Cross-Code verification and sensitivity analysis to effectively model the electrothermal instability
R. L. Masti, C. L. Ellison, J. R. King, P. H. Stoltz, B. Srinivasan
CCross-Code verification and sensitivity analysis to effectively model theelectrothermal instability
R. L. Masti a,b , C. L. Ellison b , J. R. King c , P. H. Stoltz c , B. Srinivasan a, ∗ a Virginia Polytechnic Institute and State University, Blacksburg, VA 24060, USA b Lawrence Livermore National Laboratory, Livermore, CA 94550, USA c Tech-X Corporation, 5621 Arapahoe Ave., Boulder, CO 80303, USA
Abstract
This manuscript presents verification cases that are developed to study the electrothermal instability (ETI). Specificverification cases are included to ensure that the unit physics components necessary to model the ETI are accurate,providing a path for fluid-based codes to effectively simulate ETI in the linear and nonlinear growth regimes. Twosoftware frameworks with different algorithmic approaches are compared for accuracy in their ability to simulate diffusionof a magnetic field, linear growth of the ETI, and a fully nonlinear ETI evolution. The nonlinear ETI simulations showearly time agreement, with some differences emerging, as noted in the wavenumber spectrum, late into the nonlineardevelopment of ETI. A sensitivity study explores the role of equation-of-state (EOS), vacuum density, and vacuumresistivity. EOS and vacuum resistivity are found to be the most critical factors in the modeling of nonlinear ETIdevelopment.
Keywords: electrothermal instability, MagLIF, z-pinch, vacuum resistivity, equation-of-state sensitivity
1. Introduction
The current-driven electrothermal instability (ETI)forms when the material resistivity is dependent on tem-perature, occurring in nearly all Z-pinch-like high energydensity (HED) platforms.[1] Previous work models theearly time behavior of current-driven metallic explosionsfor pulsed wire array configurations as well as for implod-ing metal liner configurations such as in the magnetizedliner inertial fusion (MagLIF) experiments.[2, 3] A num-ber of codes have been used to simulate and understandthe ETI, making it important to quantify how numericalmodeling choices influence the evolution of the instability.[1, 2, 4, 5]This work provides a series of verification cases in bothlinear and nonlinear regimes ensuring ETI-relevant unitphysics is simulated accurately. Comparing these casesacross codes highlights which differences between the codesare most important when simulating ETI such as time inte-gration schemes for diffusion, spatial differencing methods,and numerical treatment of the highly resistive vacuum inwhich the ETI target resides. Additionally, this work per-forms cross-code comparisons for simulations of nonlinearETI in regimes relevant to MagLIF and other pulsed-powerdriven HED platforms.The two codes are USim, a commercially availablemultiphysics fluid code from Tech-X [6], and Ares, a ∗ Corresponding Author
Email addresses: [email protected] (R. L. Masti), [email protected] (B. Srinivasan )
Lawrence Livermore National Laboratory (LLNL) mul-tiphysics radiation-hydrodynamics code.[7, 8, 9] USimis an unstructured-mesh-based Eulerian code while Aresis a structured-mesh-based arbitrary Lagrangian-Eulerian(ALE) code, and for this study, diffusion is temporallyhandled explicitly in USim and implicitly in Ares. Bothcodes solve the resistive magneto-hydrodynamics (MHD)equations with thermal conductivity. Obtaining similar re-sults with such different algorithmic approaches providesconfidence in the underlying discretization techniques andimplementation in both codes.This paper is structured as follows. Section 2 presentscode descriptions for Ares and USim along with detailson equation-of-state (EOS). Section 3 presents code ver-ification that ensures the magnetic diffusion is capturedaccurately relative to an analytic solution and that thelinear growth of ETI, in regimes of relevance to the Z-machine experiments [2, 4, 5], compares well with theory.Following code verification in the analytic and linear ETIregime, Section 4 presents comparisons of nonlinear ETIincluding a sensitivity study of nonlinear ETI dynamics toEOS treatment, vacuum resistivity, and vacuum density.The key contributions of this paper are two-fold. First,analytic and theory-driven unit physics cases provide ver-ification of the critical physics components necessary toaccurately model ETI. Second, the nonlinear studies high-light the sensitivity of vacuum parameters and EOS inconverged nonlinear ETI behavior. The sensitivity analy-sis shows nonlinear ETI behavior is influenced by vacuumresistivity more than vacuum density.
Preprint submitted to Journal of High Energy Density Physics February 9, 2021 a r X i v : . [ phy s i c s . p l a s m - ph ] F e b . Code Descriptions For this study, the Ares and USim codes solve the mag-netohydrodynamic equations which are given in conserva-tive form as ∂ρ∂t + ∇ · [ ρ u ] = 0 , (1) ∂ρ u ∂t + ∇ · (cid:20) ρ uu T − BB T µ + I (cid:18) P + | B | µ (cid:19)(cid:21) = 0 , (2) ∂(cid:15)∂t + ∇ · (cid:20)(cid:18) (cid:15) + P + | B | µ (cid:19) u − µ B · uB (cid:21) = S (cid:15) , (3)and ∂ B ∂t = −∇ × S E , (4)where µ , ρ , u , P , (cid:15) and B , are the magnetic permeabilityof free space, mass density, 3D velocity, total pressure, to-tal energy density, and magnetic field vector, respectively. S (cid:15) and S B represent the source terms for ohmic heatingand thermal conduction in the former, and resistive mag-netic diffusion in the latter.The codes differ in multiple ways such as in their variablestorage scheme (e.g. zone or node storage), mesh evolu-tion, diffusion evaluation, and diffusion time integration.The different treatment of the diffusion terms, representedin S (cid:15) and S B in Equations 3 and 4, respectively, signifi-cantly influences ETI growth. For this work, S (cid:15) is givenby S (cid:15) = − µ ∇ · ( ηµ ∇ × B ) , (5)where η is the electrical resistivity, and the contributionto the S B in Equation 4 is given by S B = −∇ × ( ηµ ∇ × B ) . (6)For this study, differences exist algorithmically in the tem-poral integration and spatial differentiation of these terms.Section 4 explores these differences leading to substantiallydifferent computational challenges and numerical limita-tions. USim uses finite-volume algorithms on an unstructuedEulerian grid to solve conservative equation systems. Inthe simulations presented here, USim uses the MonotoneUpwinding Scheme for Conservation Laws (MUSCL) toperform cell interface reconstruction for the computationof the Eulerian fluxes.[10] For these fluxes, USim utilizesthe Harten-Lax-van Leer-Discontinuities (HLLD) approx-imate Riemann solver given by Miyoshi and Kusano [11](5-wave) modified to incorporate the changes to the wavesintroduced by the real-gas EOS.[12] Additionally, USimutilizes hyperbolic divergence cleaning as given by Dedner et al. [13]. For hyperbolic temporal integration, USim usesa 2 nd order Runge-Kutta time integration scheme withvariable time step.For this study, USim uses super-time-stepping (STS)to handle the parabolic terms embedded in S (cid:15) and S B .STS modifies the number of Runge-Kutta stages for theparabolic terms so they can be evolved at the hyperbolictime step. [14] The number of stages is proportional tothe ratio of the hyperbolic time step to the diffusion timestep, which can be many orders of magnitude for the non-linear ETI simulations in Section 4, albeit there are lim-its to the number of stages before STS begins to impactaccuracy.