J. Plasma Fusion Res. SERIES, Vol.2 (1999) 536_540

weakly Nonlinear Theory of Rayleigh-Taylor Instability
NISHIHARA Katsunobu and IKEGAWA Tadashi
Institute of Laser Engineering, Osaka Universitv,
Suita Osaka 565-0871. Jaoan(Received: 2g December l99g / Accepted:29 July 19991

Abstract

A weakly nonlinear theory of the Rayleigh-Taylor instability has been developedfor initial
perturbations with a finite spectrum width. Theoretical result
simulation' Saturation oflinear growth and spectral broadening inagrees fairly well with hydrodynamic

a weakly nonlinear stage are discussed.

It is shown that a broad initial bandwidth leads to lower saturation amplitudes ofcompared with that of a single mode case.

the linear srowth

Keywords:
Rayleigh-Taylor instability, inertial confinment fusion, weakly nonlinear
theory,1. Introduction

finite spectrum width

but with the same assumptions as Haan made for thesaturation of the linear growth.
In this paper, we have developed, for the first time,a self-consistent nonlinear theory of the RT instability
for initial perturbations with a finite spectrum width.

An interface between two fluids is unstable when
the interface is accelerated into the heavier fluid. Thisinstability is known as the Rayleigh-Taylor (RT)instability. Recently RT instability has found a wider
spectrum of interest in such fields as geophysics andastrophysics Il] as well as inertial confinement fusion

The theory takes the third order nonlinearity intoaccount. The third order nonlinearity results in the onsetof the linear growth saturation. It will be shown that the(ICF). The RT instability in ICF target implosion occurs
both at the ablation surface in acceleration phase and at
the inner shell-fuel interface in stagnation phase. Small
perturbation grows to amplitudes so large that the shellsaturation amplitude of the linear growth thus obtained
agrees well with simulation results. Spectral broadening

in a weakly nonlinear stage is also shown. In Sec. 2,
governing equations are derived, while results of thebreaks up prior to ignition. It is therefore extremely

important for ICF to predict the linear and nonlinear
growth of the perturbation.
It is generally recognized a single RT unstablemode grows exponentially, until the amplitude is about
1/10 to 1/5 of the wavelengrh 12,31. Haan[4] was thefirst who pointed out importance of a finite spectrum
width in nonlinear RT instability. However he a priori
assumed saturation amplitude of the linear growth andnonlinear growth rate of individual modes, by using
simulation results. Ofer et a/. [5] extends Haan's modal
model by introducing the second order mode coupling,

governing equations are discussed in Sec. 3.

2. Derivation of Governing Equations
We consider a planar interface between inviscid,incompressible fluids without surface tension. It is notdifficult to extend other geometry, such as cylindricaland spherical surfaces, and also to include surfacetension. We think of the system in a gravitational field,g, with the interface z = z(x, r) between an upper fluid ofdensity p11 and a lower fluid of density pr, where x is a

Corresponding author's e-mail: nishihara@ ile.osaka_u.ac.jps36@1999 by The Japan Society of plasma
Science and Nuclear Fusion ResearchNishihara K. et al., Weakly Nonlinear Theory of Rayleigh-Taylor Instability

two-dimensional vector (.r, y), and Ps > Pr. Let z - Oand 7 = €(x, t) be the unperturbed and perturbed
surfaces, respectively. Periodic boundary conditions
apply in x and y, with box length L.
We consider a perturbation of the surface with asmall but finite bandwidth, ft = ks + 6k, where k has
discrete allowed values (2nmlL, 2nnlL), and l/c6l >ldtl. Then the perturbation of the interface can bewith each other, namelya4, a€a0"1 a€a0"1 ad"
dr dx dx l, dy dy le dz leIah,l a0.
a€ ah,l ,(6)=fr#1,.fr;,1,-El,a€

From equation of motion, the pressure in each fluid
is given by Bernoulli's equation,

expanded as- I
lao 1tvh'-s61.
p=pl+-*z€(x , 0 =2 (y(t) eik' " = eiko " 2 €ro*ar e'60 '= ((x, t1e\eo 'lot(1)tl
nalr=rrlr,

varying function in space.
We set up the problem in a standard way [6] for an
irrotational flow. Let Qt@, z, t) and Qlx, z, t) be theY'Qs=Y'Qt=0.

,"[+1, - |rv;"t'|, - rt]A)We require that @g and Qy go to zero as z goes to +6and *, respectively. In the same way as the interface
perturbation, the velocity potentials are also composedof slowly varying amplitudes as
Qs @ , z, t) = Q, (x , z, t) siko''

=Pt

(3)

where /cs = lt.l.

