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 ∂Bi ∕∂xj 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, 1982Cowley et al., 1997Jump To The Next Citation Point). Due to the solenoidal condition, ∇ ⋅ B = 0, the component of B normal to this surface must be continuous, and the discontinuity must be in the tangential components, giving rise to a surface current
K = -c-[[B ]] × ˆn, (10 ) 4π
where ˆn is the surface normal and [[B ]] ≡ B (x + 𝜖ˆn) − B (x − 𝜖ˆn) 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 ˆn ⋅ B = 0 at the surface. Such a structure may be an equilibrium provided 2 [[B ]] = 0 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.

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