The turbulent destruction of clouds - I. A k-epsilon treatment of turbulence in 2D models of adiabatic shock-cloud interactions

Abstract

The interaction of a shock with a cloud has been extensively studied in the literature, where the effects of magnetic fields, radiative cooling and thermal conduction have been considered. However, the formation of fully developed turbulence has often been prevented by the artificial viscosity inherent in hydrodynamical simulations, and a uniform post-shock flow has been assumed in all previous single-cloud studies. In reality, the flow behind the shock is also likely to be turbulent, with non-uniform density, pressure and velocity structure created as the shock sweeps over inhomogenities upstream of the cloud. To address these twin issues we use a sub-grid compressible k-epsilon turbulence model to estimate the properties of the turbulence generated in shock-cloud interactions and the resulting increase in the transport coefficients that the turbulence brings. A detailed comparison with the output from an inviscid hydrodynamical code puts these new results into context. We find that cloud destruction in inviscid and k-epsilon models occurs at roughly the same speed when the post-shock flow is smooth and when the density contrast between the cloud and inter-cloud medium is less than 100. However, there are increasing and significant differences as this contrast increases. Clouds subjected to strong ``buffeting'' by a highly turbulent post-shock environment are destroyed significantly quicker. Additional calculations with an inviscid code where the post-shock flow is given random, grid-scale, motions confirms the more rapid destruction of the cloud. Our results clearly show that turbulence plays an important role in shock-cloud interactions, and that environmental turbulence adds a new dimension to the parameter space which has hitherto been studied (abridged). Comment: 31 pages, 22 figures, accepted for publication in MNRAS

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arXiv:0807.4402v1 [astro-ph] 28 Jul 2008
Mon. Not. R. Astron. Soc. 000, 1–31 (2005)Printed 28 July 2008(MN LATEX style file v2.2)
The turbulent destruction of clouds - I. A k-ǫ treatment of
turbulence in 2D models of adiabatic shock-cloud
interactions
J. M. Pittard1⋆, S. A. E. G. Falle2, T. W. Hartquist1and J. E. Dyson1
1School of Physics and Astronomy, The University of Leeds, Leeds LS2 9JT, UK
2Department of Applied Mathematics, The University of Leeds, Leeds LS2 9JT, UK
Accepted by MNRAS, 22nd July, 2008
ABSTRACT
The interaction of a shock with a cloud has been extensively studied in the litera-
ture, where the effects of magnetic fields, radiative cooling and thermal conduction
have been considered. In many cases, the formation of fully developed turbulence has
been prevented by the artificial viscosity inherent in hydrodynamical simulations. This
problem is particularly severe in some recent simulations designed to investigate the
interaction of a flow with multiple clouds, where the resolution of individual clouds is
necessarily poor. Furthermore, the shocked flow interacting with the cloud has been
assumed to be completely uniform in all previous single-cloud studies. In reality, the
flow behind the shock is also likely to be turbulent, with non-uniform density, pressure
and velocity structure created as the shock sweeps over inhomogenities upstream of
the cloud (as seen in recent multiple cloud simulations). To address these twin is-
sues we use a sub-grid compressible k-ǫ turbulence model to estimate the properties
of the turbulence generated in shock-cloud interactions and the resulting increase in
the transport coefficients that the turbulence brings. A detailed comparison with the
output from an inviscid hydrodynamical code puts these new results into context.
Despite the above concerns, we find that cloud destruction in inviscid and k-ǫ
models occurs at roughly the same speed when the post-shock flow is smooth and when
the density contrast between the cloud and inter-cloud medium, χ∼
there are increasing and significant differences as χ increases. The k-ǫ models also
demonstrate better convergence in resolution tests than inviscid models, a feature
which is particularly useful for multiple-cloud simulations.
Clouds which are over-run by a highly turbulent post-shock environment are de-
stroyed significantly quicker as they are subject to strong “buffeting” by the flow.
The decreased lifetime and faster acceleration of the cloud material to the speed of
the ambient flow leads to a reduction in the total amount of circulation (vorticity)
generated in the interaction, so that the amount of vorticity may be self-limiting. Ad-
ditional calculations with an inviscid code where the post-shock flow is given random,
grid-scale, motions confirms the more rapid destruction of the cloud.
Our results clearly show that turbulence plays an important role in shock-cloud in-
teractions, and that environmental turbulence adds a new dimension to the parameter
space which has hitherto been studied.
<100. However,
Key words: hydrodynamics – ISM: clouds – ISM: kinematics and dynamics – shock
waves – supernova remnants – turbulence
⋆E-mail: jmp@ast.leeds.ac.uk
1 INTRODUCTION
Circumstellar, interstellar, and intergalactic environments
are inhomogeneous, with clouds of various densities and tem-
peratures embedded in a hotter, more tenuous, substrate.
This substrate is often turbulent, in part due to the interac-

