Magneto-Rayleigh-Taylor growth and feedthrough in cylindrical liners

Summary

This presentation investigates Magneto-Rayleigh-Taylor (MRT) instability growth and feedthrough in cylindrical liners relevant to the Magnetized Liner Inertial Fusion (MagLIF) concept on the Z-machine. It compares linearized ideal MHD analytical models and Hydra rad-hydro-MHD simulations against experimental Z-machine data from seeded aluminum liner implosions. The results demonstrate that early-time growth is dominated by classical Rayleigh-Taylor acceleration, while axial magnetic field compression can mitigate feedthrough and stabilize inner-surface growth at high convergence.

Title Slide

M. R. Weis, Y.Y. Lau, R.M. Gilgenbach Plasma, Pulsed Power and Microwave Laboratory Nuclear Engineering and Radiological Sciences Dept., University of Michigan Ann Arbor, MI 48109-2104 USA

M. Hess, C. Nakhleh Sandia National Laboratories Albuquerque, NM 87185 USA

Motivation

• MRT is one of the greatest challenges to success of the Magnetized Liner Inertial Fusion (MagLIF) concept

Magnetic fields introduce additional complexity over classical RTI • Feedthrough has an important role in the stability of the fuel/liner interface in MagLIF concept
Also relevant to dynamic materials experiments on Z • Analytic results provide a fast way to analyze these problems • Hydra, a rad-hydro-MHD code, provides another tool for modeling experiments on Z and other HEDP platforms
Needs benchmarking

Goal: apply these tools to a liner implosion and compare to experimental results

Magnetized Liner Inertial Fusion (MagLIF)

Magnetized Liner Inertial Fusion (MagLIF)*,** may be a promising path to high yields on Z, but liner integrity is critical

• Initial calculations suggest Z-Beamlet laser can do the preheat

Preheating the fuel reduces the required compression ratio to obtain ignition temperatures to about 30 on Z
Preheating reduces the implosion velocity needed to about 5-10 cm/μs • Axial magnetic field strength required (about 5-10 T) feasible
Similar coil design parameters to coils for dynamic materials tests • Simulations suggest 100 kJ yields on Z are possible • Success of MagLIF hinges on maintaining sufficient liner integrity

S. A. Slutz et al., “Pulsed power driven cylindrical liner implosions with magnetized and preheated fuel”, Phys. Plasmas 17 056303 (2010).

Cylindrical geometry instabilities

Cylindrical geometry instabilities:

MRT (acceleration)
Sausage / m=0
Kink / m=1 (present with no acceleration in a cylindrical current carrying plasma)

g = - d/dt (v_r) g = effective gravity in rest frame of interface • g > 0: Implosion • g < 0: Stagnation

Rayleigh-Taylor Instability (RTI)

• Interchange instability from a light fluid pushing a heavy fluid

Water on top of oil in Earth’s gravity
Deep water waters are the stable form of RTI (water supporting air) • Instability arises for: ∇p · ∇ρ < 0 • Comparison of fluid RTI (water/oil) with magnetic/liner configuration and fuel/liner interface.

Ideal MHD Equations

Mass Conservation: ∂ρ/∂t + ∇·(ρv) = 0

Momentum Conservation: ρ(∂v/∂t + v·∇v) = -∇p + J × B + ρg

Ampere’s Law: ∇ × B = μ₀J

Faraday/Ohm Law: ∂B/∂t = ∇ × (v × B)

Perturbation of equilibrium

We assume that the time scale for perturbation growth is fast compared to liner dynamics, yielding an approx. instantaneous equilibrium: dp/dr + (1/μ₀) [ B_z (dB_z/dr) + (B_θ/r) (d/dr)(rB_θ) ] = ρg

We perturb this equilibrium by a small displacement of the form: ξ_vector(r_vector, t) = (ξ_r(r), ξ_θ(r), ξ_z(r)) e^(γt + ikz + imθ)

We assume that the perturbed velocity is incompressible: ∇ · (∂ξ_vector/∂t) = ∇ · ξ_vector = 0

The growth rate, γ (or ω), is of the form: γ² ≈ kg - (k · B)²/(μ₀ρ) + C(m,k) Where C includes the effects of azimuthal and current carrying modes.

Sharp boundary model

Sharp boundary model in cylindrical geometry with constant liner density ρ_L = const, thickness Δ, inner radius r_i, outer radius r_e, and external/internal magnetic fields.

Aspect ratio: AR = r_e / (r_e - r_i) = r_e / Δ

The feedthrough of instability from the outer to inner surface for a given mode, ω, is defined as: ξ_r(r_i) / ξ_r(r_e) ≡ F(ω)

Linearized Ideal MHD Formulation

We solve the linearized ideal MHD equations: ρ ∂²ξ/∂t² = -∇p₁ + J₀ × B₁ + J₁ × B₀ + ρg · ξ = -∇p₁ + (∇ × B₀) × [∇ × (ξ × B₀)]/μ₀ + {∇ × [∇ × (ξ × B₀)] × B₀}/μ₀ + ρg · ξ

• Subject to the boundary conditions of continuity of total pressure at each interface, which is an eigenvalue problem for the eigenfunction, ξ, and eigenvalue, ω • The solution is analytically tractable for:

Constant density profiles (may be different in each region)
Constant Bz profiles (may be different in each region)
No magnetic diffusion of drive field • Otherwise the problem is solved numerically using a shooting method

Sausage and kink modes recovered

Sausage and kink modes are successfully recovered • For g = 0 and AR = 1 (solid plasma column undergoing no acceleration) give well known test problem • B_θ = 1000 T drives instability but also stabilizes m > 0 modes to some extent (bent field lines) • B_z > B_θ / √2 stabilizes

Boyd and Sanderson. The Physics of Plasmas, Cambridge Press, 2003

m = 0 modes stabilized by Bz only

m = 0 modes will be stabilized by Bz only ω² ≈ kg - (1 / μ₀ρ) [ (m/r) B_θ + k_z B_z ]²

Comparison of dispersion curves for No acceleration vs Implosion acceleration with B_θ = 1000 T, r_e = 3.47 mm, A_R = 6, and various values of B_z (10 T to 630 T).

Feedthrough reduction with Bz in AR=6 liners

AR=6 liners show feedthrough reduction with Bz as expected • Reverse feedthrough also exists for small kr

This is not present in planar results!
Increasing g reduces this effect • Plots showing feedthrough vs kr_e for no acceleration vs implosion acceleration across varying axial magnetic fields (B_z = 10 T to 630 T).

Field requirements for stabilization

For significant feedthrough and MRT stabilization, require: B_z ≈ B_θ • This is obtained by compressing the applied Bz seed field: B_z(t) = B_z0 [ r_i0 / r_i(t) ]² = B_z0 CR² • This assumes no loss of field from Nernst effect • The outer surface MRT will never be stabilized but there is hope to slow growth on the inner surface:

Minimize initial seeding from feedthrough
Stabilize growth via strong B_z • The limits for: kr << 1 will need to be examined more closely due to the peculiar behavior seen • Sausage and kink mode may complicate this stabilization

Using realistic data as input into linearized model

• Average physical quantities from 1D Hydra data in each ‘region’

Running Lagrangian zones can be used to find liner/vacuum interfaces and, hence, the boundaries for averaging • For a given wavelength we can calculate the instantaneous growth rate, ω(t) for each time step • The amplitude, η, of the instability is then determined by: d²η(t)/dt² = ω(t)² η(t) • The feedthrough between interfaces is just the ratio of the eigenfunction at the inner and outer interface: F(γ) = ξ(r_i) / ξ(r_e)

Aluminum liner experiments on Z with seeded MRT

• A 1D simulation with Hydra can be driven with the measured load current from which we can extract our averaged physical quantities • Demonstrates setup and experimental radiograph of seeded MRT on Z shot z1965 with measured load current reaching ~20 MA.

Sinars et. al. Phys. Plasmas 18, 056301 (2011)

Applying linearized model to Sinars et. al. experiments

Applying linearized model to Sinars et. al.* experiments shows good agreement while convergence is low • Aluminum liner seeded with 400 μm surface perturbation • MRT Amplitude vs Time shows close alignment between Planar, Cylindrical (m=0), Lasnex simulation, and Experimental data up to ~80 ns • Inner/outer radii dynamics show that as convergence increases, growth rate becomes more complicated

Sinars et. al. Phys. Plasmas 18, 056301 (2011)

Growth and feedthrough dominated by classical RT

While g is large and convergence is small, growth is dominated by classical Rayleigh-Taylor growth rate: ω² ≈ kg >> - (k · B)²/(μ₀ρ) + C(m,k) • If we remove g for the same problem, we see the remaining physics gives much lower growth • Feedthrough is similarly dominated by the classical expression e^(-kΔ)

Hydra modeling of Al liner implosions

Hydra has been used to model Al liner implosions with seeded MRT • A sinusoidal perturbation of λ = 400 μm was applied to the surface of an Al liner and an implosion was driven using the load current on shot z1965 in attempt to replicate the MRT growth rates shown earlier

Sinars et. al. Phys. Plasmas 18, 056301 (2011)

Z-Beamlet Laser (ZBL) radiography setup

ZBL is used to create the one or two frame 6.151 keV radiograph images • Radiograph lines of sight are ± 3° from horizontal when using two frame radiograph

This can introduce shadowing of short wavelength modes • Straight on (0°) radiographs can alleviate this but only can take one frame

Simulated vs Experimental Radiographs

• Simulated radiographs (from SPECT3D) are generated from X-ray transmission through plasma onto a submicron resolution detector and a 15 μm blur is added (ZBL resolution) • Comparing to radiographs from Sinars et. al. (2011) at t = 63.6 ns show excellent agreement both in amplitude and gross features even at 0°

Estimating growth by FFT or direct calculation

We can also estimate the growth by FFT or direct calculation • Axial FFT of result: ∫ ρ(r,z) r dr = m_L(z) m_L(k) ≈ ∫ m_L(z) e^(-ikz) dz • Direct calculation: choose 50% transmission contour to determine Bubble radius and Spike radius

Hydra agreement with experiment

Hydra shows excellent agreement • Comparison of MRT instability amplitude vs time across Lasnex, Experiment, Hydra - FFT, and Hydra - Bubble/Spike • FFT method chronically underestimates growth, possibly due to resolution issues • Later times show 400 μm peak is broadening to couple with nearby modes • Though the FFT growth calculation slows, bubble/spike shows continued growth as expected

Summary of results

• Analytic calculations show:

Early time feedthrough is minimal and classical RT dominates
At high convergence, Bz can do some good, cylindrical modes could cause problems • Hydra seems to do a good job of getting MRT correct
Amplitude growth as a function of time matches data well
Simulated radiographs match data well for most times
Tilted views tend to smooth over stranger structure and give better agreement • As non-linear MRT starts to dominate agreement with radiographs begins to degrade which could be due to any number of issues:
Insufficient resolution
Meshing issues
Missing physics (3D, Hall, etc.)

Future work

• Analyze MagLIF implosions at high convergence with analytic calculations

Analyze inner surface behavior for seeding of MRT at early times
Effect of shock propagating through liner • Use Hydra output to characterize feedthrough and compare to analytic theory
Inner interface is invisible to radiography for aluminum
Analyze inner surface stability (ET, MRT)
Feedthrough should be most important at high convergence which is difficult to image anyway • Further stress Hydra’s predictive capabilities with the latest experiments on the Z-machine