Ablative Rayleigh-Taylor instability with strong temperature dependence of thermal conductivity

Abstract

An asymptotic analysis of Rayleigh–Taylor unstable ablation fronts encountered in inertial confinement fusion is performed in the case of a strong temperature dependence of the thermal conductivity. At leading order the nonlinear analysis leads to a free boundary problem which is an extension of the classical Rayleigh–Taylor instability with unity Atwood number and an additional potential flow of negligible density expelled perpendicular to the front. The nonlinear evolution of the front is analysed in two-dimensional geometry by a boundary integral method. The shape of the front develops a curvature singularity within a finite time, as for the Birkhoff–Rott equation for the Kelvin–Helmholtz instability.

Full text

J. Fluid Mech. (2007), vol. 579, pp. 481–492. c
2007 Cambridge University Press
doi:10.1017/S0022112007005599 Printed in the United Kingdom
481
Ablative Rayleigh–Taylor instability with
strong temperature dependence of the
thermal conductivity
C. ALMARCHA
1,P.CLAVIN
1, L. DUCHEMIN
1AND J. SANZ
2
1Institut de Recherche sur les Ph´
enom`
enes Hors Equilibre, Universit ´
es d’Aix Marseille et CNRS, 49 rue
Joliot Curie, BP 146, 13384 Marseille cedex 13, France
2ETSI Aeronauticos, Universitad Politecnica de Madrid, Madrid 28040, Spain
(Received 12 September 2006 and in revised form 21 December 2006)
An asymptotic analysis of Rayleigh–Taylor unstable ablation fronts encountered
in inertial confinement fusion is performed in the case of a strong temperature
dependence of the thermal conductivity. At leading order the nonlinear analysis leads
to a free boundary problem which is an extension of the classical Rayleigh–Taylor
instability with unity Atwood number and an additional potential flow of negligible
density expelled perpendicular to the front. The nonlinear evolution of the front is
analysed in two-dimensional geometry by a boundary integral method. The shape of
the front develops a curvature singularity within a finite time, as for the Birkhoff–Rott
equation for the Kelvin–Helmholtz instability.
1. Introduction
In inertial confinement fusion (ICF) the high fuel density which is required for
nuclear burning can be obtained by imploding spherical shells by high-power laser
radiation (direct drive). Irradiation leads to surface ablation and drives the capsule
implosion. A layer of finite thickness (called the total thickness in the following)
is formed between the ablation front in the cold (dense) material and the critical
surface at low density and high temperature where the laser energy is deposited.
The main mechanism of energy transfer between the laser deposition region and the
cold material is thermal conduction. Owing to the temperature dependence of heat
conductivity, the typical length scale of temperature variation increases strongly across
the conduction region from dain the cold material to dcat the critical surface (hot
material), dadc. Such a scale separation is responsible for the existence of a thin
ablation front with a large density jump separating the cold and dense material from
the hot conduction region, see figure 1. Accelerating ablation fronts are Rayleigh–
Taylor (RT) unstable while the short-wavelength disturbances are stabilized by
ablation, see Bodner (1974). The self-consistent analyses of linear stability performed
by Sanz (1994, 1996), Bychkov, Goldberg & Liberman (1994), Betti et al. (1995), and
Goncharov et al. (1996) are too complicated to be extended into the nonlinear regime
without further approximations. Nonlinear analyses have been performed by Sanz
et al. (2002) and Clavin & Almarcha (2005) within the framework of a semi-empirical
discontinuous model, called the sharp boundary model (SBM), consisting of two
fluids of constant density separated by a front of zero thickness, see Betti, McCrory &
Verdon (1993), Piriz, Sanz & Iba
nez (1997), Piriz (2001) and Clavin & Masse (2004).

482 C. Almarcha, P. Clavin, L. Duchemin and J. Sanz
1.5 0.5 0
0
0.2
0.4
0.6
0.8
1.0
1.2
ρ/ρa
ν = 0.5
1
2.5
4
ν = 0.5
1
2.5
4
νx/da
0
0.2
0.4
0.6
0.8
1.0
(T – Ta)/(Tc – Ta)
1.0 1.5 0.5 0
νx/da
1.0
Figure 1. Density and temperature profiles for a Spitzer-type model with increasing νand
for Θc≡Tc/Ta= 50.
