Fundamentals of axis-symmetric boundary reconstruction for ideal tokamak plasmas: using toroidal harmonics to match poloidal flux measurements in the surrounding vacuum

With the development of tokamaks in the 1950s to control and sustain fusion reactions, a large amount of theory regarding associated plasmas has accumulated. Much of this theory is out of reach for the uninitiated, and applicable plasma physics texts either approach the field in too broad a path for specific reference, or they leave out many of the derivations needed for the foundational mechanics.

Here the fundamental math behind ideal tokamak plasmas is laid out to reconstruct the equilibrium boundary. This process illustrates the assumptions needed to idealize the mechanics, and thus gives wide avenues for alteration. Separation of variables is used in toroidal coordinates, and care is given to cover all the details. Optimizing the results to fit measurements from the surrounding vacuum can then produce a robust fixed boundary condition to be used in wider contexts.