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ISSN 1063 780X, Plasma Physics Reports, 2009, Vol. 35, No. 12, pp. 1032­1035. © Pleiades Publishing, Ltd., 2009. Original Russian Text © V.V. Fomichev, S.M. Fainshtein, G.P. Chernov, 2009, published in Fizika Plazmy, 2009, Vol. 35, No. 12, pp. 1114­1117.

SPACE PLASMA

A Possible Interpretation of the Zebra Pattern in Solar Radiation
V. V. Fomichev, S. M. Fainshtein, and G. P. Chernov
Pushkov Institute of Terrestrial Magnetism, Ionosphere, and Radio Wave Propagation, Russian Academy of Sciences, Troitsk, Moscow oblast, 142190 Russia
Received March 30, 2009; in final form, June 10, 2009

Abstract--The nature of the zebra pattern in continual type IV solar radio bursts is discussed. It is shown that, when a weakly relativistic monoenergetic proton beam propagates in a highly nonisothermal plasma, the energy of the slow beam mode can be negative and explosive instability can develop due to the interaction of the slow and fast beam modes with ion sound. Due to weak spatial dispersion, ion sound generation is accom panied by cascade merging, which leads to stabilization of explosive instability. The zebra pattern forms due to the scattering of fast protons by ion sound harmonics. The efficiency of the new mechanism is compared with that of previously discussed mechanisms. PACS numbers: 94.05. a, 96.60.Tf DOI: 10.1134/S1063780X09120058

1. INTRODUCTION Stripes in the radiation and absorption spectra in the form of more or less regular harmonics (the so called "zebra pattern") against the background of con tinual radiation of type IV radio bursts have been studied for nearly half a century since the first publica tion by ElgarÜy [1]. The main properties of the zebra pattern are described in reviews [2­4]. The interpreta tion of the zebra pattern has always lagged behind the accumulation of new observational data. A dozen models of the zebra pattern have been proposed, the most developed of which are based on the double plasma resonance mechanism [5, 6] and interaction of plasma waves with whistlers [7, 8]. However, these models encounter some difficulties that stimulate the search for new mechanisms. In recent years, more advantageous models based on the propagation of electromagnetic waves through regular density inho mogeneities in the solar corona have been developed [9­11]. Observations of the zebra pattern during solar bursts indicate that particles in the radio source are accelerated to relativistic velocities and different wave modes are excited. Therefore, other possible mecha nisms of wave­particle interaction should also be taken into account. For example, decay of whistlers into weakly dispersive, weakly damped ion sound har monics at frequencies much lower than the ion Lang muir frequency 0i was considered in [12]. In the present paper, we discuss an alternative mechanism related to the development of explosive instability of a weakly relativistic beam propagating in a nonisother mal plasma. Remind that explosive instabilities develop in nonequilibrium systems in which modes

with negative energies exist [13]. In this case, in the resonance triplet, the wave with the highest frequency (3) should have a negative energy, while the waves with the lower frequencies (1, 2) should have positive energies (see, e.g., [14­16]); or, vice versa, the waves 1, 2 should have negative energies, while the wave 3 should have a positive energy. When the wave 2 has a negative energy and the waves 1, 3 have positive ener gies, instability accompanied by the excitation of the highest frequency wave (up conversion) develops. In this case, the energy of the low frequency wave 1 is transferred to the higher frequency waves 2, 3, the energies of which grow by the exponential law in the given field of the wave 1 [17], as is in a conventional decay instability [18]. As will be shown below, the more efficient (from the standpoint of energy transfer) mechanism is generation of "saw tooth" ion sound due to the development of explosive instability of a weakly relativistic proton beam propagating in a highly nonisothermal plasma. Since the plasma electron temperature is much higher than the ion tempera ture, Landau damping of ion sound can be ignored [19] and ion motion can be described in terms of quasi hydrodynamic equations [19]. However, viscous damping of ion sound should be taken into account, because it is proportional to the sound wavenumber - squared and the viscosity coefficient is iiVT2i 0i2 , where VTi is the ion thermal velocity and ii is the ion­ ion collision frequency. The damping rate of ion sound is eff k2, where k is the sound wavenumber. The number of ion sound harmonics, n, is determined by two conditions: (i) ion sound dispersion should be rel

