cosmic

abundance may b e enough to make the whole cross-section of the disc turbulent, eliminating the central magnetic dead zone (Fromang et. al. 2002). This is so b ecause metal atoms generally take the charges of molecular ions, but recombine with electrons (via radiative pro cesses) at a much slower rate. If n+ and n m
+ M

are the numb er densities of molecular and metal ions, resp ectively,

the rate equations for ne and n+ can b e expressed as (e.g. Fromang et. al. 2002): m dne = n H - n e n+ - r ne n m dt and dn+ m = n H - n e n+ - t nM n+ , m m dt (3.36)
+ M

(3.35)

where is the total ionisation rate, calculated as summarised in the previous sections, and nH is the hydrogen numb er density from equation (3.29). From (3.35) and (3.36) it follows that, in equilibrium, r n e n
+ M

= t n+ nM . m

(3.37)

In the previous expressions, is the disso ciative recombination rate co efficient for molecular ions, r is the radiative recombination rate co efficient for metal ions and t is the rate co efficient of charge transfer from molecular ions to metal atoms. If it is assumed that all metals are lo cked into dust grains, which have in turn sedimented towards the midplane of the disc (i.e. n (3.36), together with charge neutrality, ne = n the rate co efficients, r = 3 â 10
-11 + M M

+ n+ , and appropriate values for m

0), equations (3.35) and

T

-1/2

cm3 s

-1

,

(3.38)


72

Chapter 3. The instability in protoplanetary discs

= 3 â 10-6 T

-1/2

cm3 s

-1

,

(3.39)

t = 3 â 10-9 cm3 s

-1

,

(3.40)

(see Fromang et al 2002 and references therein), lead to nH .

ne

(3.41)

However, as these authors p oint out also, dust grains not only absorb metal atoms, but also release them as a result of the action of x-rays. Because of this effect, the abundance of metal atoms in a disc could b e quite insensitive to the spatial distribution of dust grains. This, in turn, means that dust settling do es not necessarily lead to a severe reduction in the numb er density of metal atoms in the gas phase. If these dominate, the corresp onding ne is nH . r
M

ne that r ne n
+ M

(3.42) is such

The transition from one regime to the other is taken here to o ccur when x

> ne n+ . When this condition is satisfied, electrons are more likely m

to recombine radiatively with metal ions than they are to recombine disso ciatively with molecular ions (see equation 3.35). Using (3.37), this leads to: x > xe = 103 T t
-1/2

M M

xe .

(3.43)

Note that this minimum x

for radiative recombination to dominate is more x
+ m

stringent (by 5 orders of magnitude) than the expression given by Fromang et. al. (2002). These authors identified the transition with x x
M M

and obtained
r

> 10-2 T

-1/2

xe for metals to b e dominant. However, given that

, when

this condition is satisfied electrons will still b e more likely to recombine with molecular ions. As a result, in this study we have adopted expression (3.43) as the minimum x
M

for radiative recombination to dominate. The electron numb er density ne and, r / 300 times lower than the

hence, the magnetic coupling (see section 3.3.2) in the limit where disso ciative recombination dominates (equation (3.41)) are one obtained in the metal-dominated limit (equation (3.42)). However, the former


3.3. METHODOLOGY recombination mechanism dominates when r ne n mum x
M + M

73 < ne n+ , which gives a maxim

given by expression (3.43) with the appropriate xe for this limit (equation

mum abundance for radiative dominance. As a result, the transition b etween these in ne ­and ­ is only realised when x

(3.41)). This metal abundance is, similarly, a factor of 300 lower than the mini-

two regimes is not instantaneous: it o ccurs over a range of xM . The full reduction
M

of equation (3.43). For the fiducial mo del this o ccurs 1.5 scaleheights ab ove the

is a factor of 300 b elow the critical value

lo cation indicated by this expression. Finally, if X-rays are able to lib erate metal atoms from dust grains and the metal abundance b ecomes fairly indep endent of the dust spatial distribution (Fromang et. al. 2002), then the actual x
M

can b e up to a

factor of 1/2 larger than the value adopted in this work (see section 3.4.2). If this is the case, the transition (from radiative to disso ciative recombination regimes) will take place even higher ab ove the midplane. From the previous discussion, it is clear that the evolutionary stage ­and activity­ of the disc are imp ortant factors in the ionisation balance, as they influence the degree of sedimentation of dust grains. As the disc evolves, dust grains tend to o ccupy a thin layer around the midplane, b ecoming removed ­in any dynamical sense­ from the gas at higher vertical lo cations. This causes the ionisation fraction of the gas to increase, by eliminating recombination pathways on dust surfaces. In the present work, we have used equation (3.42) to calculate the electron (and ion) numb er density. The minimum values of x
M

for this approximation to b e valid

(equation (3.43)) have also b een computed and compared with an estimate of metal abundances in the gas phase (see section 3.4.2). Results indicate that, for the range of parameters adopted here, the abundance of metal atoms is such that radiative recombination is indeed dominant, except in the upp er regions of the disc, so the use of equation (3.42) to calculate the electron fraction is justified.

3.3
3.3.1

METHODOLOGY
Linearisation

Full details of the metho dology are describ ed in SW03. For the sake of clarity, and completeness, we summarise here the most imp ortant steps and p oint at some differences with the previous formulation. The system of equations (3.1) to (3.3),


74

Chapter 3. The instability in protoplanetary discs

(3.6) and (3.7) was linearised ab out an initial steady state where fluid motion is exactly Keplerian and the magnetic field is vertical, so J = v = E = 0 and B = B z . ^ In the initial state b oth E and J vanish, so the changes in the conductivity tensor due to the p erturbations do not app ear in the linearised equations. As a result, it is not necessary to explore how the p erturbations affect the conductivity of the fluid and only the values in the initial steady state are required. We assume that the wavevector of the p erturbations is p erp endicular to the plane of the disc (k = kz ). These p erturbations, initiated from a vertical magnetic field, are the fastest growing mo des when the fluid is in either the Hall or resistive conductivity regimes, as in these cases magnetic pressure suppresses displacements with a non-zero radial wavenumb er (Balbus & Hawley 1991, Sano & Miyama 1999). However, as p ointed out by Kunz & Balbus (2004) and Desch (2004), this is not necessarily the case for p erturbations in the ambip olar diffusion limit. The results of these studies indicate that in this regime, the fastest growing mo des may exhibit b oth radial and vertical wavenumb ers. Taking p erturbations of the form q = q0 + q(z )e
i t

ab out the initial state,

linearising and neglecting terms of order H/r or smaller, we find that the final linear system of equations that describ es the MHD p erturbations within the disc is, i vr - 2 v - i v + 1 vr + 2 i Br - c i B + c B0 J = 0 , c B0 Jr = 0 , c (3.44)

(3.45)

d E = 0, dz

(3.46)

d Er 3 + 2 Br = 0 , dz c d B , 4 dz

(3.47)

Jr = - J =

(3.48)

c d Br , 4 dz

(3.49)

Jr = 2 E r - 1 E ,

(3.50)


3.3. METHODOLOGY

75

J = 1 E r + 2 E ,

(3.51)

where E and Er are the p erturbations of the electric field in the lab oratory frame, E = E + B0 vr , and c (3.52)

B0 v . (3.53) c Note that , the comp onent of the conductivity tensor parallel to the magnetic Er = E r - field, do es not app ear in the linearised equations. This, in turn, implies that in the linear phase of the MRI and under the adopted approximations, the ambip olar diffusion and resistive conductivity regimes b ehave identically (see section 3.2.1). We express equations (3.44) to (3.53) in dimensionless form by normalising the variables as follows: z = z H v cs J = = (r, z ) 0 (r ) c E cs B 0 B = B B0 c E cs B 0

v =

E =

E =

c J = c s B 0 Here, subscript `o' denotes variables at the midplane of the disc. Note that we have used the lo cal


instead of 0 , used in SW03, to normalise and J.

This is more useful when dealing with a height-dep endent conductivity. In the following dimensionless system, we have dropp ed the asterisks and expressed the final equations in matrix form: B
r

0 -C1 A B 0 d = dz 3 Er - 2 - 0 0 0 E





0

0

C1 A

1

C1 A

2

2

C1 A3 0 0



B

r

B Er E

(3.54)


76

Chapter 3. The instability in protoplanetary discs

J = C2 v =



A A

2

-A A -2
2

3



1

E
1 2

(3.55)

1 1 + 2





- -
2

J

(3.56)

where

1 E =





1



-

1

2

J

(3.57)

=

i ,
-2

(3.58)

C1 = 1 C2 = 1 + 1+ A1 =

vA cs

C2 ,
-1

(3.59)

2

5 1 2 + 2 + 2

,

(3.60)

1 1 + 2 , 1 + 2

(3.61)

A2 =

1 2 + , and 1 + 2 1 1 1 + . 2 1 + 2

(3.62)

A3 = In the ab ove expressions,

(3.63)

Bo vA = 4 is the lo cal Alfv´ sp eed in the disc, and en
2 1 B o c c2

(3.64)

(3.65)


3.3. METHODOLOGY

77

controls the lo cal coupling b etween the magnetic field and the disc (W99, SW03, see also section 3.3.2). Non-ideal effects strongly mo dify wavemo des at, or ab ove, the critical frequency c .

3.3.2

Parameters

The following parameters control the dynamics and evolution of the fluid: 1. vA /cs , the ratio of the lo cal Alfv´ sp eed to the isothermal sound sp eed of the en gas. It is a measure of the strength of the magnetic field. In ideal-MHD conditions, unstable mo des grow when the magnetic field is subthermal (vA /cs < 1). stability is of the order of the scale height of the disc and the growth rate or ohmic diffusion dominates over the entire cross-section of the disc (SW03). However, when the fluid is in the Hall regime, with the magnetic field counteraligned with the angular velo city vector of the disc ( · B < 0), MRI unstable mo des may exist for stronger fields, up to vA /cs 3 (SW03). Under this approximation, when vA cs the minimum wavelength of the indecreases rapidly. This is also the case under the assumption that ambip olar

2. , a parameter that characterises the strength of the lo cal coupling b etween the magnetic field and the disc (equation (3.65)). It is given by the ratio of the critical frequency ab ove which flux-freezing conditions break down and the dynamical (Keplerian) frequency of the disc. If c / < 1, the disc is p o orly coupled to the disc at the frequencies of interest for dynamical analysis, which are also the interesting frequencies for the study of the MRI. 3. 1 /2 , the ratio of the conductivity terms p erp endicular to the magnetic field. It is an indication of the conductivity regime of the fluid, as discussed in section 3.2.1. We calculated the values of the three parameters ab ove at different lo cations (r , z ) within the disc using equations (3.9), (3.10), (3.17), (3.64) and (3.65). The strength of the magnetic field is a free parameter: we consider field strengths in excess of 1 mG.


78

Chapter 3. The instability in protoplanetary discs

3.3.3

Boundary Conditions

To solve equations (3.54) to (3.57) it is necessary first to integrate the system of ordinary differential equations (ODE) in (3.54). This can b e treated as a twop oint b oundary value problem for coupled ODE. Five b oundary conditions must b e formulated, prescrib ed either at the midplane or at the surface of the disc. At the midplane: We chose to assign o dd symmetry to Br and B , so they vanish at z = 0. This gives us two b oundary conditions at the midplane. Also, as the equations are linear, their overall scaling is arbitrary, and a third b oundary condition can b e obtained by setting one of the fluid variables to any convenient value. To that effect, we assigned a value of 1 to Er . The three b oundary conditions applied at the midplane are, then:

Br = B = 0, and

Er = 1 . At the surface: is inversely prop ortional to the density (equation 3.65), so if the conductivity is assumed to b e constant with height, it increases monotonically with z . This was the case in SW03, where we used this argument to prop ose that at sufficiently high z ab ove the midplane, ideal MHD conditions held. This, in turn, implies that the wavelengths of magnetic field p erturbations, given the adopted dep endency of with z (equation (3.26)), must tend to infinity when z . As a result, the amplitude of such mo des should this argument needs revisiting. In section 3.4.3 we analyse the dep endency of with height for different radial p ositions and a range of magnetic field strengths. Here, we just highlight that at the surface of the disc (z /H = 6, see b elow), the magnetic coupling is still strong. For example, for the fiducial mo del at R = 1 AU, 20. It decreases to 4.5 for R = 5 AU and to 2.5 for R = 10 AU. This is still ab ove the limit ( 1) for strong coupling (W99). When > 10 the growth rate and characteristic wavenumb er of lo cal unstable mo des differ little from the ideal limit (W99), so the same line of reasoning vanish at infinity as well. Here, however, the conductivities vary with z , so


3.3. METHODOLOGY

79

of SW03 can b e used to argue that in this case Br and B should b e zero at the b oundary as well. Although at the surface when R 5 AU is b elow 10, it is considered that the magnetic coupling there is still sufficiently strong for the ideal-MHD approximation to b e essentially valid as well. This is also confirmed by the way in which magnetic field p erturbations tend to zero at the b oundary at these radii (see results in sections 3.4 and 3.5). Consistently with this, b oth E and J vanish at the surface as well. This gives us the remaining two b oundary conditions required to integrate the system of equations (3.54). Gammie & Balbus (1994) obtained similar b oundary conditions for their `hot halo mo del', where the disc is terminated in a hot, p erfectly conducting halo at a height zh ab ove the midplane. These authors showed that the p erturb ed magnetic field in this halo is approximately force-free, so ( â B)âBo 0, and along the unp erturb ed field lines. For vanishing stress at infinity ­the condition at the surface of the disc. we are adopting here­, a 0, which recovers the conditions Br = B = 0 We chose to lo cate the b oundary at z /H = 6, after confirming that increasing this height do es not significantly affect either the structure or the growth rate of unstable mo des. Summarising, the b oundary conditions adopted at the surface are: â B = aBo . In the last expression, `a' is a function that is constant

Br = B = 0 at z /H = 6. This system of equations is solved by `sho oting' from the midplane to the surface of the disc while simultaneously adjusting the growth rate and E at the midplane. The adjustment is done via a multidimensional, globally convergent NewtonRaphson metho d, until the solution converges. The pro cedure involves supplying initial ­guessed­ values for these two variables to start the iteration. In order to avoid missing an unstable mo de ­in particular, the fastest growing mo de­ we succesively tried guessed values of Given that
max max

in the range of 0.1 to 1.0, in intervals of 0.01.