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2 J. M. Pittard, S. A. E. G. Falle, T. W. Hartquist and J. E. Dyson
tion of shocks, shells, winds and jets with these clouds. The
nature of such interactions is interesting, because the evolu-
tion and morphology of large-scale flows can ultimately be
regulated by objects of much smaller size.
In the interstellar medium (ISM), for instance, the
interaction of supernova shock waves with interstellar
clouds creates a continuous interchange of mass and en-
ergy between various thermal phases (Cox & Smith 1974;
McKee & Ostriker 1977). Several outcomes are possible.
The shocks may destroy the clouds, and mix their mate-
rial into their surroundings. Alternatively, the shocks may
trigger the collapse of the clouds and the formation of new
stars, thereby removing material (at least temporarily) from
the ISM (Elmegreen & Lada 1977). The shocks themselves
will slow and material behind them will cool to form thin
dense shells. These shells may then fragment and form new
clouds. Clouds which survive the passage of the shell sub-
sequently find themselves either inside a hot, low density
bubble (if the bubble is energy-conserving), or exposed to a
fierce, high Mach number wind (if the shell is momentum-
driven). An important process is the generation of vorticity
as the diffuse medium flows around the clouds. Understand-
ing the interaction between shocks/shells/winds/jets and in-
terstellar clouds is therefore a key step in studies of the
structure and evolution of the ISM (see the recent reviews
by Elmegreen & Scalo 2004; Scalo & Elmegreen 2004).
A similar interaction occurs in massive, early-type, stel-
lar systems where each star blows a powerful, clumpy, wind
which collides with the other. The impact of the clumps
causes the wind-wind collision region to become highly tur-
bulent (Pittard 2007a), with implications for particle accel-
eration, the timescales for equilibrium ionization and elec-
tron heating, and the physical mixing of the winds.
In this work we consider the adiabatic interation of a
shock with a cloud. An extensive literature of analytical and
numerical investigations of shock-cloud interactions now ex-
ists (see, e.g., Nakamura et al. 2006, and references therein).
The effects of ordered magnetic fields, radiative cooling,
and thermal conduction have all been considered, but not
simultaneously until recently (Orlando et al. 2008). High-
power laser experiments of shock-cloud interactions (e.g.,
Klein et al. 2003) have complemented this literature. While
thermal conduction acts to prevent hydrodynamic instabili-
ties, radiative cooling enhances them. The effect of ordered
magnetic fields is more complicated: instabilities are pre-
vented in cases where a magnetic field provides a high ten-
sion at the surface of the cloud, but the effects depend on
the strength, orientation and scale of the magnetic field. In-
stabilities and vortical motions may be prevented in some
directions, but not necessarily in others. Interestingly, in
three-dimensional simulations a magnetic field may actually
enhance the fragmentation of a cloud (Gregori et al. 1999;
Shin, Stone & Snyder 2008), though the actual mixing of
the cloud and ambient medium remains hindered. Even if
the magnetic fields are not strong enough to directly affect
the dynamics, they can significantly reduce the effects of
thermal conduction in directions normal to the field lines,
thus allowing instabilities to develop (Orlando et al. 2008).
Given the fact that in shock-cloud interactions the
Reynolds number is typically high, and the above predis-
position for instabilities to develop, the fluid velocity field
around the cloud is expected to vary significantly and irreg-
ularly in both position and time. This “turbulence” trans-
ports and mixes the cloud material much more effectively
than a comparable laminar flow. Turbulence is also effective
at “mixing” the momentum of a fluid (i.e. accelerating ma-
terial ripped off the cloud to the ambient flow speed), and at
transferring heat. In turbulent flows of high Reynolds num-
ber there is a separation of scales - large-scale motions will
be strongly influenced by the geometry of the cloud, and
control the transport and mixing of cloud material, while
the behaviour of small-scale motions is determined almost
entirely by the rate at which they receive energy from the
large scales, and by the viscosity. Hence these small-scale
motions have a universal character, independent of the flow
geometry.
While much insight has been gained from previous nu-
merical investigations of shock-cloud interactions, the arti-
ficial viscosity inherent in all such simulations has the po-
tential to prevent the formation of fully developed turbu-
lence, to limit the turbulent mixing of cloud material into
the surrounding medium, and to hinder the destruction of
the cloud. Furthermore, all previous simulations are highly
idealized in the sense that the flow behind the shock is as-
sumed to be perfectly smooth and uniform (i.e. laminar).
In reality, random inhomogenities in the ambient medium
upstream of the cloud will deform the shock, and will cause
velocity, density and pressure structures to develop in the
post-shock flow. The post-shock flow will then also be “tur-
bulent”, and it is expected that the destruction of the cloud
will be more rapid in such conditions.
In this work we address whether fully developed turbu-
lence has been prevented in all previous numerical works. For
high Reynolds number flows, the only tractable method is a
statistical approach i.e. to describe the turbulent flow, not in
terms of a velocity field u(x,t), but in terms of some statis-
tics. A model based on such statistics can lead to a tractable
set of equations, because statistical fields vary smoothly (if
at all) in position and time. To do this we use a sub-grid
turbulent viscosity model, where an attempt is made to cal-
culate the properties of the turbulence and the resulting
increase in the transport coefficients. The most widely used
is the so-called k-ǫ model, where the properties of the tur-
bulence are described by two variables, the turbulent energy
per unit mass, k, and the turbulent dissipation rate per unit
mass, ǫ. The addition of viscous and diffusive terms in the
fluid equations simulates the turbulent mixing of the cloud
and intercloud medium due to shear instabilities. In this
way the subgrid turbulence model emulates a high Reynolds
number flow. Incorporating a k-ǫ model into shock-cloud
simulations should produce more realistic results than those
from inviscid codes, where the viscosity is purely numerical
and the size of shear instabilities is determined by the res-
olution of the numerical grid. A turbulent viscosity model
differs from simply adding physical viscosity to the grid, be-
cause the turbulent viscosity is largest in shear layers and es-
sentially vanishes in regions with little shear, whereas models
with grid viscosity have the same viscosity everywhere.
The structure of this paper is as follows. The key
physics of a shock-cloud interaction is reviewed in Section 2.
Section 3 introduces the investigation and the numerical
method used. The results are presented in Section 4, where
the effects of turbulence on the cloud evolution are de-
scribed. A discussion of the relevance of our results to shock-

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The turbulent destruction of clouds3
cloud and wind-cloud observations is given in Section 5. Sec-
tion 6 summarizes the conclusions of this work, and possible
future work is noted in Section 7.
2 THE INTERACTION OF A SHOCK WITH A
CLOUD
The interaction of a shock with a cloud is a highly non-linear
and complex phenomenon which can be vastly simplified if
some assumptions are made. In this work we assume that
the magnetic field is too weak to be dynamically important
(though it must be strong enough to reduce the thermal
conductivity and effective mean-free-path), and that the in-
teraction is adiabatic. In the interstellar medium, the typical
magnetic field is ≈ 5 µG, and the magnetic pressure is typi-
cally a few times higher than the thermal pressure (e.g., Cox
2005). However, if the shock driven into the cloud is strong
and adiabatic the postshock thermal pressure becomes much
higher than the postshock magnetic pressure, justifying our
assumption that the magnetic field is too weak to be dynami-
cally important. We also assume that the magnetic field does
not prevent the full mixing of initially disparate phases (i.e.
that turbulence drives the reconnection needed to allow this
mixing).
The effectsof thermal
nored in this work. The efficiency of thermal conduc-
tion in magnetized turbulent plasmas remains highly
uncertain(see,e.g., Pittard
tions of the hot intracluster medium (Ettori & Fabian
2000; Vikhlinin et al. 2001) and theoretical considerations
(Narayan & Medvedev 2001; Asai, Fukuda & Matsumoto
2004; Chandran & Maron 2004) suggest that the coefficient
of heat conduction is at least five times lower than the
Spitzer value in the presence of tangled magnetic fields.
Furthermore, if turbulent resistivity can reconnect field
lines quickly enough, turbulent heat transport may be
more efficient than thermal conduction (Cho et al. 2003;
Chandran & Maron 2004; Lazarian 2006).
For the interaction to be adiabatic, radiative cooling
must be unimportant. This is often the case for clouds in hot,
low density environments (e.g., in planetary nebulae, in bub-
bles blown around individual or groups of massive stars, and
in starburst and superwind environments). The behaviour
of the cloud is more likely to be adiabatic if the cloud is
small. Since the assumption of adiabaticity preserves the
scale-free nature of the simulations, the interaction is then
determined by whether there is a sufficient rate of collisions
within the plasma to give it fluid properties, by the domi-
nant mechanism for the damping of hydromagnetic waves,
by the Reynolds number, and by the way the turbulence is
driven. Each of these issues is discussed below. In order that
their importance can readily be determined, two different
examples are considered. In the first scenario the cloud is
ionized, and has a radius rc = 1 pc, density nc = 0.4 cm−3,
and temperature Tc = 8000 K. In the second scenario we
consider a neutral cloud with rc = 0.46 pc, nc = 30 cm−3,
and Tc = 100 K. In both cases we imagine that the clouds
are in approximate pressure equilibrium with a surrounding
medium with nic = 3×10−3cm−3and Tic = 106K, and are
struck by a high speed Mach 10 shock with velocity vb =
1.52 × 108cm s−1. The shocked intercloud gas has a den-
conductionare alsoig-
2007b).X-ray observa-
sity ρ = 2.6 × 10−26g cm−3, temperature T = 3.2 × 107K,
and velocity uics = 0.75vb = 1.1 × 108cm s−1. The pa-
rameters noted above are typical of small interstellar clouds
(McKee & Ostriker 1977).
2.1 Collisional or collisionless?
The first consideration is whether the mean-free-path is
short enough that a collisional treatment can be adopted
(i.e. whether the plasma has fluid-like properties). If this is
the case, complications due to collisionless wave-particle in-
teractions and their associated effects such as Landau damp-
ing can be ignored (Parker 1979). The damping of hydro-
magnetic waves in a collisionless plasma is stronger than in
a collisional plasma, since collisions serve to suppress the
wave-particle interactions which damp the wave so strongly
in the collisionless case.
The Coulomb collision cross-section for electron or ion
scattering is σ ≈ 10−12/T2
ature in electron volts. For neutral material the gas atomic
cross-section is ∼ 10−16cm2. Hence the mean-free-path
within the ionized and neutral clouds is λ = 1/nσ ≈ 1012cm
and ≈ 1014cm respectively. These values are much smaller
than the cloud radii, so a fluid-like treatment is appropriate.
On the other hand, the mean-free-path in the post-shock in-
tercloud medium is λ ≈ 6 × 1020cm, which is significantly
larger than the radius of the clouds. However, even a small
B-field will significantly reduce the effective mean-free-path
(the mean-free-path is then of order the gyroradius, which
for B = 3 µG and T = 107K is rg ∼ 109cm for protons),
so the shocked intercloud gas can also be considered to be
collisional.
eVcm2, where TeV is the temper-
2.2Reynolds number, eddies and instabilities
The Reynolds number of flow past a cloud is Re = urc/ν,
where u is the average flow velocity past the cloud, rc is the
radius of the cloud, and ν is the kinematic viscosity. For a
fully ionized, non-magnetic gas of density ρ and temperature
T,
ν = 2.21 × 10−15T5/2A1/2
Z4ρ ln Λ
cm2s−1, (1)
where A and Z are the atomic weight and charge of the
positive ions, and ln Λ is the coulomb logarithm (Spitzer
1956). In a magnetized plasma, the kinematic viscosity is of
the order of the mean-free-path (i.e. the particle gyroradius)
times the typical thermal velocity.
The characteristic Reynolds number of the interaction
is higher in the cloud material than the surrounding envi-
ronment, since the cloud is considerably cooler. The shock
driven into the cloud has a speed v = vb/√χ, where χ is the
density contrast of the cloud with respect to its surround-
ings. For the ionized cloud considered in Section 2, the post-
shock temperature is 2.6×105K and ν ≈ 5×1020cm2s−1.
Hence Re ≈ 7×105. If the cloud is magnetized with a post-
shock field B ∼ 10 µG, the kinematic viscosity for protons
is ν ∼ 1015cm2s−1and the Reynolds number is propor-
tionally higher. In the neutral cloud the Reynolds number
calculated from the kinematic molecular viscosity is simi-
larly high. The Reynolds number of the flow around a cloud

