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Richtmyer–Meshkov-like instabilities and early-time perturbation growth
in laser targets and Z-pinch loads*
A. L. Velikovich,†J. P. Dahlburg, and A. J. Schmitt
Plasma Physics Division, Naval Research Laboratory, Washington, D.C. 20375
J. H. Gardner and L. Phillips
Laboratory for Computational Physics and Fluid Dynamics, Naval Research Laboratory,
Washington, D.C. 20375
F. L. Cochrana) and Y. K. Chong
Berkeley Research Associates, Incorporated, Springfield, Virginia 22150
G. Dimonte
Lawrence Livermore National Laboratory, Livermore, California 94551
N. Metzler
Physics Department, Nuclear Research Center Negev, P. O. Box 9001, Beer Sheva, Israel, and Science
Applications International Corporation, McLean, Virginia 22150
共Received 18 November 1999; accepted 19 January 2000兲
The classical Richtmyer–Meshkov 共RM兲instability develops when a planar shock wave interacts
with a corrugated interface between two different fluids. A larger family of so-called RM-like
hydrodynamic interfacial instabilities is discussed. All of these feature a perturbation growth at an
interface, which is driven mainly by vorticity, either initially deposited at the interface or supplied
by external sources. The inertial confinement fusion relevant physical conditions that give rise to the
RM-like instabilities range from the early-time phase of conventional ablative laser acceleration to
collisions of plasma shells 共like components of nested-wire-arrays, double-gas-puff Z-pinch loads,
supernovae ejecta and interstellar gas兲. In the laser ablation case, numerous additional factors are
involved: the mass flow through the front, thermal conduction in the corona, and an external
perturbation drive 共laser imprint兲, which leads to a full stabilization of perturbation growth. In
contrast with the classical RM case, mass perturbations can exhibit decaying oscillations rather than
a linear growth. It is shown how the early-time perturbation behavior could be controlled by
tailoring the density profile of a laser target or a Z-pinch load, to diminish the total mass perturbation
seed for the Rayleigh–Taylor instability development. © 2000 American Institute of Physics.
关S1070-664X共00兲93305-6兴
I. INTRODUCTION
The classical Richtmyer–Meshkov 共RM兲instability1,2
develops when a planar shock wave interacts with a corru-
gated interface between two different fluids. After 45 yr of
studies that followed Richtmyer’s theoretical discovery,1the
main features of this instability at the linear and weakly non-
linear regimes are firmly established, and the experimental
data, numerical simulation results and theoretical predictions
are in good agreement 共e. g., see Refs. 3–5 and references
therein兲.
In this paper, we discuss a wider class of instabilities
which, not being caused by the shock-interface refraction,
are still driven by exactly the same physical mechanisms as
the classical RM instability. This class includes 共but is not
limited to兲interfacial instabilities caused by the ‘‘virtual
gravity’’;6,7 instabilities produced in collisions of perturbed
fluid layers8–10 and/or of perturbed shock waves with each
other or material interfaces;11 instabilities excited in simulat-
ing the evolution of an initial discontinuity in a fluid 共the
perturbed Riemann problem12,13兲; instabilities of shock-
piston flows, such as those produced when a shock wave is
driven from a rippled surface by a uniform laser beam, or
when a nonuniform laser beam irradiating a planar surface
imprints mass perturbations into the flow.14–22 In the litera-
ture, these instabilities are sometimes referred to simply as
RM,6–10 and sometimes identified as new kinds of
instability.11 For lack of a better term, all these instabilities
will be called RM-like. Our discussion is mostly limited to
the small-amplitude, linear phase of the instability develop-
ment.
One of the most interesting features of the flows that
exhibit RM-like instabilities 关in contrast, say, with the
Rayleigh–Taylor 共RT兲unstable flows兴is their proximity to
the boundary separating stable and unstable situations. Con-
sequently, perturbation evolution is sensitive to relatively
small changes in the formulation of the problem, physical
mechanisms accounted for or ignored, initial conditions, etc.,
which can turn a stable situation into unstable, and vice
*Paper GT1 2 Bull. Am. Phys. Soc. 44, 122 共1999兲.
†Tutorial speaker.
a兲Present address: Los Alamos National Laboratory, Los Alamos, New
Mexico 87545.
PHYSICS OF PLASMAS VOLUME 7, NUMBER 5 MAY 2000
16621070-664X/2000/7(5)/1662/10/$17.00 © 2000 American Institute of Physics
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versa. This is one of the reasons that the studies of such
instabilities are an exciting playground for perfecting simu-
lation techniques. Even more important, however, are the
opportunities for laser target and Z-pinch load design that
such studies might present. Suppression of the RM-like per-
turbation growth at the early stages of laser–target accelera-
tion or a Z-pinch implosion could make a big difference in
the uniformity of imploded or stagnated plasmas.
This paper is structured as follows. In Sec. II we review
the basic features of the classical RM instability that are
relevant for further discussion, and in Sec. III present some
examples illustrating the RM-like instabilities that are dis-
tinct from the classical case of shock–interface interaction.