< 0.75 (BH91) we are confident that the most unstable mo de is

not missed, even though it can not b e guaranteed that all mo des are always found.


80

Chapter 3. The instability in protoplanetary discs

3.4
3.4.1

DISC CONDUCTIVITY
Test Mo dels

Our fiducial mo del is based on the solar nebula disc (Hayashi 1981; Hayashi, Nakazawa & Nakagawa 1985; see section 3.2.2). The structure and growth rate of MRI unstable mo des are calculated at representative radial p ositions (R = 1, 5 and 10 AU) from the central protostar. Two scenarios are explored: Cosmic rays either p enetrate the disc, although attenuated as appropriate (as given by eq. (3.30)), or they are excluded from it by the winds pro duced by the central ob ject. Unless stated otherwise, results presented using this mo del include cosmic ray ionisation. We also consider a disc mo del with an increased mass and surface density, as describ ed in section 3.2.2. For simplicity, other disc parameters remain unchanged and in this case, all calculations incorp orate the ionisation rate provided by cosmic rays. We also compare solutions obtained using different configurations of the conductivity tensor. By comparing the structure and growth rate of unstable mo des found using the commonly adopted ambip olar diffusion ­ or resistive ­ (1 = 0) approximations with those obtained with a full conductivity tensor (1 Bz > 0, 2 = 0), as well as the less common Hall limit (2 = 0, 1 Bz > 0), we can appreciate how, and in which regions of parameter space, Hall conductivity alters the prop erties of the instability. This is explored for the fiducial mo del at 1 AU only. The prop erties of the MRI in the Hall limit are dep endent on the alignment of the magnetic field and angular velo city vectors of the disc (W99). The case when these vectors are parallel (antiparallel) is characterised by 1 Bz > 0 (1 Bz < 0). In the Hall (1 Bz < 0) limit, our co de fails to converge whenever the combination of parameters is such that < 2 anywhere in the domain of integration. This is not surprising as, in this regime, when 0.5 < < 2 all wavenumb ers are unstable (W99). As a result, we explored the effect of the alignment of B and by comparing solutions obtained with a full conductivity tensor but incorp orating 1 Bz terms of opp osite sign. Even with this approach, solutions including a 1 Bz < 0 term could not b e computed at all for R = 1 AU, given that the midplane coupling at this radius is extremely low ( 10
-10

to 10

-2

for B b etween 1 mG and 1 G). Therefore, 8 mG at 5 AU and B 1.5 mG at 10

we present this analysis at 5 and 10 AU only. Solutions incorp orating a negative

Hall conductivity could b e obtained for B


3.4. DISC CONDUCTIVITY AU, where the magnetic coupling at the midplane is 1 Bz > 0 Hall term contribution. 2.

81

Unless stated otherwise, results discussed in the following sections incorp orate a

3.4.2

Ionisation Rates

Fig. 3.1 shows the ionisation rate contributed by cosmic rays, x-rays and radioactive materials as a function of height for the fiducial mo del. Results are presented at the radial lo cations of interest: 1, 5 and 10 AU. Note that x-rays p enetrate up to z /H 1.7 and z /H 0.3 for R = 1 and 5 AU, resp ectively. They reach the midplane for R = 10 AU. The x-ray ionisation rate is heavily attenuated b elow At 1 and 5 AU, the regions where x-rays are not able to p enetrate are ionised 2.2 at all studied 1.5 to 2 scaleheights from the surface, dep ending on the radius.

only by the action of cosmic rays (if present) and radioactive decay. Moreover, cosmic rays constitute the most efficient ionising agent for z /H radii. As a result, if they are excluded from the disc by the protostar's winds (i.e. Fromang et al. 2002), the magnetic coupling of the gas in these sections of the disc is exp ected to b e severely reduced. In order to explore the prop erties of MRI p erturbations under this assumption, we compare results computed including and excluding this source of ionisation for the fiducial mo del (sections 3.5.1 and 3.5.1 resp ectively). Igea & Glassgold (1999) have p ointed out that in protostellar discs, the x-ray ionisation rate can b e 103 - 105 times that of interstellar cosmic rays. Their calculations show that x-rays dominate over cosmic rays until the vertical column density, N , is 1025 , 5 â 10
23

incident photons available. The ionisation rates adopted in this study (Fig. 3.1) are in agreement with these findings. We have used eq. (3.42) to calculate the electron numb er density (section 3.2.3). This expression is valid when metal atoms dominate and electrons recombine primarily via radiative pro cesses. Here, we justify this choice. Fig. 3.2 compares the minimum abundance of metals, (eq. (3.43)), for this approximation to b e valid with an estimate of total metal abundances in the gas phase. Minimum abundances (relative to hydrogen) are shown for the radial lo cations of interest for the fiducial

N exceeds 10 cm , x-ray ionisation dies out b ecause there are no remaining

25

and 2 â 1023 cm
-2

-2

for R = 1, 5 and 10 AU, resp ectively. Once


82

Chapter 3. The instability in protoplanetary discs

& Nakano 1990). From Fig. 3.2 it is clear that the abundance of gas-phase metal

mo del. The total gas-phase metal abundance is taken as 8.4 â 10-5 2 (Umebayashi atoms is exp ected to exceed the minimum for radiative recombination pro cesses to

b e dominant at all radii of interest here, except in the upp er sections of the dics (i.e. z /H 3.5 at 1 AU). Given that the fastest growing MRI mo des are strongly is a factor of damp ed in these regions (see Fig. 3.9), and that the full impact on ne (and ) when disso ciative recombination dominates is not achieved until x
M

300 b elow the critical value of equation (3.43), (see section 3.2.3), we conclude density in this study.

that equation (3.42) is still a valid approximation to calculate the electron numb er

3.4.3

Magnetic coupling and conductivity regimes as a function of height

It is interesting to explore which conductivity terms (section 3.2.1) are dominant at different heights at the representative radial lo cations we are considering. This information will b e used in the analysis of the structure and growth of MRI unstable mo des in the next section. We recall that Hall diffusion is lo cally dominant when (SW03), < When is higher than this, but still |1 | . (3.66) 10, ambip olar diffusion dominates. In regions
cr it



where is stronger, ideal-MHD conditions hold (W99). We remind the reader than in the present formulation, the ambip olar and Ohmic conductivity regimes b ehave identically. This is the case b ecause (the conductivity comp onent that distinguishes b etween them), do es not app ear in the final, linearised system of equations (see section 3.3.1). This should b e kept in mind when analysing the results, as even though the case where 1 = 0 has b een referred to as the `ambip olar diffusion' limit ( We are interested in comparing and 2 ), it is also consistent with the `resistive' regime ( 2 ).
cr it

as a function of height for different

disc mo dels, radial lo cations and choices of the magnetic field strength. An example of the typical dep endency of these variables with z is shown in Fig. 3.3 for the fiducial mo del at R = 1 AU for B = 10 mG. Note that the conductivity term parallel to the field increases monotonically with z and is indep endent of B (see eqs.


3.4. DISC CONDUCTIVITY

83

Figure 3.1

Ionisation rate ( s

-1

) contributed by cosmic rays (curve lab eled `cr'), x-rays

(xr) and radioactive materials (rad) as a function of height, for the fiducial mo del (minimum mass solar nebula disc). Results are shown for R = 1, 5 and 10 AU. Note that at 1 AU, cosmic ray ionisation is attenuated with resp ect to the interstellar rate for z /H Also, x-rays are excluded from the disc for z /H 1.7 (at 1 AU) and z /H 2. 0.3 (at 5 2.2 in

AU). For R = 10 AU, they reach the midplane, although their ionisation rate is heavily attenuated. Cosmic rays (if present) are the most efficient ionising agent for z /H all cases.


84

Chapter 3. The instability in protoplanetary discs

Figure 3.2 x
M

Comparison of the minimum abundance of metal atoms in the gas phase,

= nM /nH , for the radiative recombination rate of metal ions ( r ) to b e the dominant

recombination mechanism, and an estimate of the total gas-phase metal abundance, as a function of height. Minimum abundances are shown for the fiducial mo del at 1 AU (solid line), 5 AU (dashed line) and 10 AU (dotted line). The estimated total gas-phase metal abundance (dot-dashed line) is from Umebayashi & Nakano (1990). This abundance exceeds the minimum required for all radii and vertical lo cations of interest in this study, except in the upp er sections of the disc. This is not exp ected to significantly affect the results presented in this study, given that the fastest growing MRI mo des are strongly damp ed in these regions.

(3.8) and (3.12)). In this case, |1 | is nearly two orders of magnitude smaller than in the resistive regime (section 3.2.1). This is consistent with findings by Wardle (in preparation), that Ohmic conductivity is imp ortant at this radius for relatively weak fields (B 100 mG) when the density is sufficiently high (e.g. n
H

2 at the midplane. As a result, near the midplane

2

|1 |, and the fluid is

1012 cm

-3

for

B = 1 mG). Both comp onents of the conductivity tensor p erp endicular to the field increase with height initially, reflecting the enhanced ionisation fraction at higher z . regime) and finally, b oth terms drop abruptly closer to the surface as a consequence of the fall in fluid density. Note that the Hall conductivity term decreases much more sharply than the ambip olar diffusion comp onent. As a result, the ambip olar diffusion term is typically several orders of magnitude greater than the Hall term in the surface regions of the disc. To reiterate this concept, we rep eat here that even in There is then a central section of the disc where |1 | 2 (the fluid is in the Hall


3.4. DISC CONDUCTIVITY regions where |1 |

85

2 , the growth and structure of the MRI in the linear regime |1 |/ (SW03).

are still dominated by Hall diffusion if diate heights, where |e | 1 while

The previous results are in agreement with the notion that |1 | > 2 at intermei

the fluid is in the ambip olar diffusion regime. In these conditions, equations (3.9) nent may dominate, dep ending on the fluid density and the strength of the magnetic field. |1 | will b e larger if the fluid density is relatively small (larger radius) and/or the field is stronger. Conversely, if the field is weak and the density is sufficiently
2

1. At higher z , typically |e |



i

1 and

and (3.10) show that |1 |

2 . On the contrary, near the midplane either comp o-

high, of
cr it

conductivity comp onents as a function of height explains, in turn, the dep endency with z : For weak fields it is
cr it

|1 | and the conductivity will b e resistive. This b ehaviour of the 1 near the midplane (
2

is sufficiently strong,

levels in the intermediate sections of the disc at 1 (|1 | sharply near the surface (
2

close to the midplane 1. It then increases with z and 2 ). Finally,
cr it

|1 |), but if B

drops

We now briefly turn our attention to the dep endency of the magnetic coupling with height. Note that first increases with z , as exp ected, in resp onse to the enhanced ionisation fraction, and lower fluid density, away from the midplane. Closer to the surface, it decreases again, as a result of the abrupt drop in the magnitudes of the conductivity comp onents. In the following subsections we explore more fully the dep endency of versus
cr it

|1 | again).

with height, which indicates which non-ideal MHD

regime is lo cally dominant, for the disc mo dels of interest.

Minimum-mass solar nebula disc Figs. 3.4 to 3.6 present curves of (solid lines) and
cr it

as a function of height for the fiducial mo del. Results are shown for R = 1, 5 and 10 AU and different choices of the magnetic field strength. Cosmic rays are either included (top panel of each figure) or excluded from the disc (b ottom panels). Bottom (solid) and leftmost (dashed) curves corresp ond to B = 1 mG in all cases. B changes by a factor of 10 b etween curves, except that for 5 and 10 AU the top (and rightmost) curves corresp ond to the maximum field strength for which p erturbations

|1 |/ (dashed lines)

grow. Fluctuations in the magnetic coupling at low z /H (lower panel of Fig. 3.5) reflect fluctuations in the X-ray ionisation rate (Fig. 3, Igea & Glassgold 1999) in


86 this region of parameter space.

Chapter 3. The instability in protoplanetary discs

We note that in all plots, the magnetic coupling near the midplane increases with B . In these regions of the disc, b oth i and e are typically when B is very strong) and from equation (3.11), the j can b e very large and cene (i - e ) , B (3.67) 1 (except p ossibly

which is indep endent of B . As a result, B 2 . On the contrary, near the surface cene (i - e ) 1 2. B e i B

in Figs. 3.4 to 3.6.