From eq.(2), the amplitudes of the velocity

potential satisfy. € ;,.,.(x, t) einko'* o,*il * "o1' r",,,&o,=-iko {o,-+l*"ol'^,

1@, t\ = L(4)

Qu@, D= 2 O[' @, z, t) einkoa, - Di ar--ler +nhz

(10)

We consider up to the third order nonlinearity inorder to describe the self modulation of the perturbation.
In eq.(10), n = 0 and n = 2 terms arise form the mode
coupling between ft6 modes and they are in the second
order of e2, while n = 3 term arises from the mode
coupling between &6 mode and 2ks mode as the thirdil.
=E'.^',". S=,"'.
ht@,t\= L Lii'@,Z,t\einko t-nbz,

where V = 0l& and we have neglected the third order
spatial derivatives of the amplitudes with respcet to.r.
From the definition of the surface perturbation {(r,d,+|-1,'r"'1,-,r] (e)In eqs.(6) and (9), the partial derivative of thevelocity potential at the interface can be calculated byTaylor series expansion with respect to the perturbation
amplitude under the assumption of ftsl{l - o(e) < 1.
Nonlinear terms in eqs.(6) and (9) generate higher
harmonics. Therefore the interface perturbation andvelocity potential can be written as-h',Q"(x,z,t)=\r(x,Z,t)sito'*h,,(8)we obtainpressibility, the potentials satisfyd€ a€ a€a0lQ)

where pg and p1 are evaluated with @g and ps or @s andps in eq.(7), respectively. Substituting eq.(7) into eq.(8),velocity potentials in the upper and lower fluids
respectively. The velocity v is given by -Vpr for z> €and -V@1 for z > (. From the assumption of incom-& o,=ifrol

At the interface the pressure should be balanced,

The amplituAe (6, r) thus introduced is a slowlyo,I##1,=-#1,, (s)where p is either Qs or Qv Since the normal componentof the fluid velocity is continuous across the interface,0(l0t evaluated with either fg of @1 should coincide

order of e3. It should be noted that n = I term contains
the third order nonlinearity due to the mode coupling,537

Nishihara K. et al., Weakly Nonlinear Theory of Rayleigh-Taylor Instability

for example, between 2fts mode and fts mode (2h - ko =h). We also assume that the bandwidth is of the order ofnumber.

e, i.e. 6k/ks - o(e), and 0$t0x - o(t2). After tedious
calculation, we obtain the governing equations as

3. Results and Discussion
First, we show the validity of the governing
equations by comparing with two dimensionalhydrodynamic simulation. To solve eqs.(lla) and (l lb)
numerically, we define the slowly varying velocityo)r',#=r?oo)-'#(*u ni QY' * tr'. d(r)**--:-' a*'- !=lh " v2 k;)lal' 6,,,

ko |zyi

potential at grid points of which number and grid sizeare chosen to be I = 32 and ksl\,x = 32, respectively.
Therefore the minimum wave number becomes 6k^i"/h= 2nlkoLL,x = 1148. We choose a Gaussian distributionfunction with the bandwidth of 6klks = l0-2 as an initial
spectrum of the velocity pelqb?lgl_gf which root
mean square(rms) equals 1q "l <(koely )2> = l0-3, and 6= 0. Two dimensional simulation is carried out with theuse of a hydrodynamic simulation code, IMPACT-2D
Ul. In the analytical model, we consider only the slowly
varying amplitude of the velocity potential, while in thehydrodynamic simulation we must use fine meshes toresolve perturbations with wavelength of the order of2nlks. We used mesh numbers of L,L, = 2048 x 300.

,n\dtkt lp, - 2pr)ap,',i'----;--------- I^r,.'l/ dtTilPn+Pt)\ Jr_ *i<pi-zpit l,-,,,l, Q,,,.
(Ps+ Pt)' I(l 1a)

The wavelength corresponding to ,t6 is chosen to be l, =I

L,/m and m = 48, so that the difference of the wave
number between the modes with m and m * I coincides
with the minimum wave number dft,1n in the analyticalmodel. Initial velocity perturbations are given for mode
numbers within n + LlZ = 48 + 16, with the same

where p(z) is given by the second order equation,a2oe) ..a ,n, . 4+. (l lb)
6u,dt''.

al =TiQ"'+ pn+pry at,

The surface perturbation amplitude is written as aamplitudes as those of the governing equations. TheAtwood number is chosen to be a = 0.82.
Figure l(a) shows the rms velocity amplitudes as afunction of time, where solid line is solution of eq.(l l),and circles are simulation results. Saturation of linear
growth occurs around time of y f = 6-7 . Normalized
velocity amplitudes of each mode from 45 to 5l are
shown in Fig. 1(b), simulation results, and Fig. l(c),solutions of eq.(l l), respectively. Dips of the veloocity
perturbations appeared in Figs. 1(b) and (c) correspondto phase change due to nonlinear interaction. As shownin Figs. 1(a), (b) and (c), solutions of eq,(ll) agree
fairly well with simulation results.
We now consider three dimensional instability, bychoosing L2 = (512)2, ks\,x = 3n and a = 0.8. The
minimum wavenumber then becomes 6k^inlko = 11768.
The initial spectrum of the interface displacement (EQ =0) + 0, 6 = 0) is assumed to be a Gaussian distributionfunction with the bandwidth of 6klks = 1O-2. Figure 2

function of the velocity potential as,

Ert ko dQ"' td1Pr- P) dPttt. dqtzt' =n ar - ry^Jp"+p) at at-/"\t.5o6[rp,-4yir,l Q$* '- Ti(Pn+P)\^lI+ktr

pr@^-3p) dQQ'. .

