Numerical Modeling of a Magnetic Flux Compression Experiment

Poster P3.083 Wednesday, February 15, 2006 1:00PM Poster Session III Rio Grande Exhibit Hall

Numerical modeling of a magnetic flux compression experiment* V. MAKHIN, B.S. BAUER, T.J. AWE, S. FUELLING, T. GOODRICH, I.R. LINDEMUTH, R.E. SIEMON, University of Nevada, Reno, U.S.A. S. F. Garanin, VNIIEF, Russia

Innovative Confinements Concepts 2006 Workshop February 13-16, 2006 Austin, Texas

Abstract A possible plasma target for Magnetized Target Fusion is a stable diffuse z pinch [1] like that of the MAGO experiments at VNIIEF [2]. The diffuse z pinch plasma resides in a toroidal cavity, e.g., between two cylindrical walls with end planes. A magnetic flux compression experiment is planned for this geometry as described in the adjacent poster. Even without injection of plasma, high-Z wall plasma is generated by eddy-current Ohmic heating from MG fields, as described recently by Garanin [3]. Results from numerical models show that the energy lost to Ohmic heating is a significant fraction of the available liner kinetic energy. Another significant inefficiency results from energy that goes into compression of liner and hard-core material according to the high-pressure equation of state for aluminum (or any metal). Despite these realistic losses, efficiency of liner compression expressed as compressed magnetic energy relative to liner kinetic energy can be close to 50%. Modeling also indicates that m=0 perturbations resulting from the Rayleigh-Taylor instability during deceleration of the liner are acceptably small if the initial perturbation level of density in the liner is around 1%. This will be an important aspect to check experimentally. Flux compression experiment is modeled with three codes: a) a semi-analytic ODE incompressible liner model, b) the Los Alamos RAVEN 1D Lagrangian code, and c) the Los Alamos 1D or 2D MHRDR Eulerian MHD simulation. 1. 2. 3.

R.E Siemon, et al., “Stability analysis and numerical simulation of a hard-core diffuse z-pinch during compression with Atlas facility liner parameters,” Nuclear Fusion 45, 1148 (2005) 2. S.F. Garanin, “The MAGO system,” IEEE Trans. Plasma Sci. 26,1230,(1998). I.R.Lindemuth et al., “Target plasma formation for Magnetic Compression/Magnetized Target Fusion (MAGO/MTF),”Phys.Rev.Lett. 75, 1953 (1995). S.F. Garanin, G.G. Ivanova, D.V. Karmishin, and V.N.Sofronov, “Diffusion of a magagauss field into metal,” J. Appl. Mech. Tech. Phys. 46, 153, (2005).

*Work supported by DOE OFES grant DE-FG02-04ER54752

Atlas power supply Summer 2005 -- began operation at Nevada Test Site

240 kV 24 MJ ~ 25 nH 30 MA

Standard person

ØD model (Matlab): Code and model features • • • •

Uses ode solver in the commercial Matlab software Realistic circuit model of Atlas is included Liner resistivity is assumed to be zero Internal current (and pressure) is computed as function of time before and after liner shorts the cavity • Liner is treated as incompressible – the “slug model” approximation that has been found useful for predicting liner dynamics • Improved model that includes approximate effects of compressibility is under development

1-D model (RAVEN): Code and model features • Lagrangian 1D code developed at Los Alamos National Laboratory • Uses Sesame tables for equation of state and resistivity (ignores thermal conductivity as appropriate for time scale of liner implosions) • Includes a realistic equivalent circuit for the Atlas capacitor bank

2-D Model (MHRDR): Code and model features • Eulerian code developed at Los Alamos National Laboratory. • Uses Sesame tables for material properties • Ohms law E+vxB=ηj • One-dimensional results similar to RAVEN • Two-dimensional results show Rayleigh Taylor instability as liner accelerates by external field and then decelerates as flux is compressed

Flux compression after flux injection z Conducting hard core

5cm

Liner

Part of Atlas current driving the liner is diverted inductively to generate the needed initial flux

~ 1.5 cm

Metal Insulator

This eliminates the need for an expensive auxiliary power supply. L1

L2

Atlas Bank Current

ØD model (Matlab): Current and liner radius vs. time 7

Outer Current and Radii vs time

x 10 2

60

Radius (mm)

