The generation of long nonlinear internal waves in a weakly stratified shear flow

Abstract

A theoretical and experimental analysis is made to understand the generation of finite amplitude internal waves in fluid systems like the oceans and atmosphere in which the Richardson number is generally much greater than ¼. The initial disturbance is assumed to be of finite amplitude with the characteristic horizontal length scale much greater than the vertical length scale. An internal Korteweg and deVries (KdV) type equation for the stream function is derived for a weakly density stratified shear flow by using a three-parameter expansion method where the three small parameters correspond to nonlinear, dispersive, and non-Boussinesq effects. The influences of basic stratification and shear to the nonlinear, dispersive, and non-Boussinesq effects are found. Numerical solutions to this KdV-type equation for a variety of different conditions are then presented to demonstrate the relative importance of nonlinearity and dispersion on the generation of large amplitude internal waves in the breakdown of internal fronts. The numerical results are also in reasonable agreement with laboratory experiments in which a two-dimensional submarine ridge is moved to create transient internal disturbances. Additional numerical calculations show that a nonlinear model accurately describes the principal features of tidal-generated large amplitude internal waves observed in Massachusetts Bay.