Green function for the Grad-Shafranov operator

Authors

  • Julio Herrera Velázquez Instituto de Ciencias Nucleares UNAM

DOI:

https://doi.org/10.31349/RevMexFisE.19.010211

Keywords:

Magnetohydrodynamic equilibrium, Grad-Shafranov equation, tokamaks, magnetostatics

Abstract

The Grad-Shafranov equation, often written in cylindrical coordinates, is an elliptic partial differential equation in two dimensions. It describes magnetohydrodynamic equilibria in axisymmetric toroidal plasmas, such as tokamaks, and yields the poloidal magnetic flux function, which is related to the azimuthal component of the vector potential for the magnetic field produced by a circular (toroidal) current density. The Green function for the differential operator can be obtained from the vector potential for the magnetic field of a circular current loop, which is a typical problem in magnetostatics. The purpose of the paper is to collect results scattered in electrodynamics and plasma physics textbooks for the benefit of students in the field, as well as attracting the attention of a wider audience, in the context of electrodynamics and partial differential equations.

References

L. D. Landau and E. M. Lifshitz, "Electrodynamics of Continuous Media" Chap. IV, Problem 2 (Addison-Wesley, 1960), p. 124-125.

J. D. Jackson, "Classical Electrodynamics" 3rd edition (Wiley, 1998), Chap. 5, Sec. 5, p. 181-182.

W. Greiner, "Classical Electrodynamics" (Springer, 1998), Chap. 9, Example 9.1, p. 206-209.

P. Bellan, "Fundamentals of Plasma Physics", 2nd edition (Cambridge, 2006), Chap. 9, Sec. 8, p. 320-328.

J. Freidberg "Ideal MHD", Chap. 9, Sec. 2 (Cambridge, 2014), Chap. 9, Sec. 2, p. 124-128.

F. F. Chen "Introduction to Plasma Physics and Controlled Fusion", 3rd edition (Springer, 2016), Chap. 10, Sec. 2, p. 357-386.

M. Ariola and A. Pironti, "Magnetic Control of Tokamak Plasmas" Appendix A.1, (Springer, 2016), p. 181-183.

S. Jardin, "Computational Methods in Plasma Physics" Chap. 4, (CRC Press, 1998), p. 109-113.

D. J. Griths, "Introduction to Electrodynamics" (Cambridge University Press, 2017), 4th edition, Appendix B, p. 582-584.

R.H. Good, "Elliptic Integrals, the forgotten functions," Eur. J. Phys. 22, 119{126.

Downloads

Published

2022-01-01