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 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). The flux function $\psi$ is both a dependent and an independent variable in this equation:

$\Delta^{*}\psi = -\mu_0 R^{2}\frac{dp}{d\psi}-\frac{1}{2}\frac{dF^2}{d\psi},$

where $\mu_0$ is the magnetic permeability, $p(\psi)$ is the pressure, $F(\psi)=RB_{\phi}$ and the magnetic field and current are, respectively, given by

$\begin{align} \vec{B} &= \frac{1}{R}\nabla\psi \times \hat{e}_\phi + \frac{F}{R}\hat{e}_\phi, \\ \mu_0\vec{J} &= \frac{1}{R}\frac{dF}{d\psi}\nabla\psi \times \hat{e}_\phi - \frac{1}{R}\Delta^{*}\psi \hat{e}_\phi. \end{align}$

The elliptic operator $\Delta^{*}$ is

$\Delta^{*}\psi \equiv R^{2} \vec{\nabla} \cdot \left( \frac{1}{R^2} \vec{\nabla} \psi \right) = R\frac{\partial}{\partial R}\left(\frac{1}{R}\frac{\partial \psi}{\partial R}\right) + \frac{\partial^2 \psi}{\partial Z^2}$.

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 slab coordinates)

In the following, it is assumed that the system is 2-dimensional with $z$ as the invariant axis, i.e. $\frac{\partial}{\partial z} = 0$ for all quantities. Then the magnetic field can be written in cartesian coordinates as

$\bold{B} = \left(\frac{\partial A}{\partial y},-\frac{\partial A}{\partial x}, B_z(x, y)\right),$

or more compactly,

$\bold{B} =\nabla A \times \hat{\bold{z}} + B_z \hat{\bold{z}},$

where $A(x,y)\hat{\bold{z}}$ 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.:

$\nabla p = \bold{j} \times \bold{B},$

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 $\bold{j}_\perp \times \bold{B}_\perp = 0$, i.e. $\bold{j}_\perp$ is parallel to $\bold{B}_\perp$.

The right hand side of the previous equation can be considered in two parts:

$\bold{j} \times \bold{B} = j_z (\hat{\bold{z}} \times \bold{B_\perp}) + \bold{j_\perp} \times \hat{\bold{z}}B_z ,$

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

$\bold{B}_\perp = \nabla A \times \hat{\bold{z}}$,

and using Maxwell–Ampère's equation, the in plane current is given by

$\bold{j}_\perp = \frac{1}{\mu_0} \nabla B_z \times \hat{\bold{z}}$.

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

Rearranging the cross products above leads to

$\hat{\bold{z}} \times \bold{B}_\perp = \nabla A - (\bold{\hat z} \cdot \nabla A) \bold{\hat z} = \nabla A$,

and

$\bold{j}_\perp \times B_z\bold{\hat{z}} = \frac{B_z}{\mu_0}(\bold{\hat z}\cdot\nabla B_z)\bold{\hat z} - \frac{1}{\mu_0}B_z\nabla B_z = -\frac{1}{\mu_0} B_z\nabla B_z.$

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:

$\nabla^2 A = -\mu_0 \frac{d}{dA}\left(p + \frac{B_z^2}{2\mu_0}\right).$

References

• 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.
• Woods, Leslie C. (2004) Physics of plasmas, Weinheim: WILEY-VCH Verlag GmbH & Co. KGaA, chapter 2.5.4
• Haverkort, J.W. (2009) Axisymmetric Ideal MHD Tokamak Equilibria. Notes about the Grad-Shafranov equation, selected aspects of the equation and its analytical solutions.
• Haverkort, J.W. (2009) Axisymmetric Ideal MHD equilibria with Toroidal Flow. Incorporation of toroidal flow, relation to kinetic and two-fluid models, and discussion of specific analytical solutions.