[14] Although USim’s STS is capable of 2 nd or-der accuracy, only the first-order accurate implementationis used for direct comparisons between codes. Section 3.1tests the magnetic diffusion contribution to S (cid:15) in Equa-tion 3 which is given by Equation 5, and contribution to S B in Equation 4 which is given by Equation 6.USim implements the divergence and curl operatorsthrough a polynomial fit approximation as this algorithmis suitable for problems with a general unstructured mesh.With this method a field variable, such as B x , is fit-ted with a multi-dimensional polynomial and the valueof the derivative is computed to obtain the differentiatedquantity; e.g. the current density.[15] Discontinuities infield variables can cause oscillations in the fitting proce-dure of the least-squares method. Circumventing theseoscillations requires using a large stencil which reduces orsmooths out the magnitude of the derivatives at the dis-continuity. As an example, S B from Equation 6, expe-riences large gradients due to the discontinuous resistiv-ity at the vacuum-liner interface during the nonlinear ETIsimulations, and a small stencil would over represent thegradient magnitude at this interface. While USim typi-cally uses a 2nd order derivative reconstruction, the largegradients in the diffusive quantities in Eq. 6 require a firstorder reconstruction due to numerical instability with thehigher order polynomial fits.Nonlinear ETI simulations in Section 4 include thermaldiffusion in addition to magnetic diffusion augmenting the S (cid:15) term. USim evolves the total energy density given inEquation 3, so an inverse EOS operation is performed atevery time step to get the temperature from the internalenergy density. For this study, USim applies thermal dif-fusion through a source term given as ∂T∂t = ∇ · ( α ∇ T ) , (7)where α is the thermal diffusivity. This equation re-quires the use of an additional EOS operation to com-pute internal energy density ( (cid:15) int ) updating the total en-ergy density ( (cid:15) ) from Equation 3 through the relation (cid:15) = (cid:15) int +1 / ρ u +1 / (2 µ ) B . For the nonlinear ETI sim-ulations in Section 4, both codes use thermal conduction, A stencil of 20 was found to be sufficient in evaluating diffusivefluxes
2s it improves numerical stability and is physically relevantin the linear ETI growth phase of the simulation.[1, 2]USim handles the multi-material setup of the nonlin-ear ETI simulation (liner-vacuum) through the use of amarker which is a unit-less identifier (-1 to 1). This markeris evolved with the normalized advective fluxes of Equa-tion 1 which follows the movement of each material respec-tively. Due to the density voids created in the nonlinearETI growth and the lack of vacuum energy conservation(see Section 4 and Figure 8), this marker is filtered suchthat a zone containing liner material above the interfacecannot transition to a vacuum zone (marker is always > > < Ares is one of LLNL’s multiphysics radiation hydro-dynamics codes specializing in inertial confinement fu-sion (ICF), high energy density (HED) physics, and en-ergetic materials [9, 7, 8]. At its core, Ares solves single-fluid multi-material multi-component Euler or Navier-Stokes hydrodynamic equations on a structured, arbitraryLagrangian-Eulerian (ALE), adaptive mesh refinement(AMR) grid. Depending on the application, additionalphysics packages are incorporated in an operator split fash-ion. Major physics packages include resistive and extendedmagnetohydrodynamics, laser ray tracing and energy de-position, single- or multi-group radiation diffusion, dis-crete ordinates ( S n ) radiation transport[16, 17], Reynolds-Averaged Navier-Stokes (RANS) turbulence models, andthermonuclear burn.For this work, all of the Ares simulations use the 2D re-sistive MHD package without AMR. The 2D MHD packageassumes that currents reside in the x − y or r − z simula-tion plane, while a single component of the magnetic fieldevolves perpendicular to the simulation plane. During theLagrange step, the zone-centered magnetic field is frozeninto the fluid. After the Lagrange step, the mesh can op-tionally be relaxed towards its initial position according tothe user’s ALE prescription. All mesh variables are theninterpolated from the post-Lagrange mesh to the relaxedmesh using conservative, finite-volume, total variation di-minishing flux-limited advection schemes. In the case ofthe magnetic field, the finite volume advection preservesmagnetic fluxes. Note that for the purposes of this work,Ares was run in full relaxation “Eulerian mode”, whichcompares nicely to USim’s Eulerian formulation. I.e., Ares evolves a single fluid velocity but multiple materialdensities and temperatures within any multi-material zones, and eachmaterial allows multiple components (equations of state) that arerequired to be in pressure and temperature equilibrium with othercomponents of the same material in the zone.
Resistive diffusion of the magnetic field and the hy-drodynamics motion are treated separately using opera-tor splitting methods. Both the magnetic diffusion equa-tion and the thermal diffusion are advanced implicitly intime using a first-order accurate backward-Euler method.Similarly, both diffusion operations employ a second-order accurate finite volume spatial discretization.[18] Thismethod is akin to a bilinear finite element discretization.Ohmic heating is applied explicitly in time after the im-plicit magnetic diffusion update. The updated magneticfield is differenced to calculate edge-centered currents ac-cording to Ampere’s Law. The ohmic heating incurredby the edge-centered currents is partitioned into the twoadjacent zones by treating the two zones as resistors inparallel.Ares handles multi-material dynamics with a volume-of-fluid approach. This approach assigns a volume frac-tion to each material present within a given zone. In ad-dition to the sub-zonal volume, each material is allowedits own sub-zonal thermodynamic state including density,temperature, and pressure. However, only a single (node-centered) fluid velocity is maintained (thus the single-fluidmulti-material designation for the code). For MHD, azone-averaged conductivity is required for magnetic diffu-sion and ohmic heating. For this study, Ares uses a mass-fraction-weighted average of the conductivities for zonesthat contain multiple materials.
Since ETI growth depends on the resistivity of a ma-terial, and the resistivity is a function of the material’sstate, an accurate EOS is important. HED simulationsof experiments often rely on tabular EOS libraries to pro-vide accurate representations of the material state acrossa wide range of densities and temperatures. These EOStables provide the P and the (cid:15) as functions of density andtemperature, including into HED regimes. In previouswork[2, 4], the SESAME EOS database was used to modelETI specifically using SESAME 3720 (SES3720) [19] foran aluminum EOS, and Sandia Lee-More based Desjarlais(QLMD) tables for aluminum transport properties.[20]The nonlinear ETI simulations in Section 4 employan analytic Birch-Murnaghan EOS (BMEOS) for ease ofcode-code comparisons. The magnetic diffusion and linearETI simulations in Section 3 employ an ideal gas EOS.Sensitivity studies in Section 4.2 assess how the nonlin-ear ETI behavior differs between BMEOS and SES3720EOS. For this study, the QLMD effective ionization tableis used in conjunction with the BMEOS to span a largestate space. BMEOS is an analytic equation of state de-termined through data regression, and this work uses thefunctional form used by McBride and Slutz [21] where the3ressure and internal energy are given by [22, 23, 24, 21] P = P + 32 A (cid:20)(cid:18) ρρ (cid:19) g − (cid:18) ρρ (cid:19) g (cid:21)(cid:20) A − (cid:20)(cid:18) ρρ (cid:19) g − (cid:21)(cid:21) , (8)where for aluminum P = (1 + Z eff ) k B ρT /m , m =4 . × − kg, A = 76 × Pa, ρ = 2700 kg m − , g = 7 / g = 5 / A = 3 .
