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Title: Linear and nonlinear evolution of azimuthal clumping instabilities in a Z-pinch wire array

Abstract

This study presents an analytic theory on the linear and nonlinear evolution of the most unstable azimuthal clumping mode, known as the pi-mode, in a discrete wire array. In the pi-mode, neighboring wires of the array pair-up as a result of the mutual attraction of the wires which carry current in the same direction. The analytic solution displays two regimes, where the collective interactions of all wires dominate, versus where the interaction of the neighboring, single wire dominates. This solution was corroborated by two vastly different numerical codes which were used to simulate arrays with both high wire numbers (up to 600) and low wire number (8). All solutions show that azimuthal clumping of discrete wires occurs before appreciable radial motion of the wires. Thus, absence of azimuthal clumping of wires in comparison with the wires’ radial motion may imply substantial lack of wire currents. Finally, while the present theory and simulations have ignored the plasma corona and axial variations, it is argued that their effects, and the complete account of the three-dimensional feature of the pi-mode, together with a scaling study of the wire number, may be expediently simulated by using only one single wire in an annular wedgemore » with a reflection condition imposed on the wedge’s boundary.« less

Authors:
 [1];  [1];  [1];  [1];  [1];  [1];  [2];  [2];  [2];  [2]
  1. Univ. of Michigan, Ann Arbor, MI (United States). Dept. of Nuclear Engineering and Radiological Sciences
  2. Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)
Publication Date:
Research Org.:
Sandia National Lab. (SNL-NM), Albuquerque, NM (United States); Univ. of Michigan, Ann Arbor, MI (United States)
Sponsoring Org.:
USDOE
OSTI Identifier:
1427015
Report Number(s):
SAND2007-0206J
Journal ID: ISSN 1070-664X; 524280
Grant/Contract Number:
AC04-94AL85000
Resource Type:
Journal Article: Accepted Manuscript
Journal Name:
Physics of Plasmas
Additional Journal Information:
Journal Volume: 14; Journal Issue: 1; Journal ID: ISSN 1070-664X
Publisher:
American Institute of Physics (AIP)
Country of Publication:
United States
Language:
English
Subject:
70 PLASMA PHYSICS AND FUSION TECHNOLOGY; 97 MATHEMATICS AND COMPUTING; plasma pinch; Lagrangian mechanics; classical field theory; Z pinch; wire array Z pinches; plasma confinement; differential equations; plasma instabilities; calculus; classical electromagnetism

Citation Formats

Tang, Wilkin, Strickler, T. S., Lau, Y. Y., Gilgenbach, R. M., Zier, Jacob, Gomez, M. R., Yu, Edmund, Garasi, Chris, Cuneo, M. E., and Mehlhorn, T. A. Linear and nonlinear evolution of azimuthal clumping instabilities in a Z-pinch wire array. United States: N. p., 2007. Web. doi:10.1063/1.2434794.
Tang, Wilkin, Strickler, T. S., Lau, Y. Y., Gilgenbach, R. M., Zier, Jacob, Gomez, M. R., Yu, Edmund, Garasi, Chris, Cuneo, M. E., & Mehlhorn, T. A. Linear and nonlinear evolution of azimuthal clumping instabilities in a Z-pinch wire array. United States. doi:10.1063/1.2434794.
Tang, Wilkin, Strickler, T. S., Lau, Y. Y., Gilgenbach, R. M., Zier, Jacob, Gomez, M. R., Yu, Edmund, Garasi, Chris, Cuneo, M. E., and Mehlhorn, T. A. Wed . "Linear and nonlinear evolution of azimuthal clumping instabilities in a Z-pinch wire array". United States. doi:10.1063/1.2434794. https://www.osti.gov/servlets/purl/1427015.
@article{osti_1427015,
title = {Linear and nonlinear evolution of azimuthal clumping instabilities in a Z-pinch wire array},
author = {Tang, Wilkin and Strickler, T. S. and Lau, Y. Y. and Gilgenbach, R. M. and Zier, Jacob and Gomez, M. R. and Yu, Edmund and Garasi, Chris and Cuneo, M. E. and Mehlhorn, T. A.},
abstractNote = {This study presents an analytic theory on the linear and nonlinear evolution of the most unstable azimuthal clumping mode, known as the pi-mode, in a discrete wire array. In the pi-mode, neighboring wires of the array pair-up as a result of the mutual attraction of the wires which carry current in the same direction. The analytic solution displays two regimes, where the collective interactions of all wires dominate, versus where the interaction of the neighboring, single wire dominates. This solution was corroborated by two vastly different numerical codes which were used to simulate arrays with both high wire numbers (up to 600) and low wire number (8). All solutions show that azimuthal clumping of discrete wires occurs before appreciable radial motion of the wires. Thus, absence of azimuthal clumping of wires in comparison with the wires’ radial motion may imply substantial lack of wire currents. Finally, while the present theory and simulations have ignored the plasma corona and axial variations, it is argued that their effects, and the complete account of the three-dimensional feature of the pi-mode, together with a scaling study of the wire number, may be expediently simulated by using only one single wire in an annular wedge with a reflection condition imposed on the wedge’s boundary.},
doi = {10.1063/1.2434794},
journal = {Physics of Plasmas},
number = 1,
volume = 14,
place = {United States},
year = {Wed Jan 31 00:00:00 EST 2007},
month = {Wed Jan 31 00:00:00 EST 2007}
}

