Cloud-Shock Interactions
Charles Danforth
Interstellar Medium
Dec 16, 1998

Introduction - The Observational Basis

Much of our basic understanding of conditions in the ISM comes from observations of shock waves and their interaction with it. Supernova Remnants (SNRs) are prime examples of ISM interactions and, while most observed SNRs are too distant to resolve individual details, a handful are near enough and large enough to show clearly different morphologies at different locations along their rims. These morphologies are indicative of different types of shock interactions with the ISM.

Roughly we see two different types of filaments in SNRs (Figure 1). Non-radiative filaments are visible primarily in Balmer transitions and in the X-ray. They are smooth in appearance and very thin and may stretch for several parsecs or more. The physical interpretation is that a fast shock is moving through very homogeneous, tenuous medium. The shock is fast enough that the post-shock gas is heated to X-ray temperatures and cools very slowly. The northern rim of the Cygnus Loop is a perfect example of this. In the optical, we see a very thin, smooth ribbon of H$\alpha$ emission. In the X-ray, we see a sharp edge dividing a bright, hot, shocked region from a dark pre-shocked region.

The other general class is the clumpy, knotted morphology characterized by short, complicated filaments and bright line emission. This is generally believed to be indicative of cloud shock interaction which bends and warps the planar symmetry of the blast wave. The SN blast wave has traveled through a denser medium which cools more rapidly and is thus brighter. Cooling is typically partly or wholly radiative and many heavy ion lines are seen (OVI, [OIII], CIV, [CIII], NV, etc). It is these interactions which we wish to explore further in this paper.

Background Physics

Before we can discuss cloud shock interactions further, we must have a basic understanding of some of the fluid processes involved.

Fluid Instabilities

Rayleigh-Taylor Instability

: Two types of fluid instabilities affect cloud-shock interactions. The Rayleigh-Taylor (RT) instability occurs any time a dense, heavy fluid is being accelerated by light fluid such as is the case with a cloud and shock system. Two completely plane-parallel layers of fluid are stable, but the slightest perturbation leads to tangential ``gravity''. Any small local minimum will quickly be amplified by material flowing down under the influence of this force. Dimples are quickly magnified into sets of inter-penetrating ``RT fingers'' as the heavy material moves down and the light material flows up. This process is evident in many terrestrial examples from boiling water to weather inversions.

RT instabilities are quelled by any sort of restoring force. In astrophysical situation, a magnetic field applies a certain amount of tension and can inhibit instability growth.

Theoretical work on RT instabilities and magnetic fields was done recently by Jun, Stone and Norman (1995) and showed that a field of

$$B \geq \frac{[g\lambda(\rho_2-\rho_1)]^{1/2}}{cos \theta}$$

would suppress instabilities where g is the local 'gravity', $\lambda$ is the wavelength of the instability and $\theta$ is the angle between B and the interface. Thus, a field may reduce the small fluff, but large wavelength RT instabilities are harder to suppress. Figure 3 shows the effects of a tangential magnetic field on the formation of turbulence.

RT fingers are especially obvious in the Crab Nebula (Hester et al.1996), Tycho's SNR (Velazquez et al.1998) and several other astronomical objects.

Kelvin-Helmholtz Instability

: The Kelvin-Helmholtz (KH) instability results from velocity shears between two media. These media need not even be of different densities. Any time there is a non-zero curvature, the flow of one fluid around another will lead to a slight centrifugal force which in turn leads to a change in pressure thereby amplifying the ripple. The most familiar example of this is wind blowing over calm water. Tiny dimples in the smooth surface will quickly be amplified to small waves and finally to frothing white-caps.

As with RT instability, any sort of surface tension will hinder KH instabilities. If there is some restoring force T_g, the instability will arise if

$$\Delta v^2 \ge \frac{2(\rho_1 +\rho_2 )}{\rho_1 \rho_2 }[T_g(\rho_2 -\rho_1 )]^{1/2}.$$

For water waves, surface tension stabilizes the interface until the wind reaches v > 650 cm s^-1 (12.5 knots). For astrophysical applications, magnetic field tension supplies a stabilizing force. The stability requirement is

$$\Delta v \leq 2 v_A = \frac{2B}{(\mu_o \rho)^{1/2}}.$$

In the evolved stages of KH instability, cyclonic 'cat-eye' structures are formed. When combined with RT instabilities, KH tends to form 'mushroom caps' on the end of RT fingers. Since the growth rate of these fingers is proportional to their cross section, KH tends to slow down finger growth.

Conductive Heating

Momentum is transfered to the cloud from the blast wave and from the streaming material. But thermal energy is also transferred by conduction. The exterior gas is diffuse and hot while the cloud gas is relatively dense and cold. Thermal conduction will heat the cloud gas and cause evaporation. This process was investigated by McKee & Cowie (1975) in great detail.

In general, the collisional mean free path for a post-shock particle is very long and thus conduction is particularly inefficient. However even a very weak magnetic field can give ions a gyroradius much shorter than the diameter of a typical ISM cloud. Heat conduction then possible.

