Regimes of the Vishniac–Ryu Decelerating Shock Instability

Here we revisit the derivation of the instability of dense shocked layers, originally developed by Vishniac and Ryu. Our motivation is that density profiles found in actual astrophysical and laboratory systems often do not match the assumptions in that paper. In order to identify the anticipated theoretical growth rates for various circumstances, one must first revisit the derivation and allow for the possibility that the density scale length differs, in magnitude and/or in sign, from the isothermal scale height. This analysis leads us to find regimes of purely convective instability and also of Vishniac stabilization of this instability, in addition to some new regimes of Vishniac behavior. We also identify a typographical error in the original paper that matters for quantitative evaluation of growth rates.


Introduction
Thin layers of matter are common in the universe. They are found wherever radiative losses of energy become large, for example, in old supernova remnants (Blondin et al. 1998) and in shocks as they emerge from supernovae (Ensman & Burrows 1992). They may perhaps also be found elsewhere, for example, in collisions of porous asteroids with dense objects. They are also found in laboratory systems that produce strongly radiating, shocked layers (Reighard et al. 2006) or blast waves in certain gases (Grun et al. 1991;Ditmire & Edens 2008). Since these layers are typically moving, they tend to accumulate mass and to decelerate. Vishniac (1983) recognized that there is an important imbalance between the internal pressure driving a thin layer and the ram pressure associated with accumulating matter. The first always acts along the normal to the surface of the thin layer, while the second acts in the direction of the incoming flow. In the presence of a modulation in the layer, this imbalance causes mass to accumulate in the lagging material. In response, the lagging material decelerates less rapidly than the leading material at lower density, causing the modulation to invert, after which the process repeats. Under various geometrical assumptions, this fundamental process can produce either power-law or exponential growth. Bertschinger (1986) explored the problem of a thin dense layer accumulating mass behind a radiative shock, solving for the internal profile of the layer. He confirmed the results of Vishniac (1983) and also showed that the layer must in fact be pressure-driven in order to have instability.
A few years later, Vishniac & Ryu (1989) examined the behavior of a thin layer with internal structure. In the following, we refer to Vishniac and Ryu as VR and to their seminal paper on instabilities in shocked, decelerating layers as VR89. In VR89, VR also examined the stability of self-similar, spherical shock waves. Our focus here will be on the evolution of thin, planar layers, which might represent a phase in the evolution of a small segment of a spherical system or the evolution of a laboratory system.
A number of simulation studies have explored the behavior of such layers. Strickland & Blondin (1995) found fluctuations growing in a cool dense layer produced by a radiative shock. Blondin et al. (1998) examined the transition from a Sedov-Taylor blast wave to a radiatively collapsed shock. Michaut et al. (2012) and Cavet et al. (2011) examined the long-term behavior of a similar system, using a distinct value of γ for the dense shell, to approximate the behavior of a cooling layer. Their work concluded, as did Mac Low & Norman (1993) for the blast-wave case, that the instability develops as anticipated by Vishniac, but then dies out in the long run as the shell thickens and the Mach number of the shock decreases, leaving behind only some internal structuring.
Some other related works are also worth mentioning. Badjin et al. (2016) have noted that care must be taken in simulations if one is to avoid the excitation of numerical instabilities that can mimic the physical instabilities of interest here. Vishniac (1994) considered shock-bounded slabs, later simulated by Blondin & Marks (1996). They found a similar instability, which has been called the Nonlinear Thin Shell Instability, but it is not of direct interest here. Robinson & Pasley (2018) have recently emphasized that ionization and other real-gas (i.e., non-polytropic) effects can enhance the onset of VR-type instabilities in systems that might otherwise be thought to be stable. This is consistent with our view that most of the relevant large-scale mechanics depend only on length scales which, for real gases, may not be well described by simple polytropic closures.