Section IV reviews the RM-like instabilities in ablatively
driven shock-piston flows. Section V shows how the RM-
like growth could be suppressed using tailored density pro-
files in both Z-pinch loads and laser targets. In Sec. VI, we
conclude with a discussion.
II. THE CLASSICAL RM INSTABILITY
Here we briefly review the basic features of the classical
RM instability in the small-amplitude regime. Let a planar
shock wave be normally incident from fluid 1 upon a contact
interface separating it from a different fluid, 2. The interface
is slightly corrugated. In the small-amplitude theory we deal
with a single-mode sine corrugation ⬀
␦
x0exp(iky), where
␦
x0is the preshock corrugation amplitude, and k⫽2
/is
the perturbation wave number. After the interaction, the in-
terface acquires a constant velocity Uin the direction of the
incident shock wave, a perturbed shock wave is transmitted
into fluid 2, and either a perturbed shock wave or a perturbed
centered rarefaction wave is reflected back into fluid 1.
Typical behavior of perturbation amplitudes and growth
rates is illustrated by Fig. 1共b兲. The fluid parameters in the
figure correspond to the ideal gas model23 developed to
simulate interaction of strong radiatively driven shock waves
with a contact interface between a solid beryllium ablator
and a foam tamper in the RM experiments on Nova laser at
Lawrence Livermore National Laboratory 共LLNL兲.24,3 Fig-
ure 1 is plotted for a strong incident shock wave, Mach num-
ber M0⫽10.8.
The displacement amplitude of the contact interface
␦
x(t) grows with time. Note that for the whole class of the
RM-like instabilities, the growth rate, which we denote by
⌫C(t)⫽(d/dt)
␦
x, has a dimensionality of velocity 共cm/s兲in
contrast with the case of RT and most of the other fluid
instabilities, which are characterized by exponential growth
rates expressed in inverse seconds. This is because the RM-
like instabilities exhibit at most linear with time, secular,
rather than exponential perturbation growth. When an insta-
bility of this class develops at a contact interface, then the
growth rate defined above is the same as the perturbation
amplitude of the axial velocity at the interface, ⌫C(t)
⫽
␦
vx(x⫽xC,t)共this is not necessarily the case, e.g., the
local velocity perturbation amplitude at the ablation front
does not coincide with the growth rate兲. The growth rate ⌫C,
after some damped oscillations at late time kUtⰇ1, ap-
proaches a constant asymptotic value, ⌫⬁; in other words
linear late-time perturbation growth. The solid line labeled S
shows the displacement amplitude at the shock front
␦
xs;
the dotted line shows relative pressure perturbation
␦
ps/p.
The transmitted shock wave is seen to be superstable, which
means that its perturbations actually decrease in time as
⬀t⫺3/2, except when the 共transmitted兲shock wave is very
strong, and they decay as t⫺1/2, see Refs. 25. If a shock wave
is reflected, its perturbations behave similarly.
The reflected centered rarefaction wave is unstable.26,27
The ripple amplitude at its leading edge remains constant in
time, whereas perturbations of its trailing edge grow linearly
in time. For the case shown in Fig. 1, the corresponding
asymptotic growth rate is about four times higher than the
interface growth rate ⌫⬁and has a different sign.
The physical factors that drive the classical RM instabil-
ity are illustrated in Fig. 1共a兲. Instant deposition of localized
vorticity at the interface during the shock refraction makes
the fluid move along the contact interface, decreasing the
pressure at the convex side of a bubble 共darker side兲, which
then makes the bubble grow. The pressure perturbations
coming from the stable shock front共s兲affect the growth, typi-
cally slowing it down 共even a full cancellation is possible,
leading to zero asymptotic growth rate, so called
freezeout12,28兲. The asymptotic growth rate is expressed via
an integral of all these pressure perturbations
␦
p(x,t), com-
ing to the contact interface C:
FIG. 1. 共a兲The flow structure at the density interface supports the RM
instability growth; vortical flow along the contact interface Ccreates a pres-
sure gradient directed from the convex side of the bubble, and sonic pressure
perturbations sare exchanged between the interface and the shock front S.
共b兲Linear RM growth for adiabatic exponents of fluids 1 and 2
␥
1
⫽1.8,
␥
2⫽1.45, respectively, and preshock Atwood number ⫺0.868. Time
histories of normalized growth rates at the interface (⌫C) and at the trailing
edge of the reflected rarefaction wave (⌫R/4) , shock displacement and shock
pressure perturbation amplitudes 共solid and dotted lines S, respectively兲;the
dotted line MB is the asymptotic interfacial growth rate predicted by the MB
prescription 共Ref. 30兲. The perturbation amplitudes and growth rates are
shown in units of k
␦
x0and kU
␦
x0, respectively.
1663Phys. Plasmas, Vol. 7, No. 5, May 2000 Richtmyer-Meshkov-like instabilities and early-time...
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⌫⬁⫽⫺ 1
*
冋
x
冕
0
⬁
␦
p共x,t兲dt
册
x⫽xC
,共1兲
where
*is the constant postshock density at the either side
of the interface x⫽xC.
