Eigenmode analysis of the sheared-flow Z-pinch
Abstract
Experiments have demonstrated that a Z-pinch can persist for thousands of times longer than the growth time of global magnetohydrodynamic (MHD) instabilities such as the m=0 sausage and m=1 kink modes. These modes have growth times on the order of ta=a/vi, where vi is the ion thermal speed and a is the pinch radius. Axial flows with duz/dr ≲ vi/a have been measured during the stable period, and the commonly accepted theory is that this amount of shear is sufficient to stabilize these modes as predicted by numerical studies using the ideal MHD equations. However, these studies only consider specific equilibrium profiles that typically have a modest magnitude for the logarithmic pressure gradient, qP≡d ln P/d ln r, and may not represent experimental conditions. Linear stability of the sheared-flow Z-pinch is studied here via a direct eigen-decomposition of the matrix operator obtained from the linear ideal MHD equations. Several equilibrium profiles with a large variation of qP are examined. Considering a practical range of k, 1/3 ≲ ka ≲ 10, it is shown that the shear required to stabilize m=0 modes can be expressed as duz/dr≥Cγ0/(ka)α. Here, γ0=γ0(ka) is the profile-specific growth rate in the absence of shear, which scales approximately with |qP|. Both C and α are profile-specific constants, butmore »
- Authors:
- Publication Date:
- Sponsoring Org.:
- USDOE
- OSTI Identifier:
- 1737847
- Grant/Contract Number:
- AC52-07NA27344
- Resource Type:
- Publisher's Accepted Manuscript
- Journal Name:
- Physics of Plasmas
- Additional Journal Information:
- Journal Name: Physics of Plasmas Journal Volume: 27 Journal Issue: 12; Journal ID: ISSN 1070-664X
- Publisher:
- American Institute of Physics
- Country of Publication:
- United States
- Language:
- English
Citation Formats
Angus, J. R., Van De Wetering, J. J., Dorf, M., and Geyko, V. I.. Eigenmode analysis of the sheared-flow Z-pinch. United States: N. p., 2020.
Web. doi:10.1063/5.0029716.
Angus, J. R., Van De Wetering, J. J., Dorf, M., & Geyko, V. I.. Eigenmode analysis of the sheared-flow Z-pinch. United States. https://doi.org/10.1063/5.0029716
Angus, J. R., Van De Wetering, J. J., Dorf, M., and Geyko, V. I.. Thu .
"Eigenmode analysis of the sheared-flow Z-pinch". United States. https://doi.org/10.1063/5.0029716.
@article{osti_1737847,
title = {Eigenmode analysis of the sheared-flow Z-pinch},
author = {Angus, J. R. and Van De Wetering, J. J. and Dorf, M. and Geyko, V. I.},
abstractNote = {Experiments have demonstrated that a Z-pinch can persist for thousands of times longer than the growth time of global magnetohydrodynamic (MHD) instabilities such as the m=0 sausage and m=1 kink modes. These modes have growth times on the order of ta=a/vi, where vi is the ion thermal speed and a is the pinch radius. Axial flows with duz/dr ≲ vi/a have been measured during the stable period, and the commonly accepted theory is that this amount of shear is sufficient to stabilize these modes as predicted by numerical studies using the ideal MHD equations. However, these studies only consider specific equilibrium profiles that typically have a modest magnitude for the logarithmic pressure gradient, qP≡d ln P/d ln r, and may not represent experimental conditions. Linear stability of the sheared-flow Z-pinch is studied here via a direct eigen-decomposition of the matrix operator obtained from the linear ideal MHD equations. Several equilibrium profiles with a large variation of qP are examined. Considering a practical range of k, 1/3 ≲ ka ≲ 10, it is shown that the shear required to stabilize m=0 modes can be expressed as duz/dr≥Cγ0/(ka)α. Here, γ0=γ0(ka) is the profile-specific growth rate in the absence of shear, which scales approximately with |qP|. Both C and α are profile-specific constants, but C is order unity and α≈1. It is further demonstrated that even a large value of shear, duz/dr=3vi/a, is not sufficient to provide linear stabilization of the m=1 kink mode for all profiles considered. This result is in contrast to the currently accepted theory predicting stabilization at much lower shear, duz/dr=0.1vi/a, and suggests that the experimentally observed stability cannot be explained within the linear ideal-MHD model.},
doi = {10.1063/5.0029716},
journal = {Physics of Plasmas},
number = 12,
volume = 27,
place = {United States},
year = {2020},
month = {12}
}
https://doi.org/10.1063/5.0029716
Works referenced in this record:
Kinetic simulations of sheared flow stabilization in high-temperature Z-pinch plasmas
journal, June 2019
- Tummel, K.; Higginson, D. P.; Link, A. J.
- Physics of Plasmas, Vol. 26, Issue 6
Linear Stability of the High Temperature, Dense Pinch
journal, April 1995
- Arber, T. D.; Russell, P. G. F.; Coppins, M.