(3.68)

Because of this, near the surface is quite insensitive to changes in B , as evidenced Note that when cosmic rays are assumed to b e excluded from the disc, there is a discontinuity in the curve of vs. z /H at R = 1 and 5 AU. This discontinuity is caused by the drop in the ionisation fraction at the height where x-rays are not able to p enetrate any further within the disc (see b ottom panels of Figs. 3.4 and 3.5). It is, therefore, not present at 10 AU, b ecause x-rays reach the midplane at this radius (see b ottom panel of Fig. 3.6). In general, when B increases, the region around the midplane where Hall diffusion coupling. At 1 AU, when cosmic ray ionisation is included, this criterion is satisfied is lo cally dominant ( < |1 |/ ) is reduced, as a result of the stronger magnetic in the inner sections of the disc for all magnetic field strengths of interest (Fig. 3.4, top panel). Ab ove this region, there is a relatively small section where ambip olar diffusion is lo cally dominant while near the surface, ideal-MHD holds for all magnetic field strengths. For R = 5 AU (Fig. 3.5, top panel), Hall diffusion dominates for z /H when B 10 mG, but for 10 mG B 2 100 mG, ambip olar diffusion is lo cally

the magnetic coupling is such that > 10 even at the midplane and the fluid is in ideal-MHD conditions over the entire cross-section of the disc. At this radius, the fluid is in the resistive regime only for very weak fields (B
12 -3

dominant in the inner sections of the disc (|1 |/ < < 10). For stronger fields,

1 mG) and very high

densities (nH > 10 cm ; Wardle, in prep.). Similarly, results at R = 10 AU (Fig. 3.6, top panel), show that ambip olar diffusion dominates in the inner sections of the


3.4. DISC CONDUCTIVITY

87

disc for B < 10 mG. For stronger fields, ideal-MHD is a go o d aproximation at all z . When cosmic rays are assumed to b e excluded from the disc, the previous results at 1 AU (b ottom panel of Fig. 3.4) are largely unchanged, except for those obtained the previous case. This is due to the sharp fall in the magnetic coupling in the region which x-rays are unable to reach. At 5 AU, the Hall regime is now dominant near the midplane for all studied magnetic field strengths, while for R = 10 AU there is now a Hall dominated central region for B B fields ideal MHD holds for all z . Finally, we observe that (as exp ected), the magnetic coupling at the midplane increases with radius in resp onse to the higher ionisation fraction in the central sections of the disc at larger radii. In fact, o increases by 3 - 4 orders of magnitude b etween R = 1 and 10 AU. On the contrary, the coupling at the surface do es not change as much with radius, decreasing only from 22 to 2.5 b etween the same radii. 100 mG, ambip olar diffusion dominates near the midplane, but for stronger 10 mG. It extends to z /H 1.3. When with B = 1 G, where Hall diffusion dominates now for z /H 1.8, up from 1 in

A more massive disc Fig. 3.7 displays curves of and
cr it

as a function of height for a more massive

disc, as detailed in section 3.2.2. The ionisation balance is calculated assuming that cosmic rays p enetrate the disc and results are shown for R = 1, 5 and 10 AU for the same range of magnetic field strengths explored in the minimum-mass solar nebula mo del. Increasing the surface density causes the ionisation fraction near the midplane to drop sharply. As a result, the magnetic coupling is drastically reduced in these regions at all radii. This is esp ecially noticeable at 1 AU, where in this case x-rays are completely attenuated for z /H < 2.6. This, together with a very weak ionisation rate (
CR

low ­ and only weakly dep endent on z ­ in this section of the disc.

10

-18

for z /H 2.6 and it is negligible at z = 0), causes to b e very

The weaker coupling at low z results in Hall diffusion b eing lo cally dominant over a larger cross-section of the disc in this mo del. We find that for R = 1 AU, there is again a Hall dominated region ab out the midplane for all studied B . This section now extends to z /H 2.3 for B = 1 G, up from z /H 1 in the minimum-mass


88

Chapter 3. The instability in protoplanetary discs

solar nebula disc. For weaker fields, these regions are also larger. For R = 5 AU, Hall diffusion is now dominant near the midplane for B < 1 G. For comparison, in the previous case Hall conductivity was lo cally imp ortant for B < 10 mG. Furthermore, ambip olar diffusion dominates over a small cross-section of the disc for all but the strongest (B B B 1 G) magnetic field strengths at this radius. This contrasts with the previous case, where ideal MHD held at all z when 100 mG. Even at 10 AU, Hall diffusion now dominates near the midplane when 10 mG. When B is stronger than this, but still 10 mG. 100 mG, ambip olar diffusion 2, while in the minimum-mass solar nebula, ideal-MHD

predominates for z /H held for B

3.4.4

Magnetic field strength

In the following sections we will discuss how the prop erties of the MRI dep end on the strength of the unp erturb ed magnetic field Bo . The origin of this field, as well as its likely configuration in protoplanetary discs, remains unclear. Two main scenarios have b een considered in the literature to explain the magnetisation of discs (e.g. Stepinski 1995): On the one hand, the field may originate in the molecular cloud from which the material that forms the protostar­disc system has collapsed. In this case the field is advected in, as well as amplified, during the collapse (Pudritz & Norman 1983, K¨ onigl 1989). This is an attractive picture, b ecause the resulting field configuration seems favourable for driving hydromagnetic winds, so commonly observed in young stellar ob jects. On the other hand, the field can b e generated within the disc itself, as a result of dynamo action (Tout & Pringle 1992, 1996; Stepinski 1995). In this scenario, the dynamo amplifies an arbitrarily small preexisting seed field. The magnetorotational instability itself has b een asso ciated with a self-sustaining dynamo mo del (Tout & Pringle 1992): The disc shear turns Br into B while the MRI creates b oth horizontal field comp onents from the vertical field. Finally, B
z

is generated from b oth Br and B by the Parker instability. It has b een argued that this mechanism can generate a quasi-steady-state dynamo with an asso ciated viscosity parameter 1 (Tout & Pringle 1992). It may also mantain a distribution of flux lo ops in all scales through an `inverse cascade' pro cess (Tout & Pringle 1995), which pro duces large lo ops from the original smaller ones via reconnection of the


3.5. MAGNETOROTATIONAL INSTABILITY field lines. As a result, the original flux lo ops of length-scale launching of centrifugally-driven winds and jets.

89 H pro duced by

the dynamo can b e transformed into lo ops of the appropriate scale ( R) for the As mentioned ab ove, a shared consensus on whether the magnetic field in protoplanetary discs is generated in the disc or in the parent cloud has not b een reached. We do not explore this issue any further, given that in this study the initial magnetic field Bo is a free parameter. In the next section, we analyse the effect of different conductivity regimes b eing dominant at different heights, in the structure and growth rate of MRI p erturbations. Different disc mo dels, radial lo cations and magnetic field strengths are discussed.

3.5
3.5.1

MAGNETOROTATIONAL INSTABILITY
Structure of the p erturbations

All unstable mo des at 1 AU Fig. 3.8 shows the structure and growth rates of all unstable MRI p erturbations for the fiducial mo del at 1 AU and B = 100 mG. Solid lines denote Br while dashed lines corresp ond to B . This notation is observed in this pap er in all plots that display the structure of the instability. The fastest growing p erturbation in this case grows at = 0.5020 and there are 15 unstable mo des with 0.1151 0.5020. Slow growing p erturbations, with < 0.1806, are active even at the midplane, while faster mo des exhibit a central magnetically inactive (dead) zone (Gammie 1996, Wardle 1997). Moreover, the extent of the dead zone increases with the growth rate and for the fastest growing growing mo des, are asymmetrical ab out zero and their averages over vertical sections these two fluid variables in equation (3.47). Most unstable mo des at different radii, including cosmic ray ionisation The weakest strength for which unstable mo des could b e computed was 1 mG. This is a computational, rather than a physical limit: the dep endency of
max

mo de it extends to z /H 1.6. We observe that Br and B , esp ecially in the slow

of the disc have opp osite signs. This app ears to b e related to the dep endency of

with B


90

Chapter 3. The instability in protoplanetary discs

Figure 3.3

Example of the dep endency of the conductivity comp onents parallel ( )
cr it

and p erp endicular (|1 | and 2 ) to the field, magnetic coupling () and and B = 10 mG. Note that for z /H 2,
2

|1 |/



as a function of height. Results shown corresp ond to the fiducial mo del for R = 1 AU |1 | and the conductivity is 2 (the fluid is in the Hall
2

resistive. There is then a central section for which | 1 | conductivity regime), while for higher vertical lo cations diffusion dominates. In the region where <
cr it

|1 | and ambip olar

, Hall diffusion dominates the structure

and growth rate of the MRI (SW03). Note also that could b e reduced by up to a factor of 300 near the surface if disso ciative ­ and not radiative ­ recombination dominates the ionisation balance at high z ab ove the midplane.


3.5. MAGNETOROTATIONAL INSTABILITY

91

Figure 3.4 Comparison of the lo cal magnetic coupling (solid lines) and

cr it

|1 |/



(dashed lines) for the fiducial mo del at R = 1 AU and different choices of the magnetic field strength. In each case, Hall diffusion is dominant in the regions where < (SW03). Ambip olar diffusion dominates where
cr it cr it

<

10. When is stronger than

this, the fluid is in nearly ideal-MHD conditions (W99). From top to b ottom (solid lines), and right to left (dashed lines), the magnetic field drops from 1 G to 1 mG. B changes by a factor of 10 b etween curves in all cases. Top panel: Cosmic rays are present. Hall diffusion dominates for z /H 3.5 to z /H 1 for B increasing from 1 mG to 1 G, resp ectively. Bottom panel: Cosmic rays are excluded from the disc by protostellar winds. Note the discontinuity in at the height b elow which x-rays are completely attenuated.


92

Chapter 3. The instability in protoplanetary discs

Figure 3.5

As p er Fig. 3.4 for R = 5 AU. Top (and rightmost) lines corresp ond here

to B = 795 mG (top panel) and B = 615 mG (b ottom panel), which are the maximum field strengths for which p erturbations grow in each scenario. When cosmic rays are included, the Hall regime dominates near the midplane for B B < 100 mG, there is an intermediate region where | 1 |/


10 mG. Similarly, for

< < 10, and ambip olar

diffusion is dominant. For stronger fields, ideal-MHD holds for all z . When cosmic rays are excluded, Hall diffusion dominates near the midplane, and ambip olar diffusion is dominant at intermediate heights, for all B of interest here.


3.5. MAGNETOROTATIONAL INSTABILITY

93

Figure 3.6

As p er Figs. 3.4 and 3.5 for R = 10 AU. From top to b ottom (and right

to left), lines corresp ond here to B = 100, 10 and 1 mG. In this case, if cosmic rays are present, Hall diffusion is not dominant lo cally for any field strength, given the strong magnetic coupling near the midplane. Ambip olar diffusion dominates when B rays are removed, the Hall regime is relevant near the midplane only for B When B 100 mG, ideal MHD holds for all z . 10 mG, 10 mG. but for stronger fields ideal-MHD holds over the entire cross-section of the disc. If cosmic


94

Chapter 3. The instability in protoplanetary discs

Figure 3.7 As p er Figs. 3.4 to 3.6 for R = 1, 5 and 10 AU for a more massive disc, such that o = 10o and o = 10o . Cosmic rays are present. From top to b ottom (and right to left), lines corresp ond to B = 1, 0.1, 10 10
-2 -2

and 10

-3

G (R = 1 and 5 AU), and B = 0.1,

and 10

-3

G (R = 10 AU). The increased surface density causes the magnetic coupling

to drop near the midplane. As a result, Hall conductivity dominates over a larger section


3.5. MAGNETOROTATIONAL INSTABILITY

95

Figure 3.8

Structure of all unstable mo des for the fiducial mo del at R = 1 AU and

B = 100 mG. In each case solid lines show B r while dashed ones corresp ond to B . The growth rate is indicated in the lower right corner of each panel. There are 15 unstable mo des, with 0.1151 0.5020. Some p erturbations grow at the midplane, particularly when < 0.1806. Mo des growing faster than this show a dead region around the midplane, whose vertical extent increases with the growth rate of the p erturbations. Note the asymmetry of Br and B ab out zero, esp ecially in slowly growing mo des.