Of note here is that VR assumed the shocked layer to be isothermal. In the presence of deceleration, this leads to an exponential density profile that decreases behind the shock front. However, there are good reasons to consider other possibilities. There are many other possible profiles. Depending on the heat transport dynamics, the profile might be adiabatic. In general, the density in the profile may be determined by physics other than simple hydrodynamics and might not be described well by any value of polytropic index, γ. If the maximum density after radiative collapse is limited by the accumulation of magnetic pressure, as discussed in Blondin et al. (1998), then the density profile might either increase or decrease behind the shock, depending on the magnetic structure within the unshocked medium. In the early phases of laboratory radiative shocks, and likely also in shocks emerging from supernovae, the radiation transport physics causes the layer density to increase away from the shock, producing a slope having the opposite sign of that produced by the isothermal case. Other possibilities arise if the shock is accelerating, as can happen in a steep density gradient (Zeldovich & Razier 1966) or if the pressure driving the shock is caused to gradually increase. The qualitative argument above would suggest that modulations might grow unstably without oscillations in such a case. In addition, one can recognize this case as a thin-shell variant on the Rayleigh-Taylor instability. We have considered both accelerating and decelerating shocks in the presence of density profiles of arbitrary sign. We find unstable behavior that differs from that discussed in VR89, for cases whose assumptions differ from theirs, including some regimes of very rapid instability. We also find a typographical error in VR89 of interest to anyone needing to do a precise evaluation. We present our calculation and discuss these results here.
In a related but separate branch of inquiry, Ryu & Vishniac (1987) examined the stability of blast waves that begin with a self-similar, Sedov-Taylor structure. By assuming that the large-scale structure of the material is governed by polytropic gas laws, they expressed the parameters of both the VRI and the Sedov-Taylor blast wave on common terms. They found that the overstability can outrun the expansion of the blast wave and appear at large scales if γ is less than about 1.2, which corresponds to an 11-fold density jump. This type of closure for the large-scale structure is not of direct interest to us here, but did lead to a variety of subsequent works (Ryu & Vishniac 1991;Nishi 1992;Mac Low & Norman 1993;Nishi & Kamaya 2000;Kushnir et al. 2005;Sanz et al. 2016) elucidating the detailed behavior of such systems. In particular, Ryu & Vishniac (1991) described a case, for a blast wave whose largescale structure is determined by a polytropic index γ, in which a convectively unstable, post-blast-wave layer could be stabilized by oblique shock relations. We discuss this work further in Section 4.3. Here we proceed by retaining mechanical length scales, the relationships between which are not closed by assuming a polytropic γ. We recover some parallel results to behaviors found in a space defined by such an assumption, but can now elucidate both their origin and behavior over a larger space of possible parameters.
In the following, we first present the definition of the problem and discuss the quasi-steady, hydrostatic structure of the layer. We then summarize the derivation of the dispersion relation in Section 3. In Section 4, we explore the unstable behavior for a wide range of conditions-decelerating or accelerating layers having density gradients that point either outward or inward. In Section 5, we examine the behavior of the modulations of the various relevant variables. Section 6 then concludes the paper. In the appendices, we rework the derivation of VR89 for the more general case. We first develop the equations that describe the dynamic behavior in Appendix B, which is followed with a discussion of the boundary conditions in Appendix C. Here in particular is where one must closely attend to the difference between the interior density scale length, L, and the isothermal scale height, h, which involves the magnitude of the deceleration. We combine the results to date in Appendix D to obtain a dispersion relation that equals that found in VR89 in the correct limit. Figure 1 shows a schematic of the system being analyzed. A shocked layer, sustained by an internal pressure p i , slowly decelerates as it accumulates mass from the swept-up external material at initial density ρ E . The entire layer decelerates at a rate V ṡ , taken to be given. The deceleration is slow enough that the layer sustains a quasi-steady hydrostatic structure. Here when we refer to the shock front, we refer to the initial density jump (technically the shock itself) combined with any rapid, subsequent density increase, generally due to radiative cooling. We take the postshock pressure, in this sense, to be V E s 2 r , for shock speed V s . In detail, the pressure should be multiplied by (1 − ρ s /ρ E ), which is very close to 1 for the large density jumps of interest here.

Problem Definition and Hydrostatic Structure
Under the influence of the deceleration, the pressure decreases through the layer, of thickness H, so that in which the areal density of the layer is σ. It is worth highlighting the fact that the entire original argument of Vishniac rests on having p i be constant along the layer, even as the layer evolves. This requires that any ripples grow and propagate at subsonic speeds. This remains true for most but not all of the circumstances explored below.
