Conjugate flows and flat solitary waves for a continuously stratified fluid

The concept of conjugate flows is used to determine the vertical structure of solitary internal waves which are horizontally uniform in their center. Continuously stratified fluids are considered and solutions obtained with and without the Boussinesq approximation are compared. Only mode-1 waves are considered. For stratifications with a single pycnocline, conjugate flow solutions are obtained provided the pycnocline is not too close to the upper or lower boundaries. The parallel shear flow in the center of a flat solitary wave is potentially unstable (minimum Richardson number less than 1/4) if the upstream pycnocline is sufficiently narrow. For stratifications with two pycnoclines, cases with three mode-1 conjugate flow solutions have been found. Some conjugate flow solutions for the two-pycnocline case do not seem to correspond to a flat solitary wave. Non-Boussinesq effects were found to be small if the surface to bottom density difference is about 4% for stratifications with one or two pycnoclines.