The two distinct contributions to the growth rate are
separated in the following exact formula for ⌫⬁:
⌫⬁⫽
1
*
␦
vy1
*⫺
2
*
␦
vy2
*
1
*⫹
2
*⫺
1
*Fs1⫺
2
*Fs2
1
*⫹
2
*⬅⌫0⫹⌬⌫s,
共2兲
derived by Wouchuk and Nishihara 共WN兲.13 Here
1,2
*,
␦
vy1,2
*
are densities and transverse velocities immediately after the
shock–interface interaction, and Fs1,2 are parameters ac-
counting for the contribution of perturbations to the
asymptotic growth rate from reflected and transmitted shock
waves 共in the case of a reflected rarefaction wave, Fs1⬅0).
The first contribution ⌫0comes from the localized vorticity
deposited at the contact interface immediately after the shock
refraction, and has an explicit analytical formula. The second
contribution ⌬⌫swhich, as mentioned above, typically has a
different sign than ⌫0, comes from the interaction of the
perturbed interface with one or two shock waves propagating
from it. Formulas for Fs1,2 are not yet available in a closed
form; the procedure of their approximate evaluation de-
scribed in Ref. 13 yields the same asymptotic growth rates as
the other methods.12,26,29 Interaction with the unstable re-
flected rarefaction wave does not contribute to ⌫⬁. The
value of ⌬⌫sin the WN formula 共2兲could be determined via
the time histories of the pressure perturbations at the shock
front.
The RM instability is often compared to 共and sometimes
is believed to be the same as兲the particular case of RT in-
stability, when the gravity acceleration, rather than being
constant 共which corresponds to the classical exponential
growth兲or slowly varying, is impulsive: g(t)⫽U
␦
(t), simu-
lating instantaneous shock acceleration. This approach yields
the well-known Richtmyer’s estimate for the asymptotic
growth rate
⌫⬁⫽
冕
⫺0
⫹0dtkg共t兲A共t兲
␦
xC共t兲⫽kUAeff
␦
xeff ,共3兲
where the instant t⫽0 corresponds to the shock–interface
interaction, and Aeff and
␦
xeff are some effective values of
the Atwood number and interfacial perturbation amplitude
that both change discontinuously when the shock wave
passes the interface. To actually use Eq. 共3兲for estimates, it
has to be replaced by some prescription. The available op-
tions
⌫⬁⫽
再
kUA*
␦
xC
*共R兲;
kUA*⫻1
2共
␦
xC
*⫹
␦
x0兲共MB兲;
kU⫻1
2共A*
␦
xC
*⫹A
␦
x0兲共VMG兲;
共4兲
have been suggested, respectively, by Richtmyer 共R兲,1Meyer
and Blewett 共MB兲,30 and Vandenboomgaerde et al.
共VMG兲.31 The prescriptions are not exact expressions for
⌫⬁, but rather heuristic formulas approximating it. In the
weak-shock limit, all the prescriptions yield the same result,
which is consistent with compressible theory. The R pre-
scription sometimes works better for the reflected shock
case, MB—for the reflected rarefaction case, and VMG—for
both cases, if the shock wave is not too strong. 共None of the
prescriptions are reliable for strong shocks.兲For the particu-
lar case of a strong shock in Fig. 1, the MB prescription
happens to be a very good approximation of the asymptotic
growth rate, while R and VMG predictions are not 共0.11 and
⫺0.38, respectively, normalized as in Fig. 1兲. For the ex-
ample of air/He shock–interface interaction at M0⫽10, none
of the prescriptions are found to work: R, MB and VMG
normalized growth rates are 0.11, ⫺0.28, and ⫺0.32, respec-
tively, whereas the actual asymptotic growth rate equals
⫺0.16.
III. SOME EXAMPLES OF RM-LIKE INSTABILITIES
Comparing 共2兲and 共3兲, we see that the former formula
describes a wider class of instabilities driven by the same
physical mechanisms. To drive this kind of instability, we
have to deposit initial localized vorticity at the contact inter-
face and/or provide sonic interaction of the interface with
one or two shock waves propagating from it. This could be
accomplished not only through classical shock–interface in-
teraction, but also in many other ways. This is how we come
to the generalization of RM-like instabilities. They are de-
fined here as the interfacial instabilities driven by the same
physical factors as the classical RM, but differing from it by
the source of the unstable configuration, and sometimes as
well by some other factors affecting the perturbation evolu-
tion.A very good example is the instability of an impulsively
accelerated fluid interface, observed using the novel experi-
mental techniques that allow the experimentalists to produce
controlled time histories of acceleration acting upon the ob-
served interface. The linear electric motor 共LEM兲built at
LLNL6provides variable electromagnetic acceleration of test
vessels. The free-falling tank technique7makes the test ves-
sel fall and then bounce off a fixed spring, producing a sharp
acceleration pulse. Both techniques allow the possibility of
incompressible fluids and the production of a RM-like insta-
bility development in the absence of any shocks. For the case
of a small-amplitude, single-mode initial perturbation, Richt-
myer’s formula 共3兲is an exact result rather than an approxi-
mation; for incompressible fluids, it is well defined, since
neither Anor
␦
xCchange during the acceleration pulse, and
no prescriptions are necessary. This exact linear asymptotic
growth rate coincides with the contribution ⌫0of instantly or
rapidly deposited interface vorticity, as defined in 共2兲. This
case could be regarded as a ‘‘more-than-classical’’ regime of
the RM-like growth, which literally corresponds to the Rich-
tmyer’s model of impulsive acceleration. Figure 2 presents
an example of experimental results obtained on LEM6for
such an acceleration regime. Here, the RM-like growth ex-
cited by a short acceleration pulse is undoubtedly a particular
case of 共and, understandably, slower than兲the RT growth
supported by continuous acceleration.
