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,$$where $$\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{align} \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{align}$$

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.

Derivation (in Cartesian coordinates)
In the following, it is assumed that the system is 2-dimensional with $$z$$ as the invariant axis, i.e. $\frac{\partial}{\partial z}$ produces 0 for any quantity. Then the magnetic field can be written in cartesian coordinates as $$ \mathbf{B} = \left(\frac{\partial A}{\partial y}, -\frac{\partial A}{\partial x}, B_z(x, y)\right),$$ or more compactly, $$ \mathbf{B} =\nabla A \times \hat{\mathbf{z}} + B_z \hat{\mathbf{z}},$$ where $$A(x,y)\hat{\mathbf{z}}$$ is the vector potential for the in-plane (x and y components) magnetic field. Note that based on this form for B we can see that A is constant along any given magnetic field line, since $$\nabla A$$ is everywhere perpendicular to B. (Also note that -A is the flux function $$\psi$$ mentioned above.)

Two dimensional, stationary, magnetic structures are described by the balance of pressure forces and magnetic forces, i.e.: $$\nabla p = \mathbf{j} \times \mathbf{B},$$ where p is the plasma pressure and j is the electric current. It is known that p is a constant along any field line, (again since $$\nabla p$$ is everywhere perpendicular to B). Additionally, the two-dimensional assumption ($\frac{\partial}{\partial z} = 0$ ) means that the z- component of the left hand side must be zero, so the z-component of the magnetic force on the right hand side must also be zero. This means that $$\mathbf{j}_\perp \times \mathbf{B}_\perp = 0$$, i.e. $$\mathbf{j}_\perp$$ is parallel to $$\mathbf{B}_\perp$$.

The right hand side of the previous equation can be considered in two parts: $$\mathbf{j} \times \mathbf{B} = j_z (\hat{\mathbf{z}} \times \mathbf{B_\perp}) + \mathbf{j_\perp} \times \hat{\mathbf{z}}B_z ,$$ where the $$\perp$$ subscript denotes the component in the plane perpendicular to the $$z$$-axis. The $$z$$ component of the current in the above equation can be written in terms of the one-dimensional vector potential as $$j_z = -\frac{1}{\mu_0} \nabla^2 A. $$

The in plane field is $$\mathbf{B}_\perp = \nabla A \times \hat{\mathbf{z}}, $$ and using Maxwell–Ampère's equation, the in plane current is given by $$\mathbf{j}_\perp = \frac{1}{\mu_0} \nabla B_z \times \hat{\mathbf{z}}.$$

In order for this vector to be parallel to $$\mathbf{B}_\perp$$ as required, the vector $$\nabla B_z$$ must be perpendicular to $$\mathbf{B}_\perp$$, and $$B_z$$ must therefore, like $$p$$, be a field-line invariant.

Rearranging the cross products above leads to $$\hat{\mathbf{z}} \times \mathbf{B}_\perp = \nabla A - (\mathbf{\hat z} \cdot \nabla A) \mathbf{\hat z} = \nabla A,$$ and $$\mathbf{j}_\perp \times B_z\mathbf{\hat{z}} = \frac{B_z}{\mu_0}(\mathbf{\hat z}\cdot\nabla B_z)\mathbf{\hat z} - \frac{1}{\mu_0}B_z\nabla B_z = -\frac{1}{\mu_0} B_z\nabla B_z.$$

These results can be substituted into the expression for $$\nabla p$$ to yield: $$\nabla p = -\left[\frac{1}{\mu_0} \nabla^2 A\right]\nabla A - \frac{1}{\mu_0} B_z\nabla B_z.$$

Since $$p$$ and $$B_z$$ are constants along a field line, and functions only of $$A$$, hence $$\nabla p = \frac{dp}{dA}\nabla A$$ and $$ \nabla B_z = \frac{d B_z}{dA}\nabla A$$. Thus, factoring out $$\nabla A$$ and rearranging terms yields the Grad–Shafranov equation: $$\nabla^2 A = -\mu_0 \frac{d}{dA} \left(p + \frac{B_z^2}{2\mu_0}\right).$$

Derivation in contravariant representation
This derivation is only used for Tokamaks, but it can be enlightening. Using the definition of 'The Theory of Toroidally Confined Plasmas 1:3'(Roscoe White), Writing $$\vec{B}$$ by contravariant basis $$(\nabla \Psi, \nabla \phi, \nabla \zeta)$$: $$ \vec{B} = \nabla\Psi \times \nabla \phi + \bar{F} \nabla\phi, $$

we have $$\vec{j}$$: $$ \mu_0 \vec{j} = \nabla \times \vec{B} = -\Delta^* \Psi \nabla \phi+ \nabla\bar{F} \times \nabla \phi \quad \text{, where}\ \Delta^* = r\partial_r(r^{-1}\partial_r) + \partial^2_\phi \text{;} $$

then force balance equation: $$ \mu_0 \vec{j} \times \vec{B}= \mu_0 \nabla p\text{.} $$

Working out, we have: $$ -\Delta^* \Psi = \bar{F} \frac{d\bar{F}}{d \Psi} + \mu_0 R^2 \frac{d p}{d \Psi} \text{.} $$