Stellarator Equilibrium Theory

 

The equilibrium (i.e., time-independent state) of the stellarator configuration is the starting point for most stellarator modeling and calculations. The information we get from the equilibrium state consists of the complete 3-dimensional structure of the magnetic field within the plasma. In order to obtain a unique stellarator equilibrium, the following data must be specified:

- the plasma pressure profile

- the plasma current profile or rotational transform profile

- the 3-dimensional shape of the outermost flux surface

The most fundamental form of the equilibrium equations consists of the plasma force balance combined with two of Maxwell's equations (obtained by neglecting electric fields and setting time derivatives to zero):

These equations are not solved directly, but rather reduced to a variational form involving the plasma potential energy:

This potential energy is then minimized by what is known as an inverse approach [VMEC model]. It is first assumed that closed flux surfaces exist within the plasma. The shape of these surfaces is represented by Fourier expansions for the major radius R (distance from the center of the torus) and the height z above the midplane of the torus:

The set of Rmn's and Zmn's that minimize the potential energy are then solved for. This is an inverse approach because we are solving for the coordinates of a specified value of magnetic field (i.e., the magnetic flux surface) rather than the more usual direct approach of solving for the magnetic field as a function of coordinates.

 

 

The above plot shows the results of a stellarator equilibrium calculation. Here a section of a stellarator is shown with the colors proportional to the magnetic field strength. The concentric (or nested) curves shown in the cross-sectional plane are the magnetic flux surfaces. As one moves around the torus and looks at other cross-sectional planes, the flux surface shapes will vary, as shown in the following figures:

Toroidal angle: z = 0

z = 90

z = 180

 

 

Animated quicktime movie showing flux surface shape as a function ot toroidal position:

 

 

S. P. Hirshman, J. C. Whitson, "Steepest-descent moment method for three-dimensional magnetohydrodynamic equilibria," Physics of Fluids 26, (December, 1983) pg. 3553.