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.^[1] 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 ${\displaystyle (r,\theta ,z)}$ as the cylindrical coordinates, the flux function ${\displaystyle \psi }$ is governed by the equation,

where ${\displaystyle \mu _{0}}$ is the magnetic permeability, ${\displaystyle p(\psi )}$ is the pressure, ${\displaystyle F(\psi )=rB_{\theta }}$ and the magnetic field and current are, respectively, given by

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

Derivation (in Cartesian coordinates)

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

or more compactly, where ${\displaystyle A(x,y){\hat {\mathbf {z} }}}$ 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 ${\displaystyle \nabla A}$ is everywhere perpendicular to B. (Also note that -A is the flux function ${\displaystyle \psi }$ mentioned above.)

Two dimensional, stationary, magnetic structures are described by the balance of pressure forces and magnetic forces, i.e.:

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 ${\displaystyle \nabla p}$ is everywhere perpendicular to B). Additionally, the two-dimensional assumption (${\textstyle {\frac {\partial }{\partial z}}=0}$) 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 ${\displaystyle \mathbf {j} _{\perp }\times \mathbf {B} _{\perp }=0}$, i.e. ${\displaystyle \mathbf {j} _{\perp }}$ is parallel to ${\displaystyle \mathbf {B} _{\perp }}$.

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

where the ${\displaystyle \perp }$ subscript denotes the component in the plane perpendicular to the ${\displaystyle z}$-axis. The ${\displaystyle z}$ component of the current in the above equation can be written in terms of the one-dimensional vector potential as

The in plane field is

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

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

Rearranging the cross products above leads to

and

These results can be substituted into the expression for ${\displaystyle \nabla p}$ to yield:

Since ${\displaystyle p}$ and ${\displaystyle B_{z}}$ are constants along a field line, and functions only of ${\displaystyle A}$, hence ${\displaystyle \nabla p={\frac {dp}{dA}}\nabla A}$ and ${\displaystyle \nabla B_{z}={\frac {dB_{z}}{dA}}\nabla A}$. Thus, factoring out ${\displaystyle \nabla A}$ and rearranging terms yields the Grad–Shafranov equation:

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 ${\displaystyle {\vec {B}}}$ by contravariant basis ${\displaystyle (\nabla \Psi ,\nabla \phi ,\nabla \zeta )}$:

we have ${\displaystyle {\vec {j}}}$:

then force balance equation:

Working out, we have:

References

Further reading

• Grad, H., and Rubin, H. (1958) Hydromagnetic Equilibria and Force-Free Fields Archived 2023-06-21 at the Wayback Machine. 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.