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).
as the cylindrical coordinates, the flux function
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
Then the magnetic field can be written in cartesian coordinates as
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
(Also note that -A is the flux function
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
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.
The right hand side of the previous equation can be considered in two parts:
subscript denotes the component in the plane perpendicular to the
component of the current in the above equation can be written in terms of the one-dimensional vector potential as
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
Rearranging the cross products above leads to
These results can be substituted into the expression for
are constants along a field line, and functions only of
{\displaystyle \nabla p={\frac {dp}{dA}}\nabla A}
and rearranging terms yields the Grad–Shafranov equation:
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
then force balance equation: