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Дата индексирования: Mon Oct 1 21:35:00 2012
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Поисковые слова: molecular cloud

Hydrodynamical Simulations of Molecular Dynamics in
Supersonic Turbulent Flow
Georgi Pavlovski (gbp@star.arm.ac.uk) and Michael D. Smith
Armagh Observatory, College Hill, Armagh, BT61 9DG, N.Ireland, U.K.
Mordecai{Mark Mac Low
Department of Astrophysics, American Museum of Natural History, Central Park
West at 79th Street, New York, NY 10024-5192, U.S.A.
Alexander Rosen
Armagh Observatory, College Hill, Armagh, BT61 9DG, N.Ireland, U.K.
June 20, 2003
Abstract. Here we present results from simulations of turbulence in star form-
ing environments obtained by coupling three dimensional hydrodynamical models
with appropriate chemical processes. We investigate regimes of decaying high-speed
molecular turbulence. Here we analyse PDFs of density for the volume, mass, molec-
ular mass and the energy distribution over the range of scales. We compare our
results to those previously obtained for isothermal turbulence and suggest possible
explanations.
Keywords: hydrodynamics { turbulence { molecular processes { clouds { ISM
1. Motivations
It is generally accepted now that supersonic turbulence is of funda-
mental importance to many processes related to the formation of stars
(Vazquez-Semadeni et al., 2000; Padoan et al., 2001; Mac Low and
Klessen, 2003). Turbulent motions redistribute energy inside molecu-
lar clouds, giving rise to their hierarchical structure and determining
cloud fragmentation (Padoan and Nordlund, 2002). In this article we
report results of our ongoing study of molecular turbulence, i.e. turbu-
lence in the uid with active chemistry and cooling appropriate for the
star-forming environments.
Molecular chemistry and cooling is critical to cloud formation and
evolution (Langer et al., 2000; Lim, 2001; Lim et al., 1999). Molecular
hydrogen forms most eфciently where the gas is warm but the grains
are cool (H 2 forms mainly when atoms combine after colliding and
sticking to dust grains). Simple molecules like OH, CO and H 2 O form
in the gas phase with H 2 as the reactive agent. These molecules are not
only important coolants, but associated emission lines provide a means
This e-mail address is available for all requests and questions
c
2003 Kluwer Academic Publishers. Printed in the Netherlands.
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2 G. Pavlovski et al.
of measuring the cloud properties. Molecules are dissociated as a con-
sequence of fast shocks, UV radiation, X-rays and cosmic rays (Herbst,
2000). We thus need to study molecular turbulence to determine the
distribution and abundances of molecular species.
1.1. Methods
Numerically we solve the time-dependent ow equations:
@
@t +r ( v) = 0; (1a)
@ ( v)
@t + (v r)v = 1
rp; (1b)
@e
@t
+ v r e = pr v + (T ; n; f) ; (1c)
@ (fn)
@t +r (fnv) = R (T ; n; f) D (T ; n; f) ; (1d)
where n is the hydrogen nuclei density, e is the internal energy density
and f is the molecular hydrogen abundance (i.e. n(H 2 ) = fn). We
consider the gas as a mixture of atomic and molecular hydrogen with
10% of helium (i.e. n(He) = 0:1 n), therefore the total particle density
is n tot = (1:1 f) n and the temperature is T = p = (k n tot ). is
internal energy loss through radiation and chemistry per unit volume,
the function consists of 13 separate parts (some of which heat the gas),
their detailed description can be found in Smith and Rosen, Pavlovski
et al. (2003, 2002). R and D are reformation and dissociation rates
of molecular hydrogen respectively (see Appendix A and B in Smith
et al., 2002).
As a basis, we employ the ZEUS-3D code (Stone and Norman, 1992).
This is a second-order in space and rst order in time, grid-based code,
which is using van Leer type advection. We study here compressible
hydrodynamics without external gravity, self-gravity or thermal con-
duction. A small amount of linear physical viscosity is modelled, and
von Neumann type of articial viscosity used to capture shocks and
determine the dissipation in the shock front.
For computation of cooling we have employed the simultaneous im-
plicit method discussed by Suttner et al. (1997) in which the time step
is adjusted so as to limit the change in internal energy in any zone to
30%. This limit implies much shorter time-steps in comparison to any
dynamical timescale, and it is one of the most restraining factors in our
simulations.
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Molecular turbulence 3
The cooling is appropriate for dense cloud material of any atomic-
molecular hydrogen mixture. We include H 2 ro-vibrational and disso-
ciative cooling, CO and H 2 O ro-vibrational cooling, gas-grain, thermal
bremsstrahlung and a steady-state approximation to atomic cooling
(see Appendix A of Smith et al., 2002 and Pavlovski et al., 2002).
We take a very basic network of chemical reactions. Time-dependent
hydrogen chemistry is included (Eqns. 2a,b,c), but C and O chemistry
is limited to the reactions with H and H 2 which generate OH, CO and
H 2 O (Eqns.3a,b,c). Equilibrium abundances are calculated, which are
accurate for our purposes within the shocks where molecules are rapidly
formed and destroyed.
H+H+ (grain) ! H 2 + (grain) (2a)
H 2 +H ! 3H (2b)
H 2 +H 2 ! 2H +H 2 (2c)
O+H 2 ! OH+H (3a)
OH+ C ! CO+ C (3b)
OH+H 2 ! H 2 O+H (3c)
2. Simulations
We have run a set of simulations (with resolution ranging from 32 3 to
256 3 ) of decaying turbulence in dierent velocity regimes: 15 km s 1 ,
30 km s 1 , 60 km s 1 . The number density has been taken to be n = 10 6
cm 3 , physical box size L = 10 16 cm, and the initial temperature has
a homogeneous distribution of T 0 = 100 K. The hydrogen was xed to
be fully molecular at the beginning: fraction f = 0:5. The high number
density was selected to create a suфciently high average column (10 22
cm 2 ), which ensures that the simulated region is optically thick. Initial
stress was introduce by perturbations applied to model velocities with
a spectrum extending over a narrow range of wave numbers 3 jkj 4
(see Fig. 1)
2.1. Results
We nd that the dynamical behaviour of the molecular turbulence is
not dramatically dierent from behaviour of the turbulence with the
isothermal equation of state.
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4 G. Pavlovski et al.
Figure 1. Initial energy distribution over the dierent scales. Thick line { total
energy, corresponds to l.h.s. y-axis and bottom x-axis; thin line { ratio of the
compressional to the solenoidal energy, corresponds to the r.h.s. y-axis and upper
x-axis.
The decay law of kinetic energy in the simulations are similar to the
decay laws of the high Mach number isothermal simulations uncovered
in Smith et al. (2000). However, in our simulations thermal energy is
not constant, and decays only a little bit slower than kinetic energy (for
full details see Pavlovski et al., 2002). This fact results in sustaining
Mach number > 1 for a longer, compared to isothermal regime, time.
Statistical analysis of the turbulence showed that probability den-
sity functions (PDFs) of fractional volume per unit density (dV=d),
fractional mass per unit density (dM=d), and fractional molecular
mass per unit density (dM=d) can be approximated by log-normal
distributions as in the isothermal case (Padoan et al., 1997; Passot and
Vazquez-Semadeni, 1998; Ostriker et al., 2001),
p V;M;M (log(x)) = 1
V;M;M
p
2
expf (log(x) j V;M;M j) 2 =(2 2
V;M;M )g;
(4)
where x = =hi, hi { volume mean of density. We nd, that modules of
means for the distribution have close values, but generally the following
is true:
j V j < j M j < j M j; V > M > M : (5)
Typical distributions of these values are presented in Figs. 2,3. The
values of the parameters of the distributions are: in the isothermal case,
V = 0:43, M = 0:47, V = 0:74, = 0:51; in the molecular case,
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Molecular turbulence 5
Figure 2. PDFs of volume and mass distributions with density in the initially high
speed isothermal turbulence. Data taken from 256 3 simulations with initial r.m.s.
velocity of 60 km s 1 after 130yr of evolution (average Mach number: M 9)
Figure 3. PDFs of volume, mass and molecular mass distributions with density in
the decaying molecular turbulence. Data taken from 256 3 simulations with initial
r.m.s. velocity of 60 km s 1 after 130yr of evolution (average Mach number: M 10)
V = 0:42, M = 0:46, M = 0:55, V = 0:71, M = 0:52, M = 0:50.
