Abstract
We evaluate the stability of electron currentflow in highpower magnetically insulated transmission lines (MITLs). A detailed model of electron flow in crossfield gaps yields a dispersion relation for electromagnetic (EM) transverse magnetic waves [R. C. Davidson et al., Phys. Fluids27, 2332 (1984)] which is solved numerically to obtain growth rates for unstable modes in various sheath profiles. These results are compared with twodimensional (2D) EM particleincell(PIC) simulations of electron flow in highpower MITLs. We find that the macroscopic properties (charge and current densities and selffields) of the equilibrium profiles observed in the simulations are well represented by the laminarflow model of Davidson et al. Idealized simulations of sheared flow in electron sheaths yield growth rates for both long (diocotron) and short (magnetron) wavelength instabilities that are in good agreement with the dispersion analysis. We conclude that electron sheaths that evolve selfconsistently from spacechargedlimited emission of electrons from the cathode in wellresolved 2D EM PIC simulations form stable profiles.
The authors are extremely grateful for the sustained support to this research by J. Porter, K. Matzen, and L. Schneider. We thank M. Dyson and S. Welch for technical editing. This work is supported by the Department of Energy through Sandia National Laboratories. Sandia National Laboratories is a multiprogram laboratory operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Co., for the U.S. Department of Energy’s National Nuclear Security Administration under Contract No. DEAC0494AL85000.
I. INTRODUCTION
II. EQUILIBRIUM ELECTRON FLOW IN SELFLIMITED MITL SIMULATIONS
III. STABILITY ANALYSIS OF LAMINAR ELECTRON FLOWS IN PLANAR MITLS
A. Equilibrium model
B. Linear stability analysis
C. Numerical results
IV. IDEALIZED MITL SIMULATION MODEL
A. Case 3sim: Stable equilibrium sheath from the PIC simulation
B. Case 3b: An example of an unstable sheath profile
V. SUMMARY AND DISCUSSION
Key Topics
 Particleincell method
 36.0
 Cathodes
 25.0
 Plasma sheaths
 16.0
 Brillouin scattering
 14.0
 Laminar flows
 14.0
Figures
Schematic of the 2D parallelplate MITL simulation model (not drawn to scale).
Schematic of the 2D parallelplate MITL simulation model (not drawn to scale).
Plots of the magnitudes of the (a) electric and (b) magnetic field profiles and (c) the sheath number density between and 65 cm after 10 ns from the full MITL simulation at 3.22 MV.
Plots of the magnitudes of the (a) electric and (b) magnetic field profiles and (c) the sheath number density between and 65 cm after 10 ns from the full MITL simulation at 3.22 MV.
(a) Average sheath density, (b) electric field, (c) magnetic field, and (d) current density as a function of distance across the AC gap for three simulations at 1.47, 3.22, and 10.57 MV. All data averaged in between 55 and 65 cm after steadystate equilibrium conditions are reached .
(a) Average sheath density, (b) electric field, (c) magnetic field, and (d) current density as a function of distance across the AC gap for three simulations at 1.47, 3.22, and 10.57 MV. All data averaged in between 55 and 65 cm after steadystate equilibrium conditions are reached .
Sample electron orbits in the equilibrium section of the sheath, launched from the cathode at for the 3.22 MV simulation. Test electrons are launched at , 50, 55, 60, and 65 cm.
Sample electron orbits in the equilibrium section of the sheath, launched from the cathode at for the 3.22 MV simulation. Test electrons are launched at , 50, 55, 60, and 65 cm.
Schematic of the MITL flow model in the planar diode configuration.
Schematic of the MITL flow model in the planar diode configuration.
Equilibrium MITL characteristics from the 3.22 MV PIC simulation (solid curves) and the analytic model of Sec. III A (dashed curves representing case 3sim from Table II) plotted as a function position between the cathode and anode . The sheath electron density is plotted in (a) along with the density from the minimum current Brillouin flow theory (dotdashed curve). The and field components are compared in (b) and (c) and the sheath current density is compared in (d).
Equilibrium MITL characteristics from the 3.22 MV PIC simulation (solid curves) and the analytic model of Sec. III A (dashed curves representing case 3sim from Table II) plotted as a function position between the cathode and anode . The sheath electron density is plotted in (a) along with the density from the minimum current Brillouin flow theory (dotdashed curve). The and field components are compared in (b) and (c) and the sheath current density is compared in (d).
(a) Sheath density, , and (b) unstable growth rate spectra for the 3.22 MV model sheath equilibria given in Table II.
(a) Sheath density, , and (b) unstable growth rate spectra for the 3.22 MV model sheath equilibria given in Table II.
Eigenfunctions for the case 3b equilibrium of Table II at (a) , (b) , and (c) .
Eigenfunctions for the case 3b equilibrium of Table II at (a) , (b) , and (c) .
(a) Sheath density profiles and (b) growth rate spectra for Brillouin flow equilibrium with (dashed curves) and the case 3c equilibrium (solid curves).
(a) Sheath density profiles and (b) growth rate spectra for Brillouin flow equilibrium with (dashed curves) and the case 3c equilibrium (solid curves).
Schematic of the 2D idealized PIC simulation geometry. The simulation box is of length with periodic boundaries at and . The anode and cathode are conducting boundaries separated by a distance . A laminar sheared electron sheath profile is preloaded into the simulation along with the initially static magnetic and electric equilibrium fields.
Schematic of the 2D idealized PIC simulation geometry. The simulation box is of length with periodic boundaries at and . The anode and cathode are conducting boundaries separated by a distance . A laminar sheared electron sheath profile is preloaded into the simulation along with the initially static magnetic and electric equilibrium fields.