9, and g = 2 / Z eff , k B , ρ , T , representing the effective ionization level, Boltz-mann constant, density [kg m − ], and temperature [K],respectively.[21] Similarly, the specific internal energy den-sity is given by (cid:15) = (cid:15) + 916 A ρ − (cid:34) A (cid:20)(cid:18) ρρ (cid:19) g − (cid:21) + (cid:20)(cid:18) ρρ (cid:19) g − (cid:21) (cid:20) − (cid:18) ρρ (cid:19) g (cid:21) (cid:35) , (9)where (cid:15) = 3 / Z eff ) k B T /m . BMEOS has a max dif-ference to SES3720 of 40%; see Appendix A.Implementing tabulated EOS or tabulated transport co-efficients requires a choice of interpolation algorithm suchas bilinear or bicubic. Section 4.2 shows the effect ofinterpolation algorithm and EOS on the Ares nonlinearETI simulation. For this study, Ares uses LLNL’s LEOS[25, 26, 27] algorithms for table interpolation, and USimuses Los Alamos’s EOSPAC interpolation library. [28]
3. Cross Code Verification
This work provides a guide to running nonlinear ETIsimulations by sequentially verifying the individual physicscomponents relevant to ETI. The first verification test isof magnetic diffusion where solutions from the two codesare compared against an analytical result. The secondverfication test is a linear ETI simulation with negligiblemagnetic diffusion relative to the ohmic heating in S (cid:15) ofEquation 3, isolating the ohmic heating and resistive feed-back mechanisms. This test case involves an x -directed magnetic field vary-ing sinusoidally along the y direction resistively diffusingdue to a constant resistivity in space and time. This testuses Cartesian coordinates with the fluid initially at rest.The magnetic field diffuses towards the steady state solu-tion of a constant field. Comparing this time evolution tothe analytically-derived solution quantifies the numericalerror.The diffusion equation in Equations 4 and 6 with con-stant resistivity reduces the curl operations to a simpleLaplacian diffusion equation in Cartesian coordinates. Inorder to isolate magnetic diffusion from the full MHD equation, the initial conditions and strength of electricalresistivity must satisfy certain conditions. These condi-tions are that any thermal pressure due to ohmic heatingbe negligble relative to the magnetic pressure, and thatany motion due to the magnetic pressure occurs at muchlonger time-scales than the magnetic diffusion time-scale.Using a plasma beta, the ratio of thermal pressure to mag-netic pressure, of unity satisfies the first condition, anda Lundquist number of unity satisfies the second condi-tion. Although only the magnetic field is needed to ana-lyze Equation 6, the codes evolve the full MHD equations;hence, a low plasma beta and a low Lundquist number arechosen to study the isolated effect of magnetic diffusion inEquations 4 and 6.The chosen simulation grid uses an x domain of 0 .
25 mand an y domain of 1 m with a resolution of 50x200grid cells, respectively. The uniform initial state is P =1 . × Pa and ρ =0 .
164 kg m − with an idealgas equation of state (EOS) using γ = . The initial mag-netic field is B = (cid:104) . πy ) , , (cid:105) T, and the initialelectrical resistivity is η =1 . × − Ω m. Given theseparameters, the characteristic magnetic diffusion rate is γ md = 4 π L y L u (cid:115) P βρ ≈ . × s − , (10)and the chosen simulation end time is t f =45 . µ s (2 /γ ). y [m] B y [ m T ] A) theoryAresUSim 0 20 40 time [ s] e rr o r [ . % ] B) AresUSim
Figure 1: Plot A on the left shows the x -direction magnetic field in[mT] along the vertical direction at the magnetic diffusion simulationend time of 45 . µ s over a subset of the simulation range. Plot Bon the right shows the L norm of the error between the simulatedmagnetic field and the analytically-derived magnetic field over time. Figure 1 presents the error of both codes as a functionof time, and an instantaneous lineout of each simulationalong with the analytically-derived solution. This configu-ration results in a maximum global error of less than 0.01%for both the Ares and USim simulations. This low errorprovides confidence in the magnetic diffusion capabilitiesof both codes, and is critical for resolving the nonlinearETI magnetic diffusion wave. The choice of initial state or EOS has no impact on this test .2. Linear ETI ETI occurs whenever ohmic heating is applied to a ma-terial with a temperature-dependent resistivity. The com-bination of the changing resistivity and ohmic heating cre-ates a positive feedback loop causing hot spots to developinternally. The linear ETI growth rate is given by[29, 2] γ = η T J z (cid:16) − α γ/γ (cid:17) − k κρ(cid:15) T , (11)where γ = 2 kη/µ ∆ r , and where k , κ , η , T , (cid:15) T , and α ,are the wavevector, the thermal conductivity, the resistiv-ity, the temperature, the partial derivative of the specificinternal energy (J kg − ) with respect to temperature, andthe angle between the wavevector and magnetic field, re-spectively. For η T ≡ ∂η∂T > α = 90 ° , resulting in a growthrate of γ = η T J z − k z κρ(cid:15) T . (12)While the magnetic diffusion test verifies the effect of theelectrical resistivity on the magnetic field evolution, thislinear ETI test verifies the effect of the electrical resistivityon the internal energy density evolution. Figure 2 showsthe problem setup. In the absence of thermal conductivity,the growth rate becomes γ = η T J z ρ(cid:15) T . (13)This form of the theoretical growth rate depends heav-ily on ohmic heating, so reproducing this analytical resultthrough simulation provides confidence in each code’s abil-ity to capture ohmic heating and the feedback of such heat-ing on the evolution of the material state. For this sim-ulation, the initial parameters (relevant current, length,and time scales) are set to reproduce ohmic heating ina typical pulsed power regime. This test uses aluminumas the conducting material following the state parametersand conductivities derived from the SES3720 and QLMD29373 tables, respectively.[19, 20].This test uses initial parameters of ρ = 2700 kg m − , T = 250 K, I = 10 MA, over an annulus with a thicknessof 500 µ m starting at a radial location of 2 .