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atively small (n 2q 2cs2 2i , where q is the wavenumber 0 of the fundamental sound mode, c s = ( Te/ M )1/ 2 is the ion sound velocity, M is the ion mass, and is the Bolt zmann constant) and (ii) the damping rate eff should be much smaller than the circular frequency of sound (i.e., the nth sound harmonic should be weakly damped). Analyzing conditions (i) and (ii), we find the number of harmonics n in the zebra pattern. 2. INITIAL EQUATIONS, CONDITIONS OF SYNCHRONISM, AND EQUATIONS FOR THE AMPLITUDES OF WAVES Let us consider a weakly relativistic monoenergetic proton beam1 propagating with the velocity V0 ~ c/3 (where c is the speed of light in vacuum) in a highly nonisothermal quasineutral plasma. The plasma elec trons in the wave field are assumed to have a Boltz mann distribution [19]. The beam ions interact with the plasma via the electric field E (here, ( 0x k j;, where kj is the wave vector of the jth mode). Beam­ plasma interaction can be described in terms of quasi hydrodynamic equations [19, 21],

2 where 2b = 4 e 2N 0bM 0-1, 0i = 4 e 2N 0M -1, is the 0 circular frequency, and k is the wavenumber. For V0 / cs 1 , N 0s/ N 0 1, we obtain from Eq. (2) the following approximate dispersion relations:

1 csk1 csmq,
3, 2 - k3, 2V0 0b + ,

(3)

1. (4) 0b Dispersion relation (3) describes an ion sound wave with a positive energy, while dispersion relations (4) describe a slow (3) and a fast (2) beam wave having a negative and a positive energy, respectively. It is easy to see that the slow beam wave (3, k3), fast beam wave (2, k2), and sound wave (, q) satisfy the synchro nism conditions [22]. Taking into account Eqs. (3) and (4), we find from the synchronism conditions that
mq 20sV0-1 .

(5)

Since ion sound dispersion is weak, the following cas cade process is possible:
mq + mq 2mq + mq 3 mq + mq. . . mnq .

E=- Vi V + Vi i t x V s t

E ; = 4e (e - s - i ) ; x x i = - e E - eff Vi; + (iVi ) = 0; t x M (1) V + Vs s = - e E + c -2Vs2E ; x M o + V = 0, ( s) ( s s) t x

Let us expand the nonlinear terms in Eq. (1) in power series in perturbed quantities and retain the sec ond order terms. Then, applying a standard technique [14, 22], we obtain the following reduced equations for the complex amplitudes aj(x, t) (j = 1, 2) and bk (k = 1, 2, ..., mn), where aj are the amplitudes of the beam modes, bk are the amplitudes of the ion sound modes, and is a small parameter (for simplicity, we consider a steady state, in which / t = 0):

V

0

where e, M0, and M are the electron charge, the rest mass of a beam ion, and the mass of a plasma ion, respectively; e = N 0 ex p( e / Te) ; is the electric potential; b = N 0b + b , Vb = V0 + Vb ; b,Vb, i, , and Vi are deviations of the beam ion density, beam ion veloc ity, plasma ion density, and plasma ion velocity from their equilibrium values N0b, V0, N0, and 0, respec tively; and = (1 - Vb2/ c 2)-1/ 2. Linearizing Eq. (1) with respect to perturbed quan tities, which are assumed to vary in space and time as ~exp(it ­ ikx), we obtain the dispersion relation for the beam­plasma system,

a1 a * = 1a2bm, V0 2 = 2a1bm, x x 1 2 = ,

(6)

b * cs m = a1 a2 x * * + i b2mbm + b3mb2m + ... bnmb(* n c

(7) b2m 2 * * = bmb3m + b2mb4m + ... bnmb(* - 2)m + bm - 4 bm, s n x b * * c s 3m = 3 b4mbm + b5mb2m + ... bnmb(* - 3)m + b2mbm - 9 b m. n x

( 2(

- 1)m

)

- bm,

)

(

)

1-

2 2 0i 2i 0b - 202 - = 0, cs k (- kV ) 2 1 - - ck 0 3ck

(

)