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  • This paper presents an analytic theory on the linear and nonlinear evolution of the most unstable azimuthal clumping mode, known as the pi-mode, in a discrete wire array. In the pi-mode, neighboring wires of the array pair-up as a result of the mutual attraction of the wires which carry current in the same direction. The analytic solution displays two regimes, where the collective interactions of all wires dominate, versus where the interaction of the neighboring, single wire dominates. This solution was corroborated by two vastly different numerical codes which were used to simulate arrays with both high wire numbers (upmore » to 600) and low wire number (8). All solutions show that azimuthal clumping of discrete wires occurs before appreciable radial motion of the wires. Thus, absence of azimuthal clumping of wires in comparison with the wires' radial motion may imply substantial lack of wire currents. While the present theory and simulations have ignored the plasma corona and axial variations, it is argued that their effects, and the complete account of the three-dimensional feature of the pi-mode, together with a scaling study of the wire number, may be expediently simulated by using only one single wire in an annular wedge with a reflection condition imposed on the wedge's boundary.« less
  • A simple model is constructed to evaluate the temporal evolution of azimuthal clumping instabilities in a cylindrical array of current-carrying wires. An analytic scaling law is derived, which shows that randomly seeded perturbations evolve at the rate of the fastest unstable mode, almost from the start. This instability is entirely analogous to the Jeans instability in a self-gravitating disk, where the mutual attraction of gravity is replaced by the mutual attraction among the current-carrying wires.
  • Experimental evidence suggests that the energy balance between processes in play during wire array implosions is not well understood. In fact the radiative yields can exceed by several times the implosion kinetic energy. A possible explanation is that the coupling from magnetic energy to kinetic energy as magnetohydrodynamic plasma instabilities develop provides additional energy. It is thus important to model the instabilities produced in the after implosion stage of the wire array in order to determine how the stored magnetic energy can be connected with the radiative yields. To this aim three-dimensional hybrid simulations have been performed. They are initializedmore » with plasma radial density profiles, deduced in recent experiments [C. Deeney et al., Phys. Plasmas 6, 3576 (1999)] that exhibited large x-ray yields, together with the corresponding magnetic field profiles. Unlike previous work, these profiles do not satisfy pressure balance and differ substantially from those of a Bennett equilibrium. They result in faster growth with an associated transfer of magnetic energy to plasma motion and hence kinetic energy.« less
  • Experimental evidence suggests that the energy balance between processes in play during wire array implosions is not well understood. In fact the radiative yields can exceed by several times the implosion kinetic energy. A possible explanation is that the coupling from magnetic energy to kinetic energy as magnetohydrodynamic plasma instabilities develop provides additional energy. It is thus important to model the instabilities produced in the after implosion stage of the wire array in order to determine how the stored magnetic energy can be connected with the radiative yields. To this aim three-dimensional hybrid simulations have been performed. They are initializedmore » with plasma radial density profiles, deduced in recent experiments [C. Deeney et al., Phys. Plasmas 6, 3576 (1999)] that exhibited large x-ray yields, together with the corresponding magnetic field profiles. Unlike previous work, these profiles do not satisfy pressure balance and differ substantially from those of a Bennett equilibrium. They result in faster growth with an associated transfer of magnetic energy to plasma motion and hence kinetic energy.« less