68 mm. Uni-formally distributing the current in the annulus resultsin a current density of J z = 1 . × A m − , and issimilar to the values from Figure 4 of Peterson et al. [4](1 × A m − to 7 × A m − ). From the growthrate defined in Equation 13, the η T and the (cid:15) T for thissimulation use values consistent with realistic solid metal-lic parameters relevant to pulsed power HED regimes.Figure 3 shows the conductivity for aluminum atsolid density over the entire range of the QLMD ta-ble in panel A) and a linear fit to a small rangeof temperatures in panel B).[20] The fit from plot Byields an η T = 1 . × − s K − (in mks: η T = r z J z = CδT δη
More Resistive = > Heat Faster Less Resistive = > Heat Slower ∂η∂T > k z perturbation Figure 2: Schematic depicting the linear ETI test in cylindrical co-ordinates wherein a spatially varying resistivity exists inside a uni-formly distributed current. The perturbed temperature (or the inter-nal energy density) perturbs resistivity, and provided the resistivityincreases with temperature, this configuration is ETI unstable asimplied by Equation 12. . × − Ω m K − ). This η T is valid for constant den-sity solid aluminum between 200 K to 900 K; note that theelectrical resistivity is more sensitive to the density thanthe temperature in this state space region. [2] T [K] l o g ( [ c m ]) A)
200 400 600 800
T [K] [ c m ]
1e 5 B) =(1.099e-08)*T+(-6.439e-07)116 K - 1.03e+03 K 8.1e-07 cm - 2.5e-05 cm Figure 3: Plot A on the left shows a constant density contour of thelogged QLMD conductivity table for aluminum for a certain temper-ature range.[20] Plot B on the right shows a subset of Plot A in thelow temperature regime, where the resistivity linearly increases withtemperature. The fitted linear region in plot B yields an estimatefor η T . The specific heat capacity, (cid:15) T , is approximately822 J kg − K − based on SES3720 at the initial state. Thistest uses an ideal gas EOS with the adiabatic index, γ , cho-sen to maintain a constant (cid:15) T . Knowing (cid:15) T , the adiabaticindex is γ = 1 + [ k B / ( m(cid:15) T )] where k B is the Boltzmannconstant and m is the atomic weight, resulting in an adi-abatic index of γ ≈ . .
54 mm to3 .
16 mm with the inner annulus radius set to 2 .
66 mm, andthe simulation axial domain is ± .
25 mm and is arbitrarybased on Equation 13 (absent k z dependence). The res-olution varies in the radial direction from 38 to 300 cellsand in the axial direction from 25 to 200 using a factor oftwo refinement levels to produce the convergence plot in5igure 4.With these parameters, the ETI growth rate from Equa-tion 12 is γ ≈ . × s − . The end time for this sim-ulation is three growth periods corresponding to 3 /γ ≈ . and the theoretical growth rate produces Figure 4using different spatial resolutions and time steps. number of axial zones e rr o r [ % ] AresAres dt/10Ares dt/100USim2nd Order
Figure 4: Plot of percent error, relative to the theoretical growthrate, of the linear ETI simulation in Section 3.2 for different radialand axial resolutions. The Ares convergence exhibits second-orderspatial convergence (error ∝ ∆ z − =zone size − ) provided the timediscretization errors are small (USim was not tested in this limit),and both codes asymptote to similar percent error when using similartime steps . The rate (order) of convergence describes how the er-ror (difference between exact solution and numerical ap-proximation) decreases for increasing spatial or temporalresolution. Figure 4 presents convergence results of USimand Ares for the linear ETI simulation. For the largesttime step, Ares asymptotes to 0.1 percent error and USimapproaches first-order spatial convergence. This differenceis due to Ares using a fixed time step and USim usingan adaptive time step. The error depicted in Figure 4has contributions from both the spatial discretization andtemporal discretization. The temporal discretization er-ror, at sufficiently small time step size, becomes small rel-ative to spatial discretization error thereby recovering theAres spatial order-of-accuracy of second order for uniformmeshes. Comparing the asymptotic errors exhibited byAres in the high resolution limit, there is reasonable agree-ment with the anticipated first-order accuracy in time out-lined in Section 2. Additionally, while Ares uses a fixedmaximum time step for this simulation, USim has an adap-tively changing time step that decreases slightly as the so-lution evolves, which likely contributes to the lower asymp-totic error.In summary, both codes accurately capture the theoret-ical growth rate to within 0.1% in the asymptotic limitgiven the maximum stable time step. Figure ?? shows Obtained by fitting an exponential growth of the temperatuedeviation from the mean (maximum - minimum) y (r)x (z) 85 µm µm ρ = 2700 . kgm ρ = ρ floor = 1e − ρ lin Liner Region: Al EOSVacuum Region: Al EOS P = 1 . × P a
Multimode: δy (interface) (cid:126)k z ( θ ) Injected (cid:126)B θ ( t ) η ( t ) = 5 . − κ = 0 . Wm K κ, η = QLMDS B θ (cid:54) = 0S (cid:15),B r,z = 0S (cid:15),B r,θ,z (cid:54) = 0 µm Figure 5: The simulation setup for the nonlinear ETI discussed inSection 4, showing the different regions, densities, source terms, andconductivities. the Ares spatial order-of-accuracy is second order, and thetemporal order-of-accuracy is approximately first order.This case shows excellent agreement between the linearETI growth rate from Equation 12 and the simulated lin-ear ETI growth across both codes.
4. Nonlinear ETI
The nonlinear ETI case explores, in planar coordinates,the effect of resolving the magnetic diffusion wave througha medium (aluminum). The subsequent current redis-tribution, and the spatially non-uniform ohmic heatingleads to nonlinear growth of ETI. This nonlinear ETI casebuilds upon the verfied magnetic diffusion from Section 3.1and the verified ohmic heating from Section 3.2 to run aphysically-relevant ETI simulation for solid cylindrical andwire explosion regimes.[1, 2, 4] This specific case, as shownin Figure 5, is based on the simulation work done by Pe-terson et al. [4].
For the current source in these simulations, Ares allowsthe user to directly specify the current as a function of timewhereas USim requires specifying the value of the magneticfield at the boundary. Within the code, Ares uses Am-pere’s law to specify the magnetic field at the boundary,and given the same coordinate system, this is the same asdirectly specifying the magnetic field at the boundary asdone in USim. Limitations in the accuracy of USim’s curloperator in cylindrical coordinates lead to using planar ge-ometry in both codes for these simulations. Although thesimulations are in planar geometry, the specified magneticfield boundary condition is consistent with the cylindricalgeometry.6 .0 0.2 0.4 0.6 0.8 1.0 1.2 time [s]
1e 71.01.52.02.53.0 r a d i u s ( m )
1e 3 0.00.51.01.52.0 c u rr e n t ( A ) r l ( t ) I ( t ) I t ( t ) Figure 6: Liner outer radius r l ( t ), normal current I ( t ), and adjustedcurrent I t ( t ) as a function of time with the adjusted current profileproviding inclusion of the prepulse phase typical of a Z-machine shot. From Slutz et al. [3], closed-form expressions for linerradius and current are empirically-derived from pulsed-power experiments conducted on the Z machine. The pur-ple dashed curve in Figure 6 represents the outer linerradius, r l , as a function of time given by r l ( t ) = r l (cid:32) − (cid:18) tt p (cid:19) (cid:33) , (14)where r l is the initial outer liner radius (2 .
92 mm to3 .