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4J. M. Pittard, S. A. E. G. Falle, T. W. Hartquist and J. E. Dyson
is also high. For a flow with B = 3 µG and T = 107K,
the kinematic viscosity ν ∼ 1017cm2s−1. In all cases, the
viscous stresses acting on the boundary layer which forms
as the shock sweeps over the cloud are negligible, and a tur-
bulent energy cascade ensues.
The largest eddies have a length scale, l, comparable in
size to the cloud, while the smallest eddies where the turbu-
lent energy is dissipated have a length scale η ∼ Re−3/4l.
Due to the nonlinear term u · ∇u in the equation of mo-
tion, large eddies, created by instabilities in the mean flow,
are themselves subject to inertial instabilities and rapidly
“break-up” or evolve into yet smaller vortices. Energy is
transferred into vortices of about one-half their size in a
time comparable to their “turnover time” (t = l/u, where
u is the characteristic velocity of eddies of size l, Davidson
2004). Provided that energy is constantly injected at large
scales, small-scale eddies are superimposed on larger eddies.
The timescale to set up a turbulent energy cascade is roughly
twice the turnover time of the largest eddies (i.e. t = 2l/u).
Since l/u ∼ rc/uics, the setup time is t ∼ 2rc/vb = tsc,
where tsc is the timescale for the shock in the intercloud
medium to sweep over the cloud. For dense clouds (χ ≫ 1),
this setup time is much shorter than the survival time of the
cloud, which is a few times the “cloud crushing” timescale,
tcc ≡χ1/2rc
vb
, (2)
for the cloud to be crushed by the initial shock that is driven
into it (Klein, McKee & Colella 1994).
Simultaneously with the top-down energy transfer from
large to small eddies, Kelvin-Helmholtz (KH) and Rayleigh-
Taylor (RT) instabilities inject energy from the bottom-up,
since the smallest scale disturbances grow fastest. The KH
and RT growth times are
tKH ∼
tcc
kλrc,tRT ∼
tcc
(kλrc)1/2, (3)
where
(Klein et al. 1994). The smallest scale of the instabilities
is set by the scale at which the damping of hydromagnetic
waves occurs, which was shown in Section 2.1 to be through
particle collisions rather than wave-particle interactions.
For the unmagnetized ionized cloud the minimum scale
due to viscous damping is ηvis ∼ Re−3/4rc ∼ 4 × 10−5rc.
However, in ionized plasmas the thermal conductivity is even
more effective at damping waves with a significant longitudi-
nal component. The thermometric conductivity K = κT/U,
where U is the thermal energy per unit volume and κ is the
thermal conduction coefficient (Parker 1979). Here we find
that K ≈ 40 ν, and thermal conduction prevents instabili-
ties with a scale smaller than ηtc ∼ 6 × 10−4rc. In magne-
tized ionized clouds the lengthscale at which instabilities are
damped is smaller if the magnetic field is sufficiently weak
to be dynamically unimportant. On the other hand, if the
magnetic field is strong enough to be dynamically impor-
tant, magnetic tension increases the minimum lengthscale
at which instabilities occur.
A disturbance of wavelength λ = 10−3rc grows in a
timescale of tKH = 1.6 × 10−4tcc and tRT = 1.3 × 10−2tcc.
With the ionized cloud parameters given above, χ = 133,
tcc = 5.8 tsc, and we obtain tKH ≈ 10−3tsc and tRT ≈
0.1 tsc. These timescales are much faster than the one to
kλ
is thewave-number ofthe perturbation
setup the turbulent energy cascade (which was shown above
to be ∼ tcc), and the (ensemble-averaged) turbulent spec-
trum will differ from the classical Kolmogorov spectrum be-
cause it is driven by energy input at both large and small
scales.
Photoionization at the surface of neutral clouds main-
tains most of the C in the form C+(e.g., Hartquist et al.
1998). An upper limit to the ratio of the ionized to neutral
mass density, ρ∗/ρ, is therefore the fractional abundance by
mass of carbon, which has a cosmic value of about 3×10−3
(e.g., Dopita & Sutherland 2003). Kulsrud & Pearce (1969)
note that for hydromagnetic waves with λ < λ2, the charged
particles move as if the neutrals were absent, while for
λ > λ1, the entire medium moves. In contrast, when λ2 <
λ < λ1, the neutrals and ions move independently and fric-
tion is strong, and waves do not propagate. λ1 corresponds
to waves with angular frequencies comparable to the rate at
which neutrals transfer momentum to ions, while λ2 is the
corresponding rate for momentum transfer from ions to neu-
trals. Assuming that the magnetic field strength within our
neutral cloud is ∼ 3 µG, we find that λ1 ≈ 4 × 1015cm and
λ2 ≈ 1.3 × 1015cm. In this case fully developed turbulence
is prevented, since λ1/rc > ηvis when Re ∼ 105, although
instabilities continue to be driven from the bottom up. Fully
developed turbulence may be obtained when the magnetic
field is weaker, since λ1 and λ2 are both proportional to B,
or when the cloud is larger.
In numerical simulations of shock-cloud interactions,
the growth of KH and RT instabilities is closely related to
the development of the slip surface around the cloud, and
perturbations with wavelengths smaller than the thickness of
the shear layer are stabilized (Nakamura et al. 2006). Thus,
small scale instabilities have been artificially suppressed in
previous work on hydrodynamic shock-cloud interactions.
Our use of a k − ǫ turbulence model in this paper attempts
to address this shortcoming.
2.3The turbulent boundary layer
Hartquist & Dyson (1988) argued that the turbulent bound-
ary layer which forms around a cloud has a thickness of
the order of rc/√Ret, where Ret is an effective “turbulent”
Reynolds number arising from the fact that the turbulence
itself gives rise to an effective viscosity. Since Ret ∼ 103
(Hartquist & Dyson 1988), the thickness of the turbulent
boundary layer is a few percent of the cloud radius. More
detailed calculations and laboratory experiments show that
the opening angle of a turbulent mixing layer in a mildly
supersonic flow (such as occurs behind a high Mach number
shock) is of order 10◦(Cant´ o & Raga 1991) - we find good
agreement with this (see Section 4.2). Convergence tests re-
veal that a minimum numerical resolution of about 120 cells
per cloud radius is needed for convergence of various global
quantities (see, e.g., Nakamura et al. 2006, although the nec-
essary resolution may to some extent also depend on the
numerical scheme). This is consistent with simulations of
this resolution and higher beginning to resolve the turbu-
lent boundary layer.

