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 [math]\displaystyle{ (r,\theta,z) }[/math] as the cylindrical coordinates, the flux function [math]\displaystyle{ \psi }[/math] is governed by the equation,

[math]\displaystyle{ \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}, }[/math]

where [math]\displaystyle{ \mu_0 }[/math] is the magnetic permeability, [math]\displaystyle{ p(\psi) }[/math] is the pressure, [math]\displaystyle{ F(\psi)=rB_{\theta} }[/math] and the magnetic field and current are, respectively, given by[math]\displaystyle{ \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} }[/math]

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

Derivation (in Cartesian coordinates)

In the following, it is assumed that the system is 2-dimensional with [math]\displaystyle{ z }[/math] as the invariant axis, i.e. [math]\displaystyle{ \frac{\partial}{\partial z} }[/math] produces 0 for any quantity. Then the magnetic field can be written in cartesian coordinates as [math]\displaystyle{ \mathbf{B} = \left(\frac{\partial A}{\partial y}, -\frac{\partial A}{\partial x}, B_z(x, y)\right), }[/math] or more compactly, [math]\displaystyle{ \mathbf{B} =\nabla A \times \hat{\mathbf{z}} + B_z \hat{\mathbf{z}}, }[/math] where [math]\displaystyle{ A(x,y)\hat{\mathbf{z}} }[/math] 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 [math]\displaystyle{ \nabla A }[/math] is everywhere perpendicular to B. (Also note that -A is the flux function [math]\displaystyle{ \psi }[/math] mentioned above.)

Two dimensional, stationary, magnetic structures are described by the balance of pressure forces and magnetic forces, i.e.: [math]\displaystyle{ \nabla p = \mathbf{j} \times \mathbf{B}, }[/math] 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 [math]\displaystyle{ \nabla p }[/math] is everywhere perpendicular to B). Additionally, the two-dimensional assumption ([math]\displaystyle{ \frac{\partial}{\partial z} = 0 }[/math]) 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 [math]\displaystyle{ \mathbf{j}_\perp \times \mathbf{B}_\perp = 0 }[/math], i.e. [math]\displaystyle{ \mathbf{j}_\perp }[/math] is parallel to [math]\displaystyle{ \mathbf{B}_\perp }[/math].

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

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

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

Rearranging the cross products above leads to [math]\displaystyle{ \hat{\mathbf{z}} \times \mathbf{B}_\perp = \nabla A - (\mathbf{\hat z} \cdot \nabla A) \mathbf{\hat z} = \nabla A, }[/math] and [math]\displaystyle{ \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. }[/math]

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

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

Further reading

• 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.