168 mm) and t p is the pulse time ( ≈
135 ns). [3, 30, 2]Current, I , as a function of time is given by I ( t ) = I x (cid:18) (cid:19) (cid:115)(cid:18) tt p (cid:19) − (cid:18) tt p (cid:19) , (15)where I x is the peak current (20 MA to 27 MA).Most high power, pulsed-power machines, such as Z,have a prepulse phase before the full pulse is delivered.On the Z machine, this prepulse lasts anywhere from 40 nsto 80 ns as evident from experimentally-measured loaddata.[3, 30, 2] Equation 15 does not have slowly risingpre-pulse behavior, which would cause premature ablationdisrupting the perturbation of the liner-vacuum interface.To incorporate this experimentally-observed initial rise,the functional form of the current drive in Equation 15is modified to obtain I t . Figure 6 depicts the original cur-rent I and the modified current I t . The adjusted form ofthe current is I t ( t ) = I ( t ) (cid:32) − exp (cid:34) − (cid:18) tt r (cid:19) (cid:35)(cid:33) . , (16)where I ( t ) is given by Equation 15 and t r is the adjustablepre-pulse time of 60 ns. Note that although the simula-tions shown in this section are early in time, this currentform is usable to accurately approximate late time phe-nomena in Z-like pulses. Though the peak current of theadjusted form is noticeably smaller than the peak of theoriginal form, it is closer to the experimentally-measuredpeak current from Figure 11b of Peterson et al. [2].To convert this current drive to a planar magneticboundary condition for USim, Ampere’s law is used in cylindrical coordinates to determine the magnetic field atthe time-varying radial location of the liner interface. ForAres, the current is set as I Ares ( t ) = I t ( t )2 πr l ( t ) , so that the magnetic field in both planar problems is thesame and representative of the magnetic field experiencedby the liner on Z.Ares solves parabolic equations implicitly, contrarily,USim solves parabolic equations semi-explicitly (as dis-cussed in Section 2). With an implicit sover, Ares canhandle large vacuum resistivity values without excessivecomputational cost. Because USim uses the semi-explicitSTS scheme, it is not computationally practical to runwith the same large vacuum resistivity as this leads toan impractically large number of STS stages making thecomputational cost significantly more expensive. Thus, amuch lower vacuum resistivity needs to be specified withcertain constraints. First, if the resistivity of the vacuumis relatively low, this results in unphysical currents in thevacuum, diverting the current away from the liner region.These currents cause unphysical ohmic heating in the vac-uum resulting in a highly restrictive time-step. Hence,the vacuum resistivity is set to a large value, and ohmicheating is neglected in the vacuum. Further numericalchallenges include the creation of density voids during ETIdevelopment due to a finite diffusion rate of the magneticfield through the vacuum. This finite diffusion rate leadsto waves in the vacuum creating low density regions wherelarge magnetosonic speeds further restrict the time-step.Avoiding this requires a large enough vacuum resistivitysuch that the vacuum magnetic diffusion transit time issmall relative to the hyperbolic time-step.Figure 5 presents the simulation setup consisting of an x domain of 85 µ m and y domain of 250 µ m with a resolu-tion of 256x640 cells, respectively. The multimode pertur-bation of the interface is of the form δ = 132 m =32 (cid:88) m =1 β m cos (cid:20) π (cid:18) m · xλ max + β m (cid:19)(cid:21) , (17)where β is a random number from 0 to 1, λ max = 200 µm isthe maximum wavelength associated with the lowest mode(see Appendix B for the coefficients used). Figure 5 showsthe initial state. This setup uses a pressure equilibrium toavoid bulk motion of the liner, as material strength modelsare not employed. Note the EOS interpolation algorithm isthe birational LEOS interpolation scheme for Ares[25, 26,27] and the birational EOSPAC interpolation scheme forUSim[28]. For thermal and electrical conductivities of theliner, the QLMD table 29373 for aluminum is used, and This violates energy conservation (at least in the vacuum), butis necessary for an accurate current rise in the liner given a finiteresistive vacuum when using an explicit or semi-explicit scheme.
30 nsUSim 40 ns 50 ns 60 ns 70 ns
25 0 2520020406080100120140
Ares
25 0 25 25 0 25 25 0 25 25 0 250.00.51.01.52.02.5 0.00.51.01.52.02.5 0.00.51.01.52.02.53.0 0.00.51.01.52.02.53.03.5 0.00.51.01.52.02.53.03.54.00.00.51.01.52.02.5 0.00.51.01.52.02.5 0.00.51.01.52.02.53.0 0.00.51.01.52.02.53.03.5 0.00.51.01.52.02.53.03.54.0 x [ m] [ g cc ] y [ m ] Figure 7: Density plots of the nonlinear ETI simulation outlined in Section 4 at different times (correspond to Figure 6) with the top rowshowing the USim results, and the bottom row showing the Ares results. The current at these times is indicated by the green markers inFigure 6 .00.51.01.52.0 USimfft( ) Ares0 20 40 600.00.51.01.52.0 fft( T ) 0 20 40 60 01234024 t [ns] k x [ m ] Figure 8: The fast Fourier Transform of y-averaged values, of density(top) and temperature (bottom), over a range of − µ m to 145 µ malong the x -direction are presented. Plots shown are different snap-shots in time from 0 ns to 70 ns of the nonlinear ETI simulationoutlined in Section 4. The colorbar corresponds to spectral energy,the x-axis corresponds to time, and the y-axis corresponds to thewave number along the x -direction. the EOS for the vacuum and liner is BMEOS, as discussedin Section 2.3.[20]Figure 7 shows snapshots in time of the simulation re-sults for USim (top) and Ares (bottom). Both codes showa similar magnetic diffusion wave and a density spike thatis propagating inward in the 40 ns, 50 ns, and 60 ns snap-shots at y ≈ µ m, y ≈ µ m, and y ≈ µ m, respec-tively. . While the USim results do not allow ablationinto the vacuum, resulting in no mixing of material re-gions, the Ares results do show ablation into the vacuum,as its multimaterial treatment handles mixing of mate-rial regions. This ablation/finger development, seen in theAres results and slightly in the USim results, is the be-ginning of the subsequent electro-choric instability (ECI)as discussed by Pecover and Chittenden [31]. Due to thedifference in multimaterial treatment between the codes this is not explored further, but note the time for whenECI begins to form (60 ns) matches the observations fromthe Pecover and Chittenden [31] simulations.Earlier in time, the interface in the USim simulation isdiffuse, showing a smoother density gradient relative to theAres result at 30 ns and 40 ns. At 70 ns, the Ares resultshows smaller wavelength growth with a sharper densitygradient at the spike interface even though the spikes pene-trate to a similar distance ( ≈ µ m). At 60 ns, both codesshow similar wavelength modes of ETI, whereas at 70 ns, Figures 7 uses a subsection of the simulation result and is whythe peak is not seen explicitly in the 70 ns plot USim currently does not have vapourisation capabilities k x [ m ]
30 ns40 ns60 ns
USimAres
Figure 9: Vertical lineouts of the temperature FFTs is shown inthe bottom row of Figure 8 at various times for both the Ares andUSim results. Note that the differences between the codes are morepronounced early-in-time (top plot) whereas the solutions agree moreclosely late-in-time (bottom plot).
