1. INTRODUCTION
In this article we report the Rayleigh–Taylor instability (RTI)
of a laser-accelerated ablative surface of a thin plasma layer in an
inertial fusion energy (IFE) target shell with an incompressible
viscous, electrically conducting plasma in the presence of a transverse
magnetic field. The RTI occurs at an ablative surface between the dense
and less dense plasma if the dense plasma is accelerated by the less
dense plasma. It is one of the physical mechanisms and limits the
performance of laser fusion IFE targets. Hence, the additional
mitigation of the RTI growth rate is needed to achieve high gain of
IFE.
At present, mechanisms like a gradual variation of density instead of
an abrupt change in a heterogeneous inviscid fluid and compressible
inviscid fluid without a foam layer have been used to reduce the growth
rate of RTI. The results of numerous experiments and numerical
simulations of the RTI growth rate of compressible inviscid fluid at
the ablation surface fit the following generalized dispersion formula
(see Kanuer et al., 2000):
where n is the growth rate, [ell ] is the wave number of the
perturbation, g is the acceleration due to gravity at the
interface, A is the Atwood number, ε is a constant
multiplying the density gradient correction term, L is the
density scale length at the ablative surface, β is a constant
multiplying the ablation stabilization term, and
va is the velocity across the ablation
surface. The first term on the right-hand side of Eq. (1) is the
classical growth rate of a classical RTI for an inviscid incompressible
fluid and the second term is the effect of compressibility. We note
that with a suitable choice of ε, L, A, and
β, we can get the compressible fluid results of Takabe et
al. (1985), Kilkenny et al.
(1994), Betti et al. (1995), Lindl (1995), and
Bychkov et al. (1994). During the
past approximately 10 years, porous IFE-relevant ablation layers with
foam have been considered by IFE researchers (Sethian et al., 1999; Batani et al., 2000; and references
therein) to reduce the RTI growth rate. To our knowledge, there is no
definite analytical formula to predict the effect of foam to reduce the
growth rate. Recently, Rudraiah (2003) has
shown that the nondeformable porous lining made up of foametal porous
material or aloxite porous material on one side and the IFE target
shell filled with viscous incompressible fluid on the other side
bounded by rigid surface reduces the RTI growth rate,
np, considerably and derived the
formula
where the subscript p denotes the quantities in the presence
of porous lining, B is the Bond number, which is a measure of
gravitational effect to surface tension effect,
σp is the permeability of the porous lining,
α is the slip parameter (see Rudraiah,
1985) vap is the velocity across
the ablative surface lined with porous lining and other quantities are
as defined in Eq. (1). From Eq. (2), Rudraiah (2003) had obtained the maximum wave number
and the corresponding maximum growth rate
nmp is
From this he had obtained
where
is the maximum growth rate in the absence of porous lining. From Eq.
(1) with β = 3, A = 0.9 Takabe et al. (1985) have obtained
where the subscript Ta refers to the results of Takabe
et al. (1985). From Eq. (7) Takabe
et al. (1985) have concluded that
the growth rate of a RTI is reduced to 45% of the classical value
(nmo)Ta. For foam
material used in the experiments of Beavers and Joseph (1967), the slip parameter α ranges from 0.1 to
4.0 and porous parameter σp ranges from 4 to
20. For α = 0.1 and σp = 4.0, Eq. (5)
reduces to nmp =
0.7857nmo. From this result Rudraiah
(2003) had concluded that the maximum growth
rate, nmp, is reduced to 78.57% of
nmo. In particular he (Rudraiah 2003) has shown that the reduction of the growth
rate is possible even up to 80% for a proper choice of the porous
parameters α and σp. Physically, this
reduction in growth rate in the presence of porous lining is due to the
contraction and expansion of flow in the pores, which absorbs some of
the energy that would go otherwise into the target.
We know that in continuum plasma the effect of a transverse magnetic
field is to suppress the perturbations by converting the kinetic energy
in to magnetic energy. In classical plasma, that is, incompressible,
inviscid, perfectly conducting fluid in the presence of a transverse
magnetic field, called the classical problem of Magneto Rayleigh Taylor
Instability (MRTI) has been extensively investigated in the literature
(see Chandrasekhar, 1961). However, the MRTI
of a laser-accelerated ablative surface of a thin target shell with
viscous, electrically conducting incompressible plasma in the presence
of a transverse magnetic field and surface tension has not been given
much attention. The study of it is the main object of the present
article with the motivation to know whether a suitable strength of a
transverse magnetic field can reduce the RTI growth rate in the
presence of viscous shear and surface tension without considering the
mechanism of porous lining. It is also the objective to derive a simple
analytical formula analogous to Rudraiah's formula (2).
To achieve these objectives, the plan of this article is as follows.
In Section 2, the basic equations, the relevant boundary and surface
conditions and approximations are given. The dispersion relation is
obtained in Section 3 using linear stability analysis. The formula for
the growth rate in the presence of a magnetic field and the absence of
a porous lining, analogous to Eq. (2) is also derived in this section.
Some important conclusions are drawn in the Section 4.
2. FORMULATION OF THE PROBLEM
We consider a thin target shell in the form of a film of unperturbed
thickness h filled with light incompressible viscous
electrically conducting plasma of constant density ρ1
bounded on one side by a rigid surface and on the other side by an
incompressible heavy viscous electrically conducting plasma of density
ρ2 of an infinite extent with an interface between the
two plasma layers subject to a transverse magnetic field and surface
tension (see Fig. 1). This assumption on
density is needed for the RTI as defined in Section 1. The fluid within
the shell is set in motion by the laser-accelerated ablative surface.
At time t, the fluctuations of the interface are amplified and
the local thickness becomes a function of the position and time
t and we have y = h +
η(x,t) where η(x,t) is
the surface displacement. We consider a rectangular coordinate system
(x,y) as shown in the Figure
1, with the x-axis parallel to the shell and the
y-axis normal to it with η(x,t) as the
perturbed interface.
The basic equations for conducting, incompressible, viscous and
electrically conducting plasma in the film are the conservation of
momentum,
and the conservation of mass for an incompressible plasma,
where q = (u,v) is the velocity,
J = σ[E +
σhq × H] is the
current density, ∇ × E =
−μh∂H/∂t,
∇·E = 0, ∇·H = 0,
E is the electric field, H is the magnetic field,
μ is the viscosity, μh is the magnetic
permeability, and σ is the electrical conductivity of fluid. These
equations must be supplemented with suitable boundary and surface
conditions. These equations are sufficient for our purpose because we
deal with electrically conducting fluid of small conductivity σ, so
that the induced magnetic field can be neglected in comparison with the
applied magnetic field.
In this article, we deal only with a linear two-dimensional RTI in
continuum plasma considering infinitesimally small disturbances
superposed on the basic state. The basic state is quiescent and the
interface is flat. Further, the following stokes and lubrication
approximations (see Babchin et al., 1983, Rudraiah et al.,
1997) will greatly simplify the analysis: (1) η <<
h. This assumption helps to ignore the variation of horizontal
velocity u with respect to x. (2) The Bond number
B = δh2/γ << 1, which
implies the gravitational effect is small compared to the surface
tension effect where γ is the surface tension and δ =
g(ϱamp;2 − ϱamp;1) is the normal
stress. (3) The Reynolds number R =
Uh2/Lν << 1, where ν is
the kinematic viscosity, which enables us to neglect inertial force
because of very viscous fluid. (4) The magnetic Reynolds number
Rm =
μhσUh << 1 because of small
electrical conductivity. This enables us to ignore the induced magnetic
field compared to the applied magnetic and electric fields. (5) The
Strouhal number
,
where t0 and U are the characteristic time and
velocity, which enables us to ignore local acceleration in the momentum
equation. These approximations, which are valid when the wavelength of
the instability of the ablative surface is large compared to the
thickness of the layer, are useful to ignore many terms, particularly
nonlinear terms in the basic Eq. (8). We also assume that heavy fluid
bounding the lighter fluid is almost static because of creeping flow
approximation, which is needed to study RT instability (see Babchin et al., 1983). Under these
approximations, the basic Eq. (8) reduces to, after making the
resulting equations dimensionless using the scales h for
length, δh for pressure,
δh2/μ for velocity, and
μ/δh for time, the form
where
is the Hartmann number and H0 is the applied
transverse magnetic field. These equations have to be solved using the
following boundary and surface conditions. The no-slip condition at the
rigid surface is
No shear at the free surface is
The dynamic condition is
For linear analysis, the kinematic condition is
3. DISPERSION RELATION
Solving Eq. (10) and, using the conditions (13) to (16), we get
Integrating Eq. (12) with respect to y from 0 to 1 and
simplifying, we get
From Eq. (16), using normal mode solution of the form η =
η0ei[ell ]x+nt and using
Eqs. (15) and (18), we get the dispersion relation of the form
where n is the growth rate, [ell ] is the wave number,
B is the Bond number, and
obtained from Eq. (19) in the limit of M → 0, and for
convenience we call it a classical value.
