A numerical study of conjugate flows and flat-centred internal solitary waves in a continuously stratified fluid

Wan, Bangjun (1997) A numerical study of conjugate flows and flat-centred internal solitary waves in a continuously stratified fluid. Masters thesis, Memorial University of Newfoundland.

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Abstract

In this thesis a theoretical model describing the limiting flow structure in the centre of a fully nonlinear, flat-centred internal solitary wave in a fluid of finite depth H has been developed using the conjugate flow concept. The conjugate flow solution gives the vertical structure of the isopycnal displacement and the fluid velocity at the centre of a flat-centred internal solitary wave as well as the propagation speed of the wave. The mode-1 internal solitary waves are calculated in a continuously stratified fluid given by hyperbolic tangent density profiles with one or two pycnoclines. Solutions obtained with and without the Boussinesq approximation are compared. The non-Boussinesq results are almost identical with the Boussinesq results if the surface to bottom density difference is 4% or less unless the pycnoclines have a thickness comparable to the total fluid depth. -- For density stratifications with a single pycnocline, conjugate flow solutions are obtained when the pycnocline is not too close to the boundary. The size of the valid solution range decreases as the thickness of pycnocline increases. When the Boussinesq approximation is applied, the magnitude of the extreme isopycnal displacement grows as the centre of the pycnocline in the undisturbed region moves away from the mid-depth: the wave propagation speed increases as the centre of pycnocline moves toward the mid-depth. If the thickness of the pycnocline is greater than 8.4% of the fluid depth, the parallel shear flow in the centre of a flat-centred internal solitary wave is linearly stable. As the pycnocline gets narrower the flow becomes potentially unstable over an increasing range of pycnocline heights. -- For stratifications with two pycnoclines multiple conjugate flow solutions may exist. When the two pycnoclines are equidistant from the mid-depth, one above and one below, there are two solutions if the pycnoclines are well separated and not too close to the boundaries. If the pycnoclines are close together there are no solutions if the Boussinesq approximation is made and one solution if the approximation is not made. If the two pycnoclines are not equidistant from the mid-depth there can be 0. 1. 2. or 3 solutions. Flat-centred wave can exist only if there is a conjugate flow solution, but the converse is not true. Having a conjugate flow solution does not necessarily mean that there is a flat-centred internal solitary wave.

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