Ares retains more shorter-wavelength modes compared toUSim.Figure 8 presents a discrete fast Fourier transform ofdensity and temperature along the x -direction, and high-lights how the mode structures change over time. Figure 8also shows the early time (30 ns, 40 ns, and 50 ns) sup-pression of the perturbation in the USim results due tonumerical diffusion at the interface. Diffusion is more pro-nounced in the FFT of temperature (bottom left plot) inFigure 8. Both codes converge later in time to lower mode(lower k ) growth, as is observed qualitatively in Figure 7and quantitatively in Figure 8.Figure 9 shows FFT of the temperature at several dif-ferent times corresponding to the decrease and increase inamplitude shown for the USim results in the bottom leftplot of Figure 8. By doing so, Figure 9 highlights the USimresult diverging from the Ares result early in time whileapproaching the Ares result later in time. This suppressionand growth of the perturbation in USim could be due toUSim’s handling of the source terms in the vacuum. Thiscould also be due to the diffusion stencil at the interfacereducing mode amplitude early in time, and increases intemperature variation caused by signifcant ohmic heatingraising mode amplitude late in time. A sensitivity analysis performed with Ares shows howdifferent choices of EOS and interpolation scheme impactthe development of ETI in the nonlinear regime. FromEquation 12, the growth rate of ETI is dependent on (cid:15) T (specific heat capacity), and this value is deduced from the9 bilinear BMEOS bicubic bicubic 1000x P bilinear 1000x P
25 0 2550050100 SES3720 25 0 25 25 0 25 25 0 25 1231234 x [ m] [ g cc ] y [ m ] Figure 10: Density evolution of the nonlinear ETI simulations ispresented at 60 ns using different EOS interpolation schemes (bilinearand bicubic) and different EOS (BMEOS and SES3720).[19] representative EOS. Choices of table interpolation, inver-sion, and monotonicity, will influence the nonlinear ETIbehavior directly through (cid:15) T (indirectly through η T ).Figure 10 shows the effect of using different EOS onthe ETI mode growth in the Ares simulations with theBMEOS (top) and the SES3720 EOS (bottom). Usingbilinear interpolation at low pressure and high density re-sults in the crash of both the BMEOS and the SES3720simulations as noted in the leftmost subplots (top forBMEOS and bottom for SES3720) of Figure 10. Thiscrash is from the evaluation of the sound speed, obtainedfrom derivatives of the pressure with respect to densityand temperature, encountering imaginary values (imag-inary time-step) for both EOS simulations. The qualita-tive differences between the bilinear interpolation BMEOSand SES3720 cases before the simulations crash (leftmostplots) are likely due to the resolution of the table at thehigh-density, low-temperature regime and/or the magni-tude of the sound speed evaluated in this region. Usingbicubic interpolation produces the same result as morehigh fidelity interpolation schemes such as birational, bi-hermite, bimonotonic, and biquintic, not shown here.The bilinear interpolation becomes suitable in the high-pressure, high-density regime, as this is a better-definedregion of the table. The two right-most columns of Fig-ure 10 reflect this by showing no difference between bicu-bic and bilinear interpolation, and the simulation is ableto run to completion. Increasing the initial pressure movesthe initial state to higher temperature (not linearly) and Better as in satisfying monotonicity, or having positive pressureand energy values. Negative values result from the imaginary statespace of the Van Der Waal’s isotherm loops into a smoother region in both tables, permitting the sim-ulation to run to completion. This smoother region has a30 percent difference between the BMEOS and SES3720,while the lower initial pressure region has 40 percent dif-ference. This difference is still large and explains the qual-itatively different result in the 2 right-most columns ofFigure 10. Initial pressure is varied instead of density be-cause the electrical resistivity is highly sensitive to densitynear reference solid density, as mentioned in Peterson et al.[2], and it would change the η T value through differencesin collisional quanities.Qualitatively the representative SES3720 and BMEOSsimulations show differences late in time, while early intime they show similar smaller wavelength mode structure(not shown here). These differences late in time highlightthe importance of the EOS on nonlinear ETI development.For the parameters surveyed in this study, the dependenceon interpolation algorithm is not as significant as long asan interpolation scheme is used with higher fidelity thanbilinear interpolation. In this work, the vacuum is treated as a separate ma-terial from the liner that has a fixed (i.e., constant) largeresistivity (5 . × − Ω m) while the liner material usestabulated electrical conductivity (see Section 2.3 and Fig-ure 5). The only difference between the treatment of theliner and vacuum regions in these Ares simulations is inthe different electrical and thermal conductivities. Themesh is initialized such that the interface contains multi-material zones which are handled with the method statedin Section 2.2. The presence of multiple-materials due tothe liner being ejected into the vacuum is evident in thelate-stage evolution presented in Section 4.1 (60 ns and70 ns).The vacuum resistivity changes the rate that the mag-netic field diffuses through the vacuum and is importantin evaluating the ohmic heating due to the spatial vari-ation in resistivity. For explicit codes, the vacuum resis-tivity should be as low as possible while still achievinga similar result to the infinitely resistive vacuum limit.To determine the role of vacuum resistivity on ETI de-velopment, the Ares nonlinear ETI simulations are re-peated while multiplying the vacuum resistivity by up to100 times the nominal value used in the preceding studies(5 . × − Ω m).Figure 11 shows the result of varying the resistivity andheight of the vacuum. As the resistivity increases in thevacuum, the solution converges as noted in the upper rowof Figure 11. Qualitatively, the 5x, 10x, and 100x η v sim-ulations show no differences, but the 1x and 2x η v simu-lations show noticeable differences. The main difference isin the magnitude of density inside the liner hotspots. Fig-ure 12 presents horizontal lineouts at y = 25 µ m for the toprow of Figure 11. These lineouts show that the magnitudeof density inside the liner hotspots varies greatly (50% at x = 5 µ m) between the 1x and 2x runs, and varies little10 vac h v vac x2 vac x5 vac x10 vac x10025 0 25050100 vac x0.25h v x0.5 25 0 25 vac
25 0 25 vac x5 25 0 25 vac x10 25 0 25 vac x100 01230123 x [ m] [ g cc ] y [ m ] Figure 11: Density evolution of the nonlinear ETI is presented at60 ns with the resistivity varying by column and the initial heightof the vacuum ( h v ) varying by row. The bottom row is for halfof the vacuum height compared to the top row. The green dashedline indicates the lineout location used for the top row to generateFigure 12.
40 20 0 20 40 x [ m] [ g cc ] vacvac x2 vac x5 vac x10 vac x100 Figure 12: Horizontal lineouts of the top row of Figure 11 is presentedfor all vacuum resistivity cases with the original vacuum height ( h v ).The horizontal lineout, indicated by the dashed green line in theupper leftmost plot of Figure 11, is at y = 25 µ m for all cases. ( <
5% at x = 5 µ m) between the 5x, 10x, and 100x runs.These results indicate that a resistivity ratio of approxi-mately 5 . × − Ω m (i.e., 10x, which has <
1% varia-tion with the 100x run and the not-shown 1000x run) issufficient in capturing ETI in the infinitely resistive limitfor this nonlinear ETI setup.To investigate the underlying mechanism for the con-verged vacuum resistivity threshold, an additional resis-tivity scan is performed in a configuration with half theoriginal vacuum height. The vacuum height is the ini-tial spatial distance in the y-direction between the liner-vacuum interface and the outer edge of the vacuum region(lower boundary in the y-direction). For the simulationspresented in Figure 11, the vacuum height, h v , is 100 µ m.The bottom row of Figure 11 shows the reduced vacuumsize simulations of varying vacuum resistivity. Reducingthe vacuum height probes the influence of the characteris-tic time for the magnetic field to diffuse through the vac-uum on ETI development. Reducing the vacuum height bya half would result in a quarter of the vacuum resistivityneeded to maintain the same vacuum magnetic diffusiontransit time. Reducing the vacuum resistivity needed for aconverged result would be beneficial for codes with explicitdiffusion algorithms.The two left-most plots of Figure 11 have the same mag-netic diffusion transit time, but show starkly different ETIgrowth. This implies the vacuum magnetic diffusion tran-sit time is not an underlying mechanism for the convergedvacuum resistivity threshold, suprisingly. The convergedvacuum resistivity is approximately 5 times the nominalresistivity from previous studies, and is the same for bothvacuum sizes. A more relevant scale parameter for the con-verged results may be the ratio of the liner resistivity to thevacuum resistivity, as this influences the reconstruction ofderivatives given the spatially-varying resistivity. Chang-ing the numerical derivatives leads to different values ofcurrent through its impact on ohmic heating, thereby pro-ducing differences in the nonlinear ETI growth.Based on these findings, to get the converged ETI result(infinitely resistive vacuum) requires a minimum vacuum-to-liner resistivity ratio of 2 × . Ares is used here be-cause of the challenges associated with performing sucha convergence study with an explicit diffusion vacuummodel, such as with USim’s STS diffusion algorithm, socare is needed when checking the convergence of vacuumresistivity using explicit codes. When simulating a vacuum using a fluid code, the vac-uum density is traditionally set relatively low. [2, 4] Notevolving ohmic heating, S (cid:15) in Equation 3, and the mag-netic acceleration of the vacuum, Equation 2 through S B in Equation 4, allows for a less restrictive time step byreducing the sound speed. For the nonlinear ETI simula-tions, the vacuum density is evolved with a floor value ofthe initial vacuum density (2 . × − g cm − ). Varying11 vac x50
25 0 25 vac x10
25 0 25 vac x2
25 0 25 vac x0.1
25 0 25 vac x0.01 x [ m] [ g cc ] y [ m ] Figure 13: Density evolution of the nonlinear ETI simulations ispresented at 60 ns varying the vacuum density. this floor value determines the effect of vacuum hydrody-namics on nonlinear ETI behavior for this particular setup.Figure 13 shows the nonlinear ETI simulation at 60 nsfor vacuum density varying from 50x to 0.01x of the basevacuum density (2 . × − kg m − ). There is no discern-able difference between the simulations in Figure 13. Allother values of vacuum density show qualitatively simi-lar results when neglecting ohmic heating and magneticacceleration of the vacuum. Running with too small ofa vacuum density leads to long simulation times due toshort time steps required to resolve potential hydrody-namic (acoustic) oscillations in the vacuum. More mod-erate values of vacuum density of approximately 2 to 3orders of magnitude lower than the reference liner densityproduce converged results. Using such a large vacuum den-sity only changes the dynamics when significant vacuuminertia is added that impedes the ablation of the liner.Including magnetic acceleration and ohmic heating wouldchange the outcome of this converged density ratio, but isnot explored here.
5. Conclusion
This work compares nonlinear ETI simulations usingtwo different codes with significantly different algorithmicapproaches to solving the resistive-MHD equations. Al-though these codes differ in many ways, the most signif-icant difference, relative to simulating nonlinear ETI, isin the spatial and temporal discretization of the diffusionterms as discussed in Section 2. The handling of theseterms directly affects the evaluation of the magnetic diffu-sion wave and ohmic heating which are both essential insimulating nonlinear ETI growth. Furthermore, the rangeof viable parameters for stable and efficient computationalresults is dictated by the discretization methods. For thesenonlinear ETI simulation comparisons, a tabulated EOSfor aluminum, BMEOS, is developed that compares wellwith the previously used SES3720 table. [2]Section 3 shows development of verification test casesevaluating each code’s diffusion capabilities for S (cid:15) and S B from Equations 3 and 4, respectively. First, the codessimulate a simple magnetic diffusion test case showingthat both codes accurately recover the analytical solution. Next, the codes recover the theoretical linear ETI growthrate shown in Equation 13. Comparing the simulationgrowth rate to the theoretical one for varying spatial reso-lution and time step size results in the convergence shownin Figure 4, where the anticipated orders of accuracy ofspatial and temporal discretization are obtained. Thesetests give confidence in each code’s ability to handle thefundamental aspects of simulating nonlinear ETI.Section 4 compares the codes for simulating nonlinearETI with the baseline setup shown in Figure 5. This sim-ulation has full coupling of both source terms S (cid:15) and S B to the full set of MHD equations, Equations 1-4, and alsoincludes the non-ideal-gas BMEOS. The simulation usesan analytic form for the current rise akin to a typical Z-machine current rise with a correct prepulse as representedin Figure 6. Figures 7-9 show qualitative and quantitativeagreement between the two codes, although interesting dif-ferences arise in the details of mode evolution.Additionally, the Ares nonlinear ETI simulation under-goes a sensitivity analysis for the vacuum conditions shownin Sections 4.3 and 4.4. The analyses show a strong de-pendence on the vacuum resistivity, but not the vacuumdensity. Specifically, the resistivity analysis shows thata vacuum-to-liner resistivity ratio of ≈ × is suffi-cient to capture the converged (i.e., the infinitely resistivevacuum limit) nonlinear ETI simulation. The EOS im-plementation is tested across different interpolation algo-rithms, EOS tables, and initial conditions, showing a largedependence of nonlinear ETI growth to the EOS table andinterpolation algorithm, specifically near the solid densityand low pressure state. These sensitivity analyses provideguidelines for how codes that explicitly integrate diffusionterms can still capture nonlinear ETI without the need fora infinitesimally dense and infinitely resistive vacuum. Acknowledgement
This work was supported through the Lawrence Liver-more National Laboratory Weapons and Complex Integra-tion (LLNL WCI) High Energy Density Fellowship, andthrough the US Department of Energy under grants DE-SC0016515, DE-SC0016531, & DE-NA0003881. In addi-tion a portion of this work was sponsored by LLNL WCIHED summer program. This research used resources ofthe National Energy Research Scientific Computing Cen-ter (NERSC), a U.S. Department of Energy Office of Sci-ence User Facility operated under Contract No. DE-AC02-05CH11231. This document has been approved for releaseunder LLNL-JRNL-812448.A portion of this work was performed under the aus-pices of the U.S. Department of Energy by Lawrence Liv-ermore National Laboratory under Contract DE-AC52-07NA27344. This document was prepared as an account ofwork sponsored by an agency of the United States govern-ment. Neither the United States government nor LawrenceLivermore National Security, LLC, nor any of their em-ployees makes any warranty, expressed or implied, or as-12umes any legal liability or responsibility for the accu-racy, completeness, or usefulness of any information, ap-paratus, product, or process disclosed, or represents thatits use would not infringe privately owned rights. Refer-ence herein to any specific commercial product, process,or service by trade name, trademark, manufacturer, orotherwise does not necessarily constitute or imply its en-dorsement, recommendation, or favoring by the UnitedStates government or Lawrence Livermore National Se-curity, LLC. The views and opinions of authors expressedherein do not necessarily state or reflect those of the UnitedStates government or Lawrence Livermore National Secu-rity, LLC, and shall not be used for advertising or productendorsement purposes.
