A class of exact two‐dimensional kinetic current sheet equilibria

Abstract

The present paper discusses a class of exact two‐dimensional kinetic current sheet equilibria. The general solution to the two‐dimensional Grad‐Shafranov equation was first obtained by Walker in 1915 in terms of the generating function g(ζ) (ζ = X + iZ), where X and Z are two dimensionless spatial coordinates. There are infinite choices of g(ζ), but not every solution yields physically meaningful or mathematically useful form. The known solutions to date with geophysical application include those by Harris [1962], Fadeev et al. [1965], Kan [1973], Manankova et al. [2000], and Brittnacher and Whipple [2002]. In this paper, these solutions are reviewed systematically, and several new solutions are proposed. These include a generalization of the Harris‐Fadeev‐Kan‐Manankova line of model, a model for an isolated X‐line alternative to that of Brittnacher‐Whipple, and finally a model which represents an isolated magnetic island.