The optical analogy for vector fields

This paper develops the optical analogy for a general vector field F(r), noting that the projection Fs of F onto any surface locus S of ∇xF can be expressed as the gradient of a scalar, Fs=-∇φ. It follows that the field lines of Fs pass across S as if refracted by an index of refraction |Fs|. A sufficiently localized maximum in |Fs| refracts the field so as to produce a gap in the field pattern in S, with the field bifurcating and passing around the sides, rather than across, the maximum. Such a disruption of the field pattern produces a discontinuity in F. The optical analogy allows the examination of certain aspects of a vector field that are not otherwise readily accessible. In particular, in the cases of a stationary Eulerian flow v of an ideal fluid and a magnetostatic field B, the vectors v and B have surface loci in common with their curls. The intrinsic discontinuities around local maxima in |v| and |B| take the form of vortex sheets and current sheets, respectively, the former playing a fundamental role in the development of hydrodynamic turbulence and the latter playing a major role in heating the x-ray coronas of stars and galaxies