1664 Phys. Plasmas, Vol. 7, No. 5, May 2000 Velikovich
et al.
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Now consider a completely different example, illustrated
in Fig. 3. In the initial state, two identical uniform gases
(
␥
⫽5/3), separated by a sine-shaped contact interface, have
identical constant mass velocities, Uand⫺Uat t⫽0, both
directed from the interface. This ‘‘anticollision’’ produces
two centered rarefaction waves of identical strength that
propagate from the interface.12 If Udoes not exceed the ve-
locity of free gas expansion into vacuum 共as in the case of
Fig. 3兲, then the perturbed contact interface continues to exist
at t⬎⫹0, separating the gas layers left behind the trailing
edges of the rarefaction waves. There is no impulsive accel-
eration here: the xvelocity of the interface is exactly zero
both at t⫽⫺0 and at t⫽⫹0. The displacement amplitude
␦
x0does not change in this interval. Similarly, there is no
uncertainty in the Atwood number, it remains exactly zero
throughout the interaction. This situation therefore has no
resemblance to the shock-excited or impulsively driven RT
instability, and formula 共3兲, as well as any of the prescrip-
tions 共4兲, predicts zero perturbation growth rate. However,
due to our discontinuous initial conditions, a known amount
of localized vorticity is deposited at the interface at t⫽⫹0.
According to 共2兲, this is enough to drive a RM-like pertur-
bation growth. Moreover, since there are no shock waves for
the contact interface to interact with, the prediction of the
WN formula 共2兲is available in a closed form, ⌫⬁⫽⌫0
⫽⫺kU
␦
x0. This result13 is shown in Fig. 3 to be in full
agreement with the linear theory;12 so far, it is the only
known example when the asymptotic growth rate of a RM-
like instability in a compressible fluid is exactly expressed by
a closed analytical formula.
Let the same problem be modified for initial conditions
corresponding to a collision rather than anticollision 共Fig. 4,
inset兲, with two identical shock waves propagating from the
contact interface after interaction. For the same reasons as in
the previous example, this case has nothing to do with the
impulsively driven RT instability, and formulas 共3兲,共4兲pre-
dict zero growth rate for it. In this case, however, the shock-
interface contribution to the growth rate ⌬⌫sis nonzero. To
verify the prediction of the linear theory for this case, we did
a numerical simulation of this problem with the FAST2D hy-
drocode developed at the Naval Research Laboratory
共NRL兲.32 Figure 4 demonstrates good agreement between the
perturbation growth predicted by the linear theory and the
simulation results. This is also an unmistakably RM-like type
of perturbation growth.
The interface does not even have to exist before the in-
teraction. Remove any half from the initial configurations of
Figs. 3, 4, and impose instead a boundary condition requiring
constant pressure at the perturbed surface 共no longer an in-
terface兲. None of the results of Figs. 3, 4 would change. We
see that any shock-piston flow driven by a pressure applied to
a free surface would excite a RM-like instability, no matter
whether it is the surface that is initially perturbed, or whether
the pressure driving it is slightly nonuniform. Alternatively,
the contact interface could be created when two uniform lay-
FIG. 2. 共a兲The acceleration profiles produced in four linear electric motor
runs, varying from approximately constant, to increasing, to decreasing, to
impulsive. 共b兲Time evolution of the bubble amplitudes for these four cases.
Most of the observed growth is in the nonlinear stage. The RM-like growth
corresponds to impulsive acceleration 共⽧兲.
FIG. 3. Perturbation growth at a contact interface for a symmetrical Rie-
mann problem producing two outgoing rarefaction waves, and U⫽0.6
•a0,wherea0is the speed of sound before interaction. Units are the same as
in Fig. 1. 共Inset兲scheme of the initial state.
FIG. 4. Same as in Fig. 3 for the case of two outgoing shock waves,
␥
⫽6/5 and U⫽2.9•a0.共Inset兲scheme of the initial state.
1665Phys. Plasmas, Vol. 7, No. 5, May 2000 Richtmyer-Meshkov-like instabilities and early-time...
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ers of different fluids collide head-on, depositing initial vor-
ticity at the interface separating them, and/or producing
rippled shock waves. The existing methods of calculating
linear time histories of perturbation growth and asymptotic
growth rates,1,12,13,26 including Eq. 共2兲, are directly appli-
cable to all these cases, provided that the fluid共s兲ahead of the
shock and or rarefaction wave共s兲propagating from the inter-
face or surface, are uniform and unperturbed.