96

Chapter 3. The instability in protoplanetary discs

in the weak-field limit, shows no evidence for a minimum required field strength for the instability to grow (see section 3.5.2). A comparison of the structure of the most unstable mo des for the fiducial mo del at R = 1, 5 and 10 AU is shown in Fig. 3.9. We display solutions for B from 1 mG, up to the maximum field strength for which unstable mo des grow for each radius. Unless stated otherwise, this criterion is followed in all plots that show the structure of the fastest growing mo des as a function of B in this study. Note that the maximum value of B is dep endent on the disc mo del, the ionising agents incorp orated and radius of interest. Both the structure and growth rate of these p erturbations are shap ed by the comp eting action of different conductivity comp onents, whose relative imp ortance change with height. We defer the analysis of their growth rate to section 3.5.2. Here, we discuss the structure as a function of z . Note first of all that at each radius, the wavelength of the p erturbations increase with the magnetic field strength, as exp ected by b oth ideal-MHD and non-ideal MHD lo cal analyses (Balbus & Hawley 1991, W99). At 1 AU (leftmost column of Fig. 3.9), for B 500 mG, the region next to the midplane is a magnetically inactive zone (Gammie 1996, Wardle 1997). The extent of this region decreases as the field gets stronger. For example, for B = 1 is no appreciable dead region, a result of the stronger magnetic coupling close to the midplane and the relatively small wavenumb er of the p erturbations. Also, at this radius, when B is relatively strong (B > 500 mG), the amplitude of these mo des increase with z , a prop erty that is typical of MRI p erturbations driven by ambip olar diffusion. Ambip olar diffusion mo des have this prop erty b ecause, as the lo cal analysis (W99) indicates, in this limit the lo cal growth of unstable mo des increases with the magnetic coupling, which (except in the surface regions) increases with z (see Fig. 3.3). As a result, the lo cal growth rate of the MRI also increases with height in this regime and is able to drive the amplitude of global unstable mo des to increase. This explains the shap e of the envelop e of these mo des. Finally, note that when the magnetic field is weak, the p erturbations' wavenumb er is very high, and Br and B are not symmetrical ab out z = 0. At 5 AU, when B 100 mG, p erturbations grow even at the midplane. For 0.5. 100 mG, is fairly weaker fields, they exhibit only a very small dead zone which extends to z /H Note also that the envelop e of these mo des, particularly for B mG it extends from the midplane to z /H 1.8, but when B > 500 mG, there


3.5. MAGNETOROTATIONAL INSTABILITY

97

flat. This is explained by recalling that the magnetic coupling at the midplane in this region is very high ( 90 for B = 500 mG, Fig. 3.5, top panel), so the the no de closest to the midplane, given that the lo cal growth rate is not a strong function of and do es not vary by much with height (see also top row of Fig. 6 of SW03, which shows similar envelop es for p erturbations obtained with o = 100 and different configurations of the conductivity tensor). On the other hand, for B 100 mG, non-ideal MHD effects are imp ortant. When B = 10 mG, ambip olar 2.3. This conductivity term is likely to drive this diffusion is dominant for z /H ideal-MHD approximation holds. Under these conditions, unstable mo des p eak at

p erturbation's structure, as evidenced by the central dead zone and the envelop e p eaking at an intermediate height. For B = 1 mG, Hall conductivity is dominant close to the midplane (z /H 2), the region where the envelop e of this p erturbation p eaks. Furthermore, the high wavenumb er suggests that its structure is determined by lo cal effects. As a result, Hall conductivity is likely to drive the structure of this mo de. Finally, for R = 10 AU there is no appreciable dead zone for B 10 mG, given the strong magnetic coupling at this radius. Even for B = 1 mG, o 3, a figure that increases to 100 for B = 100 mG. As a result, the fluid is in nearly ideal-MHD conditions (see section 3.4.3), which explains the flat envelop e of these mo des.

Most unstable mo des at different radii, excluding cosmic ray ionisation We also calculated the structure of the most unstable mo des at the same three radial p ositions under the assumption that cosmic rays are excluded from the disc by protostellar winds (Fig. 3.10). In this case the central dead zones at 1 AU (B 100 mG) extend over a larger cross-section of the disc, given that cosmic rays (when present) are the main source of ionisation near the midplane at this radius (see section 3.4.2). Without them, the electron fraction b elow z /H 1.7 (where x-rays are completely attenuated) plummets, causing the amplitude of MRI p erturbations in this section of the disc to b e severely reduced. As a result, when B 100 mG, p erturbations are damp ed for z /H 2. Another difference with the previous case is noticeable for stronger fields (B 500 mG), where b efore the MRI

was active even at the midplane. In this case, the abrupt change in the ionisation balance at the height where x-ray ionisation b ecomes active, (Fig. 3.1, top panel),


98

Chapter 3. The instability in protoplanetary discs

causes the current to b e effectively discontinuous there and pro duces the observed kink in the amplitude of these mo des. In the zone where x-rays are excluded (z /H 1.7), the main ionising agent is
-8

the decay of radioactive elements within the disc. However, they can only pro duce increases to 10 very weak magnetic coupling. For example, o 10
-3

for B = 1 mG, and it only

when B = 1 G. As a result, the amplitude of the p erturbations 500 mG), is very small at this radius.

near the midplane, even for strong fields (B

If the abundance of metals in the gas phase is reduced from the fiducial value adopted here (2 0.02) to (say) 2 â 10-3 , the zone ionised only by radioactivity b ecomes magnetically dead (as exp ected). The p erturbations' growth is only marginally affected. For R = 5 AU, we also observe a small kink in the p erturbations' amplitude for B 100 mG. In this case it o ccurs much closer to the midplane, b ecause at this radius x-rays p enetrate to z /H 0.3. Finally, for R = 10 AU, x-rays p enetrate

the entire cross section of the disc, so there is no kink in the amplitude of these mo des. However, the midplane cosmic ray ionisation rate at this radius is ab out two orders of magnitude larger than that of x-rays (in fact they dominate over xrays for z /H 2.5, see Fig. 3.1, b ottom panel), so excluding them do es reduce significantly the magnetic coupling in this region. As an illustration of this, note (see Fig. 3.6). As a result, for z /H that for B = 10 mG, o decreases from 16 in the previous case to only 1 here 0.3 the fluid is in the Hall regime while ambip olar diffusion dominates at higher z . Being a high wavenumb er p erturbation, the structure of this mo de reflects mainly lo cal fluid conditions. This explains the shap e of its envelop e (see rightmost column of Fig. 3.10, second panel from the top): flat envelop e close to the midplane where the Hall effect is dominant and amplitude increasing with z at higher vertical lo cations, driven by ambip olar diffusion. Finally, for B = 1 mG, < |1 |/ for z /H < 1.3. This, together with the high wavenumb er of the p erturbations, causes Hall diffusion to shap e the envelop e of this mo de.

Conductivity regime comparison (1 Bz > 0) It is interesting to compare the structure of the most unstable p erturbations obtained with different configurations of the conductivity tensor (assuming that different conductivity regimes dominate over the entire cross section of the disc) against the full


3.5. MAGNETOROTATIONAL INSTABILITY

99

Figure 3.9

Comparison of the structure of the most unstable mo des for the fiducial

mo del at R = 1, 5 and 10 AU for different choices of the magnetic field strength. B is indicated in the upp er right hand of each panel while the growth rate is shown in


100

Chapter 3. The instability in protoplanetary discs

Figure 3.10

As p er Fig. 3.9 assuming cosmic rays are excluded from the disc. Note 100 mG) and the kink in

the increased extent of the central dead zone at 1 AU (B

the amplitude of the p erturbations for stronger fields. This feature is caused by the sharp


3.5. MAGNETOROTATIONAL INSTABILITY

101

conductivity results discussed in section 3.5.1. This way we can explore more fully the effects of different conductivity comp onents in the overall prop erties of the instability. Fig. 3.11 presents such solutions for the fiducial mo del at R = 1 AU as a function of the strength of the magnetic field. The left column shows solutions computed with a full conductivity tensor, while the middle and right ones display mo des obtained using the ambip olar diffusion (1 = 0) and Hall regime (2 = 0, 1 Bz > 0) approximations, resp ectively. Note that, for all magnetic field strengths shown here, p erturbations computed using the ambip olar diffusion approximation have a more extended central dead zone than mo des incorp orating a full conductivity tensor. This is in agreement with the lo cal analysis (W99), which showed that the growth of MRI p erturbations in the ambip olar diffusion limit decreases steadily with the lo cal magnetic coupling. As a result, the lo cal growth of these mo des is severely restricted near the midplane. This is esp ecially effective for weaker fields, when global effects are less imp ortant due to the high wavenumb er of the p erturbations. For example, when B = 1 mG, the magnetically dead zone in mo des found using the ambip olar diffusion approximation extends to z /H 2. The thickness of the dead region decreases for stronger B , but even with B = 1 G, there is a small section (z /H not grow. Turning now our attention to Hall p erturbations, we observe that their amplitude is fairly stable, esp ecially for strong fields (B 500 mG), where they show the characteristic flat envelop e rep orted by SW03. When the field is weaker than this, p erturbations are damp ed near the midplane. This is particularly evident for B < 100 mG and is probably related to the fact that at 1 AU the magnetic coupling at the midplane in this limit is very low. Even for B = 100 mG, o in the Hall limit 10
-4

0.5) in which p erturbations do

significantly smaller than the ones shown in Fig. 3.4, which were obtained with a full conductivity tensor). Lo cal results highlight that the maximum lo cal growth rate of MRI p erturbations in the Hall limit is effectively unchanged from the ideal case the coupling is very low, the amplitude of global Hall limit p erturbations can b e damp ed close to the midplane (see rightmost column of Fig. 3.11, top three panels). from the midplane to z /H 0.5. For stronger fields, there is no appreciable dead On the other hand, there is only a small dead zone for B 1 mG, which extends when 0 (W99). However, the global analysis presented here shows that when

and it decreases to 10

-10

for B = 1 mG (note that these values are


102 region.

Chapter 3. The instability in protoplanetary discs

In general, the structure of full p erturbations reflect the contribution of Hall as well as ambip olar diffusion conductivity terms. When the magnetic field is strong (B > 100 mG), ambip olar diffusion is lo cally dominant over a more extended section of the disc, and the dead zone of p erturbations in this limit is smaller than they are for weaker fields. On the other hand, mo des in the Hall limit grow now even at the midplane and have a significantly higher wavenumb er than ambip olar diffusion p erturbations. This is reflected in the structure of full mo des in this region of parameter space: Their envelop e is shap ed by ambip olar diffusion (the amplitude increases with height), but the wavenumb er is higher and they grow closer to the midplane than pure ambip olar diffusion mo des do. This reflects the contribution of the Hall effect. These results are also in agreement with similar trends found with illustrative calculations in SW03. It was discussed in section 3.4.3 that at this radius (1 AU), Hall conductivity is lo cally dominant near the midplane for all magnetic field strengths of interest. The transition to the zone where ambip olar diffusion dominates o ccurs at a lower z for stronger fields (Fig. 3.4, top panel). This analysis reveals that the Hall effect mo difies the structure of MRI mo des even when Hall diffusion is lo cally dominant only for a small section close to the midplane of the disc. In section 3.5.2 it will b e discussed how the Hall effect also alters the growth of all fastest growing mo des at this radius. At 5 AU, ambip olar diffusion is imp ortant in the inner sections of the disc for B 100 mG (Fig. 3.5). It is exp ected that p erturbations obtained using this approximation will b e different from the corresp onding full mo des in this region of parameter space. Fig. 3.12 compares solutions obtained with different configurations of the conductivity tensor at this radius. Ambip olar diffusion mo des (left column) are indeed different from full (1 Bz > 0) ones (middle column) when B using different configurations of the conductivity tensor are alike, as exp ected. 100 mG. For stronger fields, ideal-MHD holds, so the structure of MRI mo des computed

Conductivity regime comparison (1 Bz < 0) So far we have discussed solutions obtained with a 1 Bz > 0 Hall conductivity term, which corresp onds to the case where the magnetic field and angular velo city vectors


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103

of the disc are parallel ( · B > 0). We now explore how these results are mo dified when these vectors are antiparallel. As noted b efore, we explored in this case a reduced region of parameter space. In particular, solutions at 1 AU could not b e computed at all, as
o

2 for all

B of interest at this radius, the limit b elow which all wavenumb ers grow in this regime (W99, see also section 3.4.1). On the other hand, results at 10 AU do not differ appreciably in this case from those obtained using a 1 Bz > 0 conductivity, given the strong magnetic coupling throughout the cross-section of the disc at this radial lo cation. At 5 AU, however, Hall diffusion is imp ortant for relatively weak fields (B 10 mG), and b oth sets of results should b e different in this region of parameter space. Fig. 3.12 (middle and right columns) compares the structure of MRI unstable mo des computed with a p ositive and negative Hall conductivity at this radius, as a function of the magnetic field strength. When 1 Bz < 0, no results could b e computed for B < 8 mG, as o < 2 in these cases. We find that, indeed, for B = 10 mG the most unstable mo de computed with a negative Hall conductivity has a higher wavenumb er, and a slower growth rate, than the corresp onding mo de with 1 Bz > 0.

More massive disc To finalise the analysis of the structure of MRI p erturbations, we describ e now the prop erties of unstable mo des in a more massive disc, as characterised in section 3.2.2. Results are displayed in Fig. 3.13. As exp ected, the larger column density causes unstable mo des to have a more extended central dead zone in relation to results in the minimum-mass solar nebula mo del. This is particularly noticeable for R = 1 AU, where there is now a magnetically inactive zone for B 1 G. X-rays are excluded from the midplane at this radius (they can only p enetrate up to z /H 2.6

for this disc mo del) and the cosmic ray ionisation rate at z = 0 is negligible. As a causes p erturbations to b e damp ed at low z . We observe here the same trends

result, even for B = 1 G the magnetic coupling at the midplane is 10-4 , which

discussed in the analysis of the minimum-mass solar nebula mo del in relation to the structure of these mo des. Ambip olar diffusion shap es the envelop es, particularly for strong fields, while Hall diffusion increases their wavenumb er in comparison with results in the ambip olar diffusion limit. Finally, note the kink on the amplitude of


104

Chapter 3. The instability in protoplanetary discs

Figure 3.11 Comparison of the structure of the most unstable mo des of the MRI for the fiducial mo del at R = 1 AU as a function of the field strength for different configurations of the conductivity tensor. The field strength spans from 1 mG up to 5 G, the maximum


3.5. MAGNETOROTATIONAL INSTABILITY

105

Figure 3.12

Comparison of the structure of the fastest growing mo des for the fiducial

mo del at R = 5 AU, for different configurations of the conductivity tensor. The left column corresp onds to the ambip olar diffusion approximation. The middle and right columns show solutions obtained with a full conductivity tensor when the magnetic field and angular velo city vectors of the disc are parallel ( 1 Bz > 0) and antiparallel (1 Bz < 0), resp ectively.