The jump in density (across the ablation front) is a free parameter in the SBM,
and an ad hoc assumption is required to close the problem. However, in the limit
of an infinitely large density jump, the SBM leads to a simple formulation of the
nonlinear problem that can be written in conveniently reduced variables as
Ψ−=0,Ψ
+=0,lim
ζ→−∞ Ψ−=0,lim
ζ→+∞Ψ+=ζ, (1.1a–d)
n·∇Ψ−|ζ=ζf=Uf
n,Ψ
+|ζ=ζf=0,∂Ψ−
∂τ +|∇Ψ−|2
2+|∇Ψ+|2
2ζ=ζf
=ζf,(1.2a–c)
where τand ζ=ζex+ηeyare reduced time and spatial coordinates, Uf
nis the
normal velocity of the ablation front ζf(t), and Ψ−(ζ,τ)andΨ+(ζ,τ) are the velocity
potentials of the cold and hot flow respectively. Except for the presence of Ψ+due to
the hot flow expelled from the ablation front, the problem is like the RT problem with
unity Atwood number. The uncontrolled approximations of the semi-empirical SBM
limit the physical validity of the result, all the more so since the unknown density
jump is used in the scaling of the reduced variables in (1.1) and (1.2), see Clavin &
Almarcha (2005).
The objective of the present paper is towfold. First, by using an asymptotic analysis
extending the linear study of Sanz, Masse & Clavin (2006) to the fully nonlinear
regime, it is shown that the same equations as (1.1) and (1.2) are obtained from
the basic equations for the conservation of mass, momentum and energy in a limit
involving a strong temperature dependence of thermal conductivity, see (4.1) and (4.2).
A major difference with the SBM is that the reduced variables appearing in (1.1)
and (1.2) depend on the physical parameters defining the problem, namely the Froude
number and the power index of thermal conductivity. Secondly, we obtain solutions
in two-dimensional geometry of (1.1) and (1.2) by a boundary integral method in the
spirit of the pioneering work of Baker, Meiron & Orszag (1980), as done by Duchemin,
Josserand & Clavin (2005) in revisiting the RT instability. It is then shown that the
front develops a curvature singularity within a finite time. The singularity is similar to
the one observed in the solutions of the Birkhof–Rott equation describing the Kelvin–
Helmhotz instability. The problem is formulated in §2. The orders of magnitude are
presented in §3. The asymptotic analysis leading to (1.1) and (1.2) is developed in §4.
The nonlinear solution is presented in §5.

Ablative Rayleigh–Taylor instability 483
2. Formulation
In the limit of a low Mach number and for a perfect gas, ρT =ρaTa, with a constant
specific heat Cp, the fluid equations are
∂tρ+∇·(ρu)=0,(∂t+u·∇)u=−ρ−1∇p+gex,∇·[u−(λ/CpρaTa)∇T]=0.(2.1a–c)
Here, ρand Tare the density and temperature respectively, pis the pressure, u=
uex+weyis the flow velocity, xis the longitudinal coordinate and yis the transverse
coordinates, exand eyare the unit orthogonal vectors, expointing toward the hot
material, λis the heat conductivity, and gis the constant acceleration in the frame of
the capsule. The upstream boundary condition are
x→−∞:ρ=ρa,T=Ta,u=Vaex,p=¯
p, (2.2)
where Vais the ablation velocity and pthe unperturbed pressure, d(p+ρu2)/dx=ρg,
the overline denoting unperturbed quantities. The laser intensity I, actually an energy
flux, is absorbed at the critical density ρc, much lower than the initial density,
ρcρa.InICF,λ(T)=λaΘν,Θ≡T/T
adenoting the reduced temperature. The
power index is ν=5/2 for electron heat conduction and ν=13/2 for radiative heat
conduction of a fully ionized plasma. Introducing the conduction length at the cold
side, da≡λa/ρaVa, three non-dimensional parameters characterize the problem: the
inverse Froude number Fr−1≡gda/V2
a, the temperature ratio Θc≡Tc/Ta=ρa/ρc1
and the power index ν. The conduction length at the critical surface is dc≡daΘν
cand
dc/ν is the total thickness of the thermal wave. In conditions of ICF implosion, Θc1
and the acceleration is such that 1/Θν−1
cFr−1/ν < 1. Therefore, the wavelengths of
the most unstable disturbance and of the marginal mode (zero linear growth rate)
are of the same order of magnitude 2π/kmsuch that ν/dckm1/da. The thermal
perturbations cannot reach the critical surface which remains planar. The downstream
boundary conditions then reduce to
x→+∞:T−T=0,uand p−pbounded.(2.3)
Equations (2.1) to (2.3) constitute the basic framework for studying the ablation
front during the implosion in ICF, see Atzeni & Meyer-Ter-Vehn (2004). The unper-
turbed temperature profile upstream of the critical surface is a solution to the first-
order equation µCp(T−Ta)−λdT/dx=0 where µdenotes the unperturbed ablation
rate, µ≡ρaVa=I/Cp(Tc−Ta). At the cold side, the profile of temperature (or density)
presents a sharp bend of width of order da, representative of the ablation front, see
figure 1. Choosing the origin of the x-axis at the bend, the temperature profile satisfies
a power law in the hot region where terms of order 1/Θare negligible,
Θ≡T(x)/Ta≈(νx/da)1/ν, θ (x)≡Θν(x)/ν ≈x/da.(2.4)
This outer solution is singular at the origin where the ablation front is located.
