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Analysis of the isotropic and anisotropic Grad–Shafranov equation

Part of: Focus on Fusion

Published online by Cambridge University Press:  28 September 2021

S. Jeyakumar Open the ORCID record for S. Jeyakumar [Opens in a new window] ,
D. Pfefferlé Open the ORCID record for D. Pfefferlé [Opens in a new window] ,
M.J. Hole Open the ORCID record for M.J. Hole [Opens in a new window]  and
Z.S. Qu
S. Jeyakumar*
Affiliation:
Mathematical Sciences Institute, Australian National University, Acton, ACT 2601, Australia The University of Western Australia, 35 Stirling Highway, Crawley, WA 6009, Australia
D. Pfefferlé
Affiliation:
The University of Western Australia, 35 Stirling Highway, Crawley, WA 6009, Australia
M.J. Hole
Affiliation:
Mathematical Sciences Institute, Australian National University, Acton, ACT 2601, Australia Australian Nuclear Science and Technology Organisation, Locked Bag 2001, Kirrawee DC, NSW 2232, Australia
Z.S. Qu
Affiliation:
Mathematical Sciences Institute, Australian National University, Acton, ACT 2601, Australia
*
†Email address for correspondence: [email protected]
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Abstract

Pressure anisotropy is a commonly observed phenomenon in tokamak plasmas, due to external heating methods such as neutral beam injection and ion-cyclotron resonance heating. Equilibrium models for tokamaks are constructed by solving the Grad–Shafranov equation; such models, however, do not account for pressure anisotropy since ideal magnetohydrodynamics assumes a scalar pressure. A modified Grad–Shafranov equation can be derived to include anisotropic pressure and toroidal flow by including drift-kinetic effects from the guiding-centre model of particle motion. In this work, we have studied the mathematical well-posedness of these two problems by showing the existence and uniqueness of solutions to the Grad–Shafranov equation both in the standard isotropic case and when including pressure anisotropy and toroidal flow. A new fixed-point approach is used to show the existence of solutions in the Sobolev space $H_0^1$ to the Grad–Shafranov equation, and sufficient criteria for their uniqueness are derived. The conditions required for the existence of solutions to the modified Grad–Shafranov equation are also constructed.

Keywords

fusion plasmaplasma heating
Type
Research Article
Information
Journal of Plasma Physics , Volume 87 , Issue 5 , October 2021 , 905870506
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Copyright
Copyright © The Author(s), 2021. Published by Cambridge University Press

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