106

Chapter 3. The instability in protoplanetary discs

the B = 5.85 G mo de, attributed to the sharp increase in the ionisation fraction at the height where x-ray ionisation b ecomes active. in comparison with that of p erturbations obtained using the minimum-mass solar nebula mo del. In that case, ideal-MHD conditions held throughout the cross-section of the disc for B = 100 mG, with o 20 (see section 3.5.1). On the contrary, in is lo cally dominant for 0.7 the present mo del, o 0.2 for this magnetic field strength and ambip olar diffusion z /H 2 (see Fig. 3.7, middle panel). As a result, the At 5 AU, there is also a more extended dead zone, observed up to B 100 mG,

amplitude of this p erturbation increases with z , a typical b ehaviour of ambip olar diffusion driven MRI, instead of b eing flat as b efore. Finally, for R = 10 AU, ideal-MHD conditions hold throughout the disc crosssection for B 100 mG (Fig. 3.7, b ottom panel). Because of this, the envelop es are flat in this region of parameter space (see Fig. 3.13, right column, lower two panels). On the other hand, when B 10 mG, o 1.5 and ambip olar diffusion dominates envelop e p eaks at an intermediate height. Finally, when B 1 mG, o 0.05 and Hall diffusion is dominant for z /H 2. Consequently, the central dead zone is much reduced, and this p erturbation grows closer to the midplane despite the weak magnetic coupling. up to z /H 2. As a result, this mo de exhibits a small central dead zone and the

3.5.2

Growth rate of the p erturbations

Fastest growing mo des for different conductivity regimes The dep endency of the growth rate of the most unstable mo des with the strength of the magnetic field is shown in Fig. 3.14 for different configurations of the conductivity tensor. Results corresp ond to the fiducial mo del at R = 1 AU (top panel) and 10 AU (b ottom panel). We will use Fig. 3.12 to analyse the corresp onding results at 5 AU. The overall dep endency of the follows:
max max

vs. B curve for full mo des is typically as

at 0.75, the maximum growth rate for ideal-MHD p erturbations in Keplerian discs.

initially increases with B with a p ower law dep endency. It then levels

Eventually, the maximum growth rate decays sharply at a characteristic magnetic vs. B curve

field strength at which the p erturbations' wavelength is H , the scaleheight of the disc. Note that in the weak-field limit, the dep endency of the
max


3.5. MAGNETOROTATIONAL INSTABILITY

107

Figure 3.13

Comparison of the structure of the most unstable mo des of the MRI for a

more massive disc, incorp orating cosmic ray ionisation. We present results at R = 1, 5 and 10 AU as a function of the strength of the magnetic field. The disc surface and mass density are o = 10o and o = 10o . For simplicity, it was assumed that the temp erature, sound sp eed and scale height are unchanged from those of the minimum-mass solar nebula mo del. Note that these mo des have a more extended central dead zone, esp ecially for R = 1 AU,


108

Chapter 3. The instability in protoplanetary discs

shows no evidence of a minimum field strength, b elow which mo des do not grow. This characteristic shap e of the
max

vs. B curve can b e explained in terms of the

dep endency of the p erturbations' growth rate with B when different conductivity comp onents are dominant at different heights. Examining Fig. 3.14 (top panel), it is clear that for weak fields (B p erturbations computed with a full have a
max

200 mG),

vs. B dep endency similar to that

of solutions obtained using the ambip olar diffusion approximation. This indicates that this feature is driven by ambip olar diffusion. In this limit, the lo cal maximum growth rate of MRI mo des increases with the magnetic coupling (W99), which in turn, increases with B , except when i and |e | a result,
max

1 (e.g. near the surface). As

should increase with B , as observed. On the contrary, when the

magnetic field is stronger than 200 mG, the growth rate of MRI mo des obtained this region of parameter space, the growth rate of global mo des remains unchanged

with a full conductivity tensor is practically identical to that of Hall limit mo des. In when the magnetic field gets stronger, as exp ected when Hall diffusion drives the

growth of the instability. Eventually, B b ecomes strong enough that the fastest growing mo de b ecomes the scaleheight of the disc, and the growth rate rapidly declines. These results are in agreement with previous findings by W99 and SW03. Note also that the maximum growth rate of p erturbations in the Hall limit do es not change appreciably with B , until they are damp ed for a sufficiently strong field, B 1 mG. as exp ected. In fact, Hall p erturbations grow at the ideal-MHD rate even for Hall diffusion is lo cally dominant at 1 AU in the lower sections of the disc for all

magnetic field strengths for which unstable mo des exist (Fig. 3.4, top panel). As a result, p erturbations obtained with a full conductivity tensor grow significantly faster than mo des found using the ambip olar diffusion approximation for all B . For R = 5 AU, the Hall effect increases the growth rate of full p erturbations when B 10 mG, the region of parameter space where Hall diffusion is lo cally dominant near the midplane (compare the growth rates in left and middle columns of Fig. 3.12). Moreover, full mo des computed with 1 Bz > 0 (middle column) and 1 Bz < 0 (right column) grow at different sp eeds and have different structure, as exp ected in the region where Hall diffusion dominates. Finally, Fig. 3.14 (b ottom panel) displays the growth rate of the most unstable mo des at 10 AU. Results were computed with a full conductivity tensor ­ but with


3.5. MAGNETOROTATIONAL INSTABILITY

109

In this case, Hall diffusion is lo cally unimp ortant for all B studied here, so the sign of 1 Bz do es not affect the growth of the p erturbations. On the other hand, at this radius, ambip olar diffusion is imp ortant for B than this value. 10 mG. This slows the growth of p erturbations in the ambip olar diffusion limit when the magnetic field is weaker

opp osite signs of · B ­, as well as using the ambip olar diffusion approximation.

Fastest growing mo des at different radii We also compare the growth rate of the most unstable mo des at different radii as a function of the strength of the magnetic field (Fig. 3.15). Three sets of results are displayed. The first two corresp ond to the minimum-mass solar nebula disc assuming cosmic rays either p enetrate the disc (top panel) or are excluded from it (middle panel). The last set presents results for the more massive disc mo del (b ottom panel), incorp orating cosmic ray ionisation. Note that in all three cases, MRI p erturbations grow over a wide range of magnetic field strengths. The maximum field strength for which unstable mo des exist (within each panel), is weaker at larger radii. In ideal-MHD conditions, as well as when either the ambip olar diffusion or Hall (1 Bz > 0) conductivity regimes dominate, unstable MRI H . In the Hall (1 Bz < 0) limit, unstable mo des have b een found for vA /cs 3 (SW03, alb eit using a constant conductivity profile). In any case, as b oth the gas en of the Alfv´ to sound sp eed asso ciated with a particular magnetic field strength increases with R, and as a result, the p erturbations are damp ed at a weaker field for larger radii. The maximum magnetic field strengths for which MRI unstable mo des grow, as well as the range for which
max

mo des are damp ed when vA /cs 1 (Balbus & Hawley 1991), which corresp onds to

density (eq. 3.26) and the sound sp eed (eq. 3.17) decrease with radius, the ratio

for the radii and disc structures of interest here. Note that for the fiducial mo del, we obtained unstable mo des at 1 AU for B of the ideal-MHD rate ( = 0.75) for 200 mG 8 G. The growth rate is of the order B 5 G.

the ideal-MHD rate, are summarised in Table 3.1

When cosmic rays are assumed to b e excluded from the disc (see middle panel of Fig. 3.15 and central two columns of Table 3.1), unstable mo des are obtained at 1 AU only for B 2 G. The corresp onding range at 5 AU is only slightly reduced and


110

Chapter 3. The instability in protoplanetary discs

Table 3.1 Magnetic field strengths for which MRI p erturbations grow at 1, 5 and 10 AU for different disc mo dels and sources of ionisation. For the minimum-mass solar nebula disc, two sets of results are shown, where cosmic rays are either present or excluded from the disc. For the massive disc mo del, only the former case is presented. Columns lab eled `B
max

' list the maximum value of B for which each disc supp orts unstable MRI mo des.

Similarly, the `B ( 0.75)' columns sp ecify the subset of these for which p erturbations grow at nearly the ideal-MHD rate ( 0.75).

Minimum-mass solar nebula B
CR

Massive Disc B
CR

Radius (AU) 1 5 10 B

CR

included B ( 0.75) 0.2 - 5 0.02 - 0.5 0.002-0.05

excluded B ( 0.75) 0.1 - 1 0.01 - 0.05 0.005 - 0.05

included B ( 0.75) 0.2 - 0.5 0.05 - 0.5 0.01 - 0.5

max

max

max

7.85 0.80 0.25

2.10 0.62 0.25

5.85 2.36 0.82

it is essentially the same as b efore at 10 AU. This is consistent with our exp ectation that cosmic rays are a particularly imp ortant source of ionisation at 1 AU, where x-rays are excluded from the midplane. In this case, mG B 1 G.
max

0.75 at 1 AU for 100

In the massive disc mo del (b ottom panel of Fig. 3.15 and two rightmost columns of Table 3.1), the MRI is active at 1 AU for B 6 G and
max

< B < 500 mG. For R = 5 and 10 AU, MRI unstable mo des exist in this disc for stronger fields than in the minimum-mass solar nebula mo del. This can b e explained by recalling that the larger mass and column density translates, at these radii, into a larger gas pressure and a stronger equipartition magnetic field strength at the

0.75 for 200 mG

midplane. MRI mo des are, as a result, damp ed for a stronger field than they are in the minimum-mass solar nebula mo del. On the contrary, for R = 1 AU, the range of magnetic field strengths over which p erturbations grow do es not change appreciably, given the shielding of the inner sections at this radius.

3.6

Discussion

In this pap er we have explored the vertical structure and linear growth of MRI p erturbations of an initially vertical magnetic field, using a realistic ionisation profile


3.6. Discussion

111

Figure 3.14

Growth rates of the fastest growing mo des as a function of the strength

of the magnetic field for different configurations of the conductivity tensor. Results are presented at R = 1 AU (top panel) and 10 AU (b ottom panel) for the fiducial mo del. It is evident that at 1 AU, when the magnetic field is weak (B driven by ambip olar diffusion. As a result,
max

200 mG), p erturbations are

increases with B . As the field gets even

stronger, the maximum growth rate of full and Hall limit p erturbations are practically identical, signalling that Hall conductivity determines the growth of global unstable mo des in this region of parameter space. Finally,
max

decreases rapidly when B is so strong that

the wavelength of the fastest growing mo de is the scaleheight of the disc.


112

Chapter 3. The instability in protoplanetary discs

Figure 3.15

Growth rate of the most unstable mo des of the MRI for R = 1, 5 and

10 AU as a function of the strength of the magnetic field. Top and middle panels show


3.6. Discussion

113

and assuming that ions and electrons are the sole charge carriers. This formulation is appropriate to mo del low conductivity discs (Gammie & Menou 1998; Menou 2000; Stone et al. 2000) at late stages of accretion, after dust grains have settled into a thin layer around the midplane ( 105 years, Nakagawa et. al. 1981, Dullemond & Dominik 2004) and b ecome dynamically uncoupled from the gas at higher z . of the strength of the magnetic field for different configurations of the conductivity tensor, disc mo del and sources of ionisation. We have shown that the magnetic field is dynamically imp ortant in low conductivity accretion discs over a wide range of field strengths. An example of this activity is the generation and sustaining of MRI-unstable mo des, which are thought to b e required to provide the angular momentum transp ort for accretion to pro ceed. The structure and growth rate of MRI p erturbations are a function of the disc prop erties and the strength of the field. For a particular radius and disc mo del, they are a result of the comp eting action of different dominant conductivity comp onents at different heights. mG at 5 AU), the fluid close to the midplane is in the Ohmic conductivity regime (
2

Solutions were obtained at 1, 5 and 10 AU from the central ob ject as a function

For R a few AU and relatively weak fields (B

100 mG at 1 AU or B

1

fields) Hall conductivity dominates even at z = 0 (Wardle, in preparation; see also For still higher z , the magnetic coupling is so strong that ideal-MHD holds. The p osition, are a function of the strength of the magnetic field. Note that even when

|1 |, e.g. see Fig. 3.3). Conversely, for larger radii (or stronger

section 3.4.3). Ambip olar diffusion dominates ab ove this region until 10 (W99).

heights at which these transitions take place, for a particular disc mo del and radial |1 | 2 , Hall diffusion still dominates the structure and growth of MRI mo des
cr it

when

At 1 AU, for the fiducial mo del, unstable mo des exist for B significant subset of these strengths (200 mG B

|1 |/ (SW03).