If the latter condition is not met, it does not change the
nature of the RM-like growth. However, in this case, no
linear theory is yet available to describe it. The simplest
example is the reshock of the contact interface by the trans-
mitted shock wave that is reflected from the end of a shock
tube 共or a computational box兲, and is incident back upon the
interface, carrying the perturbations that are coherent with
those already in the flow. Another example is interaction of a
rippled shock wave with a contact interface.11 In both cases,
the factors that drive the perturbation growth remain the
same as shown in Fig. 1共a兲. Calculation of the time histories
and asymptotic growth rates, where applicable, is more com-
plicated because it has to account for the perturbations enter-
ing the flow via either perturbed shock wave or perturbed
rarefaction wave. This has not been done yet. Probably the
same computational difficulty has so far prevented a theoret-
ical analysis of the classical RM instability in magnetohydro-
dynamics 共MHD兲, which for some cases has been studied
numerically in Ref. 33. Here, after interaction, three waves
共slow, Alfve
´n and fast兲propagate in each direction, and one
has to describe how the perturbations reaching the interface
interact with all of these. This case, when appropriately stud-
ied, would present abundant opportunities for benchmarking
multidimensional MHD hydrocodes: there are 189 non-
equivalent cases of perturbed Riemann problems in MHD,
compared to only five in ordinary gas dynamics 共two of
which have been illustrated above兲.
We conclude that the family of the RM-like instabilities
extends well beyond the strict limits of applicability of Rich-
tmyer’s original analogy 共3兲of the perturbed interface with a
pendulum driven by a short pulse of gravity. There might be
no gravity, and the pendulum does not have to exist before it
is kicked. Nevertheless, the analogy with a pendulum with-
out gravity remains useful for all the RM-like instabilities. It
clarifies that the instability has no external energy source to
drive the perturbation growth, only a limited deposit of per-
turbation energy left from the initial conditions or some tran-
sient phase. This would generally imply a linear rather than
exponential perturbation growth. Any physical factor capable
of dissipating this limited energy, or converting it into some
other form, can completely suppress the RM-like instability
development. For instance, even though the MHD RM prob-
lem has not yet been solved, there can be little doubt that in
any configuration with k"B⫽0 at either side of the contact
interface after interaction, its perturbations would oscillate
instead of growing linearly, because bending of magnetic
force lines requires extra energy.
IV. RM-LIKE INSTABILITIES OF ABLATIVELY DRIVEN
SHOCK-PISTON FLOW
A shock-piston flow driven by a uniform pressure ap-
plied to a rippled surface is a good example of a RM-like
unstable flow. The analogy with a pendulum without gravity
suggests that if a constant pressure is applied to a rippled
surface of a half-space filled with a uniform gas, driving a
rippled shock into it, then a linear surface perturbation
growth will follow, the asymptotic growth rate being ex-
pressed by Eq. 共2兲. However, if the pressure is nonuniform
and constant in time, then the constant force driving our
pendulum would supply it with a constant acceleration, pro-
ducing quadratic in time perturbation growth at the interface.
On the other hand, if the external nonuniform pressure acts
only for a limited time, or decays fast enough, then it is
equivalent to a single kick, producing linear growth. Apply-
ing the RM linear theory to these cases, one can see that
perturbation growth indeed occurs in all these cases just as
described above 关see Fig. 5共c兲and Ref. 20兴.
The shock-piston flow, with a shock of constant strength
driven by a constant pressure, is a good model for interaction
of the low-intensity foot of the laser beam with a direct-drive
target. During the foot of the laser pulse, the target material
is slightly shock-accelerated and somewhat shock com-
pressed to prepare it for the arrival of the main laser pulse.34
During the shock-transit interval, most of the mass perturba-
tions due to the laser beam nonuniformity and the surface
roughness of the target are imprinted into the flow. This im-
printing that proceeds essentially as a RM-like instability
development, as first identified in Ref. 14, has been studied
by many authors.14–22 Due to thermal smoothing of the laser
perturbations in the developed corona, the effect of the driv-
ing laser beam nonuniformity on the uniformity of ablative
pressure for any given wavelength lasts only for a finite in-
terval of time. Then both causes of the nonuniformity could
be expected to cause linear perturbation growth.
Surprisingly, perturbation growth due to both laser non-
uniformity and surface roughness has been observed to slow
down instead of growing linearly.15–17 It should be empha-
sized that the saturation-like slowing down was observed in
the linear, small-amplitude regime, and has nothing to do
with nonlinear saturation of perturbation growth at large am-
plitudes, like that observed in Fig. 2共b兲. Moreover,
simulations19 have demonstrated that after reaching some
maximum value the mass perturbations exhibit decaying os-
cillations, like those presented in Figs. 5共a兲,5共b兲. Having
observed decaying oscillation of our pendulum, we must
identify two factors: the effective gravity that makes it oscil-
late, and the effective friction that makes the oscillations
decay.
