caseA.den.jpg caseA.temp.jpg cci_d.1500.jpg caseB_t.density.270.jpg v444_dens.full.jpg v444_temp.red.jpg mclump.030.jpg ddd.dens.3_w.0000.jpg ddd.dens.3_w.0030.jpg ddd.dens.3_w.0060.jpg ddd.dens.3_w.0070.jpg ddd.dens.3_w.0150.jpg ddd.dens.3_w.0500.jpg ddd.mach.3_w.0000.jpg ddd.mach.3_w.0030.jpg ddd.mach.3_w.0060.jpg ddd.mach.3_w.0070.jpg ddd.mach.3_w.0150.jpg ddd.mach.3_w.0500.jpg cci_d.0500.jpg cci_d.1000.jpg cci_d.1500.jpg



Shock bound, supersonically turbulent slabs and stability of radiative shocks

by Doris Folini and Rolf Walder



Page contents

Introduction
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Stability of radiative shocks: 1D results
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Stability of radiative shocks: 2D results
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Supersonic turbulence in 2D slabs
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References
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Introduction

Supersonically colliding flows, their accompanying radiative shock waves, and associated supersonic turbulence are ubiquitous in astrophysics.

Radiative shock waves play a crucial role in connection with structure formation in space. They decisively co-determine the dynamics of such different objects as core-collapse supernova, atmospheres of hot stars, or the circumstellar matter in binary star systems. The proper understanding of the structure and stability of radiative shocks is compelling in order to trace down the dynamical variables and the abundances in supernova remnants, nova-outflows, wind-driven structures, and jets. Moreover, the stability properties of radiative shock waves are a key for the understanding of the elemental mixing in such flows.

Turbulent shock bound slabs serve as useful models for such diversified things as galactic sheets in the process of galaxy formation, molecular clouds and associated star formation, shock-bound disks in wind-accreting binary star systems, and the cold part of wind-driven structures and supernova-remnants. A proper theoretical understanding of such slabs thus greatly contributes to the understanding of these objects and related, decisive astrophysical questions.

We have investigated the structure and stability of radiative shocks in one and two space dimensions, using Euler equations and parameterized radiative cooling or an isothermal equation of state. In addition, we examine, by means of high resolution numerical simulations, what happens to the interior of such a collision zone. In particular, we investigate the supersonic turbulence in 2D shock bound slabs. The simulations are all performed with the AMRCART codes from the A-MAZE package. More details on the points below you can find in the papers given in the references.


Stability of radiative shocks: 1D results

While 1D results are mostly far from physical reality they are an ideal ground to understand some basic physics. Once basical physical mechanisms are understood one may extrapolate again (carefully!) to the actual physical object of interest. There are not too many hydrodynamical instabilities present in 1D. Most of them require at least two spatial dimensions to develop. Here we concentrate on the radiative cooling instability.

The basic structure of a flow collision zone in 1D is shown here and consists of two hot post shock zones and a cold layer of dense gas, CDL in the following. Our simulations show that for wide temperature ranges the hot post shock zones become unstable. In those temperature regions where the radiative loss function possesses a slope which is smaller than 1, runaway cooling can occur, and the hot post-shock layer can vanish entirely. As soon as this has happened, however, a hot post-shock layer is rebuilt and, after some time, becomes again thermally unstable. Phases of comparatively strong cooling alternate with phases of reduced cooling, and the spatial extension of the hot shocked zones oscillates with time. Several different types and modes of oscillation can be observed, as well as the transition between different modes and types in case of a global slow down of the interaction zone. The observed oscillation modes are in agreement with linear analysis.

The instability of the hot shocked zones induces disturbances in the layer of cold, compressed gas. Such disturbances will prove important in the 2D case, where turbulence within the cold, compressed interaction zone is partially driven by them. In our 1D simulations, we observe that density disturbances in the preshock flow can finally cause strong oscillations of the CDL.


Stability of radiative shocks: 2D results

We have performed a variety of different 2D simulations of radiative colliding flows with very high spatial resolution and on a comparatively long time scale. Common to all results is the fact that the interaction zone of the two radiative colliding flows is unstable. Depending on flow parameters and geometry different kinds of instabilities are present, interacting and dominating each other.

2D, plane parallel, asymmetric

In plane parallel geometry we have investigated the case of momentum balanced radiative colliding flows where the colliding flows have different densities and, therefore, strongly different cooling times. As indicated in the schematic setting, one of the hot post shock zones then cools efficiently and in our case is thermally unstable, while the other zone behaves nearly adiabatically at a very high temperature. For this case, which we call the asymmetric case, we observe the growth of spikes of cold, dense matter out of the cold dense layer and into the nearly adiabatically hot shocked matter. Finally, parts of these spikes breaks off and drift into the hot shocked matter. The growth of the spikes is due to Rayleigh-Taylor and Richtmyer-Meshkov instabilities, induced into the cold dense layer by the thermal instability of one of the two shock waves. The motion of the matter withing the cold dense layer itself becomes mildly supersonically turbulent.
caseA.den.jpg caseA.temp.jpg
Density. Click on image to enlarge or watch an mpeg video (7.3MB). Temperature. Click on image to enlarge or watch an mpeg video (7.7MB).


