SOLITARY WAVES ON A VORTICITY LAYER

Contour dynamics methods are used to determine the shapes and speeds o f planar, steadily propagating, solitary waves on a two-dimensional la yer of uniform vorticity adjacent to a free-slip plane wall in an, oth erwise irrotational, unbounded incompressible fluid, as well as of axi symmetric solitary waves propagating on a tube of azimuthal vorticity proportional to the distance to the symmetry axis. A continuous family of solutions of the Euler equations is found in each case. In the pla nar case they range from small-amplitude solitons of the Benjamin-Ono equation to large-amplitude waves that tend to one member of the touch ing pair of counter-rotating vortices of Pierrehumbert (1980), but thi s convergence is slow in two small regions near the tips of the waves, for which an asymptotic analysis is presented. In the axisymmetric ca se, the small-amplitude waves obey a Korteweg-de Vries equation with s mall logarithmic corrections, and the large-amplitude waves tend to Hi ll's spherical vortex.