The optical analogy for vector fields
- University of Chicago, Chicago, IL (USA). Department of Physics University of Chicago, Chicago, IL (USA). Department of Astronomy
This paper develops the {ital optical} {ital analogy} for a general vector field {bold F}({bold r}), noting that the projection {bold F}{sub {ital s}} of {bold F} onto any surface locus {ital S} of {del}{times}{bold F} can be expressed as the gradient of a scalar, {bold F}{sub {ital s}}={minus}{del}{phi}. It follows that the field lines of {bold F}{sub {ital s}} pass across {ital S} as if refracted by an index of refraction {vert bar}{bold F}{sub {ital s}}{vert bar}. A sufficiently localized maximum in {vert bar}{bold F}{sub {ital s}}{vert bar} refracts the field so as to produce a gap in the field pattern in {ital 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 {bold 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 {bold v} of an ideal fluid and a magnetostatic field {bold B}, the vectors {bold v} and {bold B} have surface loci in common with their curls. The intrinsic discontinuities around local maxima in {vert bar}{bold v}{vert bar} and {vert bar}{bold B}{vert bar} 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.
- OSTI ID:
- 5310015
- Journal Information:
- Physics of Fluids B; (United States), Vol. 3:9; ISSN 0899-8221
- Country of Publication:
- United States
- Language:
- English
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