User:Bpsullivan/GS

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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

\bold{B} = (\partial A/\partial y,-\partial A /\partial x,B_z(x,y))

or more compactly,

\bold{B} =\nabla A \times \hat{\bold{z}} + \hat{\bold{z}} B_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.

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. Note from the form of this equation that we also know p is a constant along any field line, (again since \nabla 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 \bold{j}_\perp \times \bold{B}_\perp = 0, i.e. \bold{j}_\perp is parallel to \bold{B}_\perp.

We can break the right hand side of the previous equation into 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 = -\nabla^2 A/\mu_0. . The in plane field is

\bold{B}_\perp = \nabla A \times \hat{\bold{z}} ,

and using Ampère's Law the in plane current is given by

\bold{j}_\perp = (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{j}_\perp, and Bz must therefore, like p be a field like invariant.

Rearranging the cross products above, we see that that

\hat{\bold{z}} \times \bold{B}_\perp = \nabla A ,

and

\bold{j}_\perp \times \bold{\hat{z}} = -(1/\mu_0)B_z\nabla B_z

These results can be subsituted into the expression for \nabla p to yield:

\nabla p = -[(1/\mu_0) \nabla^2 A]\nabla A-(1/\mu_0)B_z\nabla B_z.

Now, since p and B_\perp are constants along a field line, and functions only of A, we note that \nabla p = (d p /dA)\nabla A and \nabla B_z = (d B_z/dA)\nabla A. Thus, factoring out \nabla A and rearraging terms we arrive at the Grad Shafranov equation:

\nabla^2 A = -\mu_0 \frac{d}{dA}(p + \frac{B_z^2}{2\mu_0})