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The turbulent destruction of clouds5
2.4Cloud destruction and mixing
The main stages in the destruction of a cloud by a shock and
the subsequent mixing of its material into the surrounding
flow are reviewed in Section 4.1. The most disruptive KH
and RT instabilities are those with wavelengths of the or-
der of the cloud radius. However, there are a number of
ways in which the growth of KH and RT instabilities may
be hindered or amplified. For instance, clouds with diffuse
boundaries are less susceptible to KH instabilities and sur-
vive longer (Nakamura et al. 2006). Two dimensional MHD
simulations found that the growth of KH and RT instabili-
ties is strongly inhibited when there is a dynamically impor-
tant magnetic field, due to the field providing an additional
tension at the interface between the cloud and the surround-
ing flow (Mac Low et al. 1994). Somewhat surprisingly, fully
three dimensional magneto-hydrodynamic (MHD) simula-
tions of a wind-cloud interaction revealed that an ordered
magnetic field can actually enhance hydrodynamic instabil-
ities, as background field lines become trapped in deforma-
tions in the surface of the cloud (Gregori et al. 1999, though
these were low resolution calculations). Higher resolution
shock-cloud simulations recently presented by Shin et al.
(2008) show that irrespective of the field geometry, and the
morphology of the cloud fragments which are produced in
the interaction, the rate of mixing is reduced compared to
the non-magnetic case. However, sufficiently weak magnetic
fields have no dynamical influence, and only offer a potential
reduction in the collision mean-free-path. Strong thermal
conduction suppresses hydrodynamic instabilities (see Sec-
tion 2.2 and also Orlando et al. 2005; Marcolini et al. 2005),
but the degree of this effect is sensitive to the orientation of
any magnetic field (Orlando et al. 2008).
KH and RT instabilities are always stronger when radia-
tive cooling is important, and the cloud breaks up into nu-
merous dense, cold fragments (Mellema, Kurk & R¨ ottgering
2002; Fragile et al. 2004, 2005; Van Loo et al. 2007). In the
corresponding simulations the fragments appear to survive
for an appreciable time, but are poorly resolved, so the
timescales corresponding to their further evolution are some-
what uncertain. The way in which clouds are destroyed and
mixed into their surroundings is sensitive also to the density
contrast between the cloud and the surrounding medium, in
that clouds with higher values of χ suffer direct stripping
of material from their surfaces by hydrodynamic ablation
(Klein et al. 1994, also compare Figs. 4, 9 and 10 in this
work).
Cloud destruction and the mixing of the cloud mate-
rial into the surrounding flow are two distinct processes
which were not always distinguished in previous work. In
Nakamura et al. (2006), the cloud destruction timescale is
taken to be the time when the largest fragment drops be-
low a certain fraction of the initial cloud mass, while the
timescale for the mixing of former cloud material into the
surrounding medium is estimated by comparing the inte-
grated mass above a particular threshold density with the
initial cloud mass. However, even weak magnetic fields can
prevent the actual mixing of stripped material with the sur-
rounding medium, and reconnection on small-scales is nec-
essary if the plasmas are to mix fully. This may, of course,
take some considerable time to achieve.
3 THE NUMERICAL SETUP
3.1 The numerical scheme
The shock-cloud interaction is modelled by solving numer-
ically the Euler equations of inviscid fluid flow, supple-
mented by a sub-grid turbulent viscosity model as appro-
priate. When the sub-grid model is included, the continuity,
scalar, momentum, energy, turbulent energy and turbulent
dissipation equations are respectively:
∂ρ
∂t+ ∇ · (ρu) = 0,
∂ρκ
∂t
∂ρu
∂t
∂E
∂t+ ∇ · [(E + P)u − u · τ] −
(4)
+ ∇ · (ρκu) − ∇ · (µT∇κ) = 0, (5)
+ ∇ · (ρuu) + ∇P − ∇ · τ = Sp,(6)
γ
γ − 1∇ · (µT∇T) = SE, (7)
∂ρk
∂t
∂ρǫ
∂t
Here ρ is the mass density, u is the velocity, E is the total
energy density
+ ∇ · (ρku) − ∇ · (µT∇k) = Sk, (8)
+ ∇ · (ρǫu) − ∇ · (µǫ∇ǫ) = Sǫ. (9)
E =
P
γ − 1+12ρu2, (10)
P is the thermal pressure, k is the turbulent energy per unit
mass, ǫ is the turbulent dissipation rate per unit mass, and
the turbulent diffusion coefficients are
µT = ρCµk2
ǫ,
where Cµ = 0.09. κ is an advected scalar used to distinguish
between cloud and ambient material.
The momentum equation source term, Sp, is zero in
Cartesians. In cylindrical symmetry it is
µǫ =µT
1.3, (11)
Sp =



µT
?2
3r∇ · u − 2ur
r2
?
+1
r
?
P +2
3rρk
?
0


. (12)
The k and ǫ source terms are respectively
Sk= Pt− ρǫ,
and
Sǫ =ǫ
k(C1Pt− C2ρǫ),
where C1 = 1.4 and C2 = 1.94.
The turbulent production term
(13)
(14)
Pt = µT
?
∂ui
∂xj
?
∂ui
∂xj
+∂uj
∂xi
??
−2
3∇·u(ρk +µT∇·u),(15)
where the summation convention is assumed. The terms in-
volving µT are due to the rate of working of the turbulent
stresses, and the terms involving ∇ · u take into account
volume changes on the turbulence. In cylindrical symmetry,
the production term has to be complemented by an extra
geometric term
2µTu2
r
r2.(16)