2D, plane parallel, symmetric

For the symmetric case, where both shocks are radiative, we observe a very dynamical evolution of the cold dense layer, including strong global bending and rich interior structure. Again we look at a plane parallel situation as sketched here where the two colliding flows are momentum balanced. The matter within the cold dense layer then gets supersonically turbulent, causing the mean density of the layer to be substantially reduced.
cci_d.1500.jpg caseB_t.density.270.jpg
Density. Isothermal calculation. Click on image to enlarge or watch an mpeg video (8.6MB) or a quicktime movie (115MB). Density. Calculation with radiative loss function, cooling layers resolved. Click on image to enlarge or watch an mpeg video (5.3MB).


2D, axisymmetric, binary star setting

Unstable behaviour is also characteristic for radiative colliding flows in massive binary systems. As can be taken from the schematic setting the situation now is no longer plan parallel. 2D simulations assuming axial symmetry and including a radiative cooling function reveal that the interaction zone between the winds of the two massive stars forms no smooth cone at all. It is tilted and twisted instead, containing high density knots and causing large voids through shielding effects. The tilting of the interaction zone can lead to rather high post shock temperatures far away from the system center if the tilting is such that the interaction zone becomes locally nearly perpendicular again to the stellar wind.
v444_dens.full.jpg v444_temp.red.jpg
Density. Click on image to enlarge or watch an mpeg video (1.6MB). Temperature, central region. Click on image to enlarge or watch an mpeg video (1.6MB).


2D, axisymmetric, binary star setting, clumped flows

We have also performed first studies on the effect of clumped stellar winds on the wind-wind interaction zone in massive binary system. For this purpose we have added a bunch of small but high density clumps to one of the stellar winds. We find that the interaction zone again is highly unstable. In addition, our simulations also show that the high density clumps can locally lead to a substantially enhanced cooling of the post shock flow.
mclump.030.jpg Density. Click on image to enlarge.



Supersonic turbulence in 2D slabs

The 2D simulations which we have performed of shock bound slabs not only show that it takes but little for such slabs to become unstable, but also that supersonic turbulence in their interior is a natural consequence. As the turbulence develops, the mean density within the slab decreases while the mean Mach number increases. This turbulence is continuously driven by the energy input from two colliding flows. The resulting structure within the slab appears to increase in size as the slab thickness increases. This finding is of particular interest with regard to structure formation, for example in molecular clouds.
ddd.dens.3_w.0000.jpg ddd.dens.3_w.0030.jpg ddd.dens.3_w.0060.jpg ddd.dens.3_w.0070.jpg ddd.dens.3_w.0150.jpg ddd.dens.3_w.0500.jpg
Isothermal slab, density, time series, starting at t=0 (left). Click on images to enlarge or watch a mpeg-video (5.8MB) or a quicktime movie (91MB).
ddd.mach.3_w.0000.jpg ddd.mach.3_w.0030.jpg ddd.mach.3_w.0060.jpg ddd.mach.3_w.0070.jpg ddd.mach.3_w.0150.jpg ddd.mach.3_w.0500.jpg
Isothermal slab, Mach number, time series, starting at t=0 (left). Click on images to enlarge or watch a mpeg-video (2.5MB) or a quicktime movie (40MB).
cci_d.0500.jpg cci_d.1000.jpg cci_d.1500.jpg
Isothermal slab, density, As time increases (886 y, 1773 y, 2663 y) the slab becomes thicker and the scale of the density structure within the slab increases. Click on image to enlarge or watch a mpeg (8.6MB) or a quicktime movie (115MB)



References

R. Walder and D. Folini 1998
Knots, filaments, and turbulence in radiative shocks
A & A 330, L21-L24
(Available as a 47 KB gzipped ps-file, text only, plus a 1.8 MB gzipped ps-file figures only)

R. Walder and D. Folini 1996
Radiative cooling instability in 1D colliding flows
A & A, 315, 265-283
(Available as a 378 KB gzipped ps-file)

R. Walder and D. Folini 1999
The formation of knots and filaments in shocks
Astrophysics and Space Science 260, 215-224
(Available as a 761 KB gzipped ps-file)

R. Walder and D. Folini 2000
On the stability of colliding flows: radiative shocks, thin shells, and supersonic turbulence
Astrophysics and Space Science, 274, 343-352
(Available as a 0.5 MB gzipped ps-file)

R. Walder and D. Folini 1999
Radiative Shocks, Supersonic Turbulence and Structure Formation in Space
Proceedings of the Seventh International Conference on Hyperbolic Problems:
Theory, Numerics, Applications
(Available as a 2.7 MB gzipped ps-file)

D. Folini 1998
Computational approaches to multidimensional radiative transfer and the physics of radiative colliding flows
PhD thesis, ETH Zurich, No. 12606 (Available as a 3.3 MB gzipped ps-file)

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Doris Folini and Rolf Walder
Last Update: October 14, 2002