格拉德-沙弗拉诺夫方程格拉德-沙弗拉诺夫方程为理想等离子体中用角向磁通描述等离子体平衡的方程。该方程最初的形式为二维的,但也可以通过一维格拉德-沙弗拉诺夫方程来描述一维螺旋磁镜位形的等离子体平衡。 Δ ∗ ψ = − μ 0 R 2 d p d ψ − 1 2 d F 2 d ψ {\displaystyle \Delta ^{*}\psi =-\mu _{0}R^{2}{\frac {dp}{d\psi }}-{\frac {1}{2}}{\frac {dF^{2}}{d\psi }}} 其中 μ 0 {\displaystyle \mu _{0}} 为磁导率, p ( ψ ) {\displaystyle p(\psi )} 为压强, F ( ψ ) = R B ϕ {\displaystyle F(\psi )=RB_{\phi }} 磁场与电流由下式给定: B → = 1 R ∇ ψ × e ^ ϕ + F R e ^ ϕ {\displaystyle {\vec {B}}={\frac {1}{R}}\nabla \psi \times {\hat {e}}_{\phi }+{\frac {F}{R}}{\hat {e}}_{\phi }} μ 0 J → = 1 R d F d ψ ∇ ψ × e ^ ϕ − 1 R Δ ∗ ψ e ^ ϕ {\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 }} Δ ∗ {\displaystyle \Delta ^{*}} 的具体形式为: Δ ∗ ψ = R ∂ ∂ R ( 1 R ∂ ψ ∂ R ) + ∂ 2 ψ ∂ Z 2 {\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 u z z + u y y − 1 y u y = y 2 f ( u ) + g ( u ) {\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.