The density structure of the layer depends upon its history and its properties. If the layer is isothermal, then the density profile is  where the isothermal sound speed is c p s r = .
Other profiles are possible for other assumptions regarding the properties of the dense layer, as discussed above. In the present paper, we will characterize the actual density profile by a scale length, L z 1 lnr = - ¶ ¶ ( ) (with the sign anticipating a density decrease deeper into the shocked layer). We will characterize the adiabatic sound speed c s g as the value corresponding to adiabatic fluctuations of small amplitude, so that p c s 2 r g ¶ ¶ = , with the partial derivative taken at constant entropy, with γ being the usual adiabatic or polytropic index. The isothermal limit of VR89 then corresponds to γ=1 and to L=h.
We refer to unit vectors in the direction of the deceleration and in a transverse direction (see Figure 1) as zˆand x, respectively. The hydrostatic velocity profile is as required to keep the mass flux through the shocked layer constant, so that

Structure and Results of the Derivation
The derivation, detailed in Appendices A-D, begins with the Euler equations for the mass density and the momentum density. It assumes small perturbations about the hydrostatic profile and that the fluctuations of the pressure are adiabatic and thus barotropic. This produces a set of differential equations that are linear in the magnitude of the perturbations, assumed to have amplitudes e i t kx µ w + ( ) . The general solution to these equations involves three coefficients, each associated with some spatial structure. Two of these follow from the assumption that for these, one can ignore terms involving u∂ z v z , in which v z is the first-order perturbation in the z-component of velocity. The third applies at any location, such as the shock front, where terms involving u∂ z v z become very large. In this way, one finds equations for the components of the spatial structure of the system. All of these turn out to involve the density scale length L; none of them involve h.
The boundary conditions then enable one to relate the coefficients and find a dispersion relation. The first boundary condition imposes continuity of velocity across the shock. The other two boundary conditions impose the correct behavior of the pressure, which must have a known value at the shock front and which must be continuous at the internal interface. The two pressure boundary conditions involve the isothermal scale height h, but not L.
The dispersion relation is most compactly written by normalizing ω to the acoustic frequency kc s so that ω n =ω/(kc s ). The resulting dispersion relation is , where f=2L/H. The only variable that changes sign with L is f. We are also set up to use H h HV c s s 2 =˙as a parameter, and note that it too can change sign. In Equation (7), one finds the variable Q, which arises during the spatial solution and is In VR89, the factor of 4 in the middle term in Equation (8) is missing. This typographical error does not affect the structure of the solutions but does affect their numerical evaluation. We now proceed to work with Equations (5) through (8) to analyze and evaluate the unstable behavior under various assumptions.

Analysis of the Dispersion Relation
It is useful to look at the roots of Equation (5). These can at times be simply interpreted, as Π is real whenever ω n is purely real or imaginary. The four roots are Figure 2 shows these, for Ω=1, which is the case in VR89. One sees that there is a purely stable zone for −0.25Ω 2 < Π<0, so that this zone shrinks or expands with Ω. In the limit of large κ=kL, small kH, and large kh such that Ω→1, one finds kHkh hH The minimum unstable wavelength also can be written as For Π>0, the unstable roots for ω n are purely imaginary or purely real, independent of Ω, and Π is self-consistently real. The root corresponding to exponential growth in this case is If Ω<0 and of sufficient magnitude, for example, when kh=1, then the growth rate becomes kh For moderate values of Ω (and positive h), it is Π that determines the behavior, so long as ω n is purely real or purely imaginary. We can analyze the sign-determining parts of the structure term, written as , where Q is positive definite, and the first term within the square brackets is positive and >1 and symmetric in Q/f. One has This term is not readily >1, since L tends to be not much smaller than H, and kH tends to be not much greater than 1, while n 2 w | | is often a significant fraction of 1. Correspondingly, the first term in the square brackets in Equation (13) is only slightly greater than 1. Setting this term equal to 1 in Equation (7) which establishes a lower limit on λ/H for instability. In addition, the above discussion lets us see that, when f becomes small while H/h and kH do not, the 1/f term dominates both the values of Ω and Π. In this case, there is a region where in which ω is real, and there is no instability. Returning to Equation (13), one can see that positive Π occurs when f−1 + H/h, corresponding to L<0 and , requiring h>H to have instability. This case is then unstable for all Q. As k or h go to infinity, one can see in Equation (7) that Π goes to zero, and the remaining solution is ω=±ω ac =±kc s , corresponding to sound waves being all that is left in the system.