- Physical Review Letters, Vol. 74, Issue 14
Transient phenomena and secularity of linear interchange instabilities with shear flows in homogeneous magnetic field plasmas
journal, February 2001
- Tatsuno, T.; Volponi, F.; Yoshida, Z.
- Physics of Plasmas, Vol. 8, Issue 2
An attempt to produce a thermo-nuclear reaction in deuterium by means of a high current spark discharge
journal, February 1952
- Reynolds, P.; Craggs, J. D.
- The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, Vol. 43, Issue 337
A review of the dense Z -pinch
journal, June 2011
- Haines, M. G.
- Plasma Physics and Controlled Fusion, Vol. 53, Issue 9
Influence of flow shear on localized Rayleigh-Taylor and resistive drift wave instabilities
journal, December 2019
- Zhang, Yanzeng; Krasheninnikov, Sergei I.; Smolyakov, Andrei I.
- Contributions to Plasma Physics, Vol. 60, Issue 5-6
Comment on “Sheared Flow Stabilization of the Kink Mode in Z Pinches”
journal, March 1996
- Arber, T. D.; Howell, D. F.
- Physical Review Letters, Vol. 76, Issue 12
Large Larmor radius stability of the z pinch
journal, April 1994
- Arber, T. D.; Coppins, M.; Scheffel, J.
- Physical Review Letters, Vol. 72, Issue 15
Different responses of the Rayleigh–Taylor type and resistive drift wave instabilities to the velocity shear
journal, February 2020
- Zhang, Y.; Krasheninnikov, S. I.; Smolyakov, A. I.
- Physics of Plasmas, Vol. 27, Issue 2
Influence of velocity shear on the Rayleigh-Taylor instability
journal, May 1982
- Guzdar, P. N.; Satyanarayana, P.; Huba, J. D.
- Geophysical Research Letters, Vol. 9, Issue 5
Inviscid evolution of large amplitude filaments in a uniform gravity field
journal, November 2014
- Angus, J. R.; Krasheninnikov, S. I.
- Physics of Plasmas, Vol. 21, Issue 11
Sheared Flow Stabilization of the Kink Mode in Pinches
journal, October 1995
- Shumlak, U.; Hartman, C. W.
- Physical Review Letters, Vol. 75, Issue 18
Increasing plasma parameters using sheared flow stabilization of a Z-pinch
journal, May 2017
- Shumlak, U.; Nelson, B. A.; Claveau, E. L.
- Physics of Plasmas, Vol. 24, Issue 5
Gyrokinetic simulations of m = 0 mode in sheared flow Z-pinch plasmas
journal, June 2019
- Geyko, V. I.; Dorf, M.; Angus, J. R.
- Physics of Plasmas, Vol. 26, Issue 6
Evidence of Stabilization in the -Pinch
journal, October 2001
- Shumlak, U.; Golingo, R. P.; Nelson, B. A.
- Physical Review Letters, Vol. 87, Issue 20
Formation of a sheared flow Z pinch
journal, June 2005
- Golingo, R. P.; Shumlak, U.; Nelson, B. A.
- Physics of Plasmas, Vol. 12, Issue 6
The effect of sheared axial flow on the linear stability of the Z‐pinch
journal, February 1996
- Arber, T. D.; Howell, D. F.
- Physics of Plasmas, Vol. 3, Issue 2
Sustained Neutron Production from a Sheared-Flow Stabilized Pinch
journal, April 2019
- Zhang, Y.; Shumlak, U.; Nelson, B. A.
- Physical Review Letters, Vol. 122, Issue 13
Linear and nonlinear development of m=0 instability in a diffuse Bennett Z-pinch equilibrium with sheared axial flow
journal, July 2010
- Paraschiv, I.; Bauer, B. S.; Lindemuth, I. R.
- Physics of Plasmas, Vol. 17, Issue 7
Does a sheared flow stabilize inversely stratified fluid?
journal, May 2002
- Benilov, E. S.; Naulin, V.; Rasmussen, J. Juul
- Physics of Fluids, Vol. 14, Issue 5
On the possibility of producing thermonuclear reactions in a gas discharge
journal, February 1957
- Kurchatov, I. V.
- Journal of Nuclear Energy (1954), Vol. 4, Issue 2
Suppression of turbulence and transport by sheared flow
journal, January 2000
- Terry, P. W.
- Reviews of Modern Physics, Vol. 72, Issue 1
Drift-ideal magnetohydrodynamic simulations of m = 0 modes in Z-pinch plasmas
journal, July 2019
- Angus, J. R.; Dorf, M.; Geyko, V. I.
- Physics of Plasmas, Vol. 26, Issue 7
Stability of stratified flow with inhomogeneous shear
journal, February 2005
- Mikhailenko, Vladimir S.; Scime, Earl E.; Mikhailenko, Vladimir V.
- Physical Review E, Vol. 71, Issue 2
Sheared flow stabilization experiments in the ZaP flow Z pinch
journal, May 2003
- Shumlak, U.; Nelson, B. A.; Golingo, R. P.
- Physics of Plasmas, Vol. 10, Issue 5