As the cloud is immersed in hot gas (for instance, after the main blast wave has passed by) a conduction front moves into the cloud at v_cond. V_cond depends strongly on the efficiency of radiative heat loss. If radiative losses in the front are large, v_cond < c_s. If radiative losses are minor compared with the heat flux (non-radiative), v_cond > c_s and a shock-like set of jump conditions develops. It appears that older SNRs (Cygnus) have radiation-dominated conduction fronts while younger SNRs (CasA) are non-radiative (McKee & Cowie 1975).

It should be pointed out that for any reasonable set of astrophysical conditions, thermal conduction is a very slow process and thus probably does not play a major role in shaping SNR morphology. The time scale for cloud shocks and conduction fronts are related by

$$\frac{t_{cond}}{t_{shock}} = \frac{n_{cloud,preshock}v_{shock}}{n_{cloud,postshock} v_{cond}}$$

If the conduction front is radiation dominated, the time scale is even longer. In any case, the conduction time scale is much longer than the shock crossing time scale and probably longer than the lifetime of the SNR itself (McKee & Cowie 1975).

Cloud-Shock Evolution

With an understanding of the physics, we are finally able to talk about the details of cloud-shock interactions. The theory has evolved dramatically in two or three decades. Qualitatively, the most basic picture is put forward by McKee and Cowie in their 1975 paper. Their picture is of a spherical cloud being struck by a Supernova blast wave. Like any wave-barrier physics problem, part of the wave energy is transmitted and part is reflected. Given the density of the cloud medium, the transmitted shock is much slower than the external blast wave. The reflected wave propagates back upstream into the outward-streaming post-shock flow. If this flow is supersonic with respect to the cloud, then this reflected wave is set up as a bow shock. (Spitzer (1982) showed that if the Mach number of the blast wave was greater than 2.76, the reflected wave would be a shock whereas waves of less than this critical value would propagate normally.) Meanwhile, the blast wave external to the cloud continues on.

While this model is a good first order-approximation, the devil is in the details and these details of cloud-shock interactions must be simulated by numerical means. This is a computationally taxing project and so understanding in the field comes with increases in computing power and algorithmic sophistication.

Some of the first model calculations were done by Sgro in 1975. He used a two dimensional square cloud. He showed that there was a critical density above which the cloud cooled quickly after being shocked and below which cooling was inefficient. Furthermore, he speculated that the external shock diffracts around the cloud and reconnects with itself. In this reconnection region, pressures are higher and the shock travels more quickly. Thus any dimple left by diffraction around a cloud will be evened out by the time the shock travels a few cloud diameters and will have no lasting effect on the shock front morphology.

Later models such as those performed by Bedogni & Woodward (1990) were expanded to cover a wider array of shock velocities and cloud-ISM density contrasts. Furthermore they assumed spherical clouds, cylindrical symmetry and a more sophisticated numerical hydrodynamic code. These models showed much the same results but a wealth of new details as well.

Again, when the external shock first encounters cloud a transmitted and reflected shock are set up (Figure 4a). The reflected shock becomes a bow shock if M>2.76 (Spitzer, 1982). While the cloud shock propagates, the external shock wraps around the cloud setting up secondary shocks driven obliquely into the cloud along the surface. When the blast wave reaches the downstream side of the cloud, it reconnects producing a region of transient high pressure and sending a second reflected shock back into into cloud (Figure 4b). A vortex system appears, the blast wave disconnects from the cloud and propagates onward.

The two cloud shocks move toward the center causing axial collapse (Figure 4b,c,d). Meanwhile the streaming post shock gas at sides of cloud form low density, low pressure zones by the Bernoulli principle and excite Kelvin-Helmholtz instabilities. Thus while the cloud is being crushed axially, it is expanding radially. When the two cloud shocks meet and pass each other, they begin to re-expand the cloud. Rayleigh-Taylor instabilities form clumps and K-H instabilities form vortices. Finally, the cloud is completely disrupted and fragmented (Figure 4e,f)

Bedogni & Woodward's models show the different morphologies that can be achieved by varying the Mach number and density contrast of the simulations. For high density clouds seem to survive more intact than low density clouds. Low shock velocities tend to leave a cloud roughly spherical as well whereas high velocities tend to produce streamers and vortex rings trailing downstream from the cloud.

Three dimensional calculations were carried out by, among others, Stone & Norman (1992) on spherical clouds. Figure 5 shows three frames from their simulation plotted with greyscale proportional to log density. In the left image, we see the spherical cloud beginning to collapse as the external shock (incident from the top) sweeps past. In the middle frame, the blast wave has detached, the reverse cloud shock has been initiated and turbulence has started on the downstream edge of the cloud. Notice some material has been pulled along with the blast wave. In the right-hand image fragmentation is well advanced and quite a lot of filamentary, turbulent structure is seen. Notice that the cloud as a whole is accelerated in the direction of the shock.