Focusing attention on intermediate wavelengths, ν/dck1/da, equations (2.1) to
(2.3) describe a free boundary problem with two external regions: a constant-density
region of dense and cold medium (subscript −), ρ−=ρa,Θ= 1, and a hot conduction
region (subscript +), Θ≡T/T
a=ρa/ρ+1, separated by a thin ablation front located
at x=α(y,t) (to simplify notation the Cartesian equation of the front is written for a
single-valued surface, but the method applies for a multi-valued front). Density and
temperature vary in the hot region.
Let us split the flow within the hot side into two parts, u+=u(p)
++u(r)
+,whereu(p)
+≡
VadaΘ∇θis associated with the thermal flux Θν∇Θ=Θ∇θ,θ≡Θν/ν. According to

484 C. Almarcha, P. Clavin, L. Duchemin and J. Sanz
(2.1), ∇·u(r)
+= 0, and the thermal equation yields
∂t+u(r)
+·∇
Θ−1+Vadaθ =0,θ≡Θν/ν, u+=u(p)
++u(r)
+,u(p)
+≡VadaΘ∇θ.
(2.5)
In the limit kda→0, equations (2.1) lead to jumps in density, temperature Θf+,mass
flux and pressure across the ablation front while conservation of energy gives the
mass flux in terms of the local gradient of temperature,
ρu·n−uf
nx=α+
x=α−=0,µ2
fρ+px=α+
x=α−=0,u(r)|x=α+=u−|x=α−,(2.6a–c)
µf/ρaVa=(1−1/Θf+)dan·∇θ|x=α+≈dan·∇θ|x=α+,(2.7a, b)
where x=α+and x=α−denote the hot and cold side respectively, nis the unit
vector perpendicular to the front (pointing in the direction of the hot side), uf
n≡
[1 + (∂α/∂y)2]−1/2∂α/∂t is the normal velocity of the front in the frame of the
unperturbed front, and µfis the ablation rate (mass flux across the ablation front,
µf≡ρ(u·n−unf )|x=α). Equation (2.6c) comes from the conservation of the tangential
component of the flow velocity together with equation (2.1c) and the definition of
u(p)|x=α+.
3. Orders of magnitude
We consider intermediate accelerations, 1/Θν−1
cFr−1/ν 1 in two-dimensional
geometry. An asymptotic analysis is then performed using the small parameter
ε≡(Fr−1/ν)1/(2(ν−1)). At leading order in the distinguished limit ε→0, Θc→∞,
εΘ1/2
c→∞, consistent with the real conditions, the linear growth rate σof a perturba-
tion of wavelength 2π/k is
σd
a/Va=(kda)Fr−1−(kda)2−1/ν q∗
oν1/ν ,S=K−q∗
oK2−1/ν (3.1a, b)
where S>0, K>0andq∗
o=d
2˜
ϕo/dξ2|ξ=0 are quantities of order unity,
K≡kda/νε2ν=O(1),S≡(σd
a/Va)/νε2ν−1=O(1),(3.2a, b)
with ˜
ϕo(ξ) the solution to the fifth-order system describing the hot conduction region
(ξ>0),
d3˜
ϕo
dξ3−d˜
ϕo
dξ−˜
ϕo
νξ =−d˜
θo/dξ
νξ1−1/ν ,d2˜
θo
dξ2−(˜
θo−1) = −˜
ϕo
νξ1+1/ν (3.3a, b)
and satisfying to the five boundary conditions
ξ=0:˜
ϕo=0,d˜
ϕo/dξ=0,˜
θo=0; ξ→+∞:d
˜
ϕo/dξ=0,˜
θo=1,(3.4)
where ξ≡k(x−α), see Sanz et al. (2006). Equation (3.3b) is the heat transfer equation
(2.5), and ˜
θois related to the perturbation δθ,θ=θ(x−α)+δθ, while ˜
ϕorepresents the
stream function of the solenoidal flow u(r)
+. More precisely, for a perturbed front of the
form α/da=˜
αeiky+σt,˜
θoand ˜
ϕoare defined by δθo=˜
θo˜
αeiky+σt and ϕo=˜
ϕo˜
αeiky+σt,
the subscript odenoting the leading order. Introducing δu(r)
+,u(r)
+=Va+δu(r)
+,the
components of (kda)−1+1/νν−1/ν δu(r)
+/Vaare (−ϕo,−idϕo/dξ). Pressure and vorticity
∇×δu(r)
+are represented by d2˜
ϕo/dξ2−d(ξ1/n ˜
θo)/dξand ˜
ϕo−d2˜
ϕo/dξ2respectively.