The role of effective friction is played by mass
ablation.18–20 Since the ablative mass leaves the flow region
between the ablation front and the shock front, it carries
away all of its vorticity and pressure perturbations, which
drain the limited energy resource available for perturbation
development, as explained in Sec. III. If the friction was the
only stabilizing effect taken into account, then the pendulum,
after being kicked, would stop at some amplitude, implying
1666 Phys. Plasmas, Vol. 7, No. 5, May 2000 Velikovich
et al.
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saturation of perturbation growth 关curve 3 in Fig. 5共c兲兴.
There is, however, another stabilizing influence, the ‘‘rocket
effect’’35,36 that acts as effective gravity. It is based on the
fact that the ablation front is an isotherm.37 When the front
moves into a higher temperature area, its temperature does
not increase, but the temperature gradient near it does. In
other words,22 since the temperature distribution in the co-
rona is smoother than the ablation front, the tops of the
ripples that come closer to the laser receive more energy than
the bottoms. Then, as first noticed long ago 共Ref. 38, Section
‘‘Stability’’兲, the increased temperature gradient speeds up
the mass ablation, producing the dynamic pressure pushing
the front back. This mechanism acts as an effective gravity.
The theory of RM-like growth with the rocket effect term
taken into account was developed in Ref. 21 and demon-
strated oscillations qualitatively similar to those observed in
Fig. 3. It would be highly desirable to carry out a special
experiment to observe the full oscillation cycle in imprinting,
beyond the vicinity of the first peak, to test both the simula-
tion codes and the analytical models.21,22
It should be noted how different the effect of these
physical phenomena on the ablative RT and RM-like insta-
bilities is. In the developed acceleration regime, for relevant
wavelengths, neither the small friction nor the small
‘‘rocket’’ gravity can strongly affect the RT instability of the
flow, where perturbations derive their energy from a power-
ful external energy source, ‘‘gravity’’ 共ablative acceleration兲.
On the other hand, each of these mechanisms separately is
sufficient to completely suppress the RM-like growth.
Since the mass perturbations in the target should be
small 共otherwise laser fusion will not work兲, the contribution
of the surface roughness could be simulated as described
above, and shown in Fig. 5共b兲, modeling each relevant wave-
length separately and then summing it up. If the behavior of
each Fourier mode is known, then we only need to know the
spectrum of initial surface roughness to predict the total rms
contribution of this source for any instant. This is not the
case, however, for the laser imprint. A laser beam that is well
smoothed with, for example, induced spatial incoherence
共ISI兲technology39 does not impose constant phase perturba-
tions onto the target. Rather, phases and amplitudes of the
perturbations change very rapidly, so that the perturbations
in the target are essentially noise driven. We can however
apply the elementary theory of noise driven oscillators to see
if it provides any helpful insight into the actual behavior of
ISI perturbations.
Let as assume linear response of the mass perturbation to
the corresponding Fourier component of the driving pertur-
bation
␦
mk共t兲⫽
冕
0
tdt⬘Gk共t,t⬘兲
␦
Ik共t⬘兲,共5兲
where Gk(t,t⬘) is the corresponding Green function, and
␦
Ik(t) is the 共random兲Fourier component of the relative in-
tensity variation. Ensemble averaging immediately shows
that
具
␦
mk(t)
典
⫽0共the noise does not drive mass perturba-
tions in any preferential direction兲, but
具
兩
␦
mk共t兲
兩
2
典
1/2⬅
␦
mk,rms共t兲⫽
冋
c
冕
0
t
兩
Gk共t,t⬘兲
兩
2dt⬘
册
1/2,
共6兲
where the driving noise is assumed to be delta correlated:
具
␦
Ik(t)
␦
Ik
*(t⬘)
典
⫽
c
␦
(t⫺t⬘). Note that the linear relation
共5兲is by no means guaranteed. Since the laser speckle struc-
FIG. 5. 共a兲Simulated growth of mass perturbations in a DT target irradiated
by a KrF laser at 2.8⫻1013 W/cm2with single-mode, constant phase 10%
laser intensity perturbation; 共b兲same for initial surface ripple; 共c兲for the
same conditions and 10% laser intensity perturbation: no thermal smooth-
ing, no mass ablation 共1兲; thermal smoothing, no mass ablation 共2兲; thermal
smoothing and mass ablation 共3兲; thermal smoothing, mass ablation and the
’’rocket effect’’ 共4兲.
1667Phys. Plasmas, Vol. 7, No. 5, May 2000 Richtmyer-Meshkov-like instabilities and early-time...
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tures driving the perturbations are not small perturbations
共all the smallness comes from the time averaging兲, higher-
order combinations of
␦
Ik(t) could contribute to
␦
mk. The
linear relation can be only validated in the limit of large
bandwidth, and verified by the inverse-square-root relation
between
␦
mkand ⌬
. For a finite bandwidth there are con-
tributions of higher orders, which makes establishing the link
between constant phase and ISI simulations difficult. Let us
assume that the Gaussian full width half maximum band-
width ⌬
of the driving noise is high enough, so that 共5兲
holds.
