Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://star.arm.ac.uk/preprints/401.ps Äàòà èçìåíåíèÿ: Wed Sep 17 19:19:00 2003 Äàòà èíäåêñèðîâàíèÿ: Mon Oct 1 21:34:31 2012 Êîäèðîâêà: Ïîèñêîâûå ñëîâà: molecular cloud

Mon. Not.
R. Astron. 339, 133--147 (2003)
The instability fast shocks molecular clouds
Michael Smith and Alexander Rosen
Armagh Observatory, College Hill, Armagh BT61 9DG
Accepted 2002 October
7. Received 2002 September original form October
ABSTRACT
report
on discovery moderately shocks dense molecular clouds with low
transverse magnetic fields likely
to
be unstable. The instability
is triggered promoted
cooling that results the formation
of carbon monoxide water molecules extended
warm shock section. Numerical methods employed demonstrate the absence
magnetic fields, instability regime restricted
to densities above
0
, velocities
between 30--70 km
s
, and
C abundances above #10
, that cooling from reforming
molecules dominates warm without being suppressed ultraviolet dissociation. The
result either quasi­periodic chaotic collapse re­establishment
of warm shock
a typical time­scale
of
6 with variations shorter time­scales and
changes
in period being possible. Infrared emission from unstable region, including
2 lines, exhibit orders magnitude variability. Atomic H# display constant
fluxes undergo radial velocity variations.
words: hydrodynamics instabilities molecular processes
-- shock waves ISM: lines
bands ISM: molecules.
NTRODUCT
Fast shock waves alter dynamical, physical chemical
erties dense interstellar clouds (Hollenbach &McKee 1979).
shocks driven internally externally
a variety sources,
including supernova blast waves, protostellar jets, cloud--cloud
lisions, stellar winds planetary nebula. define
a
one capable destroying molecules path.
a dense cloud,
many
of molecules subsequently reform
in cooling
layer compressed accelerated material (Hollenbach &McKee
1989; Neufeld &Dalgarno 1989a). compressed cloud may
form stellar systems fragment disperse. Hence,
behaviour shocks critical
to distributions
stars interstellar gas.
Here investigate time­dependent properties shocks
through one­dimensional (1D) numerical simulations.
ture signatures steady fast shocks was studied
in
by
Hollenbach
& McKee (1989), Neufeld
& Dalgarno (1989a,b)
Wolfire K˜onigl (1991).
a shock
is unstable, however,
reconsider only observed emission properties
of radiative
shocks also transfer turbulent energy cloud,
fragmentation destruction cloud, result­
ing shock­induced molecular abundances and accuracy
of
numerical simulations not resolve individual shocks.
Hydrodynamic shocks destroy molecules
at shock
possess speeds excessvadjust #23
-1 (Kwan
E­mail: mds@star.arm.ac.uk (MDS); rar@star.arm.ac.uk
Hollenbach
& McKee 1980; Smith 1994). dense molecular
shielded ultraviolet (UV) radiation, shocks continuous
(C­type)
in which
a fraction leads drift
of the magnetic
through neutral case, molecules largely survive
in shocks with speeds
to #30
s
if magnetic field
parallel (Smith 1993)
s
is transverse
(Draine, Roberge &Dalgarno 1983). Below these speeds, shock
cooling layer overlap. Above these speeds, however,
shock classified fast molecules abruptly disso­
ciated shock front, followed
a wide zone radiative
cooling. demonstrate moderately shocks
in dense molec­
clouds are thermally dynamically unstable. Thermal insta­
bilities small scales expected cooling layer follows
shock front provided cooling unit mass increases
as temperature (McCray, Kafatos Stein 1975). This
is
cause
a small region within the with
a lower temperature than
surroundings faster, resulting
a growing density contrast
Smith 1989).
To dynamically unstable, cooling must increase
sufficiently rapidly temperature falls (Langer, Chanmugam
Shaviv 1981; Chevalier
& Imamura 1982).
a certain time­scale,
a cooling shocked layer collapse simultaneously
as shock
shrinks back, reducing shock strength. shock
velocity promotes faster cooling, which leads even faster
collapse
of cooling layer. Thus,
a resonance occurs and entire
collapse `catastrophically'. The lowered shock velocity
leads
a lower pressure. Hence,
a resisting medium confin­
can decelerate retreat
of shock
C 2003134 Smith and Rosen
period increasing shock velocity
a cooling layer increas­
ing temperature shock layer inflates. higher temperature
gas efficiently,
so expands outwards, driving
shock
to even higher speeds. The shock perform
ing periodic oscillations
to high­amplitude quasi­periodic
oscillations Imamura, Wolff Durisen 1984; Strickland
&
Blondin 1995).