2Y1@^+ P')'

*4! lprrrlzt dd(rr.
dt
Yi'dt

Q'12,0.d)2(r2)where we omit the bar for the slowly varying amplitudes
and the subscript H for Q. y, = "[a\g and y, =',[zakog

are the linear growth rates corresponding to thewavenumber ft6 and that of the second harmonics,

shows the time evolution of rms amplitude, andindividual mode amplitudes (0, 0), (110, tlo), (120,+20), and (130, t30). Exponential growth of rms

respectively, and d,= (pu- n)l(n + pr) is the Atwood

amplitude starts to saturate when it reaches
538Nishihara K. et aI., Weakly Nonlinear Theory of Rayleigh-Taylor Instability

approximately to {<(k06)2> - 0.03 around time y1- 4.
This saturation amplitude of the linear growth is muchless than that of a single mode case. The broader initial
bandwidth leads to the lower saturation rms amplitude.
As shown in Fig. 2, the individual modes, (ft6, +mLk^in, ks, t n\.kai) within Inl < 10, Inl < l0 also
starts to saturate at much lower amplitudes. The detailsof the saturation of linear growth will be discussed

In the weakly nonlinear stage, the large amplitude
modes (lzl < 5, Inl < 5) grow linearly in time, whilemuch higher modes (lml > 20, lnl > 20) grows very
rapidly due to nonlinear interaction as shown in Fig. 2.
Until time y rt = 6, the rms amplitude grows muchslowly compared with the linear growth. After that time,
many modes become comparable to each other. Aroundtime yJ = 8, the normalized rms amplitudes become

elsewhere.

l^lt^
lsg.9

100

l* ro-,

101o(a)(U-o100!t3

Eo .lrto-10-1E(Ua ic-4

aJ'10-6'6oo02466
time f4

610Fig. 2 Normalized amplitudes of displacement ko(, asfunctions of normalized time 7't Solid line is root

time Ttmean square amplitude, and others lines
represent amplitudes of each ko( mode. (ko, +m5k^in, ko, + n6k^6!., (m, n) are indicated in thefigure.

100

1o-2

104

0.015060.0125

1o-1000.0100-2

0.007510104

106

time yt'-)u-"'*Fig. 1 {a) Root mean square of velocity perturbationsnormalized by kolyt, as functions of normalized
time ylt; solid line is analytical result, while circles

Fig. 3 Spectrum in weakty nonlinear stage, ko6(m' n)'are simulation results. Normalized velocity

integers in row and col indicates lm, nl = lko,+

perturbations of each mode from modes 45 to 51,(b) simulation results and (c) analytical results.

m6k-r", ksu + n6k-;.).
539

Nishihara K. et al., Weakly Nonlinear Theory of Rayleigh-Taylor Instability

greater than one, and the weak nonlinear theory breaks.

However it is found that if y2 = 0, the weakly nonlinear
stage continues much longer time, namely .EG6)r>- <I upto yJ - 13. The condition of y2= 0 can be realized

shown that a broad initial bandwidth leads to low
saturation root mean square amplitude of the linear

growth. This is quite different from that previously
expected in the modal model.

by the mass ablation effect [8] at the ablation surface inICF or the surface tension. The long duration of theweakly nonlinear stage with Tz = 0 predicts thereduction of the nonlinear RT growth at the ablation

In a weakly nonlinear regime, large amplitude
modes grow linearly in time in the same way as a single

mode case. However many modes are excited very
rapidly due to nonlinear interaction, and broadening of

surface in ICF, not only in the linear growth rate.

In the weakly nonlinear stage, the broadening of
the band width occurs as expected. The bandwidth
becomes 1.5 times greater than its initial value at timef { = 6. The spectrum of the interface displacement atf ( = 7 is shown in Fig. 3.

the bandwidth occurs. The mass ablation may reduce not

only linear growth but also weakly nonlinear growthalthough further investigation is required.

References[]

4. Gonclusions

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We have developed a weakly nonlinear theory of
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width. In the modal model [4,5], they a priori assume
that the linear growth starts to saturate when the root

Geophys. Res.96, 193 (1991).

[2] As a review paper, see D.H. Sharp, physica D 12,3
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t3l J.W. Jacobs, and I. Cauon, J. Fluid Mech.187.329

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