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. 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.
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 z-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 Ampère's Law 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 Bz must therefore, like p be a field like invariant.
Rearranging the cross products above, we see that that
- ,
and
These results can be subsituted into the expression for to yield:
Now, since p and are constants along a field line, and functions only of A, we note that and . Thus, factoring out and rearraging terms we arrive at the Grad Shafranov equation: