Grad-Shafranov equation

!MHD
!Plasma

The Grad-Shafranov equation (H. Grad and H. Rubin (1958) 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). Interestingly the flux function $psi$ is both a dependent and an independent variable in this equation:

$Delta^\{*\}psi\; =\; -mu\_\{0\}R^\{2\}frac\{dp\}\{dpsi\}-Ffrac\{dF\}\{dpsi\}$

where $mu\_0$ is the magnetic permeability, $p(psi)$ is the pressure, $F(psi)=RB\_\{phi\}$

and the magnetic field and current are given by

$vec\{B\}=frac\{1\}\{R\}\; ablapsi\; imes\; hat\{e\_\{phi+frac\{F\}\{R\}hat\{e\}\_\{phi\}$

$mu\_0vec\{J\}=frac\{1\}\{R\}frac\{dF\}\{dpsi\}\; ablapsi\; imes\; hat\{e\_\{phi-frac\{1\}\{R\}Delta^\{*\}psi\; hat\{e\}\_\{phi\}$

The elliptic operator

:$Delta^\{*\}$ is given by$Delta^\{*\}psi\; =\; Rfrac\{partial\}\{partial\; R\}left(frac\{1\}\{R\}frac\{partial\; psi\}\{partial\; R\}\; ight)+frac\{partial^2\; psi\}\{partial\; Z^2\}$.

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:
To begin we assume that the system is 2-dimensional with z as the invariant axis, i.e. $partial\; /partial\; z\; =\; 0$ 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 $abla\; A$ 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.:

:,

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 $abla\; p$ is everywhere perpendicular to B. Additionally, the two-dimensional assumption ($partial\; /\; partial\; z$) 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 $perp$ 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$j\_z\; =\; -\; abla^2\; A/mu\_0.$. The in plane field is
:,
and using Ampère's Law the in plane current is given by
:.
In order for this vector to be parallel to as required, the vector $abla\; B\_z$ must be perpendicular to , and $B\_z$ must therefore, like $p$ be a field like invariant.

Rearranging the cross products above, we see that that
:,
and
:
These results can be substituted into the expression for $abla\; p$ to yield:
:$abla\; p\; =\; -\; [(1/mu\_0)\; abla^2\; A]\; abla\; A-(1/mu\_0)B\_z\; abla\; B\_z.$

Now, since $p$ and $B\_perp$ are constants along a field line, and functions only of $A$, we note that $abla\; p\; =\; (d\; p\; /dA)\; abla\; A$ and $abla\; B\_z\; =\; (d\; B\_z/dA)\; abla\; A$. Thus, factoring out $abla\; A$ and rearraging terms we arrive at the Grad Shafranov equation:
:$abla^2\; A\; =\; -mu\_0\; frac\{d\}\{dA\}(p\; +\; frac\{B\_z^2\}\{2mu\_0\})$

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 equilibrum in a magnetic field, "Reviews of Plasma Physics", Vol. 2, New York: Consultants Bureau, p. 103.