(2)

Here, = (VTi/ ii)q . field 1, 2 bk can Analysis shows that i.e., Eqs. (6) and (7) [15].2
2

The solution to Eq. (7) in a given be found in [12] (see also [23]). 1, 2 , and have the same sign, describe a "stabilized explosion"

1

The beam can be considered monoenergetic under the condi 1, where Nb, VTb, and V0 are tion [20] (Nb/N0)1/3(V0/VTb)2 the beam density, the beam thermal velocity, and the equilibrium velocity of beam particles, respectively. PLASMA PHYSICS REPORTS Vol. 35 No. 12 2009

It should be noted that conventional beam instability can be ignored if its growth rate is much smaller than the inverse explo

0i / ( a ) . Moreover, beam sion time, i.e., if (N 0b/ N 0) instability is suppressed by a strong field [24].

1/3


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FOMICHEV et al.

3. QUALITATIVE ESTIMATES Let us take the following parameters of the beam­ plasma system as applied to the solar corona: N0 ~ 5 â 109 cm­3, Nb/N0 ~ 10­3, V0 c/3 ~ 1010 cm/s, Te ~ 106 K, and i ~ 105 K. In this case, we have s ~ 107 cm/s and vii ~ (0.5­1) s­1. The maximum number of sound harmonics can be estimated from conditions (i) and (ii) (see Introduction) and the conditions for radiation to escape from the solar corona, t > 0 (which corre sponds to the circular frequency of the scattered elec tromagnetic wave, t 634 MHz). Then, we have 7 â 102 mn 15 â 103. The frequency 3 ~ 0i is assumed to be given,3 while the quantities 2, k2, k3, , and q are determined from Eqs. (3) and (4) and the synchronism conditions. For a constant magnetic field of ~30 G, the growth rate of ion sound w in a given field of whistlers was estimated in [12] (w ~ |aw |2w, where |aw | is the whistler amplitude and w is the matrix interaction coefficient). In a given field of beam waves, |ab|, the growth rate of ion sound is b ~ |ab |2, where 4 â 109em/(M0c) and the quantity (see Eq. (6)) is ~102em/(M0c). In this case, the ampli tude of the fundamental ion sound harmonic increases as b1 ~ bx. Estimates show that b/w ~ 6 â 105|ab |2/|aw |2. Thus, the proposed mechanism of ion sound generation is much more efficient than that described in [12]. In the above estimates for and , the wavelength and initial frequency of ion sound are assumed to be ~100 m and ~1.0 kHz, respectively (see Eqs. (3), (5)). The circular frequency of the slow beam mode is 3 ~ 7 â 1020i (which corresponds to ~10 GHz). The generated ion sound is scattered by fast pro tons moving with a velocity of V ~ V0 ~ 1010 cm/s. According to the mechanism described in [12],4 the frequency of radiation emitted by the source is t mqnV and the frequency spacing between stripes is t = mqV. Taking into account Eq. (5), for a radia tion frequency of 634 MHz and the given parameters N0 ~ 5 â 109 cm­3 and Nb/N0 ~ 10­3, we find that m = 15. Hence, the frequency spacing between neighbor ing stripes is t 15 MHz. This value of t corre sponds to the observed period of the zebra pattern in the decimeter wave band [4]. The proposed mechanism also provides the observed increase in t (for the same nth harmonics,
3

