Mercier--Cotsaftis and Grad--Shafranov equations for anisotropic plasma

Abstract

In this brief review, the historical aspects of the generalization of the Grad--Shafranov equation to the case of anisotropic plasma are discussed.

Figures (2)

• Figure 1: The theory of axially symmetric tokamaks uses a cylindrical coordinate system $\left\{R,\phi,Z\right\}$, in which the symmetry axis $Z$ is directed along the major axis of the torus, and the coordinate $R$ is along the major radius of the torus. The components of the magnetic field in a symmetric tokamak can be expressed through the poloidal magnetic flux $\psi(R,Z)$ and the poloidal current $i_{p}(R,Z)$ through a circle of radius $R$ in the plane $Z=\mathop{\mathrm{const}}\nolimits$ using the formulas \ref{['01.01:001']}. In a symmetric tokamak, the magnetic field components $\mathbf{B}=\left\{B_{R},B_{\phi},B_{Z}\right\}$ do not depend on the toroidal angle $\phi$, as do the functions $\psi$ and $i_{p}$. The toroidal magnetic field $\mathbf{B}_{t}=\left\{0,B_{\phi},0\right\}$ is directed along the major circumference of the torus. The poloidal magnetic field $\mathbf{B}_{p}=\left\{B_{R},0,B_{Z}\right\}$ lies in the plane of section $\left\{R,Z\right\}$, perpendicular to the minor axis of the torus $R=R_{0}$, $Z=0$. The internal magnetic surfaces $\psi=\mathop{\mathrm{const}}\nolimits$ of the minor axis of the torus are shifted toward the outer circumference of the torus more strongly than the external ones. This phenomenon is called the Shafranov shift.
• Figure 2: In a symmetric torus, the toroidal magnetic field $B_{t}$ decreases inversely proportional to the distance $R$ from the $Z$ axis, since according to Stokes's theorem $2\pi R B_{t}(R) = \mu_{0}I_{p}$, where $I_{p}=2\pi i_{p}$ is the poloidal current flowing through a circle of radius $R$. Inside the torus, $I_{p}=IN=\mathop{\mathrm{const}}\nolimits$, where $I$ is the current in the coil wound on the torus, and $N$ is the number of coils. Outside the torus, $I_{p}=0$. In the theory of mirror traps, a cylindrical coordinate system $\{r,\theta ,z\}$ is used (where $z\approx R_{0}\phi$ and $R_{0}\to\infty$), see equation \ref{['00:00']}; or $\{R,\phi,Z\}$ with $R_{0}=0$; see problem 24.5 in Kotelnikov2025V2e.