Numerical simulation of ablative Rayleigh–Taylor instabilityJohn H. Gardner, Stephen E. Bodner, and Jill P. Dahlburg Citation: Physics of Fluids B: Plasma Physics (1989-1993) 3, 1070 (1991); doi: 10.1063/1.859835 View online: http://dx.doi.org/10.1063/1.859835 View Table of Contents: http://scitation.aip.org/content/aip/journal/pofb/3/4?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Weakly nonlinear ablative Rayleigh–Taylor instability at preheated ablation front Phys. Plasmas 16, 102104 (2009); 10.1063/1.3236746 Threedimensional multimode simulations of the ablative Rayleigh–Taylor instability Phys. Plasmas 2, 2453 (1995); 10.1063/1.871270 Simulation of the Rayleigh–Taylor instability in colliding, ablatively driven laser foils Phys. Fluids B 3, 3485 (1991); 10.1063/1.859726 Ablative stabilization in the incompressible Rayleigh–Taylor instability Phys. Fluids 29, 2067 (1986); 10.1063/1.865593 Vortex simulations of the Rayleigh–Taylor instability Phys. Fluids 23, 1485 (1980); 10.1063/1.863173

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Numerical simulation of ablative Rayleigh-Taylor instability John H. Gardner, Stephen E. Bodner, and Jill P. Dahlburg L! S. Naval Research Laboratory, Washington, D. C. 20375

(Received 5 July 1989; accepted 3 December 1990)

Numerical simulations over a wide range of parameters of the linear stability of an ablating laser-produced plasma show a linear decrease in the growth rate with increasing ablation velocity. Simulations in planar and spherical geometries, with red and blue laser light, at high and low intensities, and at high and low accelerations, all seem to nearly follow a consistent law y=O.9 e - 3kv,, when v, is defined as the mass ablation rate divided by the peak density, in agreement with the eigenvalue analysis of Takabe and co-workers [Phys. Fluids 26, 2299 (1983); 28, 3676 (1985)].

1. INTRODUCTION

Because of the Rayleigh-Taylor fluid instability, there is an upper bound on the aspect ratio (R/AR) of laser fusion pellets. High aspect ratios are desirable, and proba- bly even necessary, because they lead to a much more ef- ficient implosion and because they reduce the required la- ser intensity and therefore the risk of laser-plasma instabilities.’ If the Rayleigh-Taylor instability grew at the “classical” growth rate in laser fusion, then the efficiency of the implosion would be too low for any conceivable laboratory driver, and the laser-plasma instabilities would produce unacceptable numbers of super-thermal electrons. All laser fusion schemes therefore require a significant re- duction of Rayleigh-Taylor growth rates below the classi- cal values.

Reduced growth rates have been widely reported in numerical simulations2-5 and in analytic calculations,“t’ with some models predicting total stability for sufficiently short perturbation wavelengths. There is also at least one direct experimental measurement of this reduced growth rate.” The general belief is that the reduction is due to mass ablation: the instability perturbations are convected away from the ablation surface where the source of the instability is the baroclinic term VpxVp which we have observed in our simulations to be localized near the abla- tion surface. If the perturbations are convected sufficiently far from the source generation region in a growth time then the growth rate will be reduced. This stabilization by mass ablation was first suggested by Bodner in 1974;6 he used a relatively simple analytic model. Since then, more sophis- ticated models have been developed, especially those of Takabe et al.,* Ku11 and Anisimov,” and K~ll.‘~ All these models use various simplifying assumptions, especially in the boundary conditions. Takabe et al. solve an eigenvalue problem about a numerically integrated solution obtained by quadrature. The model requires that the perturbations be localized within the sonic (Mach 1) points. Ku11 avoids some of the boundary condition problems, but only at the expense of making an isobaric assumption in the energy equation. Ku11 reproduced the primary results of Takabe et al., and extended the domain of the solutions. While none of these approximations may seriously affect the actual growth rates, nevertheless, because of the limitations of the

1070 Phys. Fluids B 3 (4), April 1991

analytic and semianalytic models, it is desirable to have a completely numerical integration of the equations govern- ing the Rayleigh-Taylor instability to corroborate these findings. In addition, the numerical model allows a more straightforward coupling of the laser parameters to the ac- celerated target parameters and does not require laser en- ergy to be deposited at the critical surface or arbitrary assumptions to be made about the location of the sonic point with respect to the critical density or absorption sur- face in order to relate the growth parameters to a given set of laser and target parameters.

