The Parabolic Umbilic and Atmospheric Fronts
Abstract
This paper describes how to calculate the Legendre transformation of the beaktobeak form of the parabolic umbilic. The result is a multivalued surface in three dimensions, with multiplicity of up to five, which contains cusped edges of regression and lines of selfintersection. The calculation is selfcontained and is elementary in method but intricate in detail. The ideas in it will be available in other such calculations of similar complexity, and for any required context. Our underlying motivation is the study of atmospheric fronts in semigeostrophic theory. We show how a front may be regarded as a selfintersection line terminating at a swallowtail point on the Legendre transform surface. A physical stability argument requires excision of certain parts of the surface to leave a jointly convex and singlevalued surface having a gradient discontinuity along the front. A numerical illustration is presented.
 Publication:

Proceedings of the Royal Society of London Series A
 Pub Date:
 October 1988
 DOI:
 10.1098/rspa.1988.0111
 Bibcode:
 1988RSPSA.419..337C