the factor m increases with increasing frequency). Dis crete radiation stripes can exist if the width of the stripes is smaller than the frequency spacing between them. This condition imposes restrictions on the velocity spread of the beam ions. According to esti mates made in [12], the proton beam should be highly monoenergetic (V0/V0 < 10­3 in our case). The dips between the radiation strips of the zebra pattern can be associated with the disruption of loss cone instability (which is responsible for continuum radiation) due to additional injection of fast particles into the loss cone. This mechanism was proposed in [25] to explain negative bursts. 4. CONCLUSIONS The above mechanism based on the stabilization of explosive instability accompanying a cascade increase in the amplitudes of ion sound harmonics proves to be more efficient than the mechanism associated with the decay of whistlers into ion sound harmonics [12]. The proposed mechanism provides a greater number of zebra pattern harmonics with a frequency spacing that is independent of the ratio between the plasma and cyclotron frequencies in the source and increases with increasing frequency (in accordance with observa tions). In this case, no additional rigid constraints are imposed and there is only the conventional require ment related to the generation of monoenergetic par ticle beams, which usually occur in any strong burst. In this context, it should be noted that each improvement of the existing models is usually accom panied by imposing new restrictions on the parameters of the plasma and fast particles in the source. For example, it was shown in [26] that the double plasma resonance provides a larger number of the zebra pat tern harmonics only if fast particles possess a steep power law energy spectrum. In the whistler model [4, 8], the number of stripes in the zebra pattern is large only if the magnetic trap is filled with whistler wave packets in the regime of wave induced quasilinear dif fusion of fast particles. New models of stripe forma tion in the zebra pattern by electromagnetic waves propagating through regular density inhomogeneities in the solar corona assume the presence of small scale inhomogeneities with a spatial period of several meters. All these models and their capabilities of explaining new observational data are described in more detail in review [27]. It should be noted that the results obtained in the present study can also be applied to explain the gener ation of high power low frequency radiation in the atmospheres of stars and pulsars, estimate the effi ciency of plasma heating in ICF devices (heating of plasma targets by particle beams), and create high power generators (amplifiers) of low frequency radia tion in laboratory gas discharge and solid state plas mas.
PLASMA PHYSICS REPORTS Vol. 35 No. 12 2009

For given parameters N0, N0b, V0, and cs, from Eq. (2) with allowance for Eqs. (3) and (5), we readily obtain /0i ~10 â (7 + /(30b)), where is the quantity entering into Eq. (4).
2

4

Here, we mean stimulated wave scattering by particles (by anal ogy with Eq. (11) in [22]), in which case we have

- = (k - q)V .

t

t


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ACKNOWLEDGMENTS This work was supported by the Russian Founda tion for Basic Research, project nos. 08 02 00270 and 06 02 39007. REFERENCES
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14. J. Weiland and H. Wilhelmsson, Coherent Nonlinear Interaction of Waves in Plasmas (Pergamon, Oxford, 1976; ènergoizdat, Moscow, 1981). 15. S. M. Fainshtein, Zh. èksp. Teor. Fiz. 71, 1021 (1976) [Sov. Phys. JETP 44, 534 (1976)]. 16. S. M. Fainshtein and E. A. Chernova, Zh. èksp. Teor. Fiz. 109, 821 (1996) [JETP 82, 442 (1996)]. 17. M. I. Rabinovich and S. M. Fainshtein, Zh. èksp. Teor. Fiz. 63, 1672 (1972) [Sov. Phys. JETP 36, 884 (1972)]. 18. V. N. Oraevskii and R. Z. Sagdeev, Zh. Tekh. Fiz. 32, 1291 (1962) [Sov. Phys. Tech. Phys. 7, 955 (1963)]. 19. V. L. Ginzburg, The Propagation of Electromagnetic Waves in Plasmas (Nauka, Moscow, 1967; Pergamon, Oxford, 1970). 20. A. B. Mikhailovskii, Theory of Plasma Instabilities (Atomizdat, Moscow, 1970; Consultants Bureau, New York, 1974), Vol. 1. 21. L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields (Nauka, Moscow, 1973; Pergamon, Oxford, 1975). 22. V. N. Tsytovich, Nonlinear Effects in Plasmas (Nauka, Moscow, 1967; Plenum, New York, 1970). 23. V. M. Galkin, I. A. Kanygina, and S. M. Fainshtein, Izv. Vyssh. Uchebn. Zaved., Radiofizika 22, 1255 (1979). 24. Yu. M. Aliev and V. P. Silin, Zh. èksp. Teor. Fiz. 48, 901 (1965) [Sov. Phys. JETP 21, 601 (1965)]. 25. V. V. Zaitsev and A. V. Stepanov, Astron. Astrophys. 45, 135 (1975). 26. A. A. Kuznetsov and Yu. T. Tsap, Solar Phys. 241, 127 (2007). 27. G. P. Chernov, Center EUR Astrophys. Bull., No. 4 (2010) (in press).

Translated by E.V. Chernokozhin

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