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

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