Ablative Rayleigh–Taylor instability 485
Let us now consider the orders of magnitude. According to (3.3) and (3.4), ˜
θois of
order unity, δθo(ξ)=O(α/da). The perturbed ablation rate δµfis evaluated from its
definition (2.7) with ∇δθ =O(kδθ), δµf/¯
µ=O(kα). The order of magnitude δu(r)
+/Va=
O(ε−2kα˜
ϕo) is obtained from the expression of δu(r)
+/Vain terms of ϕoby using
(kda/ν)1/ν =O(ε2), see (3.2). Downstream of the ablation front, in the region where
ξ=O(1), the unperturbed temperature Θ(ξ) is, according to (kda/ν)−1/ν =O(ε−2),
of order 1/ε2, see (2.4). The order of magnitude δu(p)
+(ξ)/Va=O(ε−2kα)isobtained
from the definition u(p)
+≡VadaΘ∇θwith ∇δθ =O(kδθ). In the hot conduction region
a quasi-steady approximation applies because ρ+∂t=O(ερaVa∂x), as a consequence of
(3.2), σ/Vak=O(ε−1), together with ρ+/ρa=1/Θ+=O(ε2). The pressure is evaluated
from ρaVa∂xδu+≈−∂xδp+, and the orders of magnitude in the hot side are
δθo(ξ)=O(α/da),δµ
f/¯
µ=O(kα),δu(p)
+(ξ)/Va=O(ε−2kα),(3.5)
δp+(ξ)/ρaV2
a=O(ε−2kα),δu(r)
+(ξ)/Va=O(ε−2kα˜
ϕo).(3.6)
In the cold region upstream of the ablation front where the density is constant, the
pressure is evaluated from (2.1) by using σ/Vak=O(ε−1), ∂xδu−∼kδu−,∂xδp−∼kδp−,
so that ρaσδu
−∼−kδp−,andδu−/Va=O(δp−/ρaV2
a). The order of magnitude of
the pressure being the same on both sides of the ablation front, the perturbed flow
velocity in the cold side is smaller than in the hot conduction region by a factor ε,
but larger than the ablation velocity by a factor 1/ε:
δp−/ρaV2
a=O(ε−2kα),δu−/Va=O(ε−1kα),δu
nf −|δu−|ξ=0 =O(ε|δu−|ξ=0).(3.7)
4. Nonlinear analysis
In the limit we consider from now on, ν1, the last term on the left-hand side of
equation (3.3a) together with the right-hand side of (3.3b) become negligible so that
θis a solution of Laplace equation. An explicit solution of (3.4) is
d˜
ϕo/dξ=−e−ξ∞
0
x1/ν e−2xdx+e
ξ∞
ξ
x1/ν e−2xdx,
showing that ˜
ϕoand d˜
ϕo/dξare of order 1/ν,δu(r)
+=O(δu(p)
+/ν) see (3.5), (3.6). This
also indicates the existence of a boundary layer at ξ= 0 where the vorticity ∇×δu(r)is
localized, ξ1: d˜
ϕo/dξ≈ξ(1−ξ1/ν ), d2˜
ϕo/dξ2≈1−ξ1/ν ;ξ=O(1): d2˜
ϕo/dξ2=O(1/ν).
This suggests performing the nonlinear analysis in the distinguished limits
ν→∞,ε
2≡(Fr−1/ν)1/(ν−1) →0,Θ
c≡ρa/ρc→∞,ε
2Θc→∞,(4.1)
for amplitudes of the wrinkled front of same order as the wavelength,
αk =O(1),α/d
a=O(ε−2νν−1),(4.2)
see (3.2). The parameter Θcwill no longer appear in the rest of the analysis, since the
last two conditions in (4.1) are there only to ensure the validity of (2.3), kmdc/ν 1,
while ε→0 ensures kmda1, see Sanz et al. (2006).
Considering length and time scales of the order of the most linearly amplified
wavelength and its linear growth, the useful non-dimensional variables are, according
to (3.2),
ζ=(r/da)ε2Fr−1,τ=(tVa/da)εFr−1with ε2Fr−1=νε2ν,(4.3)
see (4.1). In the system of coordinates, ζ=(ζ,η), ζand ηdenote the longitudinal
and transverse coordinates respectively, and the coordinates of the ablation front

486 C. Almarcha, P. Clavin, L. Duchemin and J. Sanz
will be denoted by ζf. The orders of magnitude of the fully nonlinear regime are
obtained by introducing (4.2) into (3.5), (3.6) and (3.7) after using (3.2). Hence the non-
dimensional quantities of order unity in the limit (4.1) for characterizing the ablation
rate µ ≡µ−µ, the normal velocity of the front uf
nand the flow u±≡u±−u±,
π±≡π±−π±with π±≡p±−ρ±gx are M(ζf,τ), Uf
n(ζf,τ), U±(ζ,τ), Π±(ζ,τ),
defined as follows:
µ/µ≡M, uf
nVa≡Uf
nε, (4.4a, b)
u−/Va≡U−/ε, π−/ρaV2
a≡Π−/ε2,u+/Va≡U+/ε2,π+/ρaV2
a≡Π+/ε2.
(4.5)
Equation (4.4b) is obtained from (3.2), σ/Vak=O(1/ε), ∂α/∂t =O(σ/k).
Let us now show that a constant-density approximation holds at leading order
downstream of the front. According to (2.4) and (3.2a), one has Θ=(ξ/K)1/ν ε−2,
with, according to (3.5) and (4.2), δθ =O(ε−2ν/ν). The perturbations of θ,pand up
+
decreasing to zero exponentially fast with increasing ξ, the perturbation of ρvanishes
outside the region ξ=O(1) where ρis constant and δρ is smaller than ρat leading
order in the limit (4.1),
ξ=O(1): ρo+(ξ)/ρa=1/Θo=ε2,δρ
o+(ξ)/ρa=−δθoΘν+1
o=O(ε2/ν).(4.6)
In addition, according to the comment above (4.1), the flow is potential in this region,
uo+=u(p)
o+=Vada∇θo/ε2.
The vicinity of the origin ξ= 0 is more tricky, and requires special attention. The
boundary conditions (3.4) at ξ= 0 together with the expression q∗
o=d
2˜
ϕo/dξ2|ξ=0 are
obtained from (2.6) and (2.7) for a matching region on the hot side of the ablation
front corresponding to Θ=O(1/ε). There, the variation of the dynamical pressure
δ(µ2
f/ρ)issmallerthanδp by a factor ε, so that relation (2.6b)is[δpo]+
−=0. Not
only is the vorticity localized inside the boundary layer 1/ε 6Θ<1/ε2, but also the
variations of dynamical pressure are of order 1/ε in this layer, while the variation of
pressure is of order unity. However the thickness (measured with the ξ-variable) of
the layer shrinks to zero in the limit ν→∞. Therefore the analysis may be carried
out in the limit (4.1) by incorporating the layer ε>ρ+/ρa>ε
2into the ablation front
where µ2
f/ρ must be retained in the jump in pressure. The ablation front becomes a
vortex sheet, separating two regions of constant density ρ−=ρa,ρ+=ε2ρawhere the
flow is in irrotational motion.
The space variables ξand ζbeing obtained from the original space variable x/da
with the same scaling, see (3.2) and (4.3), the leading-order Euler equations can then
be written in non-dimensional form
∂U−/∂τ +(U−·˜
∇)U−=−˜
∇Π−,∂U+/∂ζ +(U+·˜
∇)U+=−˜
∇Π+,(4.7)
with ˜
∇·U−=0 and ˜
∇·U+=0,˜
∇denoting the gradient with respect to ζ. Here, the
approximation of a negligible unperturbed velocity in the cold flow, Va/u−=O(ε)see
(4.5), and a steady-state approximation in the hot flow, ∂t=O(εu+∂x), has been used.
Let us now go back to the jump conditions across the front. According to (4.4) and
(4.5), uf
n=O(|u−|), |u−|=O(Va/ε), the normal velocity of the ablation front is larger
than the ablation velocity Va, but smaller than the perturbed velocity of the expelled
hot fluid, |u−|=O(ε|u+|). Mass conservation, equation (2.6a), then leads to
n·U−|ζ=ζf=Uf
n,n·(ex+U+)|ζ=ζf=1+M, (4.8a, b)

Ablative Rayleigh–Taylor instability 487
where Uf
nis the velocity normal to the front ξf(t) in the system of coordinates
(4.3). Equation (4.8a) expresses that the front moves with the cold fluid, and (4.8b)
that the flow velocity of the hot flow is controlled by the ablation rate only. In
other words, the ablation rate is negligible in the cold side, |u−|/Va=O(1/ε)and
µ/¯
µ=O(1), while the front velocity is negligible in front of the flow velocity in the
hot fluid, uf
n=O(ε|u+|). By using (2.4), dan·∇θ|ζ=ζf=n·ex, equation (4.8b)together
with equation (2.7a) leads to n·U+|ζ=ζf=dan·∇(θ−θ)|ζ=ζf. This can be written in
terms of a velocity potential Φ+of order unity, n·U+|ζ=ζf=n·˜
∇Φ+|ζ=ζf, where by
definition Φ+≡(θ−θ)Fr−1=O(1), see (4.3) and (4.6). According to the leading order
of (2.4) and (2.5), θ=0,θ = 0, and the velocity potential Φ+is a solution to Laplace
equation. On the other hand, the hot fluid velocity being larger than the cold fluid
velocity, |u+|=O(|u−|/ε), conservation of transverse momentum implies that the hot
fluid is expelled quasi-perpendicularly to the front, t·(ex+U+)|ζ=ζf=0, where tis the
unitary vector tangent to the front. Viewed from the hot side, the front is an isotherm,
t·∇θ|ζ=ζf= 0 equivalent to t·(ex+˜
∇Φ+)|ζ=ζf=0, so that t·U+|ζ=ζf=t·˜
∇Φ+|ζ=ζf.
Hence the boundary condition of the hot flow is U+|ζ=ζf=˜
∇Φ+|ζ=ζf.Thisisinfull
agreement with an incompressible and a potential flow and slaved to the temperature,
as in the linear approximation, U+=˜
∇Φ+.
The flows U−and U+being irrotational, (4.7) can be written as ∂U−/∂τ =
−˜
∇{Π−+|U−|2
/2}and ∂U+/∂ζ =−˜
∇{Π++|U+|2
/2}, giving
∂Ψ−/∂ τ =−Π−−|˜
∇Ψ−|2
/2,∂Φ
+/∂ζ =−Π+−|˜
∇Φ+|2
/2,(4.9)
where Ψ−is the velocity potential of the cold flow, U−=˜
∇Ψ−, with limζ→−∞ Ψ−=0
and n·˜
∇Ψ−|ζ=ζf=Uf
n, see (4.8). According to (2.3) and (2.4), the boundary conditions
for Φ+are limζ→+∞Φ+=0 and Φ+|ζ=ζf=−ζf. The dynamics of the ablation front
ζf(τ) is then obtained by solving two Laplace equations, ˜
Ψ−=0 and ˜
Φ+= 0, with
the four boundary conditions just mentioned above, plus a fifth one at the front for
conservation of longitudinal momentum, obtained from equation (2.6b),
[π−−π+]x=α=(1+M)2/ε2−(α/da)Fr−1.(4.10)
The first term on the right-hand side is the dynamical pressure and the second
is the acceleration. According to (4.1), (4.2) and (4.3), (α/da)Fr−1=ζf/ε2,sothat
the three terms in (4.10) are of same order, see (4.5) and the discussion after (4.6)
concerning the vicinity of ξ= 0. Therefore, according to equation (4.8b) together with
(4.9), the conservation of longitudinal momentum across the front takes the following
non-dimensional form:
∂Ψ−
∂τ −∂Φ+
∂ζ +|˜
∇Ψ−|2
2−|˜
∇Φ+|2
2ζ=ζf
=1−[n·(ex+˜
∇Φ+)|ζ=ζf]2+ζf.(4.11)
This equation is similar to that obtained in the semi-empirical SBM of Clavin &
Almarcha (2005) for Fr−1=O(1) in the limit of a large jump in density across the abla-
tion front. According to the present analysis, equation (4.11) is obtained in a systematic
way in the limit (4.1). Moreover, according to (4.6), the unknown jump in density
appearing in the scalings of the SBM is now determined: ρ+/ρa=(Fr−1/ν)1/(ν−1) .
