Plasma physics / Grad Shafranov equation

[vibe3]

I am trying to understand the Grad Shafranov equation, and in particular its inputs and outputs.

I understand that the equation determines the conditions under which plasma may be in equilibrium. In particular, the MHD equations of equilibrium state:

[tex]\nabla p = J \times B[/tex]

and the solutions to the Grad Shafranov equation state the possible values of pressure p, current J and magnetic field B which will satisfy the above equation.

However, my question has to do with imposing an ambient magnetic field B_0 like you would, in say, a tokamak.

If you impose an ambient field B_0 and the plasma is able to reach an equilibrium, it will then create a diamagnetic current

[tex]J_{dia} = B \times \nabla p / B^2[/tex]

which has its own magnetic field B_dia. So the total field is then B_0 + B_dia.

So my question is, the Grad Shafranov equation seems to provide self-consistent pressure functions, currents and magnetic fields, but how do you incorporate a pre-existing ambient field B_0?

I haven't been able to find a discussion of this

 

[Crackmonkey74]

Your first equation [tex]\nabla p = J \times B[/tex] is only a general balance equation, the name 'GS equ.' actually only applies to this one:

[tex]\Delta^{*}\psi = -\mu_{0}R^{2}\frac{dp}{d\psi}-\frac{1}{2}\frac{dF^2}{d\psi}[/tex]
It is derived from the balance equation, but assumes axisymmetry.As you mentioned tokamaks, I will only show how the GS equations is actually used in real life. In tokamaks, you can measure things like the current, particle density, electromagnetic potentials, temperatures etc. You cannot directly measure [tex]\psi,[/tex] but you can reconstruct it using numerical tools.
So in real life, the GS equations solves for [tex]\psi(R,Z),[/tex] where the latter can be viewed as a label for the so-called flux surfaces (FS). In tokamaks, assuming a perfectly conducting plasma, equilibria are 'composed of' an infinite number of nested surfaces on which pressure and current density are constant (== FS), and the shape of these surfaces are described by [tex]\psi = const \rightarrow R(\psi), Z(\psi).[/tex] The time-independent equilibrium is basically determined by knowing [tex]\psi,[/tex] as you can calculate all other quantities from it within the framework of the underlying theoretical plasma model: ideal magnetohydrodynamics (MHD). There is a quite comprehensive and well-known book on this subject by Jeffrey Freidberg. The (compact) lecture script can be found here:
http://ocw.mit.edu/courses/nuclear-engineering/22-615-mhd-theory-of-fusion-systems-spring-2007/lecture-notes/"

So the pressure gradient [tex]\frac{dp}{d\psi}[/tex] and [tex]F[/tex] are not solutions of the GS equations, they are required as INPUT! So you have to know them from somewhere else, e.g. experiments, transport codes etc. The quantity F is related to the magnetic field resp. the current (density) via the Maxwell equations. So if you want to work with the magnetic field, you have to know it completely BEFORE (and diamagnetic currents are but one of many many phenomena you have to take into account!) you solve the GS equation.
On the other hand, if you e.g. use experimentally observed pressure and current density profiles as input, you can then solve for [tex]\psi[/tex] and afterwards compute the complete magnetic field via
[tex]\vec{B}=\frac{1}{R}\nabla\psi\times \hat{e}_{\phi}+\frac{F}{R}\hat{e}_{\phi}.[/tex]

So the magnetic field B in your balance equation already is the whole magnetic field of the plasma, including background fields. The GS equation only gives you the time-independent equilibrium, but provides no answer on how the plasma evolves. This is a completely different topic and requires actual time-dependent evolution equations for the plasma quantities, be it MHD or gyrofluid or gyrokinetics etc.
Hopefully this helped (?)