In this paper we present the results of a number of two-dimensional 2-D numerical simulations with various plasma parameters and geometry. In Sec. II we describe the model, and in Sec. III we present the results of the simulations. We have found that the model of Takabe et al, is excellent fit to our simulations. Shortly after we obtained these results, we discovered that some of our colleagues at Lawrence Livermore National Laboratory had obtained the same conclusion using a different computer simulation model. Their work is described in a separate paper.i2 In Sec. IV we describe the implications of our results on the design of high-aspect-ratio laser fusion targets.

II. MODEL

The calculations are based on a two-dimensional finite- difference hydrodynamics code, FASTRT. This is essentially the same as our FASTZD code,13 with geometry modifida- tions so that it can run in 2-D planar, cylindrical, or spher- ical geometries. The equations are for a single, one- temperature, ideal, compressible fluid in two dimensions. To simplify the model, we use a fully preionized plasma and an ideal, perfect-gas model to close the equation of state. Comparisons with a complete equation of state model in one dimension show that with low-Z materials, such as those used here, there are only minor differences from neglecting the ionization energy in the equation of state. The plasma hydrodynamics terms are integrated with a time-accurate, time split, finite-difference algorithm which has fourth-order phase and an overall second-order accuracy on the uniform part of the grid. The numerical grid is Cartesian with a fixed grid spacing. In planar (spherical) geometry the problem is oriented so that the

1070

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FIG. 1. The xg mesh as a function of position. The ablation front is located in the finely gridded region near x=0.

laser is incident from the x[r]= + 03 direction and the target is aligned along the JJ[~] coordinate. Periodic bound- ary conditions are used in the ~$01 direction, and zero- gradient outflow conditions are imposed in the *x[r] di- rections. Forty or more finely spaced grids (Ax=O.25 pm) are clustered around the steep gradient at the ablation sur- face. The grids are stretched smoothly to about Ax=7 pm on either side of the finely gridded region. With this mesh, both the steep gradients in the target ablation region and the energy deposition in the plasma absorption region are numerically adequately resolved. One-dimensional steady- state models predict that the density gradient scale length in the ablation region scales as T5’2 and becomes of the order 10 - 2 pm at the ablation surface itself. It was found that resolution to this scale was unnecessary to compute the growth rates adequately. This was numerically con- firmed by comparing runs with (Ax = 0.125 pm) gridding to those with (Ax = 0.25 pm). It was found that there was essentially no difference in the calculated growth rates with these two grid resolutions. Grids are further stretched near the fx[r] boundary to further isolate the boundary condi- tions from the main region of interest. The orthogonal ~$01 grid is everywhere uniformly spaced. Twenty grids are used for modes of wavelength 30 pm or less. For longer wavelength modes the number of grids is increased to maintain the aspect ratio of the grids in the ablation region. Care is taken so that the grid ratio Ax/Ay never exceeds 8 nor is less than $ in any region of the calculation except near the ( Ax) boundaries. This is necessary to maintain the accuracy of the two-dimensional flows in the ablation region. Figure 1 shows the distribution of the grids points. We employ the sliding zone version of the flux-corrected- transport (FCT) algorithm; in this version, the numerical mesh is moved with the target velocity, and the density peak remains centered in the finely gridded region. The simulation is stopped if the amplitude of the perturbations extend beyond the finely gridded region.

The equations of motion solved by the FASTRT code are equations for the conservation of mass, momentum, and energy in conservative form:

ap at= - vpv,

apv at f v*pvv= - vp,

aE at + WE +ph + V*q=O,

where p is the fluid density, v the fluid velocity, p the fluid pressure (electron plus ion), E=p(~v~v + E), E is the in- ternal energy per unit mass ( = 1.5p/p), and q is the heat flux vector. Thermal transport in FASTRT is treated with a two-dimensional, classical, nonlinear thermal conduction equation based on the model of Spitzer and HHrm.i4 The Spitzer logarithm, In A, can be either constant or variable. Constant In A, implemented to reduce computational ef- fort, is a reasonable approximation; simulations run with variable In A show m inor differences in details of the den- sity and temperature profiles near the foot of the thermal front, but do not significantly affect the growth rates. Time splitting is used to integrate the equations where first the equations are integrated forward in time with the heat flux term set to zero using the explicit (FCT) algorithm in strict conservation form. The energy equation is then again integrated forward in time including the heat conduction term but holding the density and velocity constant using an implicit solution to the resulting temperature equation. This works adequately for quasistationary solutions where the temperature, density and velocity do not change too rapidly in a given cell in a single time step. This is ac- counted for in determining the overall time step. In order to maintain the steep gradient at the foot of the thermal front, the nonlinear temperature dependence is explicitly calculated in the gradient, giving an equation of the form

c aT,$7.Kvp/2 + s vat 7 Laser*

with T the temperature, K the constant part of the thermal conduction coefficient, and C, = ( aE/aT)p,v=c,,nst.

III. RESULTS

Growth rates were determined by first producing a steady-state profile, then perturbing this profile and mea- suring the growth of the mass perturbations during the exponential phase. An analytic model was used as an initial guess for the density profile.15 The effect of initial transient shocks was m inimized by initializing the laser intensity at 1% of its peak value, and then ramping up to the desired peak intensity by means of a sin4(rr t/to) rise, where to ranged from 2 to 4 nsec depending on the initial thickness of the target. In one-dimensional simulations, this rise pro- duces only m inimal shocks and leads rapidly to the estab- lishment of a hydrodynamic steady-state profile. In most cases, where initial target thicknesses <O( 1OO ,um), a good steady-state profile is established l-2 nsec after the con- stant peak intensity has been reached. The ablation profile from the one-dimensional solution was then expanded 20 to 40 transverse grid zones depending on the wavelength of the perturbation to be simulated. Typically we varied the wavelengths from 10 to 100 pm for planar targets or from I = 210 to 1570 in spherical geometry using the same one-

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dimensional solution as the initial condition. A density per- turbation with one full sinusoidal mode across the trans- verse direction was superimposed on the one-dimensional profile to provide a seed for the Rayleigh-Taylor simula- tion. This perturbation, with an amplitude of 0.05 g/cm3, was imposed over 2.5 pm (ten zones) in the laser direction behind the peak of the density profile. This produced a 0.3% mass perturbation. Simulations done with 0.005 g/cm3 simply reduced the amplitudes by an order of mag- nitude and increased the time before nonlinear saturation,

In the course of this study it was found that the results of Ref. 16 were in error because of an improper application of boundary conditions on the side of the foil opposite the laser. This difficulty was eliminated here by increasing the number of zones in the low-density region opposite the laser. Consistency was checked by comparing the acceler- ation of the shell trajectory with that determined from the ratio of the ablation pressure to the shell mass.

The baseline calculation was a planar 100 pm plastic (CH) target irradiated with 0.264 pm light at 3 X lOI W/cm’. The target acceleration was g= 3.5 X 1015 cm/set’, with a mass ablation rate of rit =4X lo5 g/cm2/sec. The peak density was 6 g/cm3 and the ablation pressure was 35 Mbars. This corresponds to a temperature at the ablation surface of 8.8 eV, an ablation velocity of 6.7X lo4 cm/set, and an ablation Mach number of 0.03. This also corre- sponds to the instability parameter l? (Ref. 10) of 0.04, which, in turn, corresponds to Kull’s convective stabiliza- tion regime. The in-flight shell thickness defined as the distance between the l/e points of the maximum density was 12 pm. The integrated mass in the shell from the rear boundary to the ablation front was 98 g/cm2 at the start of the run.

