Strong dispersive effects on internal nonlinear waves in a sheared, stablystratified fluid layer

Internal solitary waves are widely believed to propagate due to a balance b etween nonlinearity and dispersion. The expansion procedure introduced by B enney (J. Math. Phys. 45 (1966) 52-63) for weakly nonlinear, planar waves i n sheared, stratified flows in shallow layers, approximates the motion by t he Korteweg-de Vries equation (KdV) when the Ursell number Ur = epsilon/mu( 2) approximate to 1, where epsilon is the ratio of the amplitude of the wav e to the height of the waveguide and mu is the ratio of the same height to the wavelength. However, the scaling group Ri = N-2/gamma(2) which is the s quared ratio of the buoyancy frequency to the shear rate, is left as a free parameter. In the limit of high relative shear as Ri down arrow 1/4, the l eading order dispersion coefficient in the KdV equation becomes vanishingly small and the coefficient of nonlinearity becomes unbounded. Conversely, i n relatively strong stratifications as Ri --> infinity, the coefficient of nonlinearity becomes vanishingly small. Thus, higher order terms and other mechanisms need to be considered. This paper focuses on the role of higher order dispersion which permits consideration of short-wave disturbances. In the representative case of Couette shear and constant buoyancy frequency, estimates of the higher order dispersion coefficients are made in closed fo rm, allowing the truncation of the nonlinear wave equation at the appropria te level for short nonlinear waves via the estimation of the radius of conv ergence of the phase velocity in wavenumber for linear waves. (C)1999 Elsev ier Science B.V. All rights reserved.