Grad–Shafranov equation

Definition

Use cylindrical coordinates $(R, \varphi, Z)$ on $\R^3$, and let $\mu_0$ be the magnetic permeability. Consider a static equilibrium of ideal MHD, given by a pressure $p$ and a magnetic field $\mathbf{B}$ with $$\nabla p = \mathbf{J} \times \mathbf{B}, \qquad \mu_0 \mathbf{J} = \nabla \times \mathbf{B}, \qquad \nabla \cdot \mathbf{B} = 0.$$ Assume that $p$ and $\mathbf{B}$ do not depend on $\varphi$. Write the field as $$\mathbf{B} = \frac{1}{R} \nabla \psi \times \mathbf{e}_\varphi + \frac{F}{R} \mathbf{e}_\varphi$$ with the poloidal flux function $\psi(R, Z)$ and $F = R B_\varphi$. Then $p$ and $F$ are functions of $\psi$ alone, and the Grad–Shafranov equation is $$\Delta^* \psi = -\mu_0 R^2 \frac{dp}{d\psi} - F \frac{dF}{d\psi}, \qquad \text{where } \Delta^* \psi = R \frac{\partial}{\partial R} \left( \frac{1}{R} \frac{\partial \psi}{\partial R} \right) + \frac{\partial^2 \psi}{\partial Z^2}.$$