Sample instability growth rate calculation result for case 3b perturbed at . The solid circles are the electric field amplitude associated with the perturbed wavelength at different times. The line indicates the growth rate calculated from the stability model.
Sample instability growth rate calculation result for case 3b perturbed at . The solid circles are the electric field amplitude associated with the perturbed wavelength at different times. The line indicates the growth rate calculated from the stability model.
Normalized (a) real and (b) imaginary frequencies as a function of dimensionless wavenumber . The black curves are from the dispersion analysis applied to the case 3b sheath density profile. The individual points are from idealized simulations of the case 3b equilibrium perturbed at different wavelengths.
Normalized (a) real and (b) imaginary frequencies as a function of dimensionless wavenumber . The black curves are from the dispersion analysis applied to the case 3b sheath density profile. The individual points are from idealized simulations of the case 3b equilibrium perturbed at different wavelengths.
(a) Sheath density and (b) magnetron instability growth rate spectra for equilibria corresponding to Eq. (2) with , , , and and five values of the slope parameter : (Brillouin flow) and .
(a) Sheath density and (b) magnetron instability growth rate spectra for equilibria corresponding to Eq. (2) with , , , and and five values of the slope parameter : (Brillouin flow) and .
Values of the high cutoff as a function of the inverse scale length parameter for the equilibria of Fig. 13(a) with .
Values of the high cutoff as a function of the inverse scale length parameter for the equilibria of Fig. 13(a) with .
(a) Sheath density and (b) unstable growth rate spectra for the 1.47 MV model sheath equilibria given in Table II. In (b), no unstable modes were found for case 1sim, the fitted equilibrium PIC simulation profile.
(a) Sheath density and (b) unstable growth rate spectra for the 1.47 MV model sheath equilibria given in Table II. In (b), no unstable modes were found for case 1sim, the fitted equilibrium PIC simulation profile.
(a) Sheath density and (b) unstable growth rate spectra for the 10.57 MV model sheath equilibria given in Table II. In (b), no unstable modes were found for case 10sim, the fitted equilibrium PIC simulation profile.
(a) Sheath density and (b) unstable growth rate spectra for the 10.57 MV model sheath equilibria given in Table II. In (b), no unstable modes were found for case 10sim, the fitted equilibrium PIC simulation profile.
Comparison of the sheath number density profiles. The initial sheath profile from case 3b is shown as a thin solid line. After 3.7 ns of simulation time, this profile evolves to the thick solid curve. This sheath profile is consistent with the “stable” sheath profile that arises selfconsistently in longMITL simulations (dashed curve) described in Sec. II.
Comparison of the sheath number density profiles. The initial sheath profile from case 3b is shown as a thin solid line. After 3.7 ns of simulation time, this profile evolves to the thick solid curve. This sheath profile is consistent with the “stable” sheath profile that arises selfconsistently in longMITL simulations (dashed curve) described in Sec. II.
Measured voltage (dashed line) near the beginning of the transmission line from Ref. 37 and the voltage (solid line) at from the LSP simulation.
Measured voltage (dashed line) near the beginning of the transmission line from Ref. 37 and the voltage (solid line) at from the LSP simulation.
Comparison of the measured and simulated anode (solid) and cathode (dashed) currents at five positions along the 10mlong MITL. The lefthand side of the shaded boxes in frames (a)–(d) indicates the arrival time of the retrapping wave launched when the selfinsulated wave front reaches the load in the experiment. The measured waveforms shown here are adapted from Fig. 3 of Ref. 37.
Comparison of the measured and simulated anode (solid) and cathode (dashed) currents at five positions along the 10mlong MITL. The lefthand side of the shaded boxes in frames (a)–(d) indicates the arrival time of the retrapping wave launched when the selfinsulated wave front reaches the load in the experiment. The measured waveforms shown here are adapted from Fig. 3 of Ref. 37.
Tables
Comparison of the anode , cathode , and electron sheath currents in the PIC simulations with the results of the MCB model. The ratio of the PICtoMCB currents is also given.
Comparison of the anode , cathode , and electron sheath currents in the PIC simulations with the results of the MCB model. The ratio of the PICtoMCB currents is also given.
Fitting parameters for the sheath equilibria used in the 3.22 MV stability analysis. Case 3sim corresponds to the equilibrium sheath profile in the PIC simulation result at 3.22 MV (see Sec. II). Cases 3a–3c are hypothetical sheath profiles as discussed in the text.
Fitting parameters for the sheath equilibria used in the 3.22 MV stability analysis. Case 3sim corresponds to the equilibrium sheath profile in the PIC simulation result at 3.22 MV (see Sec. II). Cases 3a–3c are hypothetical sheath profiles as discussed in the text.
Fitting parameters for the sheath equilibria used in the 1.47 and 10.57 MV stability analyses. Cases 1sim and 10sim correspond to the equilibrium sheath profile in the PIC simulation results at 1.47 and 10.57 MV, respectively. Cases 1a–1c and 10a–10c are hypothetical sheath profiles as discussed in the text.
Fitting parameters for the sheath equilibria used in the 1.47 and 10.57 MV stability analyses. Cases 1sim and 10sim correspond to the equilibrium sheath profile in the PIC simulation results at 1.47 and 10.57 MV, respectively. Cases 1a–1c and 10a–10c are hypothetical sheath profiles as discussed in the text.
Comparison of the anode , cathode , and electron sheath currents in the PIC simulation with the results of the MCB model for the Di Capua–Pellinen experiment at 1.6 MV. The average peak measured values for the anode and cathode currents are also given.
Comparison of the anode , cathode , and electron sheath currents in the PIC simulation with the results of the MCB model for the Di Capua–Pellinen experiment at 1.6 MV. The average peak measured values for the anode and cathode currents are also given.
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