We then independently varied a series of parameters with respect to this baseline calculation, to determine their effect upon the growth rate. The results are summarized in Fig. 2. The acceleration was reduced by increasing the tar- get thickness to 200 pm. This had the effect of reducing g by a factor of 0.47 for this run while the other parameters remained essentially the same. The peak density was re- duced from 6 to 1.5 g/cm3 by artificially raising the initial isentrope of the cold target. The ablation pressure for this simulation was 30 Mbars, the target acceleration was 4.2 x lOI5 cm/sec2, and the mass ablation rate was 4.1 X IO5 g/cm2/sec. The target mass was somewhat reduced as it took longer to establish a good steady state for this case. At the start of the perturbed part of the run, the initial mass was 70 g/cm2 and the I/e width of target was 45 pm. The temperature at the ablation surface was 30 eV. The laser deposition was changed to a dump-all of the energy at the critical density. This increased the ablation pressure to 45 Mbar, the mass ablation rate to 1.23 X lo6 g/cm2/sec, and the acceleration to 6.4~ 1015 cm/sec2. The peak density was reduced from the baseline case to 4.6 g/cm3, thus increasing the ablation velocity to 2.7X lo5 cm/set. Next the laser wavelength was increased to 1 pm but using the full inverse bremsstrahlung absorption. This reduced g to 2.6~ lOI cm/sec2, mass ablation rate to 3 X 10’ g/cm2/sec and the peak density to 4.0 g/cm3. The laser intensity was

1072 Phys. Fluids B, Vol. 3, No. 4, April 1991

0.8

0.6 LL YCI

0.4

0.2

H const in A ]

J.4

FIG. 2. Rayleigh-Taylor growth rates obeying a nearly universal law: y = 0.9 Gg - 3.0kvab,. Unconnected symbols, defined in the legend, are results from the FASTRT simulations. The K = const refers to a run where the conduction coefficient was held constant at the 100 eV level for tem- peratures below 100 eV (see text).

then reduced to 1.3 X 1OL3 W/cm2 also with red light. This reduced g to 6.8 X 1014 cm/sec2, and r+r to 5.6 X lo4 g/cm2/ set, with the peak density 4.8 g/cm3.

To assess the importance of spherical divergence ef- fects the geometry was also varied from planar to spheri- cal, with an initial shell radius of 2.5 mm and an initial shell thickness of 100 pm. These spherical parameters were chosen to resemble those of an imploding, high-gain, high- aspect-ratio pellet during the critical portion of the shell implosion when the shell thickness is minimum. Since the aspect ratio of these spherical targets was large and the perturbation wavelengths were small compared to the pel- let radius, we did not expect to find a large effect and there was none. We also modified the thermal conduction law near the ablation front in two different ways to investigate the effects of the uncertain energy transport processes. In the first method, we changed In A, by letting it either vary with a minimum value of 2, or by letting it have a constant value of 5. In the second method, we used a linear conduc- tion law (i.e., the conduction coefficient independent of temperature) for temperatures below 100 eV (K = const results in Fig. 2). From the T5” scaling for steady-state ablation, we would expect that we should be sufficiently resolved to see changes in the density profile for tempera- tures greater than about 40 eV. Both of these methods modified the density gradient at the ablation surface, but neither had much effect on the Rayleigh-Taylor growth rates. Again, this is not unexpected since the density gra- dient scale lengths near the ablation surface are still short compared to the mode wavelengths.

The data in Fig. 2 can all be reasonably well fitted by the linear relation

Y -7G

Gardner, Bodner, and Dahlburg 1072

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where y is the growth rate, k is the perturbation wave number, g is the acceleration, r)l is the mass ablation rate, and P&l is the peak density at the ablation front. Alterna- tively, one could use the ablation velocity in the above equation, with t&i,] = Ijl/p,bt. The values of a = 0.9 and /3 = 3 seem to provide the best fit to the data in Fig. 2. The ablation parameters resulting from these simulations rep- resent a limited range of all the possible parameter spaces for quasisteady ablation profiles, however, they represent the region of parameter space accessible for laser and tar- get parameters that are of most interest for the design of high gain laser fusion pellets.

This equation is nearly the same as the one first pro- posed by Takabe et aL8 obtained by fitting a numerical solution of an eigenvalue equation. The equation is also identical to the one of Tabak et aLI2 Tabak et al. solve the hydrodynamic equations with a much different Lagrangian code. Our solution was obtained from the FCT Eulerian code described above. Since all three techniques have ob- tained the same linear relation as a fit to their solutions, the formula can be considered to be quite reliable, within the constraints of the physics that was included in the analysis. Takabe et al. found a weak dependence of the coefficients a and fl on the nondimensional acceleration and on the ratio of the ablation to sonic density found from their steady ablation model. It is difficult to compare these parameters with our calculations since we use a continuous absorption of laser energy and our sonic point lies well outside the critical surface, in contrast to the Takabe et al. steady-state model which assumed the energy is absorbed outside the sonic point. Using our sonic point in any case gave a non- dimensional G (Ref. 8) on the order of unity for our spher- ical runs. The range of G for the parameters of our runs is insufficient to verify the dependence of a or /3 on G. Ku11 also found a similar dispersion relation but found a depen- dence of the cutoff wave number (where the growth rate becomes zero) on his stability parameter I?. The range of I available with the laser parameters relevant to high-gain pellet design is rather limited. For the parameters run here I was about 0.05. The cutoff wave number given by our formula agrees reasonably well with that given by Ku11 for l? = 0.05. We also note that it is not too far from the parameters used by Takabe et al. (see Ref. 10).

IV. DISCUSSION AND CONCLUSIONS

for A=Vp/p at the ablation front. The combination of a reduced A and a reduced p&l may provide sufficient stabi- lization for a high-gain implosion. We note that this should have the same effect as raising l? in Kull’s analysis. Thus radiation would be used to control the Rayleigh-Taylor instability by controlling the peak density and the density gradient, just as we have used the IS1 optical smoothing technique” to control the nonuniformity of laser illumina- tion. Of course we do not yet know if the above equation remains unchanged in form and unchanged in the constant coefficients when the.effects of radiation are added. We are currently investigating the effects of radiation upon the Rayleigh-Taylor instability.

The above equation for the growth rate of the Rayleigh-Taylor instability has both positive and negative implications. The good news is that the growth rate can not only be less than the “classical” value, but that it can even be zero for sufficiently large mass ablation rate, large perturbation wave number, or low peak density. The com- plete stabilization of the shortest perturbation wavelengths, which have the largest classical growth rates, is very im- portant to prevent mixing of the hot corona with the cold fuel shell. We were able to “dial in” this total stabilization by raising the isentrope of the cold fuel and thereby reduc- ing the peak density P&,1.

ACKNOWLEDGMENTS

We thank Dr. D. L. Book and. Dr. M. H. Emery for helpful conversations. We also thank the referee for a num- ber of helpful suggestions.

This research was supported by the U.S. Department of Energy and the U.S. Office of Naval Research.

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The bad news is that when one inserts the parameters 5R. G. Evans, A. J. Bennet, and G. J. Pert, Phys. Rev. Lett. 49, 1639 for a high-gain laser fusion pellet in the above equation, the (1982).

growth rate is too large for a successful implosion. For example, with a 50 pm perturbation wavelength (compa- rable to the in-flight thickness of a high-gain pellet), the growth rate is only reduced 20% below the classical value. Even a moderately thick-shell pellet, as advocated by some of our colleagues, would fail during the implosion.

The form of the equation, however, suggests alterna- tive methods to reduce the Rayleigh-Taylor growth rate. By reducing the peak density at the ablation front, the growth rate can be greatly reduced. We demonstrated this by purposely raising the initial isentrope of the cold fuel. In a high-gain pellet one cannot increase this cold fuel isen- trope, and some other method must be used to lower the peak density. We propose that this may be accomplished by using a controlled deposition of radiation that would penetrate just far enough to reduce the density at the ab- lation front without increasing the isentrope of the main body of the DT fuel. This may be possible by choosing a mixture of ablator materials, with the materials chosen to provide a broad spectrum of x-ray lines and a variety of depths of penetration of the radiation. (A CH ablator does not have this broad spectrum of lines.) In addition, this radiation would then produce a more gentle density gradi- ent at the ablation surface-quite different from the very sharp ablation front that exists without radiation. Density gradients would then modify the first term in the above equation. The equation for the growth rate might then become

Y l-x- Eli? ~=0*9dk-3~~dg~ a

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Gardner, Bodner, and Dahlburg 1074

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