Then we can evaluate the Green function from a series
of computer simulations, where a small-amplitude, constant
phase single-mode perturbation
␦
Ik(t)⫽⌰(t⫺t⬘) is ap-
plied subsequently at t⬘⫽n
,n⫽0,1,2,...,producing the re-
sponses
␦
mk(t;t⬘). Then, by definition 共5兲,G(t,t⬘)
⫽⫺(1/)(
/
t⬘)
␦
mk(t,t⬘). Interpolating this function to
substitute it into 共6兲, we can predict the ISI imprint at a given
wavelength produced with the given bandwidth.
Figure 6 shows comparison of this prediction with an
actual ISI simulation performed for a 1
4
m laser light, 1 THz
bandwidth, 3⫻1012 W/cm2, on a solid plastic target. The
correspondence is quite reasonable, and demonstrates that
our qualitative results remain valid with the laser beam
smoothing taken into account. Improvement will be needed
to remove the small discrepancy at early time.
It is natural to inquire whether we can reduce the laser
imprint. The pendulum model suggests that we have to in-
crease the friction, or gravity, or both, so that the pendulum
swings less under the same drive. Since the effective friction
scales as kVa, and effective gravity due to the ‘‘rocket ef-
fect’’ as kVa
2, where Vais the ablation velocity, the natural
way of decreasing the imprint is to increase Va, which could
be done by decreasing the density
0of the target. Indeed, as
we have demonstrated in Ref. 20 for plastic targets, the peak
imprint amplitude and the time of growth both scale approxi-
mately as
0
1/2 . The imprint, however, could be further re-
duced, if we add some more gravity independent of the
‘‘rocket effect.’’40 This is discussed in Sec. V.
V. SUPPRESSION OF THE RM-LIKE INSTABILITIES
DUE TO DENSITY TAILORING
We have noticed in Sec. III that a collision of two
slightly nonuniform surfaces excites a RM-like perturbation
growth. This kind of growth has been recently observed in
LLNL experiments simulating interaction of a supernovae
ejecta with circumstellar gas.9In these experiments, how-
ever, a relatively slow RM-like growth rapidly evolved into
much faster RT growth. Once the conditions for the bulk
convective instability ⵜpⵜ
⬍0 were satisfied in the stag-
nated material driving the interface, exponential perturbation
growth followed. This is quite natural—inverted gravity
would drive our pendulum exponentially from its equilib-
rium.
The opposite however is also true—appropriately di-
rected gravity could quench the RM-like instability develop-
ment in colliding fluids, producing oscillations instead of the
linear growth. This effect was identified in an early simula-
tion of Z-pinch implosions with double shells10 and seems to
contribute to the enhanced stability observed in most of the
Z-pinch implosions with nested shells,41–43 including dy-
namic hohlraum experiments.44 To illustrate how this stabi-
lization mechanism works, consider the experiment where an
argon double-puff load on the 4 MA Double Eagle generator
in Maxwell Physics International was imploded in a long-
pulse regime.42 The simulation illustrated by Fig. 7 was per-
formed with the NRL radiation-MHD code PRISM 共which
stands for plasma radiating imploding source model, see
Refs. 45, 46兲for realistic initial density profiles in the col-
liding shells. For tutorial purposes, a single-mode mass per-
turbation was introduced in this run at the instant of colli-
sion, to avoid contamination of the flow by the RT
perturbations developing at the outer boundary of the pinch.
At the instant of impact, the inner shell has a higher density
than the outer.
We observe that the contact interface remains virtually
nonperturbed throughout the implosion, while in the outer
part of the pinch the RT instability actively develops. Near
the interface, however, sufficient mass is located where the
density gradient is positive, and the local gravity—positive,
therefore remaining stable up to the stagnation. After the
stagnation 关Figs. 7共d兲–7共e兲兴, the density peaks at the axis and
the pinch is decelerated 共local gravity is negative兲. This leads
to oscillation of the whole structure, shifting the density
peaks by a half-period 关cf. Figs 7共e兲and 7共f兲兴 and straight-
ening the outer pinch boundary.
Specifically designing the density profile of a pinch in
such a way that the magnetic field/plasma interface is always
decelerated, we can in principle fully suppress even the RT
instability of implosion.46,47 Full stabilization, however,
comes at the price of having a shock wave present through-
out the implosion. This might be tolerable for a Z pinch used
as a plasma radiation source, but not for a laser target. With
a laser target, however, we might pursue a less ambitious
goal of reducing the imprint by designing the target to pro-
duce additional, favorable acceleration during the shock tran-
sit time, as discussed in Ref. 40. For this, we should make
the target density increase in the direction where the shock
FIG. 6. Perturbation growth for ⫽30
m obtained in an ISI simulation
共solid line兲and predicted for this case from the simulations imposing a small
constant phase single-mode perturbation.
1668 Phys. Plasmas, Vol. 7, No. 5, May 2000 Velikovich
et al.