Thus we see in Figure 2 that, when Π=0, there are only the acoustic solutions at ω n =1. When Π>0, one sees a purely real branch above ω n =1, which corresponds to an "augmented acoustic mode," and a purely imaginary branch, which corresponds to L<0. The negative Π cases correspond to "classical" Vishniac modes, with both the minimum unstable wavelength feature and the compound real and imaginary parts when it is unstable.
It is straightforward, using a computational mathematics program, to find and display the largest unstable root of Equation (5), working with whichever variables may be most convenient. (In most regimes there is only one unstable root, but at times there are two.) As usual, care must be taken to find the correct root across the domain of interest. We proceed to consider several cases. Because the values and ratio of L and h have substantial effects on the resulting solutions, we consider variations in these parameters across some orders of magnitude. The index γ only varies from 1 to 5/3. The only case we identified in which its effects are significant is that in VR89, in which one constrains L to equal h. We discuss why below. In most cases below, we show frequencies and growth rates only for γ=5/3. The reader can readily obtain solutions for other values of γ, if needed.

A Decelerating Layer with Positive L
We first consider the case VR89, in which L=h and γ=1. Using Equation (7), we found the unstable root and displayed it as follows. We show L/H=f/2 on the abscissa and λ/H=2π/(kH) on the ordinate. To give the growth rate a common scale across the plot, we plot the ratio of w -( ) I to the acoustic frequency for λ=H, 2πc s /H. This produces Figures 3(a) and 4, showing growth rate and real frequency, respectively. One sees that, across the unstable region, the absolute growth rate peaks for small λ/H and for LH, but remains significant over a large region. Figure 3(b) shows the growth rate for L=h and γ=5/3, although this is not selfconsistent. Here the threshold on the left, at lower values of L/H, is seen to have increased in comparison to the γ=1 case. We show why this happens below. In addition, one sees the effect discussed in the introduction, that for fixed λ, an increase of H over time will eventually stabilize the modulations. One sees in Figure 4 that the real frequency is near kc s across the unstable region, increasing above kc s as λ/H increases.
Turning to the effect of having different L and h, but still with L>0, we show the growth rate for various parameters in ] traces out the stability boundary in detail; it also shows that this quantity remains below 1 throughout the rest of the plot. As h/H varies, the unstable region displayed in this plane moves with the dashed curve, primarily by moderate up-and-down motion of the lower boundary. Here again the normalization by kc s obscures the fact that the largest absolute growth rates are in the middle of the plot, for λ not far above the threshold value. Figure 5(b) shows the effect of variations in h. One sees that, for fixed λ/H, the growth rate depends much more strongly on the rate of deceleration (through h/H) than it does on the density scale length (through L/H). The upper boundary of the unstable region is as expected based on Equation (15). By contrast, as one approaches the lower boundary of Equation (15), the unstable roots move onto the imaginary axis so that ω n becomes purely imaginary and quite large. Physically, the smallness of h/H, corresponding to large decelerations, implies that the dense part of the layer becomes quite thin, so increased instability is not a surprise. Here the modulations grow without propagating and have the potential to strongly affect the structure of the layer. However, once the modulation amplitude becomes large enough, its growth becomes supersonic and so would drive shocks into the interior. We did not allow for this case in our analysis above; it seems likely that the actual behavior would weaken the growth from that point forward.
We can also best explain the sensitivity to γ seen in Figure 3 by referring to Figure 5(b). One can see that, at small L/H, the  growth rate contours for low growth rates become nearly parallel to the thin dashed curve showing L=h. Small changes to the overall contours cause the intersection of the threshold with the L=h curve to move a comparatively long distance. What happens mathematically is that the increase in γ causes the value of Ω to increase above 1, which turns out to imply that Π must become more negative to produce instability. In no other case discussed in the present paper does the variation in γ cause significant changes to the contour plots. Results for any specific value of γ are straightforward to obtain.