This dynamical overstability relevant atomic shocks.
For velocities greater than approximately
s
, strong atomic
cooling near
K predicted initiate overstability
Innes, Giddings
& 1987; Gaetz, Edgar Chevalier 1988).
instability expected, however, molecular shocks
all
individual cooling functions positive functions
of tempera­
ture, shown Fig.
1.
therefore surprised
to find
a dynamical instability
in
numerical simulations. version
a three­dimensional
(3D) molecular hydrodynamic code, weperformedone­dimensional
high­resolution code improves Suttner
et
al.
(1997) including
H
2 chemistry cooling
Appendices). previous code not signs dynami­
instability. fact,
it production
2 Omolecules
cooling layer
to
of catastrophic cooling.
The formation
of molecules causes instability
by altering
cooling function
is time­dependent
as moves down­
stream. Note (1983) recognized that rapid formation
water molecules could
to thermal instabilities molecular
clouds. The fundamental condition dynamical instability
derived through numerical solutions specific cases, especially
for
monatomic with
a cooling proportional square
of
density, with atomic shocks being considered (Chevalier
& Imamura
1982; Chanmugam, Langer
& Shaviv 1985; Dgani Soker 1994).
Numerical simulations demonstrate instability leads
amplitude oscillations (Strickland Blondin 1995). Section
2,
present hydrodynamic equations discuss criteria
for
which molecular shocks should
be unstable, unstable modes
and oscillation periods. also present cooling
tion temperature steady­state shocks equilibrium
C
and
O chemistry employed
in numerical calculations
for
which time­dependent oxygen carbon chemistry
is practical
Figure cooling line) cooling functions
a
with indicated fixed density
2 fraction. cooling components
are:
1, gas--grain;
2,
2 ro­vibrational;
3, atomic;
4,
H
2 rotational;
2
O
vibrational
2
;
6,
2 vibrational
H dissociation;
8,
H
2
reformation heating (thin double line);
9, rotational; vibrational
with
H
; COvibrational with H;12,
I] structure; rotational.
to follow. compare cooling functions steady­
shocks non­equilibrium chemistry conclude
instability will probably present, simulations
time­dependent chemistry ionization needed verify this
conjecture determine minimum abundances necessary. Note
molecular hydrogen formation rate remains
certain influence instability regime.
In Section present numerical results demonstrate
nature instability yield time­scales. strong influ­
instability observational properties, emphasis
infrared emission
is then discussed (Section 4.1).
Several conditions must fulfilled instability oper­
First, sufficient
2 gas below
#8000 Hollenbach McKee (1989) have shown and
water cooling indeed dominate steady shocks
in high­density
regions. consider carbon oxygen depletions that
stabilize shock. Ameliorating factors also include thermal con­
duction and depletion. Furthermore, sufficiently strong trans­
magnetic cushions shock inhibits dynamical
instability (T’oth Draine 1993).
that radiation will influence location which
molecules reform (Hollenbach McKee 1989). shocks with
speed
v
#
80
s
, radiation will inhibit formation
of
2 molecules. However, calculation Neufeld
Dalgarno (1989a) shows cooling then replaces cooling,
providing
a similar steep increase
in cooling temperature falls.
Nevertheless, restrict results shocks below speed.
shock instabilities operating. These mainly
distort fragment dense gas accumulates either
between shocks
or between
a single shock medium
high­thermal pressure. The instabilities include Vishniac thin­
linear instability (Vishniac 1983; Mac Low
& Norman 1993),
non­linear thin­shell instability (Vishniac 1994) trans­
acceleration instability (Dgani, Buren
& Noriega­Crespo
1996). They require two­dimensional studies, whereas
cillatory instability present even one­dimensional simulations.
Moreover, instability occurs
in cooling layer rather than
accumulated cold possessing distinct signatures.
