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 physical space to amplitude
space. For steadily propagating waves the evolution equation reduces to the steady
version of the generalized Korteweg–de Vries equation. An analysis of this equation
is presented for a wide range of 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 of the shear profile in amplitude space. Up to three possible
inflection points for this function are considered, resulting in solitary waves and
kinks (dispersionless bores) which can have up to three characteristic lengthscales.
The stability of these waves is generally found to decrease as the complexity of the
waves increases. These solutions suggest that kinks and solitary waves with multiple
lengthscales are only possible for shear profiles (in physical space) with a turning
point, while instability is only possible if the shear profile has an inflection point.
The unsteady evolution of a periodic initial condition is considered and again the
solution is found to depend on the inflection points of the amplitude representation
of the shear profile. Two characteristic types of solution occur, the first where
the initial condition evolves into a train of rank-ordered solitary waves, analogous
to those generated in the framework of the Korteweg–de Vries equation, and the
second where two or more kinks connect regions of constant amplitude. The unsteady
solutions demonstrate that finite-amplitude effects can act to halt the critical
collapse of solitary waves which occurs in the context of the generalized Korteweg–de
Vries equation. The two types of solution are then used to qualititatively relate
previously reported observations of shock formation on the internal tide propagating
onto the Australian North West Shelf to the observed background current shear.
HIGHER ORDER KORTEWEG-DE VRIES MODELS FOR INTERNAL WAVES
