The Grad–Shafranov equation (Vitalii Dmitrievich Shafranov (1957); H. Grad and H. Rubin (1958)) 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 , θ , z ) {\displaystyle (r,\theta ,z)} as the cylindrical coordinates, the flux function ψ {\displaystyle \psi } is governed by the equation,where μ 0 {\displaystyle \mu _{0}} is the magnetic permeability, p ( ψ ) {\displaystyle p(\psi )} is the pressure, F ( ψ ) = r B θ {\displaystyle F(\psi )=rB_{\theta }} , and the magnetic field and current are, respectively, given by B = 1 r ∇ ψ × e ^ θ + F r e ^ θ , μ 0 J = 1 r d F d ψ ∇ ψ × e ^ θ − [ ∂ ∂ r ( 1 r ∂ ψ ∂ r ) + 1 r ∂ 2 ψ ∂ z 2 ] e ^ θ . {\displaystyle {\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}}}
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