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}}
磁导率 ，
p
(
ψ
)
{\displaystyle p(\psi )}
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 ^{*}}
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
编辑
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
编辑
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. 