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wave propagates, so that the shock will decelerate, and so
will the ablation front. The ‘‘virtual gravity’’ due to inertia
adds to that caused by the rocket effect, thus making our
pendulum stiffer. We illustrate this by some of the results of
Ref. 40.
We compare the mass perturbations imprinted by a 0.2%
constant phase laser nonuniformity, in four planar plastic tar-
gets that have the same mass, 68.2
m g/cm3. Target 1 fea-
tures a parabolic density profile where density increases in
the direction of the laser beam from 0.05 g/cm3to the solid
plastic density 1.07 g/cm3within 120
m, which is followed
bya20
m solid plastic payload. In target 2, the mass con-
centrated in the profile is uniformly distributed over 120
m,
the payload remaining the same; in target 3, the whole mass
is uniformly distributed over the whole thickness, 140
m.
Finally, target 4 is made of solid plastic, and is thinner than
the other three. 共Here, we deal with premanufactured density
profiles, although one might consider producing them dy-
namically, by irradiating the target surface by the laser
prepulse, and then releasing the pressure.48兲In Fig. 8 we
compare the rms mass perturbations imprinted into these tar-
gets by three perturbation wavelengths: 15, 30, and 60
m.
Averaging over the three perturbation wavelengths is seen to
smooth out most of the peaks due the oscillations of the
separate modes. For target 1, some oscillatory behavior is
still visible as a result of substantially increased gravity 关os-
cillations of the three modes, similar to those see in Fig. 5共a兲,
are shown as thin lines兴.
FIG. 7. Density maps in a PRISM simulation of a double puff implosion on Double Eagle. Figures 共a兲–共e兲correspond to times 190, 220, 230, 242, 250, and
270 ns.
FIG. 8. Total rms perturbation growth due to 15, 30 and 60
m perturba-
tions, driven by constant phase 0.2% laser nonuniformity in four compared
targets 共thick lines兲and oscillations of the three modes in the tailored den-
sity target 1.
1669Phys. Plasmas, Vol. 7, No. 5, May 2000 Richtmyer-Meshkov-like instabilities and early-time...
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It is not surprising at all that in all cases some extra
gravity in the right direction helps stabilize a pendulum and
makes its swing less sensitive to external drive perturbations.
Whether laser targets making use of this property could be
manufactured or are consistent with high-gain direct drive
fusion remains to be seen.
VI. CONCLUSIONS
We have demonstrated that the RM instability is just the
best-known member of a family that includes numerous non-
classical RM-like instabilities. The main features of the clas-
sical RM instability at the linear and weakly nonlinear re-
gimes are firmly established in the experiments, simulations
and theory.3–5 This level of understanding has yet to be ex-
tended to cover the whole family. Some of the outstanding
basic issues are listed below.
共1兲For the RM-like instability of colliding fluid layers,
the small-amplitude phase is described by a simple applica-
tion of the analytical techniques developed for the classical
RM instability. The same must be true for a weakly nonlin-
ear theory, but this still has to be shown. Much more detailed
numerical studies are needed. An experimental observation
of this instability with colliding laser- or Z-pinch-driven tar-
gets is very difficult, since uniformity of the densities in both
plasma layers is hard to maintain after collision. A clean
relevant experiment might require massive solid plates col-
liding at a relative velocity on the order of 10 km/s.
共2兲There are numerous RM-like instabilities that in-
volve perturbations coming to the contact interface through
perturbed shock/rarefaction waves 共instability of a planar in-
terface hit by a rippled shock wave, instabilities involving
reshocked and externally driven unstable interfaces, as well
as the whole host of respective MHD problems.兲For these
instabilities, the accurate compressible theory has yet to be
developed, even for the small amplitude case. Although a
considerable amount of experimental data and numerical
simulation results is already available, more experiments 共in-
cluding specially designed shock tube experiments兲are nec-
essary. Developing an analytical theory of shock interaction
with an interface at various stages of mixing is an issue of
much interest and importance.
共3兲The RM-like instability of ablative acceleration has
been extensively studied numerically and theoretically. A di-
rect experimental validation of these findings is now becom-
ing a focus of interest. Observation of the whole oscillation
cycle of the mass perturbation amplitude during the shock
transit time, although a serious challenge for the experimen-
talists, nevertheless appears to be feasible.
Progress in the theoretical and numerical studies of the
RM-like instabilities of the ablative shock-piston flow is a
major development toward reliable modeling of the mass
nonuniformity produced in a laser target by interaction with
surface roughness and by nonuniformity of laser beams. The
understanding gained could be helpful in designing imprint-
resistant laser targets. Similarly, better understanding of the
RM-like instability in a collision might help in the design of
multiple-shell loads in Z pinches providing improved radia-
tive performance, particularly for long current pulses.
ACKNOWLEDGMENTS
Stimulating discussions with and help of S. E. Bodner,
R. H. Lehmberg, R. P. Drake, B. Fryxell, H. Sze, R. B.
Baksht, H. Azechi, and J. G. Wouchuk are gratefully ac-
knowledged. This work was supported by the Defense Threat
Reduction Agency, U.S. Department of Energy and the Of-
fice of Naval Research.
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