Page 6

6J. M. Pittard, S. A. E. G. Falle, T. W. Hartquist and J. E. Dyson
The turbulent stress tensor, τ, is
τij = µT
?
∂ui
∂xj
+∂uj
∂xi
?
−2
3δij(µT∇ · u + ρk). (17)
All computations were performed in 2D cylindrical sym-
metry using an Eulerian adapative mesh refinement (AMR)
hydrodynamic code, with a linear Godunov solver and piece-
wise linear cell interpolation (see Falle 1991). Although it is
of lower order than the piecewise parabolic method (PPM),
it performs well in multi-dimensional problems, as it is only
partially operator split, and it performs better than PPM
for problems where there is rapid advection across the com-
putational grid (Runacres & Owocki 2005)
The entire computational domain is covered by the two
coarsest grids, G0and G1. The solution at each position
is calculated on all grids that exist there, and the differ-
ence between these solutions is used to control refinement
(note that refinement criteria based on the local gradients
of only selected variables (e.g., density) do not properly re-
solve turbulent flow - see Iapichino et al. 2008). Finer grids
are dynamically added where they are needed, and removed
where they are not. Each refinement level increases the res-
olution in each of the spatial directions by a factor of 2,
and the refinement is done on a cell-by-cell rather than on a
patch basis. The time-step on grid Gnis ∆t0/2n, where ∆t0
is the time-step on G0, in order to ensure Courant number
matching at the boundaries between coarse and fine grids.
The k-ǫ model is designed to model the mean flow
in fully developed, high Reynolds number, turbulence. It
has been calibrated by comparing the computed growth of
shear layers with experiments of high Reynolds number flows
(Dash & Wolf 1983). Although the flow in our problem is
somewhat more complicated than in these experiments, it
is described by exactly the same equations. Of course, any
turbulence model can only be approximately correct, but it
should give more reliable results than an inviscid calculation.
In the sub-grid model, turbulent energy is generated
by the action of the turbulent viscosity on the mean flow
and is converted to heat at the dissipation rate, ǫ. Since
the turbulent energy resides mainly in large eddies, while
the dissipation occurs in the small ones, one can think of
k and ǫ as describing the large-scale and small-scale turbu-
lence respectively. Since the aim of the sub-grid model is
to mimic a three dimensional turbulent flow, the effects of
the turbulence (such as enhanced transport coefficients) are
treated correctly, even though the grid is cylindrically sym-
metric. However, the turbulent motions that are resolved on
the grid are actually vortex rings, and not eddies. Further
details of the model implementation can be found in Falle
(1994).
3.2 Initial and boundary conditions
We consider a Mach 10 shock interacting with a cloud with
a density contrast χ of either 10, 102or 103, and compute
simulations with different spatial resolutions using either an
inviscid code or one which includes a k-ǫ turbulence model.
The effect of different levels of turbulence in the post-shock
gas is also explored. Table 1 summarizes the calculations
performed. Most computations are for clouds with steep
density profiles, but we have also examined the effect of a
Table 1. Summary of the shock-cloud simulations performed.
The resolution is the number of cells per cloud radius on the finest
grid. Models with “sh” in their name were computed for clouds
with a shallow density gradient. Model c1rtb32 was computed
using an inviscid code with grid-scale turbulence.
Modelχp1
resolution turbulence
c1no
c1lo
c1hi
c2no
c2lo
c2hi
c3no
c3lo
c3hi
c2nosh
c2losh
c2hish
c3nosh
c3losh
c3hish
c1rtb
101
101
101
102
102
102
103
103
103
102
102
102
103
103
103
101
10
10
10
10
10
10
10
10
10
1
1
1
1
1
1
10
32,64,128
32,64,128
64,128
32,64,128,256
32,64,128,256
16,32,64,128,256
16,32,64,128
16,32,64,128
32,64,128
64
64
64
64
64
64
32
no
low
high
no
low
high
no
low
high
no
low
high
no
low
high
grid-scale
shallower density profile on the resulting evolution (see Sec-
tion 4.4). A calculation using an inviscid code with grid-scale
post-shock turbulence is also presented (see Section 4.5).
The calculations are computed on an r−z cylindrically
symmetric grid, with a domain of 0 ? r ? 24, −94 ? z ? 6
when χ = 10, 0 ? r ? 24, −120 ? z ? 6 when χ = 102, and
0 ? r ? 48, −594 ? z ? 6 when χ = 103. This ensures that
the cloud is well dispersed and mixed into the post-shock
flow before the shock reaches the edge of the numerical grid.
In this way the global quantities detailed in Section 3.4 are
accurately computed. All calculations are for an ideal gas
with γ = 5/3, and are scaled so that the fluid variables have
values reasonably close to unity.
Several additional parameters must be specified when
using a turbulence model. One of these is the maximum
eddy size, which here is set equal to the cloud radius. An-
other choice concerns the initial level of turbulence in the
gas, and two extreme cases are considered. In the first case
(hereafter identified by “low k-ǫ”, or “lo” in the model name)
the postshock gas initially has an extremely low level of tur-
bulence, with a ratio of turbulent energy density to thermal
energy density etb/eth ∼ 10−6. In the second case (identi-
fied by “high k-ǫ”, or “hi” in the model name) this ratio is
0.13 (values much higher than this cause the shock to ac-
celerate too much and the interaction does not occur at the
intended Mach number). High levels of post-shock turbu-
lence may arise if the shock is propagating through an inho-
mogeneous medium (e.g., when there are density variations
further upstream). As we shall see, this ratio is similar to
the turbulent energy fraction attained by the cloud material
after the shock encounter (see Section 4.6.7), so there is a
degree of self-consistency in these models. In both cases, ini-
tially etb/eth= 0.04 within the cloud (this is set low enough
to not affect the dynamics - cf. the initial development of the
interaction in models c3no128 and c3lo128), and the cloud
and intercloud medium are set in pressure equilibrium.

Page 7

The turbulent destruction of clouds7
0
20
40
60
80
100
0 0.5 1 1.5
r (rc)
2 2.5 3
Density (ρamb)
p1=1
p1=10
n=2
n=24
Figure 1. Comparison of cloud density profiles obtained using
Eq. 18 (p1 = 1 and 10) and Eq. 1 in Nakamura et al. (2006)
(n = 2 and 24) with χ = 102.
3.3 Cloud density profile
Clouds in the ISM do not have infinitely sharp edges (see,
e.g., the discussion in Nakamura et al. 2006). The density
profile adopted in this work is
ρ(r) = ρamb[ψ + (1 − ψ)η],
where
η =1
2α + 1
(18)
?
1 +α − 1
?
, (19)
α = exp {min[20.0,p1((r/rc)2− 1)]}, and r is the distance
from the centre of the cloud of radius rc. ψ is adjusted to
obtain a specific density contrast for the centre of the cloud
with respect to the ambient medium, χ = ρmax/ρamb. The
parameter p1controls the steepness of the profile at the edge
of the cloud. Eq. 18 tends to give a flatter density profile
within the centre of the cloud, and a steeper profile as the
cloud merges into the ambient medium, than profiles ob-
tained using Eq. 1 in Nakamura et al. (2006). For p1∼
ψ ≈ χ. Clouds with reasonably sharp edges are obtained
with p1 = 10 (similar to those from Eq. 1 in Nakamura et al.
(2006) with n = 24), while p1 = 1 produces a much more ex-
tended cloud which is closer to the Nakamura et al. (2006)
profile with n = 2 (see Fig. 1).
>5,
3.4Global quantities
The cloud evolution is studied through various integrated
quantities (see Klein et al. 1994; Nakamura et al. 2006). Av-
eraged quantities ?f?, are constructed by
1
mβ
κ?β
?f? =
?
κρf dV,(20)
where the mass identified as being part of the cloud is
mβ=
?
κ?β
κρ dV.(21)
κ is an advected scalar, which has an initial value of
ρ/(χρamb) for cells within a distance of 2.25rc from the cen-
tre of the cloud, and a value of zero at greater distances.
Hence, κ = 1 in the centre of the cloud, and declines out-
wards. The above integrations are performed only over cells
in which κ is at least as great as the threshold value, β.
Setting β = 0.5 probes only the densest parts of the cloud
and its fragments, while setting β = 2/χ probes the whole
cloud including its low density envelope, and regions where
only a small percentage of cloud material is mixed into the
ambient medium.
To measure the shape of the cloud, effective radii normal
to and along the axis of symmetry are defined respectively
as
?1/2
In inviscid calculations a measure of the turbulence of the
cloud is obtained from the velocity dispersions in the radial
and axial directions, defined respectively as
a =
?5
2?r2?,c =?5??z2? − ?z?2??1/2. (22)
δvr =?v2
The mean density is defined as
?ρ? =mβ
r
?1/2, δvz =??v2
z
?− ?vz?2?1/2.(23)
Vβ,(24)
where Vβ is the volume of the region with κ ? β.
All quantities computed with β = 0.5 are identified with
the subscript “core” (e.g., acore), while those computed with
β = 2/χ are given the subscript “cloud” (e.g., acloud).
3.5 Timescales
Several timescales are obtained from the simulations. The
characteristic radial expansion timescale, tm, is defined as
the time at which the cloud radius normal to the axis of
symmetry, a, has increased to 90 per cent of its maximum
value. The time for the cloud velocity relative to that of
the postshock ambient flow to decrease by a factor of 1/e is
defined as the “drag time”, tdrag. The “mixing time”, tmix,
is defined as the time when the mass of the core of the cloud,
mcore, reaches half of its initial value. The zero-point of all
the time measurements quoted in this work occurs when the
intercloud shock is level with the centre of the cloud.
3.6Convergence tests
It is important to demonstrate that the calculations are per-
formed at spatial resolutions that are high enough to resolve
key features of the interaction. Increasing the resolution in
inviscid calculations leads to smaller scales of instabilities.
Quantities which are sensitive to these small scales (such
as the mixing rate between cloud and ambient gas) may
not be converged, while quantities which are insensitive to
gas motions at small scales (e.g., the shape of the cloud)
are more likely to show convergence. Previous studies (e.g.,
Klein et al. 1994; Nakamura et al. 2006) have indicated that
about 100 cells per cloud radius are needed for convergence
of the simulations. Here we carry out a similar study for
calculations which use a subgrid turbulence model.
Fig. 2 shows the evolution of the core mass and mean
cloud velocity as a function of spatial resolution for both
inviscid and k-ǫ calculations. These parameters are a good
test of the convergence, since both the rate of mixing and the
momentum transfer between the cloud and the ambient flow