8 G. For a

5 G), p erturbations grow at

ab out the ideal-MHD rate (= 0.75 in Keplerian discs). At this radius, Hall diffusion is lo cally dominant near the midplane for all magnetic field strengths for which MRI unstable mo des exist. As a result, mo des computed with a full conductivity tensor have a higher wavenumb er, and grow faster, than p erturbations obtained using the ambip olar diffusion approximation. On the other hand, ambip olar diffusion shap es the envelop es of these mo des (esp ecially for strong fields), causing them to p eak at


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Chapter 3. The instability in protoplanetary discs

an intermediate height, instead of having the characteristic flat envelop e that mo des in the Hall limit have for these field strengths. Cosmic rays are an imp ortant source of ionisation at 1 AU, given that x-rays do not reach the midplane at this radius (see Fig. 3.1). Consequently, if they are assumed to b e excluded from the disc, the extent of the magnetically inactive `dead zone' ab out the midplane increases and unstable mo des grow only for B strong initial magnetic field (e.g. B 2.1 G. Furthermore, under this assumption, p erturbations obtained with a relatively 500 mG), exhibit a kink in their amplitude. This feature is attributed to the sharp increase in the ionisation fraction at the height where x-ray ionisation b ecomes active. Finally, for the massive disc mo del (o = 10o and o = 10o , incorp orating cosmic ray ionisation), the central dead zone extends to a higher z at 1 AU, as the larger column density asso ciated with this mo del causes the ionisation fraction at low z to drop sharply (the disc is shielded from x-rays b elow z /H 2.6 and even the cosmic ray ionisation rate at the midplane is negligible). MRI p erturbations grow in this case for B 6 G, a range not

significantly different from that of results in the fiducial mo del. This is the case b ecause the effective surface density is largely unchanged, given the shielding of the inner sections. At 5 AU, for the fiducial mo del, MRI-unstable mo des grow for B When 20 mG B 800 mG. 100 500 mG, the growth rate is 0.75. At this radius, Hall

diffusion increases the growth rate and wavenumb er of unstable mo des for B

mG, but for stronger fields, p erturbations obtained with different configurations of the conductivity tensor have similar structures and growth rates, a signal that idealMHD conditions hold (Fig. 3.12). We also conducted an analysis of the effect of the alignment of the magnetic field and the angular velo city vector of the disc. Results indicate that, at this radius, the sign of 1 Bz is imp ortant for B 10 mG, the range of magnetic field strengths for which Hall diffusion is lo cally dominant at low z . As in the 1 AU case, excluding cosmic rays reduces the range of magnetic field strengths for which MRI mo des grow (unstable mo des are found for B 600 mG in this case) 100 mG. Finally, and there is a kink in the amplitude of the p erturbations for B

when the surface density of the disc is increased, the central dead zone o ccupies a larger cross-section, as exp ected, and unstable mo des are found for stronger fields than in the fiducial mo del. This is a result of the stronger equipartition magnetic field strength for this disc mo del at this radial p osition.


3.6. Discussion At 10 AU, the MRI is active for B to the ideal-MHD rate for 2 mG B

115 250 mG and the growth rate is close 50 mG. Furthermore, for B 10 mG,

p erturbations obtained with a full conductivity tensor grow significantly faster than mo des in the ambip olar diffusion limit, which reflects the contribution of the Hall effect (Fig. 3.14, b ottom panel). At this radius, when cosmic rays are excluded, the range of magnetic field strengths for which unstable mo des exist is not affected, given that x-rays are able to p enetrate to the midplane. In the more massive disc, the extent of the dead zone increases, esp ecially for weak fields (B 10 mG) and p erturbations grow for stronger fields than in the minimum-mass solar nebula disc. The results just presented demonstrate how the prop erties of global (in the z direction) MRI unstable mo des are dep endent on the comp eting action of different conductivity regimes dominating at different heights. When the wavenumb er is high, global effects ­ driven by the stratification of the disc ­ are less imp ortant and the mo des' structure is largely determined by lo cal fluid conditions. In agreement with our previous pap er (SW03), we find that in this case, p erturbations driven by Hall diffusion p eak closer to the midplane than those where ambip olar diffusion is the dominant diffusion mechanism. Conversely, when global effects are imp ortant, the envelop e of the p erturbations is shap ed by ambip olar diffusion while the Hall effect, which is typically imp ortant close to the midplane, increases the wavenumb er and growth rate of the p erturbations. In section 2.1 it was discussed why the MRI is one of the most promising mechanisms to generate and sustain the angular momentum transp ort required for disc material to accrete. The angular momentum flux asso ciated with the fastest growing mo de in the linear regime is given by, Re( Br )Re( B ) , 4

L = Re( vr )Re( V ) -

(3.69)

where the two terms in the right hand side corresp ond to transp ort via Reynolds and Maxwell stresses, resp ectively. The vertical profile of this quantity turns out to b e a go o d indicator of the vertical sections where angular momentum is transp orted once the instability b ecomes non-linear. This is so b ecause the fastest growing mo de in the linear regime is also an exact solution to the full non-linear fluid equations in the incompressible limit (Go o dman & Xu 1994). Given this, and the imp ortance of this transp ort mechanism for accretion, we present in Fig. 3.16 the angular momentum


116

Chapter 3. The instability in protoplanetary discs

flux (equation 3.69) asso ciated with the mo des shown in Fig. 3.9. Note that the regions where no angular momentum is b eing transp orted corresp ond well with the `dead zones' (understo o d as the regions where Br B 0) identified in that figure. This is exp ected, given that the MRI transp orts angular momentum mainly via magnetic stresses. At 1 AU, a magnetically inactive zone is observed for B 1

G, while for larger radii even regions close to the midplane are active. These results are consistent with angular momentum b eing transp orted radially outwards by the MRI, in the inner regions of discs, for a wide range of magnetic field strengths and fluid conditions. In the present formulation, it was assumed that ions and electrons are the only charge carriers. As discussed earlier, this is valid in late evolutionary stages of accretion, when dust grains have sedimented enough towards the midplane that they can b e neglected when studying the dynamics of the gas at higher z . If grains are well mixed with the gas, however, recombinations on dust surfaces are exp ected to b e dominant and n
e

ni (e.g. Wardle 2003, Desch 2004). The settling of

dust grains is, consequently, a crucial factor in the overall equilibrium structure of discs. The timescale for this settling to o ccur is exp ected to b e affected by MHD turbulence. Although in quiescent discs dust grains may quickly settle into a thin sub-layer ab out the midplane, the vertical stirring caused by MHD turbulence could p otentially transp ort them back to higher vertical lo cations, preventing them from settling b elow a certain height (Dullemond & Dominik 2004 and references therein). It is also likely that the transition b etween vertical sections where dust grains are well mixed with the gas phase and those completely depleted of grains by settling is not `extremely sharp' (Dullemond & Dominik 2004). The efficiency of this stirring is dep endent on the disc b eing able to supp ort MHD turbulence in the vertical lo cations where dust grains are present. On the other hand, dust grains can reduce the abundance of free electrons, and the efficiency of MHD turbulence itself, by providing additional recombination pathways on their surfaces. This, in turn, will reduce the vertical extent of the magnetically active regions, and the angular momentum flux through the disc. Ultimately, the equilibrium structure, and magnetic activity, of accretion discs will reflect the complex interplay b etween all these pro cesses. It is exp ected that the structure and growth of MRI unstable mo des in such environments will b e affected by the dep endency of the ionisation balance with height in


3.7. Summary

117

the presence of chemistry taking place on grain surfaces. The study of the prop erties of the MRI in a disc where dust dynamics and evolution are determined consistently, and where Hall conductivity is taken into account, is essential to understand more fully the presence and efficiency of MRI-driven angular momentum transp ort in accretion discs. Non-ideal MHD simulations that explore the MRI non-linear stages as outlined ab ove are of particular interest. Nevertheless, from the results presented in this pap er, it is clear that Hall diffusion is crucial for the realistic mo delling of the few AU from the central ob ject. More generally, Hall conductivity is an imp ortant factor when studying the magnetic activity of low conductivity discs at these radii. magnetorotational instability in protostellar discs, particularly at a distance of a

3.7

Summary

We have presented in this pap er the vertical structure and linear growth of the magnetorotational instability (MRI) in weakly ionised, stratified accretion discs, assuming an initially vertical magnetic field. Both the density and the conductivity are a function of height. Moreover, the conductivity is treated as a tensor and obtained with a realistic ionisation profile. Two disc mo dels were explored: The minimum-mass solar mebula disc (Hayashi 1981; Hayashi, Nakagawa & Nakazawa 1985) and a more massive disc, with the mass and surface density increased by a factor of 10. This formulation is appropriate for the study of weakly ionised astrophysical discs, where the ideal-MHD approximation breaks down (Gammie & Menou 1998; Menou 2000; Stone et al. 2000). The ionisation sources relevant here, outside the inner 0.1 AU from the central ob ject, are non-thermal: Cosmic rays, x-rays and radioactive decay. For the minimum-mass solar nebula mo del we compare solutions obtained including all three sources of ionisation with those arrived at assuming that cosmic rays are excluded from the disc by the protostar's winds. Recombination pro cesses are taken to o ccur in the gas-phase only, which is consistent with the assumption that dust grains have settled into a thin layer ab out the midplane, and ions and electrons are the only charge carriers. Perturbations of interest have vertical wavevectors (k = kz ) only, which are the most unstable mo des (when initiated from a vertically aligned magnetic field) in b oth the Hall and Ohmic regimes (Balbus & Hawley 1991; Sano & Miyama 1999). This is not necessarily the case in the


118

Chapter 3. The instability in protoplanetary discs

Figure 3.16

Angular momentum flux L (see equation 3.69) asso ciated with each of 1

the mo des presented in Fig. 3.9. At 1 AU, there is a central `dead zone' for B

G. For larger radii, even regions close to the midplane are actively transp orting angular


3.7. Summary

119

ambip olar diffusion limit (Kunz & Balbus 2004, Desch 2004), where the fastest growing mo des can have radial as well as vertical wavenumb ers. Under the adopted approximations, the prop erties of the MRI in the ambip olar diffusion and Ohmic conductivity limits are identical. Three parameters were found to control the dynamics and evolution of the fluid: (i ) the lo cal ratio of the Alfv´ to sound sp eed (vA /cs ); (ii ) The lo cal coupling b een tween ionised and neutral comp onents of the fluid (), which relates the frequency at which non-ideal effects are imp ortant with the dynamical (Keplerian) frequency of the disc; and (iii ) the ratio of the comp onents of the conductivity tensor p erp endicular to the magnetic field (1 /2 ), which characterises the conductivity regime of the fluid. These parameters were evaluated at R = 1, 5 and 10 AU for a range of magnetic field strengths. The linearised system of ODE was integrated from the midplane to the surface of the disc under appropriate b oundary conditions and solutions were obtained for representative radial lo cations of the disc as a function of the magnetic field strength and for different configurations of the conductivity tensor. The main results of this study are summarised b elow. 1. For the minimum-mass solar nebula mo del, incorp orating cosmic ray ionisation (the fiducial mo del): At 1 AU, unstable MRI mo des exist for B 8 G. When 200 mG B 5

diffusion dominates the structure and growth rate of unstable mo des for all magnetic field strengths for which they grow. For strong fields, ambip olar diffusion shap es the envelop e of the p erturbations, which p eak at an intermediate height. Finally, at this radius, a magnetically dead zone (Gammie 1996, Wardle 1997) exists when B < 1 G. As exp ected, the vertical extent of this zone decreases when the magnetic field gets stronger. It increases when a more massive disc mo del is used. At 5 AU, MRI-unstable mo des grow for B is close to the ideal MHD rate for 20 mG 800 mG and the growth rate B 500 mG. Perturbations 100 mG. When B

G, the most unstable mo des grow at the ideal-MHD rate (= 0.75). Hall

incorp orating Hall conductivity have a higher wavenumb er and grow faster than solutions in the ambip olar diffusion limit for B 10 mG, the structure and growth of full p erturbations are dep endent on


120

Chapter 3. The instability in protoplanetary discs the alignment of the magnetic field and angular velo city vectors of the disc. fields, a small dead region exists. Unstable mo des grow even at the midplane for B 100 mG but for weaker 250 mG. The growth rate is close to the 10 mG, p erturbations

At 10 AU, the MRI is active for B ideal-MHD rate for 2 mG B

50 mG and when B

obtained with a full conductivity tensor grow significantly faster than mo des in the ambip olar diffusion limit. Mo des show only a very small dead region when B 1 mG. 2. When the magnetic field is weak (e.g. B growth rate of unstable MRI mo des (
max

200 mG at 1 AU), the maximum ) increases with the strength of the

magnetic field, a feature driven by ambip olar diffusion. 3. When cosmic rays are assumed to b e excluded from the disc, unstable mo des at 1 AU grow only for B 2.1 G. Results at 5 AU only change slightly, while solutions at 10 AU are not affected at all, as exp ected. 4. For the massive disc mo del (o = 10o and o = 10o , incorp orating cosmic ray ionisation), MRI p erturbations grow for stronger fields at 5 and 10 AU, in relation to the minimum-mass solar nebula mo del. Results at 1 AU are unchanged, given that in this case the effective surface density is not significantly different. MRI p erturbations grow in protostellar discs for a wide range of fluid conditions and magnetic field strengths. Hall diffusion largely determines the structure and growth rate of p erturbations at radii of order of a few AU from the central protostar. This indicates that, despite the low magnetic coupling, the magnetic field is dynamically imp ortant in low conductivity astrophysical discs and will impact the dynamics and evolution of these discs.


