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 is both a dependent and an independent variable in this equation:
where is the magnetic permeability, is the pressure,
and the magnetic field and current are given by
- is 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 and as well as the boundary conditions.
Derivation (in slab coordinates):
To begin we assume that the system is 2-dimensional with z as the invariant axis, i.e. for all quantities.
Then the magnetic field can be written in cartesian coordinates as
or more compactly,
- ,
where 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 is everywhere perpendicular to B. (Also note that -A is the flux function 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. Note from the form of this equation that we also know p is a constant along any field line, (again since is everywhere perpendicular to B. Additionally, the two-dimensional assumption () 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 , i.e. is parallel to .
We can break the right hand side of the previous equation into two parts:
- ,
where the subscript denotes the component in the plane perpendicular to the -axis. The 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 as required, the vector must be perpendicular to , and must therefore, like be a field-line invariant.
Rearranging the cross products above, we see that
- ,
and
These results can be substituted into the expression for to yield:
Now, since and are constants along a field line, and functions only of , we note that and . Thus, factoring out and rearraging terms we arrive at the Grad–Shafranov equation:
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.