### 2.3 Magnetic discontinuities

Unless otherwise noted we will assume that the magnetic field is spatially continuous in the sense that
each of its spatial derivatives is defined and finite everywhere in space. The most important
exception occurs at a surface of magnetic discontinuity. The general theory of such discontinuities is well
documented in basic texts and reviews (see, e.g., Priest, 1982; Cowley et al., 1997). Due to the
solenoidal condition, , the component of normal to this surface must be continuous,
and the discontinuity must be in the tangential components, giving rise to a surface current
where is the surface normal and is the discontinuity across the surface
at a particular point.
Magnetic discontinuities occur across both fast shocks and slow shocks. In each shock there is a non-zero
normal component so each field lines contains an angular bend. Such a discontinuity may be removed by
local continuous deformation (essentially rounding the corners), and therefore is not an essential element of
the field’s topology. The third possibility, called a tangential discontinuity (TD) is one where
at the surface. Such a structure may be an equilibrium provided so that there is
pressure balance across the sheet (Cowley et al., 1997). The occurrence of TDs under various
circumstances is one of the key elements of topological field models. These are the most prevalent
examples of equilibrium current sheets, and so the term “current sheet” is sometimes used to mean
TD.