Chapter 4 Impact of dust grains
Impact of dust grains

4.1

Introduction

The role magnetic fields are able to play in the dynamics and evolution of low conductivity discs is largely determined by the degree of coupling b etween the field and the neutral gas. This magnetic coupling reflects the equilibrium b etween ionisation and recombination pro cesses taking place in the disc. In protostellar discs, ionisation pro cesses outside the innermost sections (R 0.1 AU) are non-thermal, driven by interstellar cosmic rays, x-rays emitted by the central protostar and the decay of radioactive materials (Hayashi 1981; Glassgold, Na jita & Igea 1997; Igea & Glassgold 1999; Fromang, Terquem & Balbus 2002). On the other hand, free electrons are lost through recombination pro cesses which, in general, take place either in the gas phase (through the disso ciative recombination of electrons with molecular ions and the radiative recombination with metal ions) or on grain surfaces (e.g. Nishi, Nakano & Umebayashi 1991). Dust grains affect the level of magnetic coupling in protostellar discs when they are well mixed with the gas (e.g. in relatively early stages of accretion and/or when turbulence prevents them from settling towards the midplane). They do so in two ways. First, they reduce the ionisation fraction by providing additional pathways for electrons and ions to recombine on their surfaces. Second, charged dust particles can b e imp ortant charged sp ecies in high density regions (Umebayashi & Nakano 1990; Nishi, Nakano & Umebayashi 1991). For example, at 1 AU in a disc where 0.1µm 121


122

Chapter 4. Impact of dust grains

grains are present, p ositively charged grains are the most abundant ionised sp ecies within two scaleheights from the midplane (Wardle in prep.). As dust particles generally have large cross sections, collisions with the neutrals are imp ortant and they b ecome decoupled (or partially decoupled) to the magnetic field at densities for which smaller sp ecies, typically ions and electrons, would still b e well tied to it. Both of these mechanisms act to lower the conductivity of the fluid, esp ecially near the midplane where the density is high and ionisation pro cesses are inefficient (in the minimum-mass solar nebula disc, for example, x-rays are completely attenuated b elow z /H 1.7; see Fig. 3.1). As a result, in the inner sections of the disc field. On the contrary, near the surface, the coupling is generally adequate as the density is significantly lower and the ionisation fraction is higher. This accentuates the `layered accretion' scenario, in which magnetic activity is supp orted only in the surface regions of the disc (Gammie 1996, Wardle 1997). Finally, it is worth noting that, in general, dust grains can affect the structure and dynamics of accretion discs via two additional mechanisms: Dust opacity can mo dify the radiative transp ort withing the disc ­which, in turn, can dramatically alter its structure­, and dust particles may b ecome dynamically imp ortant if their abundance is sufficiently high. In this study b oth effects are small b ecause the disc is vertically isothermal and grains constitute only 1% of the mass of the gas (see b elow). When dust grains are well mixed with the gas, they are exp ected to mo dify the magnetic activity of low conductivity discs through their impact on the conductivity. Therefore, a realistic study of the prop erties of the MRI in these discs must incorp orate a consistent treatment of dust dynamics and evolution (unless they are assumed to have settled, a go o d approximatiion to mo del relatively late accretion stages, as in chapter 3). This analysis is further complicated b ecause dust grains have complex spatial and size distributions (Mathis, Rumpl & Nordsieck 1977, Umebayashi & Nakano 1990), determined by the comp eting action of pro cesses involving sticking, shattering, coagulation, growth (and/or sublimation) of ice mantles and settling to the midplane (e.g. Weidenschilling & Cuzzi 1993). Previous results (chapter 3) highlight the imp ortance of incorp orating in these studies a full conductivity tensor as well (Cowling 1957, Norman & Heyvaertz 1985, Nakano & Umebayashi 1986, Wardle & Ng 1999), as Hall diffusion largely determines the growth and structure of MRI p erturbations, particularly at distances of the order of a few AU from the the magnetic coupling is typically insufficient to couple the neutral particles to the


4.2. Disc mo del central protostar.

123

We present in this chapter preliminary results on the vertical structure and linear growth of the MRI in a disc where dust grains are well mixed with the gas over the entire vertical dimension of the disc. For simplicity, we assume here that all particles have the same radius a = 0.1µm and constitute 1% of the total mass of the gas, a typical assumption adopted in studies of molecular clouds (Umebayashi & Nakano 1990). This fraction is constant with height, which means that we have assumed that no sedimentation has o ccurred. Although this is a very simplified picture, the results illustrate the imp ortance of dust particles in the delicate ionisation equilibrium of discs, and consequently, on their magnetic activity. This chapter is organised as follows: the adopted disc mo del is describ ed in section 4.2, including a discussion of the typical dep endency of the comp onents of the conductivity tensor and magnetic coupling with height with, and without, grains. Section 4.3 then presents the structure and growth of MRI mo des at a representative distance (R = 10 AU) from the central protostar, and compares solutions incorp orating different configurations of the conductivity tensor. These results, and p ossible implications for the dynamics and evolution of low conductivity discs, are discussed in section 4.4.

4.2

Disc model

We adopt the minimum-mass solar nebula mo del (Hayashi 1981, Hayashi et al 1985) as our fiducial disc, and assume that it is geometrically thin and isothermal. Our formulation incorp orates the disc vertical stratification, but neglects radial gradients. Under these assumptions, the equilibrium structure of the disc is the result of the balance b etween the vertical comp onent of the gravitational force exerted by the central ob ject and the pressure gradient of the fluid. The vertical profile of the gas and H is the scaleheight of the gas. Finally, the neutral gas is assumed to b e comp osed of molecular hydrogen and helium, such that n nH (r, z ) = (r, z )/1.4mH . As discussed ab ove, the magnetic coupling in protostellar discs is exp ected to b e low, given that ionisation pro cesses are generally ineffective ( except p ossibly in the surface regions), while recombination is accelerated by the high fluid density
He

density is then (r, z ) = o (r ) exp (-0.5z 2 /H 2 ), where o is the midplane density

= 0.2nH2 , which gives


124

Chapter 4. Impact of dust grains

and the presence of dust grains. As a result, magnetic activity is likely to o ccur near the surface, but it is exp ected to b e suppressed in the inner sections of the disc (Gammie 1996, Wardle 1997). In regions where 10 ideal-MHD conditions |1 |/ , ambip olar hold and the particular configuration of the conductivity tensor has little effect on the b ehaviour of the MRI. When is weaker than this but diffusion dominates (Wardle 1999, see chapter 3). Finally, in the vertical sections

where < |1 |/ , Hall diffusion mo difies the structure and growth of MRI unstable (see section 3.5.1).

mo des, provided that the magnetic coupling is sufficient for unstable mo des to grow Fig. 4.1 shows the comp onents of the conductivity tensor ( , |1 | and 2 ),

magnetic coupling () and ratio |1 |/ as a function of height for the chosen radial p osition (R = 10 AU) and B = 10 mG. Two cases are considered: Dust grains fully mixed with the gas phase over the entire vertical extention of the disc (b ottom panel). Note that the field-parallel ( ) and Pedersen (2 ) conductivities are always p ositive, as all charged sp ecies contribute p ositively to them. On the contrary, the contribution to the Hall term (1 ) by a particular charged sp ecies, dep ends on the sign of the charge (Wardle & Ng 1999). In this region, For higher z , When no grains are present (top panel), |1 | dominates over 2 for z /H
2

have either settled into a thin layer ab out the midplane (top panel), or they are

2.5.



|1 |
2

and the fluid is in the Hall conductivity regime.

Hall conductivity term decreases more sharply than the Pedersen conductivity in

|1 | and ambip olar diffusion dominates. Note that the

resp onse to the fall in fluid density. As a result, |1 | is typically several orders of diffusion is not exp ected to play an imp ortant role in the lo cal prop erties of the MRI for this magnetic field strength. This picture is dramatically changed when dust grains are assumed to b e well mixed with the gas (Fig. 4.1, b ottom panel). The conductivities shown here were taken from Wardle (in prep.), who mo delled the ionisation balance following Umebayashi & Nakano (1990); Nishi, Nakano & Umebayashi (1991) and Sano et al. (2000), but allowing for higher charge states that are likely to o ccur in discs. Note that in this case, all conductivity comp onents drop drastically in relation to the previous results. For example, the Pedersen conductivity drops by ab out 5 orders of magnitude at the midplane.

magnitude smaller than 2 in the surface regions. As > |1 |/ for all z , Hall


4.3. Results

125

We also note that the Hall conductivity comp onent shows characteristic `spikes', at the heights where it changes sign. In particular, 1 is p ositive when 0 and 2.5 z /H z /H 1.6 4.1. It is negative at all other vertical lo cations. The change in

sign of 1 corresp onds to a change in the direction of the magnetic field at the height where particular sp ecies b ecome decoupled to it by collisions with the neutrals. This, in turn, changes the contribution of different charged sp ecies to this comp onent of the conductivity tensor. In order to explore the contributions to 1 in each of the vertical sections in which it has a different sign, we describ e the b ehaviour of the charged sp ecies at four representative heights, namely z /H = 1 and 3 (for which 1 > 0), and z /H = 2 and 5 (where 1 < 0). This discussion is based in calculations by Wardle (in prep.). At z /H = 1, the density is sufficiently high for electrons to stick to the grains. As a result, they reside mainly on dust particles, as do ab out a third of the ions. The contribution of (negatively charged) grains and ions to the Hall conductivity term are very similar, with a small p ositive excess, which determines the sign of 1 in this region. On the contrary, at z /H = 2, ions are the dominant p ositively charged sp ecies, while the negative charges are still contained in dust grains, which drift together with the neutrals. Consequently, 1 is negative in this section of the disc. At z /H = 3, ions and electrons are the main charge carriers and ions dominate the contribution to the Hall term, which makes 1 p ositive. Finally, for z /H = 5, the dominant contribution to the Hall conductivity term comes from the small p ercentage of remaining negatively charged grains. As a result, 1 is negative (and very small). Finally, note that
2

is unimp ortant at this radius, as exp ected, given the relatively low fluid density (e.g. Wardle in prep.). Nonetheless, Hall diffusion is imp ortant within three scaleheights

|1 | for all z . This confirms that Ohmic diffusion

of the midplane ( < |1 |/ ) and would determine the prop erties of the MRI, provided that the coupling is sufficient to supp ort unstable mo des. At higher z , the


ionisation fraction is such that |1 |/



10 and ambip olar diffusion dominates.

4.3

Results

Fig. 4.2 compares the structure and growth of the fastest growing MRI mo des at the radius of interest for different choices of the magnetic field strength. The left


126

Chapter 4. Impact of dust grains

Figure 4.1 Comp onents of the conductivity tensor ( , |1 | and 2 ), magnetic coupling () and |1 |/


as a function of height for R = 10 AU and B = 10 mG. Top panel

corresp onds to the case where dust grains are settled, so charges are carried by electrons and ions only. Bottom panel displays results assuming a p opulation of 0.1µm grains is present. In the region where < |1 |/ , Hall diffusion determines the growth rate and structure of the MRI.


4.3. Results

127

column displays solutions obtained assuming that grains have settled, while the right one shows results including the single size grain p opulation describ ed ab ove. The prop erties of MRI p erturbations at this radius, asuming that dust grains are settled, were discussed in detail in section 3.5. They are summarised again b elow for greater clarity. Note that when ions and electrons are the sole charge carriers (left panel), Hall diffusion is imp ortant for B 1 mG (see Fig. 3.6). This explains the high wavenumb er, small central dead zone and fast growth of the p erturbation for B = 1 mG. On the other hand, ambip olar diffusion is dominant close to the midplane for 1 mG B 10 mG. This comp onent of the conductivity tensor typically drives the lo cal growth rate (and amplitude) of MRI unstable mo des to increase with height (see sections 3.5.1 and 3.5.2). As a result, the envelop e of the p erturbation for B = 5 mG p eaks at an intermediate z . Finally, for stronger fields (B 10 mG), idealMHD holds at all heights, which causes the flat envelop e, and growth at ab out the ideal-MHD rate, of the p erturbation obtained with B = 10 mG. Unstable mo des are found in this scenario for B 250 mG (see Fig. 3.15, top panel). As the magnetic field is coupled to the gas even at the midplane, these p erturbations are active at z = 0, with only the mo de for B = 1 mG exhibiting a very small central dead zone. The corresp onding p erturbations obtained under the assumption that dust grains are present are displayed in Fig. 4.2 (right panel). Note how the growth rate, wavenumb er and range of magnetic field strengths for which unstable mo des exist are all drastically diminished in relation to the previous case. The low magnetic coupling (see Fig. 4.1, b ottom panel), esp ecially within three scaleheights of the midplane, causes the amplitude of all p erturbations in this section of the disc to b e severely reduced. Unstable mo des were found here only for B ions and electrons are the only charge carriers. Finally, Fig 4.3 compares results obtained using the full conductivity tensor (left column), and the ambip olar diffusion approximation (right column), as a function of the magnetic field strength for the radius of interest. For B = 1 mG, full p erturbations grow faster and are active closer to the midplane, than mo des obtained in the ambip olar diffusion limit. This is the result of the contribution of the Hall conductivity term, given that at 1 mG Hall diffusion is imp ortant and the magnetic coupling is sufficiently strong to supp ort MRI unstable mo des. Conversely, for B = 5 10 mG, a much reduced range compared with the few gauss for which unstable mo des exist when


128

Chapter 4. Impact of dust grains

and 10 mG, the structure of p erturbations in b oth limits is very similar. This is exp ected, as these p erturbations p eaks at a height (z /H 3) where Hall diffusion is no longer lo cally dominant (see b ottom panel of Fig. 4.1).

