INTERNAL SOLITARY WAVES MOVING OVER THE LOW SLOPES OF TOPOGRAPHIES

Internal solitary waves moving over uneven bottoms are analyzed based on the reductive perturbation method, in which the amplitude, slope an d horizontal lengthscale of a topography on the bottom are of the orde rs of epsilon, epsilon(5/2) and epsilon(-3/2), respectively, where the small parameter epsilon is also a measure of the wave amplitude. A fr ee surface condition is adopted at the top of the fluid layer. That co ndition contains two parameters, delta and Delta, the first of which c oncerns the discontinuity of the basic density between the outer layer and the inner one; the second concerns the discontinuity of the mean density between them. An amplitude equation for the disturbance of ord er epsilon decomposes into a Korteweg-de Vries (KdV) equation and a sy stem of algebraic equations for a stationary disturbance around a topo graphy on the bottom. Solitary waves moving over a localized hill are studied in a simple case where both the basic flow speed and the Brunt -Vaisalla frequency are constant over the fluid layer. For this case, the expression for the amplitude of the stationary disturbance contain s singular points with respect to basic flow speed. These singularitie s correspond to the resonant conditions modified by the free surface c ondition. The advancing speeds of solitary waves are changed by the in fluence of bottom topography, in a case where the long internal waves propagate in the direction opposite to the basic flow, but their wavef orms remain almost unchanged.