Conjugate flows for waves with trapped cores

Previous numerical simulations have shown that in a bounded fluid, internal solitary waves with trapped cores become horizontally uniform in their center and have a maximum amplitude that is dependent on the details of the recirculating flow inside the core [K. G. Lamb, “A numerical investigation of solitary internal waves with trapped cores formed via shoaling,” J. Fluid Mech. 451, 109 (2002)]. The flow in the horizontally uniform center of these waves is called a conjugate flow. Here a theoretical model for conjugate flows with a trapped core (also called a vortex core) is presented. The flow in the core is assumed to be steady with constant density. General expressions for arbitrary nonuniform core vorticity are developed which depend on the surface velocity and the mean squared horizontal velocity in the core. Analytic results for cores with constant vorticity are presented for a two-layer fluid with constant density in the lower layer and constant buoyancy frequency in the upper layer. Comparisons of the theory with time-dependent numerical simulations of waves with flat-centered trapped cores in exponential density profiles are made. The theoretical results are shown to accurately predict the amplitude and core thickness of the conjugate flows when the values of the surface velocity and mean squared velocity are taken from the simulations.