While these results do not directly apply to spherical expansions, they should not be too far off for the behavior of comparatively short wavelengths within a small-enough segment of a spherical shell. RV87 considered power-law expansions having radius R∝t α , in both planar and spherical geometry. For such expansions, one can show that both H and h∝t α , so the highest unstable mode number does not change. If one approximates H as the thickness of a layer that can hold all of the swept-up mass at constant density, then one finds h∼3H. In addition, one can see by examining typical cases of such adiabatic blast waves that LH. It would appear from Figure 5 that such systems will access a comparatively small region in parameter space, near these values. The growth is relatively weak there, and can be stabilized entirely if L/H is small enough, at the actual value of h/H. Such adiabatic systems seem unlikely to be able to access the strongly unstable zone seen in Figure 5(b) for h<H.
In addition, when ω n becomes purely imaginary and quite large, the assumption Q Q tanh f f [ ] becomes invalid and h/H must decrease further to reach Π=1/4, as can be seen in Figure 5(c). In this regime, Ω=0 and one reaches the condition identified above where i kh 2 n w  -( ). Aside from the extreme behavior we find here for unusually small h, the primary effect of separating L and h in the analysis is to open up the possibility for instability for a much larger range of L, and higher L/λ, than one finds in the VR case shown in Figure 3.

A Decelerating Layer with Negative L
Under some circumstances, a decelerating layer may contain a density profile in which the density increases from the shock location toward the boundary with the interior matter. One example is that of the early evolution of a radiating, shocked layer that can become optically thick, occurring in shock breakout from supernovae and in laboratory radiative-shock experiments. Another example is that of a collapsing shock in an old supernova remnant that is traversing a region of increasing average magnetic field.
Because the shell is denser than the matter interior to it, the inner surface of the shell remains Rayleigh-Taylor stable in this case. Within the shell, in contrast, the pressure gradient is parallel to the entropy gradient and opposed to the density gradient, so that internal gravity waves are expected to be convectively unstable. The characteristic frequency of these oscillations, within thick layers such as Earth's atmosphere, is the Brunt-Väisälä frequency (Väisälä 1925;Brunt 1927), which, for acceleration g, is g ln , which here is 18 implying that the modes are purely unstable when h>0 and L<0. In the case of interest in the present section, the mechanisms of convective instability and those of the Vishniac instability are both at play. By studying the roots of Equation (5), we explored this regime. Figure 6 shows the growth rate in (a) and real frequency in (b) for this case with h=10H. Near the upper boundary on the top plot and toward the left, one has Ω=0 and finds large growth, while the real frequency has become zero. The growth rate increases toward the upper-left corner and is ( ), and we recover the standard convective growth rate, but note that what is growing here are ripples of the entire layer combined with mass clumping. The small value of L localizes most of the matter near the inner boundary, creating a layer that is effectively thinner than H. Here, the modulations grow without propagating and have the potential to strongly affect the structure of the layer. However, as kL becomes =1, the solutions may change because the terms we dropped involving u/L start to become significant and may alter the behavior. (This caution also applies below.) In addition, once the modulation amplitude becomes large enough, its growth becomes supersonic and so would drive shocks into the interior. We did not allow for this case in our analysis above; it seems likely that the actual behavior would weaken the growth from that point forward.