Stone and Norman manage to reproduce nearly all the details seen in earlier two-dimensional work and found that, what are seen as 'vortex rings' in the 2-D simulations, are unstable in three dimensions. Instead vortex filaments are formed. Significant turbulent mixing and filamentary structure was produced just as in the 2-D models. Furthermore, they found that the degree of fragmentation and the size of the fragments seen was directly correlated with their simulation resolution.

Conclusions

Armed with these numerical simulations, perhaps we can make some sense of some observations. The Cygnus Loop is one of the largest (apparent) SNRs in the sky and presents many fascinating morphological variations for comparison to theory. Of particular interest is the region in the NE quadrant known as XA (Hester & Cox, 1986), a cloud in the early stages of being shocked (Danforth et al.1999); and the Southeast Cloud, a fine example of a post-shock cloud.

Fesen, Kwitter and Downes (1992, hereafter FKD) observed several areas of the Cygnus Loop including the Southeast Cloud (Figure 6). The similarity with the theories is striking. On the left we see a sharp, dimpled blast wave (visible in H$\alpha$ but no other bands). The cloud is in the center right and shows evidence of crushing and instabilities. FKD interpret the sharp filament on the right as the reverse shock propagating back into the remnant, but recent analysis (Levenson, Blair, priv. comm.) suggests that this is simply another portion of the blast wave oriented edge-on to our line of sight and not related to the clod system.

A more complicated system is XA, also in the Cygnus Loop but located in the east-northeastern region at the southern end of a large complex of radiative filaments (Danforth et al.1999). To first inspection, Figure 7 appears similar to that investigated by FKD; we see a bright knot of emission surrounded by a series of sharp, curving filaments. The similarity to Figure 4a is striking. However, we see that the sharp curved filaments on the east (left) are emitting strongly in [OIII], a sure sign of a radiative shock. According to our theories, SN blast waves in free space are supposed to emit weakly in the Balmer lines only. Shock velocity diagnostics give conflicting signals and there appears to be far too much column depth of high-ionization ions.

The conclusion drawn by Danforth et al.is that this cloud is a dense region in an ambient medium which is less diffuse than the typical ISM in the region. What we are seeing could be the shock wave encountering a the wall of a cavity blown by the SN precursor.

Clearly, the whole topic of cloud-shock interactions bears a great deal more investigation. Simulations keep improving with more advanced code and faster computing time allowing better resolution (both in time and space) and more accurate physics. Observationally, we need to keep investigating regions of the Cygnus Loop, the Crab Nebula, Vela, Puppis-A and the rest of the large SNRs looking for examples of fragmentation, evaporation and clouds in different stages of iacceleratednteraction with shocks.

References

Bedogni & Woodward 1990, A&A, 231, 481.
Danforth, C., Blair, W. P. & Raymond, J. C. 1999, in prep.
Fesen, R. A., Kwitter, K. B., and Downes, R. A. 1992, AJ, 104, 719.
Hester, et al. 1996, ApJ, 456, 225.
Jun, Stone, & Norman, 1995, ApJ, 453, 322.
McKee & Cowie 1975, ApJ, 195, 715.
McKee, Cowie, & Ostriker 1978, ApJ, 219, L23.
Sgro, A. G. 1975, ApJ, 197, 621.
Spitzer, L. 1982, ApJ, 262, 315.
Stone & Norman 1992, ApJ, 390, L17.
Valzquez, et al. 1998, A&A, in prep.

 

Figure: A composite image of the Cygnus Loop H$\alpha$ courtesy of Nancy Levenson. The Northeast Filament is a classic non-radiative filament. XA and the Southeast Cloud are two good examples of cloud shock interactions.

 

Figure: a) Rayleigh-Taylor instability - a dense medium is accelerated by a light one. b) Kelvin-Helmholtz instability - velocity sheer leads to interface turbulence. Time advances to the right in each figure.

 

Figure: Calculations done by Jun, Stone and Norman (1995) show the inhibition of RT instabilities by magnetic fields. Notice the small-scale structure has vanished while the large-scale fingers are still present.

 

Figure: Six frames from one of the models of Bedogni & Woodward (1990). A spherical cloud is being struck from the left by a planar shock. In this case M=40 and $\rho_{cloud}/\rho_{ISM}=18.3$. The contours plotted are natural log of density.

 

Figure: A three dimensional numerical simulation by Stone and Norman (1992) for intermediate conditions (M=10, $\rho_{cloud}/\rho_{ISM}=10$). Displayed is a 3-D projection of log density at times equal to 0.5, 2.0 and 4.5 times the shock crossing time. See the text.

 

Figure: H$\alpha$ observation of the SE cloud in the Cygnus Loop (FKD, 1992).

 

Figure: Color image of XA in the Cygnus Loop. Red is H$\alpha$, green is [OIII], blue is the ultraviolet B5 band (1550$\AA$). Shock is incident from the right.

About this document ...

This document was generated using the LaTeX2HTML translator Version 98.1p1 release (March 2nd, 1998)

Copyright © 1993, 1994, 1995, 1996, 1997, Nikos Drakos, Computer Based Learning Unit, University of Leeds.

The command line arguments were:
latex2html cloudshock.tex.

The translation was initiated by Charles Danforth on 1999-03-24