2 THE FLOW EQUAT
I ONS AND
ANALYT
I CAL PRED
I ON
The equations
model basic radiative shocks: hydrodynamic flows
dimension. Numerically, will solve time­dependent flow
equations: (#v)
x
(#v)
t (#v
2
0
#(ev) -#(T,
f
)
f
f nv)
n,
f
)
- D(T,
),
where
n
is hydrogen nuclei density,
e internal energy
density and
f
is molecular hydrogen abundance
2
f take
a helium abundance n(He)
= Therefore, total
particle density
n (1.1
-
f and temperature
C
# 2003 RAS, MNRAS 133--147

Shocks molecular clouds 135
p/(kn
tot internal energy loss through radiation chemistry
per volume
is dissociation reformation rates
of
molecular hydrogen
by
D and
R, respectively.
The internal energy related pressure p/(#
1)
where effective specific ratio taken
as
5.
5
f
3.
3
-
f
(5)
which assumes
2 possesses two rotational degrees
of freedom.
We define
#
tot
p
=
1.
4
1. f.
(6)
2.2 Power­law cooling predictions
Shock stability studies have been mainly restricted flows
power­law cooling
of
0
#
#
(7)
(Chevalier Imamura 1982; Wolff, Gardner Wood
Strickland Blondin 1995) superpositions power­law
ing Saxton
et 1998 further references). Thermal instabil­
ities interstellar medium first analysed detail
(1965). Smith (1989) investigated small­wavelength isobaric
condensation modes within shock waves. The instability condition
(equation
of Smith 1989) reduces
to
# 1)/(3
-
#
)
shock and approaches condition
<
# downstream.
Note that
#
=
2, instability condition remains close
<
whereas
1 condition more stringent.
The oscillatory instability condition
is even more stringent,
pending upon cooling processes boundary conditions.
For
# oscillations unstable fundamental
mode
#
0.
4 and
in higher modes
c where values
c exceed (Chevalier Imamura 1982). Note, however,
apparently
a range values
#
< which non­linear
effects critical (Strickland Blondin 1995).
In this range,
ondary shocks form near wall restrict, not
collapse. fundamental
is thought most signifi­
cant since long­wavelength perturbations would least damped
motions transverse direction. Numerical simulations
demonstrate, however, higher­order modes generate
lations
#
0. (Strickland
& Blondin 1995).
applications
to shock propagation through molecular
we
should consider
#
=
7
5 (fully molecular)
# (fully molec­
ular cent helium). Furthermore, unity several
cooling processes high­density CO rotational
2
ra­
diative cooling. dissociative shocks, however,
hydrogen almost completely atomic during cooling
down below that instability
is caused
2 vibrational cooling (with molecules excited collisions
with which
# these respects, existing analyses
provide relevant criteria present conditions.
2.3 steady­state model
The cooling function
a uniform constant density
and chemistry displayed Fig.
1. component cooling
tions listed Appendix
A chemical reactions Appendix
They were selected problems associated shocks
in
dense molecular clouds. assumes equilibrium chemistry
described Appendix conclude cooling functions
Figure cooling functions corresponding
to dissociative shock waves
prove
to unstable, assuming equilibrium chemistry
for
C with
standard conditions
as shown abundances
(
5
â
=
-4 panel)
#
=
â
2
(lower panel) Alfv’en speed
s
. panel corresponds
to unstable depleted
O case. individual cooling/heating components
labelled
positive functions temperature with power­law index
being above oscillatory instability critical value each
section. This holds only indicated parameters under
much wider conditions.
unstable shocks that motivated this analysis involve com­
multicomponent time­dependent cooling function Im­
mediately following shock front,
is zone
of steep cooling,
dominated
2 radiative and dissociative cooling atomic cool­
Therefore, expect with
T
> 8000
be stable.
that thin section maintained throughout, moving
accommodate oscillating cool layer, signals trav­
elling quickly through signal propagates slowly through
following cool layer, which refer `infrared layer', and
determines dynamical instability properties.
steady­state model solves hydrodynamic equations
post­shock (with partial derivatives being dropped).
shock
is replaced discontinuity satisfying
Rankine--Hugoniot conditions, attention being spe­
heat ratio. pre­shock parameters represented
p
0
,
v
,
0
,
#
0
,
#
0
,
f
0
2
=
(#
v
2
/#
p
0 Since molecular
dissociation requires
a finite time, immediate post­shock values
f
f
,
1
0
,
C 2003 MNRAS 339, 133--147

136 Smith and Rosen
S
1
0
0
1
0 1)M
2
0 1)M
2
0
(8)
p
1
0
=
1
1
M
(9)
and
1
0
=
p
0.
The radiative
on eliminating
e, described
=
1
1
p
2
1
1
v
and, substituting,
#[p/(#
-
1)]
+
#
p
= -#(n,
f
),
which, using equation substitute from
##
#x
-4. (3.
3
)
2
#
and
#
f
1.
p
#
1
1
leave single first­order equation that solved numerically
without difficulty.
Before running steady shock model simulations, added
equilibrium oxygen carbon chemistry addition hydro­
gen chemistry. basic reactions form
OH molecule were chosen (Appendix Previously, had
abundances follow molecular hydrogen fraction
not included
2 simplest approximation that allowed
us
model molecular
in dimensions (Suttner 1997).
Here, however, assume conversion oxygen
and and
O described equilibrium
dances through reactions
2
C. This especially critical
large­scale simulations since advantageous
to avoid
introduction
of variables.
H
2
form ahead
2 reformation since small fraction
2 sufficient instigate complete conversion
oxygen. Note OH oxygen structure cooling included
steady­state slab models, excluded from
lations. find cooling contributions neglected
present shock simulations densities exceeding
3
4
,
as demonstrated hydrogen nucleon density
6
-3
Fig.
2.
Time­dependent chemistry included steady­state
calculations
to test equilibrium assumption. Equilibrium chem­
istry found
to provide reasonable estimate cooling function
high densities. demonstrated
by comparing panels
of Figs
2
and
3, cooling generated
H
O formation
is weaker
equilibrium This would influence oscillatory structure
do believe that sufficient remove instability.
instability present when either
C
or depleted
an
order magnitude, yielding more gradual increase
in
ing (e.g. lower
of Fig.
2 where
2 bump
is
Figure cooling functions dissociative shock waves non­
equilibrium chemistry standard conditions described
individual cooling/heating components labelled
greatly reduced). We note predict detailed structure,
dynamical processes
as critical chemical effects. Two­
dimensional simulations full time­dependent chemistry will
needed.
a density
4
, however, non­equilibrium chemistry
is certainly required
to reproduce important contributions
oxygen fine structure water cooling below 8000 shown
in non­equilibrium chemistry, increase
in
vibrational cooling cools below
K
is longer
mirrored
in cooling.
Dynamical instabilities may occur when molecular hydrogen
fraction inversely related temperature, such possible
in cooling layer shock,
as now discussed. cooling
is presented volume,
#,
in since cooling
functions simple power­law functions
of density. Hence,
to predict which these states would unstable, need inter­
critical temperature indices discussed Section terms
of index
#
by
#
f
(T
#
T
#
T
, which follows
a fluid parcel through cooling
at almost constant pressure.
interpretation contains uncertainty, however, since chem­
(and, hence, cooling)
is time dependent. Nevertheless,
that
a high negative value
#
is present temper­
regime between 3000 8000 standard non­depleted
O abundances. Moreover, width shock
is mainly
determined
by temperature which rapid cooling about
to
K confirmed Fig. This loca­
adjacent cooling zone possesses
a inverse
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# 2003 RAS, MNRAS 133--147

Shocks molecular clouds 137
Figure comparison
of equilibrium non­equilibrium chemistry
cooling functions within dissociative shock waves density
of
4
standard conditions described individual
cooling/heating components labelled
as
dependence temperature.
us
to conclude
shock
is,
at least, linearly unstable.
example
C strongly depleted confirms
cause instability. Low oxygen and carbon abundances permit
little
2 CO OH form. shown
in
should stable since
#
# -1.39 except narrow temperature
range,
as confirmed simulations (lower panel
of
Section 3.6).