This is in agreement with the results discussed in Ostriker et al. (2001).
2.2. Molecular hydrogen evolution
The molecular fraction `f ' is displayed in Fig. 4. The three initial states
correspond to three distinct physical regimes. With a r.m.s. velocity
of 15 km s 1 , dissociative shocks are not present but some localised
dissociation still occurs. With 30 km s 1 , a few per cent of the molecules
are dissociated whereas at 60 km s 1 , the gas becomes over 80% atomic.
Reformation of molecular hydrogen is unexpectedly rapid. The ex-
pected H 2 reformation time at 20 K and 10 6 cm 3 is t R = 3; 200 yr (see
details in Pavlovski et al., 2002) a factor of 5 larger than the simulation
time. At 100 yr, the temperature is 80 K, predicting a reformation
time of t R = 2,000 yr. Yet, reformation is occurring over 400 yr. This
speed up is caused by the turbulence itself: the molecules preferentially
reform in the denser and cooler locations. As weak shocks propagate
through the gas, dierent regions are compressed and expanded. Hence
the reformation time is not only controlled by the `average' reformation
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6 G. Pavlovski et al.
Figure 4. Evolution of the average molecular fraction with time in the runs with
dierent initial r.m.s. velocity. Data from the 256 3 runs.
time, but also by the strength of the turbulence. Given a turbulent
dynamical timescale shorter than the average reformation timescale,
then we can expect reformation to be accelerated.
3. Summary
We have presented the properties of a specic model for molecular
turbulence. We carried out three dimensional hydrodynamical simula-
tions of decaying supersonic turbulence in molecular gas. We included
a detailed cooling function, molecular hydrogen chemistry and equilib-
rium C and O chemistry. We studied three cases in which the applied
velocity eld straddles the value for which wholesale dissociation of
molecules occurs. The parameters chosen ensure that for the high-speed
turbulence, the molecules are initially destroyed in shocks and gradually
reform in a distinct phase.
We nd the following.
An initial phase of slow dissipation and shock formation.
An extended phase of power-law kinetic energy decay, as in the
isothermal case.
The thermal energy, initially raised by the introduction of turbu-
lence, decays only a little slower than the kinetic energy.
The reformation of hydrogen molecules, as the fast turbulence
decays, is several times faster than expected from the average
density.
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Molecular turbulence 7
Figure 5. Distribution of energy over dierent scales at the end of the simulation
(600 yr: 60 km s 1 , 256 3 ). Thick line { total energy, corresponds to l.h.s. y-axis
and bottom x-axis; thin line { ratio of the compressional to the solenoidal energy,
corresponds to the r.h.s. y-axis and upper x-axis. Steeper than Kolmogorov's (-11/3)
energy cascade as indication of strong dissipation.
The molecular fraction increases quite uniformly, so that density
and molecular density are almost identically distributed at any one
time.
We mainly wish here to emphasise the insight these simulations pro-
vide into how molecular chemistry and supersonic dynamics combine.
We have found that isothermal simulations are indeed very useful, not
only for the rate of energy decay but also to trace the molecules.
A simple reason for the fast decay is that a suфcient number of
strong shocks survive. As shown by Smith et al. (2000), the rate of
energy decay in decaying turbulence is dominated by the vast number
of weak shocks. These shocks are less eфcient at energy dissipation.
A second possible reason is that the curved shock structures create
small scale vorticity, which leads to enhanced dissipation of kinetic
energy. The latter argument is supported by the fact that the energy
distribution is steeper than that of Kolmogorov's spectrum, see Fig. 5
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