However, as L | | increases, which might be a consequence of convective instability, and for λ/H∼10 to 100, the frequency of the modulations soon develops a real component, and the waves begin to propagate. The dashed curve in Figure 6(b) shows where, at some value of L | |, the parameter Ω becomes zero so that the dispersion is now n 4 w = P, which always has a negative imaginary (i.e., unstable) root. This is a new regime of the Vishniac instability, apparently unidentified previously, in which one always finds instability. This regime with h>0 and L<0 features propagating modes having a real frequency typically near kc s and having a growth rate near 0.1kc s , occurring at a wavelength near h h L 2 1 l p = + | |. Figure 6(a) shows that the modes are stabilized as the wavelength decreases. The dashed curve again shows where Π=−1/4 from Equation (16). As h/H varies, the unstable region displayed in the plane of Figure 6(a) moves with the dashed curve, primarily by moderate up-and-down motion of the lower boundary. Here again, within the region where Π<−1/4 of Equation (16), the growth rate remains insensitive to L H | | at large values and the waves have a real frequency so they propagate. The growth rate decreases as λ/H does, reflecting a decrease in P | | to small values. Along its lower boundary, Ω is near 1 and the modulations become stable near the dashed curve. We identify this as Vishniac stabilization of the convective instability, in which the thin-shell modulations prevent the development of unstable convection. The Vishniac dynamics appear to be smoothing out short-wavelength density maxima faster than they can grow. Ryu & Vishniac (1991) described a related case, where a convectively unstable post-blast-wave layer could be stabilized by the oblique shock relations. They defined the large-scale structure of the adiabatic blast wave using a polytropic γ and described their results in terms of conditions on γ. Our results provide a much clearer interpretation of the stabilization they find. The appearance of the stabilized region of parameter space here reveals that this stabilization is completely disconnected from the usual Vishniac stabilization for blast waves of 1.2  g , as the latter depends on the large-scale, Sedov-Taylor dynamics which are not present here. Furthermore, by assuming neither blast-wave dynamics nor polytropic gas closures, we open this stabilization mechanism to more complex possibilities, including non-adiabatic mechanisms, which could fix the length scales relative to one another. Radiative transport in the presence of opacity or material composition gradients is just one possibility that could align L and h such that either the layer is Vishniac-stabilized or the region of propagating modes near Ω=1 is realized.
In this case we have found instability for all values of L and λ above a threshold. There is an extremely unstable case when L/H becomes small, which we identify as standard (Brunt-Väisälä) convection. There is a regime with Ω∼0 in which one always finds propagating and growing waves, and there is what we describe as Vishniac stabilization of the convective mode at short wavelengths. This range of effects may apply to the early phases of radiatively collapsing shocks, depending upon the details and changing as they evolve.

Accelerating Layers
As discussed in the introduction, there also are cases in which a thin, shocked layer might accelerate, notably including shock breakout from supernovae. This reverses the hydrostatic pressure gradient, so that the inner surface of the dense layer becomes Rayleigh-Taylor unstable. Here our separation of the density scale length and the isothermal scale height enables us to examine the combination of Rayleigh-Taylor dynamics with the compressible dynamics of the thin shell. To our knowledge this has not been looked at previously. It is not surprising, though, to find purely growing modulations in this case. These have the potential to disrupt the shell, and likely more so than pure Rayleigh-Taylor modulations without mass clumping.
In terms of the mathematics, this case corresponds to h<0. Examining Equation (7), one can see that Π>0 in this case, unless f=2L/H is very small and negative. Here the instability will weaken to unimportance wherever Π becomes small. As one would expect from the basic physics and also from Figure 2, the unstable roots of Equation (9) lie on the imaginary axis throughout the entire regime of interest, having no real part. Figure 7 shows the growth rate for both signs of L.
Here there are no stability thresholds as such, but Π becomes quite small, and Ω does not become large and negative, near the lower axis in both images and near the left axis in Figure 7(b). . In addition, since the absolute growth rates found here are of order kc s , one finds that the ratio of the thin-shell growth rate to the Rayleigh-Taylor growth rate at an embedded interface, k V s |˙| , is kh , becoming large for weak decelerations when h?λ. Overall, for the case of accelerating layers, we find instability for a very large range of conditions, with a growth rate within a factor of a few kc s .

Structure within the Layer
It may be helpful to display some aspects of the structure corresponding to these conditions. Figure 8 shows the relevant amplitudes for λ/H=2π/(kH)=10, L=10H, and h=10H. Shown in the figure is δ/B − , v x /(c s B − ), v z /(c s B − ), and Δz/(HB − ), as indicated. Since B − is orders of magnitude smaller than 1, the physical amplitudes shown in the plot are all small. Panel (a) shows the modulations at the shock front, and panel (b) shows them at the interior interface.
Attending first to Δz (the gray curve) and v z (the shortdashed curve), one can see that these variables are out of phase, as they should be. One can also see, by comparing the two parts of the figure, that the layer is thickening and thinning across the wave cycle. Comparing Δz and the density increase δ, one sees that these are close to but not exactly in phase at the interior boundary and nearly π radians out of phase at the shock. The regions of largest density do occur where the layer is thickest. Examining the long-dashed curve showing v x , one can see that the mass flow converges and then diverges near alternate zeros of Δz and δ. Considering the sense of these variations, it becomes clear that the modulations are propagating to the left, as is consistent with our imposed structure.