Page 8

8J. M. Pittard, S. A. E. G. Falle, T. W. Hartquist and J. E. Dyson
Table 2. Dependence of the global cloud and core properties on the level of turbulence and the density contrast of the cloud (see Table 1).
In each case the cloud was hit by a Mach 10 shock. The time-dependent quantities are evaluated at t = tmixrather than t = tm (c.f.
Nakamura et al. 2006), since acloudcontinues to rise in some simulations. Values in parentheses are obtained from integrations over the
“core” mass rather than the “cloud” mass. Model names containing “sh” were computed using a shallow density profile (p1= 1). Model
c1rt32 was computed using an inviscid code and grid-scale post-shock turbulence.
Modeltdrag/tcc
tmix/tcc
tm/tcc
a/rc
c/rc
c/a?ρ?/ρmax
?vz?/vb
c1no128
c1lo128
c1hi128
c2no128
c2lo128
c2hi128
c3no128
c3lo128
c3hi128
c2nosh64
c2losh64
c2hish64
c3nosh64
c3losh64
c3hish64
c1rt32
1.03 (1.10)
1.04 (1.11)
0.72 (0.86)
3.06 (3.10)
3.08 (3.09)
2.47 (2.65)
6.58 (9.48)
6.84 (7.15)
4.58 (4.93)
4.05 (5.51)
4.00 (4.33)
3.08 (3.85)
6.72 (9.84)
7.56 (14.44)
6.55 (8.36)
0.66 (0.72)
(6.82)
(5.96)
(4.37)
(5.05)
(4.86)
(5.73)
(8.59)
(7.83)
(5.95)
(7.92)
(7.37)
(4.87)
(7.86)
(8.99)
(4.95)
(5.84)
3.82
3.57
3.52
4.53
4.82
-
10.23
-
7.25
8.60
9.50
-
-
5.16
11.82
-
1.74 (1.72)
1.66 (1.74)
1.88 (1.90)
4.36 (4.21)
3.92 (3.35)
4.05 (4.27)
2.60 (1.22)
4.01 (2.82)
6.84 (7.79)
3.09 (2.75)
2.89 (2.21)
2.49 (1.46)
5.16 (2.42)
3.89 (1.05)
2.63 (2.13)
2.21 (2.58)
2.82 (2.92)
2.20 (2.40)
1.36 (1.30)
5.07 (3.58)
4.12 (3.05)
6.24 (1.66)
75.9 (12.6)
49.1 (6.88)
38.2 (5.57)
11.7 (2.33)
10.8 (2.02)
6.50 (0.98)
48.6 (8.64)
64.2 (6.12)
22.8 (2.94)
1.29 (0.90)
1.62 (1.70)
1.33 (1.38)
0.72 (0.68)
1.16 (0.85)
1.05 (0.91)
1.54 (0.39)
29.2 (10.3)
12.2 (2.44)
5.58 (0.71)
3.80 (0.85)
3.74 (0.91)
2.61 (0.67)
9.41 (3.57)
16.5 (5.86)
8.67 (1.38)
0.58 (0.35)
0.65 (0.87)
0.68 (1.03)
0.56 (0.66)
0.073 (0.130)
0.075 (0.142)
0.058 (0.078)
0.0081 (0.0401)
0.0078 (0.0248)
0.0064 (0.0122)
0.104 (0.392)
0.103 (0.446)
0.114 (0.476)
0.0124 (0.145)
0.0121 (0.085)
0.0241 (0.144)
0.61 (0.80)
0.634 (0.685)
0.579 (0.629)
0.621 (0.651)
0.635 (0.606)
0.593 (0.554)
0.606 (0.590)
0.394 (0.189)
0.430 (0.271)
0.584 (0.578)
0.455 (0.402)
0.441 (0.377)
0.379 (0.313)
0.370 (0.216)
0.302 (0.150)
0.204 (0.155)
0.614 (0.666)
0
0.2
0.4
0.6
0.8
1
0 2 4 6 8 10 12 14 16
mcore (mc)
t (tcc)
a)
c3no32
c3no64
c3no128
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0 2 4 6 8 10 12 14 16
<vz, cloud> (vb)
t (tcc)
b)
c3no32
c3no64
c3no128
0
0.2
0.4
0.6
0.8
1
0 2 4 6 8 10 12 14 16
mcore (mc)
t (tcc)
c)
c3lo32
c3lo64
c3lo128
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0 2 4 6 8 10 12 14 16
<vz, cloud> (vb)
t (tcc)
d)
c3lo32
c3lo64
c3lo128
Figure 2. Convergence tests for a shock-cloud interaction with χ = 103and p1= 10 using an inviscid (panels a and b) and k-ǫ (panels c
and d) code. The time evolution of the core mass (panels a and c) and the mean velocity of the cloud (panels b and d) are shown. Note
the much tighter correlation of mcore and ?vz,cloud? from the k-ǫ code.

