Magnetic Fields and Magnetic Diagnostics for Tokamak Plasmas

Magnetic fields and tokamak plasmas
Alan Wootton
6. TOKAMAK EQUILIBRIA
6.0. AN INTUITIVE DERIVATION OF TOKAMAK EQUILIBRIUM
Introduction
After having described how to measure the plasma current and loop voltage, the next most
important parameter to measure is the plasma position. We will show how we determine both
this, and certain integrals of the pressure and field across the plasma cross section (specifically β I
+ l i /2), in section 7. The basic idea is that we want an expression for the fields outside the plasma
in terms of plasma displacement ∆ and (β I + l i /2). We can only do this by knowing a solution to
the plasma equilibrium, i.e. we must solve the Grad Shafranov equation. I deal with this later in
this section, but first we can gain a physical picture of tokamak equilibrium by considering the
various forces acting on a toroidal plasma. Also note that there are techniques to measure plasma
position without recourse to equilibrium solutions, the so called “moments” method. However,
the interpretation of this method (i.e. what has been measured) itself requires a knowledge of the
equilibrium.
The total energy of our system must be made up of 3 parts,
W = W p + W 1 B + W 2
B
6.0.1
where W P is the energy stored as pressure, W 1
B is the energy stored in toroidal fields (poloidal
currents), and W 2
B is the energy stored in poloidal fields (toroidal currents). Once we have
calculated expressions for these terms, we can obtain the required forces: the minor radial force
F a = ∂W/∂a, and the major radial force F R = ∂W/∂R. By setting the net force = 0 we will obtain
the conditions necessary for equilibrium. We work with a circular cross sectioned plasma with
major radius R and a minor radius a, and a/R