40

1

Liner inner radius 20

0 0

0

0.5

1

1.5

2

Current (amps)

Atlas Current

-1 2.5 -5

Time (seconds)

x 10

Note: zero resistivity and incompressible liner

ØD model (Matlab): B and gap vs. time BatR1 and rin-R1 15

Gap between liner and hard core

B (T)

900

800

700

rin-R1, (mm)

B, (T)

600

10

500

400

300

200

Magnetic field on hard core

5

Gap (mm)

1000

100

0

1.1

1.2

1.3

1.4

1.5 time, (s)

1.6

Time (seconds)

1.7

1.8

0 2

1.9

-5

x 10

Note: zero resistivity and incompressible liner

1-D model (RAVEN): Radius vs time

Radius (m)

Liner radius Outer Inner

Hard core radius

Time (s) Aluminum properties; compressible liner

Energy (J)

1-D model (RAVEN): Energy 10-cm-long liner Total Energy Including heat and compression

Kinetic Energy Magnetic Energy

Time(s)

2-D Model (MHRDR): 15.9 μs Contours ρ,T,B,P Liner initially has 1% random density perturbations

Slightly before peak compression B = 2.2 MG Liner-hard core gap ~ 1.5 mm

2-D Model (MHRDR): 16.5 μs Contours ρ,T,B,P Liner initially has 1% random density perturbations

Maximum B = 2.8 MG Liner-hard-core gap has vapor and RT spokes of metal gap ~ 1 mm

Comparison of 1D simulation results for flux diffusion at a surface RAVEN MHRDR

Garanin

Non-linear diffusion B(x) after 1 μs. Field rises to 5 MG linearly in 1 μs. Copper surface is at x=0 at t=0.

Table summarizes main differences in modeling choices. The equation of state (EOS) means internal energy e and pressure p as functions of density ρ and temperature T. Also important is resistivity η(T, ρ). Choices

Garanin

MHRDR

RAVEN

EOS e(ρ,T); p(ρ,T)

A

B

B

Resistivity

A

B

B

Radiation modeling

A

off

on

Minimum zone size

0.1 micron

1 micron

1 micron

Choice A represents a model for EOS, η(T, ρ), and radiation transport coefficients [Garanin et al., Russian J. Appl. Mech. and Tech. Phys. 1, 30 (1990). Choice B for MHRDR means EOS and resistivity using Lindemuth selection of Sesame tables. MHRDR radiation is set to zero. RAVEN includes radiation using Sesame tables for opacity.

Temperature sensitive to modeling details

MHRDR

ρ (g/cm3)

Garanin Garanin

RAVEN

MHRDR RAVEN

Copper density profile at 1 μs for same conditions as above.

Copper temperature profile at 1 μs for same conditions as above

Temperature (eV)

RAVEN result for T(t) at vacuum boundary

0.1

0.5

0.3

0.7

0.9

Time (µs) vacuum cell 2 cell 4 cell 6

cell 1 cell 3 cell 5 cell 7

Stability of metal substrate is important experimental limitation 2D MHRDR simulation shows m=0 mode in metal substrate.

t = 4.0 μs

t = 5.5 μs

Figure shows two-dimensional density contours for an 8-mmradius aluminum rod driven with the Atlas 20-MA z current. Vertical direction is z (0-30 mm); horizontal direction is radius (020 mm).

t = 7.0 μs

t = 8.5 μs

Disruption begins at about peak magnetic field (6 μs).

Ruden’s ideas to sharpen current pulse zZ configuration

zθ configuration Surface to observe

Surface to observe

Liner

tinitial tfinal

Flux injection

Main current

0D simulation of zZ

Inner and outer liner radii (curves A), external liner current (B), and internal current (C) vs. time. Cavity length = 30 cm. Hard core radius = 10 mm.

Spacing (Delta) between liner surface and hard core vs. time (curve A). Magnetic field at hard-core surface (curve B). Time scale expanded around peak compression time.

Summary • Atlas offers an exciting possibility to advance the understanding of high-energy-density metal liners with application to MTF • Experimental behavior has been modeled numerically using three types of codes: 0D, 1D, and 2D. • Surface response to MG fields involves fascinating physics issues including EOS, resistivity, and radiation-MHD transport • This work involves collaboration between UNR, LANL, and AFRL in the USA and VNIIEF in Russia