# 格拉德－沙弗拉诺夫方程

${\displaystyle \Delta ^{*}\psi =-\mu _{0}R^{2}{\frac {dp}{d\psi }}-{\frac {1}{2}}{\frac {dF^{2}}{d\psi }}}$

${\displaystyle {\vec {B}}={\frac {1}{R}}\nabla \psi \times {\hat {e}}_{\phi }+{\frac {F}{R}}{\hat {e}}_{\phi }}$

${\displaystyle \mu _{0}{\vec {J}}={\frac {1}{R}}{\frac {dF}{d\psi }}\nabla \psi \times {\hat {e}}_{\phi }-{\frac {1}{R}}\Delta ^{*}\psi {\hat {e}}_{\phi }}$

${\displaystyle \Delta ^{*}}$ 的具体形式为：

${\displaystyle \Delta ^{*}\psi =R{\frac {\partial }{\partial R}}\left({\frac {1}{R}}{\frac {\partial \psi }{\partial R}}\right)+{\frac {\partial ^{2}\psi }{\partial Z^{2}}}}$.

## 二维方程

${\displaystyle u_{zz}+u_{yy}-{\frac {1}{y}}u_{y}=y^{2}f(u)+g(u)}$

## 参考文献

• Grad, H., and Rubin, H. (1958) Hydromagnetic Equilibria and Force-Free Fields. Proceedings of the 2nd UN Conf. on the Peaceful Uses of Atomic Energy, Vol. 31, Geneva: IAEA p.190.
• Shafranov, V.D. (1966) Plasma equilibrium in a magnetic field, Reviews of Plasma Physics, Vol. 2, New York: Consultants Bureau, p. 103.