Recalling that constant terms are irrelevant and that the hot flow is orthogonal to
the ablation front, the right hand side of (4.11) may be written in a simpler form
|n·(ex+˜
∇Φ+)|ζ=ζf=|ex+˜
∇Φ+|ζ=ζf.

488 C. Almarcha, P. Clavin, L. Duchemin and J. Sanz
Ω
Z = f (z)
ζ
ξ
f (z) = e–(i)z
M
Figure 2. Conformal map used to transform the periodic surface into a bounded domain.
Considering infinitesimal disturbances in the form eSτ±iKη, the linear analysis leads
to the linear growth rate S=√K−K2, in full agreement with (3.1), since the solution
to (3.3) and (3.4) shows that limν→+∞[d2˜
ϕo/dξ2|ξ=0] = 1, see Sanz et al. (2006). When
the velocity potential of the total flow Ψ+≡Φ++ζis introduced into (4.11), the
free boundary problem takes the simpler form (1.1), (1.2) describing an extension of
the RT problem for an inviscid fluid above a vacuum without surface tension. The
numerical solution of this problem is discussed in the next section.
5. Results and discussion
Accurate numerical solutions to (1.1) and (1.2) are obtained in periodic two-
dimensional geometry with an extension of the boundary integral method of
Duchemin et al. (2005). We first briefly explain the method. Thanks to the potential
flow approximation, all the useful information is concentrated on the interface. The
power of boundary integral methods consists of reducing the dimension of the problem
by one: from two dimensions to one dimension in our case. The method utilizes a
highly non-uniform set of collocation points placed on the front and redistributed at
each time-step in order to ensure a good accuracy in regions where the curvature of the
front is large. Particular attention is paid to the redistribution in order to be sure that
it does not introduce spurious effects. At each time-step, knowing Ψ−and Φ+=Ψ+−ζ
on the front (i.e. on the collocation points), the Laplace equation is solved to find the
velocities of the fluid particles on each side of the front. This computation is performed
using the conformal map Z=e
−iz,z=ξ+iζ, to transform the periodic domain into a
bounded domain where the Cauchy theorem applies, see figure 2. The discrete version
of the Cauchy theorem gives us the streamfunctions corresponding respectively to
Ψ−on the cold side and Φ+on the hot side. Knowing the streamfunctions and the
velocity potentials on each side, we deduce the normal and tangential gradients of Ψ−
and Φ+. The points on the front are advected using (1.2a) while the velocity potential
Ψ−is updated using the boundary condition (1.2c), after introducing the material
derivative ∂Ψ−/∂ τ +|∇Ψ−|2/2. Finally the time-stepping method used to obtain the
evolution of the front pattern is a fourth-order Runge–Kutta scheme with a time-step
decreasing according to the minimum acrlength between two successive points on the
surface.

Ablative Rayleigh–Taylor instability 489
π
–π
–ππ
–2π
0
0
τ = 10.4
8.29
6
Figu re 3. Numerical solution of (1.1), (1.2) (solid line) at various times for a sinusoidal
disturbance with the wavelength of the most linearly amplified mode, initial amplitude: 0.1.
The curvature singularity is spontaneously formed at τ=8.29. A smooth solution (dashed
line) is obtained when a small surface-tension-like term, 0.004 curvature, is added from the
beginning in (1.2c).
An example of solution is shown in figure 3. Unlike the RT instability with unity
Atwood number (A= 1) and zero surface tension, the problem does not possess a
smooth solution for more than a finite time, as for the solution of the Birkhoff–Rott
equation studied by Moore (1979) and also for the RT instability with A<1, see
Baker, Caflisch & Siegel (1993) and Matsuoka & Nishihara (2006). The shape of
the front develops a singularity within a finite time, the inclination of the tangent
to the front stays finite and a continuous but infinite curvature develops. The points
of maximum and minimum curvature (of opposite sign on both sides of the zero
curvature point) collapse in finite time, while the absolute value of the curvature blows
up. Using the boundary integral method, we were able to approach the singularity
with great accuracy. Figure 4 shows the time evolution of the following quantities: the
derivative of the curvature with respect to the arclength sat the inflection point κ(s),
the maximum and minimum curvatures around the inflection point κmax and κmin and
the distance in arclength between these last two points smax −smin. The critical time
τ0at which the singularity develops is found by requiring that the behaviour of these
quantities in powers of τ0−τis observed for a large number of decades. A remarkable
behaviour in powers of τ0−τis observed independently of the wavenumber. The
scaling laws derived from numerical fitting in figure 4 are self-consistent since a
variation of the curvature (τ0−τ)−1on an arclength distance (τ0−τ)4/3gives a
curvature derivative in (τ0−τ)−7/3. The exponents are accurate within a few percent.