This Eq. (19) clearly shows that the effect of the magnetic field is
to reduce the growth rate of a RTI considerably compared to the one in
the absence of a magnetic field. The physical reason for this reduction
is that the transverse magnetic field suppresses the flow by converting
the kinetic energy into magnetic energy.
4. DISCUSSION AND CONCLUSIONS
In the present article, a self-consistent analytical approach is used
to study linear RTI of ablatively laser-accelerated targets filled with
an incompressible electrically conducting viscous plasma in the
presence of a transverse magnetic field. The RTI growth rate formula
given by Eq. (19) is analogous to the one given by Eq. (1) for a
compressible fluid, and Eq. (2) for a porous lining.
Setting n = 0 in Eq. (19), we obtain the cutoff wave number
[ell ]ct, above which the RTI mode is stabilized and
is found to be
The maximum wave number [ell ]m obtained from Eq.
(19) by setting ∂n/∂[ell ] = 0 is
Eqs. (21) and (22) are true even for the case in the absence of a
magnetic field (i.e., M = 0) given by Eq. (20) and for
convenience we call them a classical result. The maximum growth rate,
nm, for the corresponding
[ell ]m given by Eq. (22) is
Similarly n0m, from Eq. (20) using Eq.
(22), is
From these, we get the ratio of maximum growth rate
nm to n0m,
given by
Relation (19) is plotted in Figure 2 for
the growth rate n versus the wave number [ell ] for M
= 1 and for different values of B. We see that the
perturbations of the interface having a wave number smaller than
[ell ]ct are amplified when δ > 0 (i.e.,
ρ1 < ρ2) and the growth rate decreases
with a decrease in B, implying an increase in surface tension.
That is, an increase in surface tension makes the interface more
stable. Similar behavior is observed even for M = 10 and we
found that an increase in M decreases the growth rate
considerably. To know the amount of reduction in the growth rate caused
by a magnetic field compared to that in the absence of magnetic field,
Eq. (25) is numerically computed for different values of M
ranging from 10−2 to 102 and the results
are tabulated in Table 1 and are also plotted
in Figure 3 with
Gm versus M. We see that the
decrease in the growth rate compared to the classical one is very steep
for M in the range of 10−1 to 101,
and the ratio Gm becomes independent of
M for values M > 10 tending to the value 0.0003.
For M = 1, we find that Gm =
0.71522, that is, the maximum growth rate is reduced to 71.52% of the
classical value n0m. However, at
M = 10 we find that the maximum growth rate is reduced by
97.3% of the classical value n0m.
Similarly for M = 100 and above, we find that the maximum
growth rate is reduced to 2.7% of the classical value
n0m. From this we conclude that an
increase in the value of the magnetic field, that is M,
reduces considerably the growth rate compared to the classical value.
This information is useful in the extraction of IFE efficiently by
maintaining the symmetry of the target.
Ratio Gm for maximum growth rate for different M
The growth rate n versus wave number [ell ] for M =
1 and for different Bond numbers B.
Ratio of maximum growth rate Gm versus
M.
ACKNOWLEDGMENT
The work reported here is supported by the Department of Science and
Technology (DST), New Delhi under the project No.
SP/12/PC-03/2000. The authors gratefully acknowledge the
financial support of DST, New Delhi.
