The effect of weak shear on finite-amplitude internal solitary waves

A finite-amplitude long-wave equation is derived to describe the effect of weak current shear on internal waves in a uniformly stratified fluid. This effect is manifested through the introduction of a nonlinear term into the amplitude evolution equation, representing a projection of the shear from p hysical space to amplitude space. For steadily propagating waves the evolut ion equation reduces to the steady version of the generalized Korteweg-de V ries equation. An analysis of this equation is presented for a wide range o f possible shear profiles. The type of waves that occur is found to depend on the number and position of the inflection points of the representation o f the shear profile in amplitude space. Up to three possible inflection poi nts for this function are considered, resulting in solitary waves and kinks (dispersionless bores) which can have up to three characteristic lengthsca les. The stability of these waves is generally found to decrease as the com plexity of the waves increases. These solutions suggest that kinks and soli tary waves with multiple lengthscales are only possible for shear profiles tin physical space) with a turning point, while instability is only possibl e if the shear profile has an inflection point. The unsteady evolution of a periodic initial condition is considered and again the solution is found t o depend on the inflection points of the amplitude representation of the sh ear profile. Two characteristic types of solution occur, the first where th e initial condition evolves into a train of rank-ordered solitary waves, an alogous to those generated in the framework of the Korteweg-de Vries equati on, and the second where two or more kinks connect regions of constant ampl itude. The unsteady solutions demonstrate that finite-amplitude effects can act to halt the critical collapse of solitary waves which occurs in the co ntext of the generalized Korteweg-de Vries equation. The two types of solut ion are then used to qualititatively relate previously reported observation s of shock formation on the internal tide propagating onto the Australian N orth West Shelf to the observed background current sheaf.