HYDROCODE
I MULAT
I ONS
3.1 numerical model: ZEUS­3D
We employ ZEUS­3D one­dimensional mode (Stone
&
Norman 1992) provide basic hydrodynamics. This
is second­
order Eulerian finite­difference code. Here study compressible
hydrodynamics without gravity, self­gravity thermal conduction.
physical viscosity
is modelled, numerical viscosity remains
present,
a Neumann artificial viscosity determines
sipation shock front.
module molecular chemistry molecular atomic
cooling added. functions rates listed
in
Appendices. ultimate goal develop
a reliable code
which tackle three­dimensional molecular dynamics,
adding gravity, magnetic fields, ambipolar diffusion radiation.
Figure Profiles temperature, density, pressure molecular fractions
(H
,
H demonstrating collapse shocked region.
display profiles from simulation covering stages
of cycle.
shows collapse
in temperatures
of #500--
initiated
by pressures (seen
as
the
in density
collapse) region.
restrict cooling chemistry
to just those items
essential
to dynamics. employed simultaneous
method discussed Suttner (1997)
in which time­
adjusted
in
to limit change
in internal energy
in any
present primarily two­shock simulations. addition, have
examined single shocks, reflection boundary condition simu­
lating
a wall. Symmetric inflow boundary conditions chosen
ends grid
in two­shock simulations. Hence, dense
cold material accumulates centre grid. then
behaviour each shock independent accu­
mulated material behaves exactly wall. Initially, however,
is effective wall shock.
a direct result, twin shock
pattern might display oscillatory instability since waves
brought resonance through reflection. Thermal instabil­
present, however, and rapidly generate some structure and
asymmetry.
C 2003 MNRAS 339, 133--147

138 Smith and Rosen
Figure cooling functions within
a stable dissociative shock
with abundances
#
2
â
#
=
10
ditions
as individual cooling/heating components labelled
3.2 Resolution convergence
Once central dense shocks usually
begin behave independently dynamical instability
in.
Table
1 includes data detailed resolution study
exploration parameter space standard two­shock case,
shock--wall example. define shock resolution
as
shock cooling length zone length.
Fig.
7 displays locations
of shock defined
positions
of maximum temperature) one shock
a
function time. This demonstrates presence
of instability
at
resolutions. however,
at resolutions above
4
10
variation time­scale amplitude approach
a consistent
behaviour. remark that oscillations change period
at
irregular intervals. propose these changes, encountered
Table
1. Shock parameters.
a logn
v
s 1­/2­shock unstable Period
b
8(10)
6
60
2(10)
6
60
3.2(11)
6
40
1.6(11)
6
40
8(10)
6
40
6D40­C 8(10)
6
40
t
< 500
S: 8.7,
<
t
< 1000 35.6, 9.4--16.6,
1000
<
t
yr:
M: 49.8,
<
t
< 750
S:
6D40­O 8(10)
6
40
M: 0.27,
6E40­CO 8(10)
6
40
4(10)
6
40
2(10)
6
40
6F40­1 2(10)
6
40 0.42, 1.3,
8(10)
6
20
1(10)
6
20
3.2(11)
5
40
1.6(11)
5
40
2.56(12)
4
40
1.28(12)
4
40
a designation simulation order, number density,
a representation resolution, pre­shock
velocity
-1 letters resolution
are associated number zones
in cooling length,
as
follows: 8--16; 16--32; 32--64; 64--128;
E, 128--256; and
F, 256--512.
b period years determined using indicating
a weak, moderate strong peak, respectively.
in atomic shocks, caused nature cooling, which
possesses
a time dependence through molecular abundances.
typically require shock resolution
of #180
to obtain physical
instability.
It significant the instability generates some chaotic
haviour, different cooling components able
to produce dif­
ferent resonances. thus find, similarly studies
of turbulent
that behaviour converges structure,
as
resolution increased.
notable points
as follows.
(i) When
a low resolution employed, shock
is unstable
unphysical long­term high­amplitude variations found.
increased period
at which shock width recorded
100 Along with change, more short­period variations
become apparent. Hence, spiky structure that appears during
shock evolution curves caused display
resolution rather than numerical resolution.
behaviour does converge
a single pattern
passes through various cyclic patterns. may caused
by
presence
of different harmonic modes interaction through
complex cooling function.