Page 9

The turbulent destruction of clouds9
0.01
0.1
1
10
10 100
Error (t = 6 tcc)
Resolution
a)
acloud
ccloud
<vz,cloud>
acore
ccore
<vz,core>
mcore
0.01
0.1
1
10
10 100
Error (t = 8 tcc)
Resolution
b)
acloud
ccloud
<vz,cloud>
acore
0.01
0.1
1
10
10 100
Error (t = 6 tcc)
Resolution
c)
<vz,cloud>
acore
ccore
<vz,core>
mcore
0.01
0.1
1
10
10 100
Error (t = 8 tcc)
Resolution
d)
acloud
ccloud
<vz,cloud>
acore
Figure 3. Relative error versus spatial resolution (number of cells per cloud radius on the finest grid) for a number of global quantities
measured from a shock-cloud interaction with χ = 103and p1 = 10 at t = 6 tcc (left panels) and 8tcc (right panels). The top panels
show results from the inviscid code, while the bottom panels are the results from the k-ǫ code with “low k-ǫ” conditions (models c3lo).
are sensitive to small scale instabilities. It is therefore not
too surprising to see that these quantities are poorly con-
verged in the low resolution inviscid calculations (see also
Shin et al. 2008). In contrast, the subgrid turbulence model
leads to results which are much less dependent on the spatial
resolution. This is also demonstrated in Fig. 3 which shows,
for a number of parameters, the relative error defined as the
fractional difference between the value measured at resolu-
tion N and the value at the finest resolution, f:
∆QN =|QN − Qf|
|Qf|
. (25)
The convergence is much better for the k-ǫ calculations (see
Fig. 3), leading to much less variation with resolution as seen
in Fig. 2. Both inviscid and k-ǫ calculations demonstrate
better convergence at χ = 102(not shown).
Figs. 2 and 3 support claims from previous studies that
of order 100 cells per cloud radius are needed for conver-
gence. However, the neglect of the effect of turbulent eddies
on the mean flow means that it is not clear that inviscid
simulations, particularly at high values of χ, are actually
converging to a physically realistic solution.
4 RESULTS
4.1 Stages
The main stages of the interaction of a shock of velocity vb
with a uniform cloud of density contrast χ, in the adiabatic,
un-magnetized, non-conducting case are: (1) an “initial tran-
sient stage”, where the incident shock propagates into the
cloud with velocity vs = vb/χ1/2, and a bow shock or bow
wave propagates upstream into the ambient medium; (2) the
“compression stage”, where the cloud is compressed mainly
in the z-direction by the transmitted shock and by a shock
driven into the back of the cloud; (3) the “expansion stage”,
where the highly-pressured cloud expands downstream and
laterally; and (4) the “destruction stage”, where the cloud
is destroyed and its material mixed into the surrounding
flow. In other situations, for example when there is efficient
cooling, the evolution can be significantly different (see Sec-
tion. 2.4). In all work to date the cloud is destroyed by the
shock. The addition of gravitational forces is likely to be
needed if the cloud is to survive.
4.2 Cloud morphology and turbulence
In Fig. 4, snapshots of the density distribution at different
times are shown for an inviscid calculation with 128 grid

Page 10

10J. M. Pittard, S. A. E. G. Falle, T. W. Hartquist and J. E. Dyson
Figure 4. Snapshots of the density distribution from an inviscid calculation of a Mach 10 adiabatic shock hitting a cloud with a density
contrast of 103with respect to the ambient medium and a density profile with p1= 10 (model c3no128). The resolution is 128 cells per
cloud radius. The evolution proceeds left to right and top to bottom with t = 0.0,0.1,0.3,0.49,0.87,1.35,1.83,2.31,2.79, and 3.75 tcc.

Page 11

The turbulent destruction of clouds11
Figure 5. As Fig. 4 but for a k-ǫ calculation with low initial postshock turbulence (model c3lo128). The times of the snapshots are
t = 0.1,0.49,0.87,1.83, and 3.75 tcc.
Figure 6. As Fig. 4 but for a k-ǫ calculation with high initial postshock turbulence (model c3hi128). The times of the snapshots are
t = 0.1,0.49,0.87,1.83, and 3.75 tcc.

Page 12

12 J. M. Pittard, S. A. E. G. Falle, T. W. Hartquist and J. E. Dyson
Figure 7. Comparison between the inviscid (model c3no128, left panel) and k-ǫ calculations with low and high initial postshock turbulence
(models c3lo128 and c3hi128, middle and right panels, respectively) at t = 5.66 tcc.
cells per cloud radius (model c3no128). The evolution of
the cloud broadly follows the stages outlined above. The
first two stages last until t ≈ tcc (i.e. the top 5 panels in
Fig. 4). The expansion of the cloud in stage 3 is supersonic
with respect to the sound speed within the cloud, and a low
density interior surrounded by a higher density shell forms
(see the snapshot at t = 1.83 tcc). The high density shell
then collapses in on itself (see the snapshot at t = 2.31 tcc).
Material is continuously ablated off the surface of the cloud
by the fast-flowing surroundings, and a turbulent wake with
prominent RT and KH instabilities forms. Fig. 7 shows that
at later times the mass-loss from the cloud resembles a single
tail-like structure (this is in contrast to models with lower
values of χ - see Section 4.3).
Fig. 5 shows that the initial interaction of the shock
with the cloud in the low k-ǫ case (model c3lo128) is similar
to the inviscid case, since the initial post-shock turbulence
is low and k and ǫ are small. However, the viscosity intro-
duced by the sub-grid turbulence model prevents the sub-
sequent development of the resolution-dependent RT and
KH instabilities seen in Fig. 4. Instead, the loss of material
from the cloud occurs much more smoothly. This is exactly
as expected given that the purpose of the k-ǫ model is to
approximate the time-averaged flow.
Simulations with a high initial level of post-shock turbu-
lence are different again, as seen in Fig. 6 where the results
of model c3hi128 are displayed. The transport/diffusion co-
efficients are considerably higher in this simulation, and
this leads to a faster rate of ablation from the cloud, and
ultimately its more rapid destruction. The high level of
upstream turbulence also smooths/broadens the bowshock
ahead of the cloud and the tailshock which forms down-

Page 13

The turbulent destruction of clouds 13
Figure 8. Snapshots of the turbulent energy per unit mass, k, from the k-ǫ calculations with χ = 103and p1= 10. The three left-most
panels show the evolution with low initial postshock turbulence (model c3lo128) proceeding left to right with t = 0.1,0.3, and 0.87 tcc.
The two right-most panels show the evolution with high initial postshock turbulence (model c3hi128) proceeding left to right with t = 0.1
and 0.87 tcc. The white regions in the middle panel are artifacts of the plotting routine.
stream, so that their time-averaged positions are repre-
sented.
Fig. 7 compares the morphology of the clouds in these
3 simulations at t = 5.66 tcc. The global features of mod-
els c3no128 and c3lo128 are reasonably comparable at this
time, but the cloud in model c3hi128 is clearly at a more
advanced stage of destruction (cf. Table 2 and Fig. 17). In-
terestingly, material stripped from the cloud lies off-axis in
model c3hi128. This develops from the off-axis density peak
seen at t = 1.83 and 3.75 tcc shown in Fig. 6, though the
initial disturbance occurs at even earlier times. Clearly the
turbulence in this model affects the properties of the shocks
driven into the cloud and the slip surface that forms around
it, with small differences at early stages being amplified dur-
ing the subsequent evolution.
The development of the sub-grid turbulence in model
c3lo128 is shown in the three left-most panels of Fig. 8,
where the turbulent energy per unit mass, k, is displayed.
k is created in regions of high shear, particularly in a thin
turbulent boundary layer along the slip surface around the
cloud (see the left-most panel of Fig. 8). The turbulent inten-
sity quickly saturates at a level that is almost independent
of its initial value.
The ∇ · u terms in the production term for k (see
Sec. 3.1) means that k is also generated behind shocks,
as can be seen in 4 specific regions in the snapshot at
t = 0.1 tcc: behind the incident shock sweeping through
the ambient medium; behind the reflected shock formed as
the incident shock converges on the axis behind the cloud;
behind the bow shock formed upstream of the cloud; and
behind the slow shock driven into the cloud.
The reflected shock on the axis behind the cloud be-
comes increasingly oblique as the point of convergence moves
away from the rear of the cloud, and interacts with the in-
cident shock to create a double Mach reflected shock that
propagates along the axis (Klein et al. 1994). A powerful
supersonic vortex ring forms just behind the Mach reflected
shock, in which a region of high turbulence is generated (see
the second from left panel of Fig. 8). While the turbulence
generated behind shocks decays very rapidly, the turbulence
associated with the vortex ring is much more persistent, as
is the turbulence generated at the slip surface around the
cloud.
Fig. 8 shows that at later times the turbulence gen-
erated at the slip surface proceeds to develop into a highly
turbulent wake with a radius comparable to the initial cloud
radius. The setup time for the wake is ∼ tcc, and the sub-
grid turbulent energy of the cloud material grows and then
dissipates as the cloud is mixed into its surroundings (see
Figs. 17g and 22). Note that the core of the cloud has very
little turbulence associated with it.
The finite timescale for the development of significant
turbulence means that simulations with the k-ǫ subgrid tur-
bulence model with a low initial level of postshock turbu-
lence produce similar morphologies to those obtained from
inviscid calculations at early times (t∼
the increase in the transport coefficients in regions of high
turbulence leads to increasing divergence from inviscid cal-
<0.5 tcc). However,