4.4

Discussion

In this chapter we have examined an illustrative example of the effect of dust grains in the magnetic activity of low conductivity discs. Solutions were computed for R = 10 AU, assuming a single size, 0.1µm grain p opulation is well mixed with the gas phase at all z . The results indicate that the p erturbations' wavenumb er and growth rates are significantly reduced when grains are present. The central dead zone, which was practically nonexistent when grains were settled, extend now to maximum field strength corresp onds well to the equipartition field at z /H 3.7, the height at which the p erturbation for this field strength p eaks (see lower right panel of Fig. 4.2), as exp ected. These results are preliminary and partial. They illustrate the impact of dust particles on the dynamics ­and evolution­ of low conductivity discs, using a single-size grain p opulation. Solutions at 10 AU were computed more or less easily. However, even with this simplified formulation, the stiffness of the equations increases considerably for smaller radii. As a result, time constraints were eventually a limiting factor that prevented further investigation of this imp ortant topic closer to the protostar. This is material to b e develop ed in future studies. These results can b e used, however, to estimate the maximum magnetic field strength to supp ort magnetic activity at 1 AU. The magnetic coupling at this radius, sufficiently couple with the gas. (Wardle, in prep.). Assuming that the maximum (z /H 3.5), we can roughly estimate that the MRI should b e active for B in a disc including 0.1µm grains, is to o weak b elow z /H 3.3 to allow the field to z /H 3. Unstable p erturbations were found in this case for B 10 mG. This

field strength for the MRI to grow is also the equipartition field at ab out this height 400 mG at 1 AU. This is a smaller range than the several gauss for which unstable mo des exist when dust grains are not present. However, MRI mo des still grow in this case for a wide range of field strengths. The results just presented were obtained assuming that all grains have the same size and are well mixed with the gas at all z . More realistic spatial, and size, distri-


4.4. Discussion

129

Figure 4.2 Structure and growth rate of the fastest growing MRI mo des for R = 10 AU and different choices of the magnetic field strength. The left column shows the case where dust grains have settled, while the right one displays results assuming single size 0.1µm grains are well mixed with the gas. The growth rate is indicated in the lower right corner of each panel. The strength of the field app ears in the top right corner. Note the reduced wavenumb er and growth rate, as well as the extended dead zone of the p erturbations obtained when dust grains are present.


130

Chapter 4. Impact of dust grains

Figure 4.3

Structure and growth of the fastest growing MRI mo des as a function of

height for different configurations of the conductivity tensor: Full (left column) and the ambip olar diffusion approximation (right column).


4.4. Discussion

131

butions must incorp orate the effects of dust dynamics and evolution within the disc. Observations of the mid ­and far­ infrared sp ectra of discs have provided credible indications that dust prop erties in discs are indeed different from those of particles in diffuse clouds (e.g. Van Disho eck 2004 and references therein). Two asp ects of this dust evolution have b een clearly identified. First, dust grains coagulate from 0.1µm to 1µm particles. Second, (silicate) material b ecomes crystallised. It is b elieved that this crystallisation o ccurs in the disc, given that crystalline silicates are absent from the interstellar medium. Furthermore, the presence of this material at radial lo cations where the temp erature is to o low to pro duce them, suggests significant radial mixing takes place as well (e.g. Van Disho eck 2004 and references therein). Similarly, simulations of dust dynamics and evolution also suggest that in quiescent environments, they tend to settle and agglomerate into bigger particles (e.g. Weidenschilling & Cuzzi 1993) and efficiently coagulate and grow icy mantles (Ossenkopf 1993). The effect of this settling in the sp ectral energy distribution (and optical app earance) of protostellar discs has b een investigated in recent studies (Dullemond & Dominik 2004). How quickly, and to what height, dust particles are able to settle is an imp ortant, and largely unanswered, question. According to Nakagawa, Nakazawa & Hayashi from 10 (1981), the mass fraction of 1 - 10µm grains well mixed with the gas, diminishes
-1

to 10

-4

the timescale for dust grains to sediment all the way to the midplane may exceed

in a timescale of ab out 2 â 103 to 105 years. Moreover, although

the lifetime of the disc, they may b e able to settle within a few scaleheights from the midplane in a shorter timescale (Dullemond & Dominik 2004). This is complicated even more by the exp ectation that the transition b etween sections were dust grains are well mixed with the gas, and those completely depleted of them, o ccurs gradually (Dullemond & Dominik 2004). All these pro cesses mo dify the surface area of dust grains and impact the recombination rate on their surfaces and the way they drift in resp onse to magnetic stresses. MHD turbulence may itself b e an imp ortant factor for the settling of dust particles. It may, in particular, pro duce sufficient vertical stirring to prevent settling b elow a certain height (Dullemond & Dominik 2004 and references therein). However, this is contingent on the disc b eing able to generate and sustain MHD turbulence in the vertical sections where dust is present. This is not guaranteed, even if turbulence exists in other regions, as dust grains generally reduce the ionisation fraction of the


132

Chapter 4. Impact of dust grains

gas, as discussed ab ove. As a result, the efficiency ­and even viability­ of MHD turbulence in the presence of dust grains, is an imp ortant topic that merits careful investigation.


Chapter 5 Conclusions
In this thesis we have examined the linear growth and vertical structure of the magnetorotational instability (MRI; Balbus & Hawley 1991, 1998; Hawley & Balbus 1991) in low conductivity discs. The MRI is the most promising candidate to generate and sustain angular momentum transp ort away from the central ob ject, so disc material can b e accreted. In addition, this is an imp ortant example of the magnetic activity of such discs. This work is relevant for the study of weakly ionised accretion discs, such as protostellar and quiescent dwarf novae systems (Gammie & Menou 1998, Menou 2000, Stone et al. 2000), where the low ionisation fraction causes the flux freezing approximation to break down. Our metho d incorp orates the effects of the magnetic coupling, the conductivity of the fluid and the strength of the magnetic field, which is initially vertical. Perturbations were restricted to have vertical wavevectors only, which are the most unstable mo des for this field geometry when the fluid is in either the Hall or resistive conductivity regimes (Balbus & Hawley 1991, Sano & Miyama 1999). This is not necessarily the case in the ambip olar diffusion limit (Kunz & Balbus 2003), where the fastest growing mo des can have radial wavevectors as well. We treat the conductivity as a tensor, which is a function of lo cation (r, z ). This is the first study to explore the effect of Hall diffusion in a stratified disc. Using this formulation we examine the impact of different conductivity regimes dominating at different heights on the prop erties of the MRI. This is imp ortant, as Hall diffusion is relevant over a wide range of fluid conditions in low conductivity discs and is exp ected to mo dify the structure and growth of unstable mo des, esp ecially in regions where the magnetic coupling is low. When Hall diffusion dominates, lo cal MRI 133


134

Chapter 5. Conclusions

p erturbations grow for levels of magnetic coupling b elow the minimum required for mo des in the ambip olar diffusion limit to exist (Wardle 1999). The conclusions of this study are summarised as follows: From the analysis of the structure and growth of the MRI in parameter space (chapter 2), we found that the envelop es of short-wavelength p erturbations are determined by the action of comp eting lo cal growth rates at different heights, driven by the vertical stratification of the disc. Ambip olar diffusion p erturbations p eak consistently higher ab ove the midplane than mo des including Hall conductivity. When the magnetic coupling is weak ( < vA /cs ), p erturbations calculated with the full conductivity tensor grow significantly faster, and are active over a more extended cross section of the disc, than those obtained adopting the ambip olar diffusion approximation. Similarly, when the magnetic coupling is reduced, the maximum growth rate regime of the fluid. In particular, p erturbations in the Hall limit grow at ab out the
2 ideal rate when the midplane magnetic coupling (o ) exceeds vA /c2 . In the ams

decreases from the ideal rate ( 0.75) in a way that dep ends on the conductivity

bip olar diffusion regime, the corresp onding limit is o > vA /cs . These results are in agreement with findings of a lo cal analysis (W99). We also derived an approximate criterion ( discs. We then explored the linear phase of the MRI using a realistic ionisation mo del, under the assumption that dust grains have settled towards the midplane (chapter 3). Solutions were presented at representative radial lo cations (R = 1, 5 and 10 AU) as a function of the magnetic field strength for different disc mo dels, configurations of the conductivity tensor and sources of ionisation. For the minimum mass solar nebula mo del, when cosmic rays are assumed to p enetrate the disc, unstable mo des exist at 1 AU for B mo des still grow for B 8 G. When 200 mG B 5 G, the growth rate is of order the ideal-MHD rate (0.75). When cosmic rays are excluded from the disc, unstable 2 G. Hall diffusion dominates the structure and growth of the instability at this radius for all magnetic field strengths for which unstable mo des exist. Solutions obtained with a full conductivity tensor grow faster, and act over a wider vertical cross-section of the disc, than p erturbations in the ambip olar diffusion limit. Finally, when the magnetic field is strong, ambip olar diffusion shap es the p erturbations' envelop es, which p eak at an intermediate height, instead of having of the MRI. This is satisfied over a broad range of conditions in low conductivity |1 |/ for when Hall diffusion dominates the structure and growth


135 the fairly flat envelop e mo des in the Hall limit have in this region of parameter space. We conclude that, despite the low magnetic coupling, the magnetic field is dynamically imp ortant for a broad range of conditions in low conductivity discs. At radii of order 1 AU, it is crucial to incorp orate Hall conductivity in studies of the magnetic activity of these discs, and in particular, in the analysis of the MRI. In chapter 4 we explored the impact of dust grains on the solutions just discussed. They affect the dynamics of discs if they are well mixed with the gas. This is generally the case in early stages of accretion, and/or when turbulence prevents them from settling towards the midplane. Dust particles can b ecome an imp ortant charged sp ecies, but their drift is impaired by collisions with the neutrals. They also op en up additional mechanisms for more mobile charged particles to recombine. As a result, grains are exp ected to lower the ionisation fraction, and the conductivity, of the gas. For simplicity, we assumed that all particles have the same radius (0.1µm) and constitute 1 % of the total mass of the gas (Umebayashi & Nakano 1990). We presented solutions for R = 10 AU and compared them with the results obtained when grains were absent. The growth rate, wavenumb er and range of magnetic field strengths for which unstable mo des exist drop sharply, as exp ected. Similarly, the central magnetic dead zone, which was practically nonexistent in the previous case, extends from the midplane up to z /H 3. We find that Hall diffusion mo difies fields (B the structure and growth of the MRI when B 1 mG. However, for stronger the prop erties of full conductivity mo des as well. This is to b e exp ected, as these p erturbations p eak at a vertical lo cation where Hall diffusion is no longer lo cally dominant. Perturbations grow for B 10 mG, a much reduced maximum strength from the 250 mG for which the disc was active when grains were assumed to height for which p erturbations p eak, as exp ected. Time constraints prevented further investigation, given the long runtime associated with finding mo des for R 5 AU. This is left for future work. However, 3 the results at 10 AU can b e used to estimate the activity of the disc at 1 AU. At this radius, the magnetic field is also not sufficiently coupled to the gas for z /H (Wardle, in prep.). Therefore, MRI unstable mo des are exp ected to p eak at a higher z . Assuming that the maximum field strength to supp ort unstable MRI mo des is

5 mG), p erturbations in the ambip olar diffusion limit are similar to

have settled. This maximum field strength (10 mG), is the equipartition field at the


136

Chapter 5. Conclusions

the equipartition field at that height (as was the case at 10 AU), we can roughly estimate the maximum field strength for which unstable mo des are exp ected to grow at this radius. Using z /H = 3.5, we find that the magnetic field should b e dynamically imp ortant for strengths up to 400 mG. These solutions were obtained assuming a single-size grain p opulation was present.

In general, however, grains have a complex size distribution (e.g. Mathis, Rumpl & Nordsieck 1977; Umebayashi & Nakano 1990). Also, although in quiescent environments they settle towards the midplane and b ecome dynamically uncoupled to the gas at higher z , MHD turbulence may prevent them from sedimenting b elow a certain height (Dullemond & Dominik 2004). On the other hand, grains may diminish the efficiency of MHD turbulence itself, by reducing the gas magnetic coupling. The equilibrium structure of low conductivity discs, and their magnetic activity, are the result of a complex interplay b etween many physical pro cesses. We have explored here, in the context of an analysis of the linear phase of the MRI, the impact of some of these pro cesses in the dynamics of discs: the magnetic field strength, level of magnetic coupling, magnetic diffusion, disc mo del, ionising sources and presence of dust grains. Other pro cesses, notably the mechanism leading to the saturation of the MRI, can only b e studied using a non-linear formulation. This is an imp ortant pro cess, given that the saturated level of the Maxwell stress (Br B /4 ) largely determines how much angular momentum is transp orted radially outwards by the MRI. The prop erties of the MRI in the saturated state seem, in turn, to b e dep endent on the characteristic length-scale of the instability in the linear phase (Sano, Inutsuka & Miyama 1998, Sano & Inutsuka 2001, Sano & Stone 2002b). This is so b ecause the fastest growing mo de in the linear regime is also an exact solution to the full non-linear equations in the incompressible limit (Go o dman & Xu 1994). Numerical studies of the saturation of the MRI (Hawley & Balbus 1992; Matsumoto & Ta jima 1995; Stone et. al. 1996; Sano, Inutsuka & Miyama 1998; Sano & Miyama (1999); Fleming, Stone & Hawley 2000; Sano & Inutsuka 2001; Sano & Stone 2002a, 2002b; Sano et. al. 2004) have not yet mo delled the case when different conductivity regimes are dominant at different heights. The results of this study suggest that such a formulation is crucial for the realistic mo delling of the saturated state of the MRI. In particular, we have seen that Hall diffusion tends to increase the wavenumb er of unstable mo des in relation to that of p erturbations in the ambip olar diffusion limit for the same field strength. This suggests that non-linear mo dels incorp orating a


137 height-dep endent conductivity, as well as Hall diffusion, are necessary to study the mechanism leading to the saturation of the MRI and, ultimately, the efficiency of MRI-driven angular momentum transp ort in low conductivity discs. More generally, all these pro cesses are also likely to affect other asp ects of the activity of discs, such as planet formation and migration, launching of jets and dynamo action. Hop efully this study will draw attention to the imp ortance of Hall diffusion and, more generally, to the link b etween the micro ­ and macro ­ physics in low conductivity accretion discs.


138

Chapter 5. Conclusions


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