Conclusion
Herein, we have revisited the classic derivation of the instability of dense, shocked layers supported from within, which was first provided by Vishniac & Ryu (1989). Our aim has been to separate the role of the density scale length from that of the isothermal scale height, for the case of thin planar layers. We find that the density gradient enters into the spatial structure determined by the basic differential equations, while the isothermal scale height appears in the boundary conditions. This opens up the possibility of having distinct behavior depending on the sign of the density gradient and on whether the layer is accelerating or decelerating. All four kinds of structure may occur, both in nature and in laboratory astrophysics experiments. We have explored these behaviors for relevant ranges of density scale length, isothermal scale height, and ratio of unstable wavelength to layer thickness. We find that the unstable growth rate is often insensitive to the density scale length over large ranges, but is almost always sensitive to the isothermal scale height (and thus to the rate of deceleration of the shocked layer). The growth rate itself is typically within an order of magnitude of the acoustic frequency.
The unstable modulations most often propagate along the layer and involve both concentrations of matter and thickening of the layer. But there are some parameter regimes in which there is no propagation and the modulations grow in place. The combination of pure growth and mass clumping has the potential to disrupt the shell. For decelerating shocks, these purely growing modes occur under two circumstances. They appear when the isothermal scale height becomes smaller than the layer thickness and the density profile decreases away from the shock, or when the density scale length becomes smaller than the layer thickness. Purely growing modes also appear for all circumstances when the shocked layer is accelerating, in which case one has a compressible, thin-shell variant of the Rayleigh-Taylor instability.
The results found here may be useful based on the observation of long-wavelength structures. Any thin, decelerating shells that do not develop clumps may lie in the stable regime seen in Figure 6 above. The observation of large, stationary modulations or of bullet formation from decelerating thin shells would be indicative of the convective regime discussed in Section 4.3, while if the shell is accelerating, one would expect the thin-shell Rayleigh-Taylor mode, in which mass clumping by the Vishniac mechanism could accentuate the breakup of the shell. By contrast, propagating structures that move laterally along the shock surface might be described as Vishniac clumps.
The results and formulae herein should also be useful in assessing the potential importance and actual growth rate of the Vishniac-Ryu instability in a variety of experimental and computational applications.
We close with a speculation about one potential implication of these results. As the shock moves outward through a corecollapse supernova, each density transition produces a dense shell between a reverse shock and a composition interface (Guzman & Plewa 2009). This shell, or its outer surface, may develop structure from the Rayleigh-Taylor instability as well. During this phase the star remains optically very thick and the Here the solid black curve shows δ, the thick gray curve shows Δz/H, the shortdashed curve shows v z /c s , and the long-dashed curve shows v x /c s . pressure in the shocked matter is almost entirely due to radiation. Later, as the density of a given shell becomes small enough, the radiation it contains will escape to the surface. In response, one would expect the shell to become much thinner because the pressure driving it will be sustained from within the star. Calculating the detailed behavior during this phase is complex; it appears not to have received much attention in simulations. Depending on the specifics, such dense shells could become violently unstable via the mechanisms described here. This would have the potential to be the missing link that could explain the degree to which remnants like Cassiopeia A (Hughes et al. 2000;Fesen 2001) which is quite small typically, as u is quantitatively small, despite formally being a "zeroth-order" variable. r r g d ¶ + ¶ + ¶ = - ¶ r ( ) Defining δ=δ ρ /ρ and seeking modulations proportional to e i t kx w + ( ) , as do VR, we obtain, also using Equation (23) (22) and (23), we see that the two terms involving u/L are no larger than second order in the perturbed quantities (even as kL drops below 1). We drop them going forward. These equations fundamentally are third order, leading one to expect a solution involving three independent coefficients, whose relationship must satisfy the boundary conditions. VR first find two approximate solutions to these equations, by also taking all terms containing u∂ z to be small. Because u is very small, these terms are effectively second order except where . Sketch to aid thinking about boundary conditions. This is not to scale since in reality kH<1. Note that the definition of the modulations implies that this propagates in the x -ˆdirection.