Page 14

14J. M. Pittard, S. A. E. G. Falle, T. W. Hartquist and J. E. Dyson
culations at later times, and ultimately to a faster destruc-
tion of the cloud.
In contrast, a high level of environmental turbulence
immediately affects the evolution of the cloud, since the
transport coefficients around the cloud are also high. The
two right-most panels of Fig. 8 show the highly turbulent
post-shock flow engulfing the cloud in model c3hi128. Both
the limb of the cloud and the bowshock upstream of the
cloud become broader and less distinct as the high level of
turbulence leads to strong diffusion across these boundaries
(see Fig. 6). Another major difference compared to model
c3lo128 is that the turbulence downstream of the cloud at
t = 0.87 tcc is roughly as strong as that in the post-shock
flow. Hence the turbulent wake which is seen so clearly in
the centre panel of Fig. 8 is indistinguishable from the sur-
roundings in the right panel of Fig. 8. Note also that the
value of k created downstream of the incident shock (see
the 2nd from right panel of Fig. 8) is much smaller than
the initial post-shock value. This reflects the fact that such
high levels of post-shock turbulence are not naturally gener-
ated by a shock sweeping through a perfectly homogeneous
medium.
The two rightmost panels in Fig. 13 show the turbulent
energy per unit mass, k, on a linear scale at t = 1.81 tcc
for low k-ǫ calculations with χ = 103and different density
profiles (models c3lo128 and c3losh64). This highlights the
fact that the strongest turbulence is generated where the
shear velocity is high, that the central region of the wake
immediately behind the cloud has a somewhat lower level
of turbulence, and that further downstream the turbulence
has penetrated throughout the wake. The opening angle of
the turbulent layer in model c3lo128 is estimated as ≈ 13◦,
which is in good agreement with experimental results (see
Cant´ o & Raga 1991, and references therein), where an open-
ing angle of ≈ 11◦is obtained for a Mach 1.3 flow past a
stationary medium (the gas behind a Mach 10 shock has a
Mach number of 1.31). Future work will examine whether
this agreement with experiment persists as the Mach num-
ber is varied.
4.3 Dependence on cloud density contrast
A range of density contrasts between the cloud and the
ambient medium is expected. For instance, χ ∼ 102for
cold atomic clouds embedded in the warm neutral or pho-
toionzied medium where T ∼ 104K, or for warm clouds em-
bedded in the coronal gas where T ∼ 106K. For molecular
clouds embedded in warm gas, χ ∼ 103, while cold atomic
clouds embedded in coronal gas have χ ∼ 104.
Fig. 9 shows the destruction of a cloud with a density
contrast χ = 10, while Fig. 10 shows the corresponding case
for a cloud with χ = 102, both computed with an inviscid
code. The colour scaling in both figures is identical to that in
previous figures for easier comparison. The timescale for the
cloud material to mix into the ambient flow scales roughly
with the cloud crushing timescale, tcc, in agreement with
previous works (Klein et al. 1994; Nakamura et al. 2006).
However, the normalized growth timescale for RT and KH
instabilities decreases with increasing χ, as is apparent from
a comparison of Figs. 4, 9 and 10. The normalized drag time,
tdrag/tcc, increases with χ, as does the axial stretching of the
cloud, c/a (see Table 2). Figs. 15-17 also reveal that the ratio
of the velocity dispersion in the axial to normal directions,
δvz/δvr, increases with χ. These results are all in agreement
with earlier works.
The effect of a highly turbulent post-shock flow is great-
est at high χ (e.g., compare the values of the mean cloud and
core velocities in Table 2 for the “high k-ǫ” models against
the “low k-ǫ” and inviscid models as a function of χ). This
is because clouds with a high density contrast survive for a
considerable time after the initial passage of the shock, and
thus are subject to considerable “buffeting” by the highly
turbulent postshock environment, whereas at lower values of
χ, the cloud is destroyed relatively quickly after the initial
passage of the shock. This is also manifest in the increasing
disparity with χ in the evolution of various global quantities
from the “hi” models on the one hand, and the “no” and
“lo” models on the other hand, as shown in Figs. 15-17.
4.4Dependence on cloud profile
In Fig. 11, snapshots of the density distribution at different
times are shown for an inviscid calculation of a Mach 10
shock hitting a cloud with a shallow density gradient (model
c3nosh64; see Table 1). While the resolution is such that rc
is equal to the width of 64 cells on the finest grid, the cloud
in fact extends to r∼
similar to the previous models (see Fig. 1).
The interaction of a shock with a smooth cloud was pre-
viously studied by Nakamura et al. (2006) for the case where
χ = 10. In this case the cloud offered little impediment to
the oncoming shock, with the result that the transmitted
shock and the intercloud shock had similar mean speeds.
As a result, the intercloud shock did not converge on the
z-axis behind the cloud, and the shock compression from
the downstream side was weak, leading to a slow lateral ex-
pansion of the cloud. In contrast, the cloud is a much more
robust obstacle when χ = 103, and we find that it maintains
many aspects of the evolution seen in sharper-edged clouds
(c.f. Figs. 4 and 11). Fig. 12 compares the density structure
at t = 2.77 tcc for inviscid and low k-ǫ calculations (models
c3nosh64 and c3losh64). At later times the material stripped
from the cloud resembles a single tail-like structure, as was
also the case for a cloud with sharper edges.
Nevertheless, the shallower density gradient does lead
to a milder interaction. This is also manifest as a slower
growth of turbulence around the cloud (c.f. Figs 8 and 13
for k-ǫ models with low initial turbulence), with the turbu-
lent wake not completely forming until t ≈ 2 tcc. An exact
comparison of the respective timescales is complicated by
the fact that the cloud with the shallow density gradient
is also larger and more massive, and so the true value of
tcc will be different between the models. In fact, we find
that the forward and rear shocks driven into the cloud con-
verge at t ≈ 0.8 tcc in model c3no128, and at t ≈ 1.0 tcc
in model c3nosh64 (we continue to calculate tcc using Eq. 2
with χ = 103and rc = 1, despite this equation being ap-
plicable only to clouds with sharp edges). Since these times
are not too discrepant, it becomes clear that the growth
of turbulence around the smoother cloud is indeed slower,
in agreement with the statement by Nakamura et al. (2006)
that it takes more time to form the slip surface around the
cloud. The maximum turbulent energy per unit mass (i.e.
kmax) is also higher in model c3lo128.
>2 rc, so the effective resolution is

Page 15

The turbulent destruction of clouds 15
Figure 9. Snapshots of the density distribution from an inviscid calculation of a Mach 10 adiabatic shock hitting a cloud with a density
contrast of 10 with respect to the ambient medium and with a density profile specified by p1= 10 (model c1no128). The resolution is
128 cells per cloud radius. The evolution proceeds left to right with t = 0.0,0.49,0.97,1.93, and 3.85 tcc.
Figure 10. Same as Fig. 9 but for a density contrast of χ = 102(model c2no128). The evolution proceeds left to right with
t = 0.0,0.46,0.91,1.82, and 3.65 tcc.