Elimination of electromagnetic radiation in plasma simulation: The Darwin or magneto inductive approximation

For many astrophysical and magnetic fusion applications, the purely electromagnetic modes generated by real as well as simulation “plasma” fluctuations are a source of high frequency radiation that is often irrelevant to the physics of interest. Unfortunately, a numerical CFL stability limit prevents either making c infinite or Δt large while using the usual explicit Maxwell's equations for the fields. A modification of Maxwell's equations, which provides implicitly the field components, circumvents this problem. The solution is to neglect retardation effects so that the electromagnetic propagation speed is effectively infinite. The purely electromagnetic modes in this limit evolve “instantly” to a time-asymptotic configuration about the macroscopic plasma configuration at each new time level. The Darwin or magnetoinduetive approximation effectively provides infinite propagation speeds for purely electromagnetic modes by converting Maxwell's equations from hyperbolic to elliptic in character. In practice, this is accomplished by neglecting the solenoidal part of the displacement current. The elimination of the CFL time step constraint more than offsets the substantially more complicated field solution that is required. The details of a numerical implementation of this model will be presented. Numerical examples will be given and extentions of the Darwin field solution to other plasma models also will be considered. This work was performed under the auspices of the U. S. Department of Energy by the Lawrence Livermore National Laboratory under Contract No. W-7405-Eng-48.