References [1] V. I. Oreshkin, Thermal instability during an electrical wireexplosion, Physics of Plasmas 15 (2008) 092103.[2] K. J. Peterson, D. B. Sinars, E. P. Yu, M. C. Herrmann, M. E.Cuneo, S. A. Slutz, I. C. Smith, B. W. Atherton, M. D. Knud-son, C. Nakhleh, Electrothermal instability growth in magnet-ically driven pulsed power liners, Physics of Plasmas 19 (2012)092701.[3] S. A. Slutz, et al., Pulsed-power-driven cylindrical linerimplosions of laser preheated fuel magnetized with an axialfield), Physics of Plasmas 17 (2010). URL: http://scitation.aip.org/content/aip/journal/pop/17/5/10.1063/1.3333505 .doi: http://dx.doi.org/10.1063/1.3333505 .[4] K. J. Peterson, E. P. Yu, D. B. Sinars, M. E. Cuneo, S. A.Slutz, J. M. Koning, M. M. Marinak, C. Nakhleh, M. C. Her-rmann, Simulations of electrothermal instability growth in solidaluminum rods, Physics of Plasmas 20 (2013) 056305.[5] K. J. Peterson, T. J. Awe, E. P. Yu, D. B. Sinars, E. S. Field,M. E. Cuneo, M. C. Herrmann, M. Savage, D. Schroen, K. Tom-linson, C. Nakhleh, Electrothermal instability mitigation byusing thick dielectric coatings on magnetically imploded con-ductors, Phys. Rev. Lett. 112 (2014) 135002. URL: https://link.aps.org/doi/10.1103/PhysRevLett.112.135002 . doi: .[6] J. Loverich, S. C. Zhou, K. Beckwith, M. Kundrapu, M. Loh,S. Mahalingam, P. Stoltz, A. Hakim, Nautilus: A tool for mod-eling fluid plasmas, in: 51st AIAA Aerospace Sciences Meetingincluding the New Horizons Forum and Aerospace Exposition,Grapevine, Texas, 2013.[7] B. E. Morgan, J. A. Greenough, Large-eddy and unsteady ranssimulations of a shock-accelerated heavy gas cylinder, ShockWaves 26 (2016) 355–383. URL: https://doi.org/10.1007/s00193-015-0566-3 . doi: .[8] C. L. Ellison, H. D. Whitley, C. R. D. Brown, S. R. Copeland,W. J. Garbett, H. P. Le, M. B. Schneider, Z. B. Walters,H. Chen, J. I. Castor, R. S. Craxton, M. Gatu Johnson, E. M.Garcia, F. R. Graziani, G. E. Kemp, C. M. Krauland, P. W.McKenty, B. Lahmann, J. E. Pino, M. S. Rubery, H. A.Scott, R. Shepherd, H. Sio, Development and modeling of apolar-direct-drive exploding pusher platform at the nationalignition facility, Physics of Plasmas 25 (2018) 072710. URL: https://doi.org/10.1063/1.5025724 . doi: . arXiv:https://doi.org/10.1063/1.5025724 .[9] R. M. Darlington, T. L. McAbee, G. Rodrigue, A study ofale simulations of rayleigh–taylor instability, Computer PhysicsCommunications 135 (2001) 58–73.[10] B. Van Leer, Towards the ultimate conservative differencescheme. v. a second-order sequel to godunov’s method, Journalof computational Physics 32 (1979) 101–136.[11] T. Miyoshi, K. Kusano, A multi-state hll approximate riemann solver for ideal magnetohydrodynamics, Journal of Computa-tional Physics 208 (2005) 315–344.[12] J. R. King, R. Masti, B. Srinivasan, K. Beckwith, Multidimen-sional tests of a finite-volume solver for mhd with a real-gasequation of state, IEEE Transactions on Plasma Science 48(2020) 902–913.[13] A. Dedner, F. Kemm, D. Kr¨oner, C.-D. Munz, T. Schnitzer,M. Wesenberg, Hyperbolic divergence cleaning for the mhdequations, Journal of Computational Physics 175 (2002) 645–673.[14] V. Alexiades, G. Amiez, P.-A. Gremaud, Super-time-steppingacceleration of explicit schemes for parabolic problems, Commu-nications in numerical methods in engineering 12 (1996) 31–42.[15] D. Mavriplis, Revisiting the least-squares procedure for gra-dient reconstruction on unstructured meshes, in: 16th AIAAComputational Fluid Dynamics Conference, 2003, p. 3986.[16] K. D. Lathrop, Ray effects in discrete ordinates equations, Nu-clear Science and Engineering 32 (1968) 357–369.[17] J. I. Castor, Radiation hydrodynamics, 2004.[18] G. Pert, Physical constraints in numerical calcula-tions of diffusion, Journal of Computational Physics42 (1981) 20 – 52. URL: . doi: https://doi.org/10.1016/0021-9991(81)90231-X .[19] S. D. Crockett, Al-13, LA-UR-04-6442 (Aug. 28). URL: .[20] M. Desjarlais, J. Kress, L. Collins, Electrical conductivity forwarm, dense aluminum plasmas and liquids, Physical ReviewE 66 (2002) 025401.[21] R. D. McBride, S. A. Slutz, A semi-analytic model of magne-tized liner inertial fusion, Physics of Plasmas 22 (2015) 052708.[22] F. Birch, Finite elastic strain of cubic crystals, Phys. Rev.71 (1947) 809–824. URL: https://link.aps.org/doi/10.1103/PhysRev.71.809 . doi: .[23] F. D. Murnaghan, Finite deformations of an elastic solid,American Journal of Mathematics 59 (1937) 235–260. URL: .[24] F. D. Murnaghan, The compressibility of media un-der extreme pressures, Proceedings of the NationalAcademy of Sciences 30 (1944) 244–247. URL: . doi: . .[25] R. More, K. Warren, D. Young, G. Zimmerman, A new quotid-ian equation of state (qeos) for hot dense matter, The Physicsof fluids 31 (1988) 3059–3078.[26] D. A. Young, E. M. Corey, A new global equation of statemodel for hot, dense matter, Journal of applied physics 78(1995) 3748–3755.[27] F. N. Fritsch, The LEOS Interpolation Package, Technical Re-port, Lawrence Livermore National Lab., CA (US), 2003.[28] C. W. Cranfill, EOSPAC: A subroutine package for accessingthe Los Alamos SESAME EOS data library, Technical Report,Los Alamos National Lab., NM (USA), 1983.[29] D. Ryutov, M. S. Derzon, M. K. Matzen, The physics of fast zpinches, Reviews of Modern Physics 72 (2000) 167.[30] D. B. Sinars, et al., Measurements of magneto-rayleigh-taylor instability growth during the implo-sion of initially solid metal liners, Physics of Plas-mas 18 (2011) 056301. URL: http://aip.scitation.org/doi/abs/10.1063/1.3560911 . doi: . arXiv:http://aip.scitation.org/doi/pdf/10.1063/1.3560911 .[31] J. Pecover, J. Chittenden, Instability growth for magnetizedliner inertial fusion seeded by electro-thermal, electro-choric,and material strength effects, Physics of Plasmas 22 (2015)102701. ppendix A. Comparison of BMEOS andSES3720 T [ l o g ( K )] [log (kg/m )] % D i ff [ % ] Figure A.14: Percent difference of the specific internal energy densitybetween the BMEOS and SES3720 in the state space relevant to thenonlinear ETI evolution in HED regimes. The plot on the right isan expanded scale of the left plot to highlight the region of largestdifference.
Appendix B. Coefficients of the multimode per-turbation mode (i) β ii