Grad–Shafranov equation

The Grad–Shafranov equation (H. Grad and H. Rubin (1958); Vitalii Dmitrievich Shafranov (1966)) is the equilibrium equation in ideal magnetohydrodynamics (MHD) for a two dimensional plasma, for example the axisymmetric toroidal plasma in a tokamak. This equation takes the same form as the Hicks equation from fluid dynamics. This equation is a two-dimensional, nonlinear, elliptic partial differential equation obtained from the reduction of the ideal MHD equations to two dimensions, often for the case of toroidal axisymmetry (the case relevant in a tokamak). Taking $(r,\theta ,z)$ as the cylindrical coordinates, the flux function $\psi$ is governed by the equation,

${\frac {\partial ^{2}\psi }{\partial r^{2}}}-{\frac {1}{r}}{\frac {\partial \psi }{\partial r}}+{\frac {\partial ^{2}\psi }{\partial z^{2}}}=-\mu _{0}r^{2}{\frac {dp}{d\psi }}-{\frac {1}{2}}{\frac {dF^{2}}{d\psi }},$

where $\mu _{0}$ is the magnetic permeability, $p(\psi )$ is the pressure, $F(\psi )=rB_{\theta }$ and the magnetic field and current are, respectively, given by ${\begin{aligned}\mathbf {B} &={\frac {1}{r}}\nabla \psi \times {\hat {\mathbf {e} }}_{\theta }+{\frac {F}{r}}{\hat {\mathbf {e} }}_{\theta },\\\mu _{0}\mathbf {J} &={\frac {1}{r}}{\frac {dF}{d\psi }}\nabla \psi \times {\hat {\mathbf {e} }}_{\theta }-\left[{\frac {\partial }{\partial r}}\left({\frac {1}{r}}{\frac {\partial \psi }{\partial r}}\right)+{\frac {1}{r}}{\frac {\partial ^{2}\psi }{\partial z^{2}}}\right]{\hat {\mathbf {e} }}_{\theta }.\end{aligned}}$

The nature of the equilibrium, whether it be a tokamak, reversed field pinch, etc. is largely determined by the choices of the two functions $F(\psi )$ and $p(\psi )$ as well as the boundary conditions.