The collapse re­expansion
display excerpt
in from recorded
a time
resolution displayed short variations
time­scale superimposed variations
a time­scale
of Although most collapses expansions exhibit smooth
haviour, collapses place high speed. This
is displayed
explicitly
9 where plot distribution shock
velocity,
as calculated between each consecutive pair.
asymmetry demonstrates that collapses often,
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# 2003 RAS, MNRAS 133--147

Shocks molecular clouds 139
Figure dependence instability spatial resolution. position
of maximum temperature clarity,
of shock)
a function
time simulations (from down) 6C40, 6D40, 6E40 6F40 span different spatial resolutions.
,
is displayed
in each
panel. The frequency
of displayed identical each panel, show datum
for
the thereafter one
yr.
Figure Expanded view short­term oscillations
in high­resolution
case. position maximum temperature
a function
shocks, where
is shown
a dotted high­resolution
simulation 6F40. positions plotted shows
the
similar,
if somewhat time, behaviour independent shock.
exclusively,
at high speed,
in range
s
probable shock front velocity
is larger average
shock speed
-1
. Note these speeds
an
order magnitude smaller shock speed, they
large­amplitude variations shock length.
The distances shock fronts the centre
displayed Since investigating instability,
reason shock location should coincide.
Figure probability distribution shock velocities high­
resolution relative fraction velocities, which deter­
mined consecutive times positions, shocks
high­resolution simulation 6F40, portion
of simulation
0.1­yr intervals. The
of velocity
-1
Shock profiles
displays profiles temperature, density, pressure and
molecular fractions
at different phases during collapse. The
upper panel demonstrates collapse indeed coincides with
shrinking warm infrared layer. This layer determines
C 2003 MNRAS 339, 133--147

140 Smith and Rosen
length
of shock cools efficiently temperatures
below until some COmolecules formed. end
of
collapse, remaining infrared layer consists #8000
K
gas. other hand, both occupy
regions. Note pressure trough appears when shock wide.
low pressure results shock collapse.
In this produces
a
weak secondary shock returns upstream.
The logarithmic profile hydrogen molecular fraction
tains simple shape since
2 reforms infrared
Fast reformation does occur cooled below
300 which close
to interface between shocks.
shape
of
H
2
O fraction follows
2 quite closely.
contrast, fully reforms infrared layer. Note generation
secondary shock
x =-3â
12
in figure) during
collapse that
is sufficiently strong
to destroy most reforming
COmolecules. Although shock
in comparison
mary shock,
it occur regime where
2 and chemistry
most sensitive temperature.
3.5 Density velocity
Fig. shows that instability
is present densities consid­
ered here. code assumes equilibrium chemistry remains
a
valid approximation other cooling mechanism
neglected. density
of
10
, however, assumption
of
equilibrium cooling invalid slow formation CO
bined enhanced
O structure cooling, leads
a
flatter cooling function.
standard
-1 that highest ampli­
tude greatest variation time­scale occur lowest density.
The time­scale dependence expected since lower densities
have longer cooling times. apparent from difference
in
shock width shown Fig. Quasi­periodic oscillations
shock found some stages simulations.
Figure The dependence dynamical instability
on density. position maximum temperature
as function time different densities
plotted simulations 6E40, 5F40, 4F40
the down). Fig.
7 displays two­shock simulations, show position
of only
shocks. Here, interval
yr
in 6E40, every
in every 0.25
yr
for
all
t
in simulation 5F40, every
4F40.
the shock
is inversely proportional density, vertical different. time­scale oscillations
increases density
is decreased.
employed Fourier transform (FFT) highest­resolution
and present
in
1 some periods that appear.
oscillatory pattern changes periods that show
in analysis very weak (<3#
, denoted
W
in
Table). Still, these values show inverse relationship between
period the pre­shock density. Specifically, periods
approximately
6 where
6 n/10
6
, range
densities simulations shown
in Fig.
A study dependence shock width
on shock speed
is presented density
6
. The
instability present speeds above km
s
, where
drogen molecules destroyed immediately behind shock
rapid reformation leads
to dynamical instability. have
investigated shocks
-1 since,
at speeds,
radiation from gas delays formation
of molecules
infrared layer. Nevertheless, instability could
deeper infrared layer. includes radiative transfer
effects
be necessary confirm
that altered time­step display
at 100
In two­shock simulations, allows shocks progress
point where
is independent one another and enables
us
to display structure that w