Figure 5 shows the rescaled curvature as a function of the rescaled arclength for

490 C. Almarcha, P. Clavin, L. Duchemin and J. Sanz
10–4 10–3 10–2 10–1 100
10–6
10–4
10–2
100
102
104
106
108
1010
(τ0 – τ)4/3
(τ0 – τ)–7/3
(τ0 – τ)–1
κ′(s)
κmax
–κmin
smax – smin
τ0 – τ
Figure 4. κ(s), κmax ,−κmin and smax −smin as a function of τ0−τin a log-log plot.
–6 –4 –2 0 2 4 6
–0.8
–0.6
–0.4
–0.2
0
0.2
0.4
Figure 5. Rescaled curvature according to (τ0−τ)−1as a function of the arclength rescaled
according to (τ0−τ)4/3.
various time-steps. The very good collapse of these curves exhibits a self-similarity
solution of the problem close to the critical time τ0.
The nonlinearity introduced by ablation |˜
∇Φ+|2plays a key role in the singularity
formation, since such a singularity is not observed in the RT problem with A=1 for
which Φ+does not exist, see Duchemin et al. (2005). There is however an indication
that the singularity is smoothed out at following orders of the asymptotic expansion
when the finite thickness of the ablation front is taken into account. If a small

Ablative Rayleigh–Taylor instability 491
surface-tension-like term is introduced into the right-hand side of (1.2c) for modelling
the finite thickness effect of the curved front, as for wrinkled flames, see Pelc´
e & Clavin
(1982), the arclength between the extrema of curvature still decreases to zero but in
infinite time, exponentially, while the curvature modulus increases exponentially. The
rest of the front shape is not influenced sensitively by the small surface-tension-like
effect. For periodic conditions and infinitesimal initial disturbances, the tip of the
bubble reaches the same constant curvature and velocity in the long time limit as
in the RT problem investigated by Taylor (1950), but with an overshoot. This is
confirmed by a local analysis with a single-mode approximation of the Layzer-type,
see Layzer (1955), but with a different wavenumber for Ψ−and Ψ+, showing that
the contribution of |˜
∇Ψ+|2in (1.2) gives a constant term in the long time limit. The
spike continues to accelerate while its radius increases with time, unlike the radius of
curvature of the RT spike which decreases in time like 1/τ3, see Clavin & Williams
(2005) and Duchemin et al. (2005).
To conclude, the free boundary problem (1.1), (1.2) is mathematically ill-posed for
general classes of initial conditions, since a curvature singularity appears within a
finite time, but it provides a simple and useful framework for studying the nonlinear
growth of the hydrodynamical instability of ablation fronts in ICF, at least before
the sudden formation of the singularity. With our regularizing scheme, no roll-up
is observed after the singularity, see figure 3, unlike in Kelvin–Helmholtz instability.
A very accurate numerical method is required to observe the singularity. An open
question is whether the smooth solution of equations (1.1), (1.2) modified by a surface,
tension-like term is physically relevant for the long-term evolution, beyond the time at
which the singularity spontaneously appears in the absence of the surface-tension-like
term. The vortex produced by front wrinkling could become no longer negligible
as soon as a strong increase of curvature develops locally, and a vortex sheet could
appear in the hot fluid. If this were the case, the singularity would indicate a transition
in the dynamics of the pattern, beyond which the present analysis would fail. This
seems to be the case in the direct numerical simulation presented by Betti & Sanz
(2006) who investigated the role of the vorticity for ν=5/2, Fr−1=0.2 and a real
density profile. In this respect the influence of higher-order effects and the physical
relevance of both the singularity and the limit of strong temperature dependence of
thermal conductivity remain to be investigated.
This research was supported by the French program “Instabilit ´
es Hydrodynamiques
intervenant en FCI en lien avec le LMJ” sponsored by the CEA-DIF and the CNRS.
We would like to thank Professor Yves Pomeau for fruitful discussions and careful
reading of the manuscript.
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