Trapped internal waves over topography: Non-Boussinesq effects, symmetry breaking and downstream recovery jumps

Abstract

It is well-known that in certain parameter regimes, the steady flow of a density stratified fluid over topography can yield large amplitude internal waves. We discuss an embedded boundary method to solve the Dubreil-Jacotin-Long (DJL) equation for steady-state, supercritical flows over topography in an inviscid, stratified fluid. The DJL equation is equivalent to the full set of stratified steady Euler equations and thus the waves we compute are exact nonlinear solutions. The numerical method presented yields far better scaling with increase in grid size than other iterative methods that have been used to solve this equation, and this in turn allows for a more thorough exploration of parameter space. For waves under the Boussinesq approximation, we contrast the properties of trapped waves over hill-like and valley-like topography, finding that the symmetry of freely propagating solitary waves when the stratification is reflected across the middepth is not present for trapped waves. We extend the derivation of the DJL equation to the non-Boussinesq case and discuss the effect of the new, non-Boussinesq terms on the structure of the trapped waves, finding that the sharp transition between large and small amplitude waves observed under the Boussinesq approximation is much more gradual when the Boussinesq approximation is relaxed. Finally, we demonstrate the existence of asymmetric steady states over hill-like topography where the flow is subcritical upstream of the topography but transitions to supercritical somewhere over the hill. Waves in this new class of exact solutions are related to so-called downstream recovery jumps predicted on the basis of hydrostatic (shallow water) theories, but when breaking does not occur the recovery jump does not stop propagating downstream and an asymmetric state across the topography maximum is reached for long times.

Topographically generated internal waves and boundary layer instabilities

“…As described by Soontiens et al, 15 large trapped waves over topography can be expected in inviscid, supercritical flow over depression topography when the pycnocline is centered above the mid-depth and in flow over elevated topography when the pycnocline is centered below the mid-depth. These large-amplitude cases have motivated simulations 2DL1 and 2DLVa which examine flow over elevated and depression topography, respectively.…”

“…These simulations were motivated by large amplitude solutions to a steady state inviscid problem. 15 Next, we examine how the viscous boundary layer modifies trapped wave formation. In the case of depression topography, the presence of the viscous boundary layer can greatly alter the wave generation process, as demonstrated in Figure 5 where we plot the density field from inviscid steady theory and time-dependent simulations using the parameters in simulations 2DLVa and 2DLVb.…”

“…The inviscid results are solutions to the DJL equation, a single nonlinear partial differential equation for the isopycnal displacement that is equivalent to the steady Euler equations of motion. The solution procedure is described by Soontiens et al 15 who found that very large trapped waves over depression topography can exist if the background speed U 0 is close to the conjugate flow speed c j . Large waves over elevation topography also exist but do not reach the same amplitudes.…”

“…Previous work using a steady inviscid theory has shown that very large amplitude trapped internal waves exist over both elevated and depression topography in flow conditions where no upstream propagating modes exist. 15 These large wave states have motivated the cases considered in this study which investigates wave formation in the presence of a viscous boundary layer. Simulations in both two and three dimensions are considered.…”

“…The mechanisms that lead to the generation of these large internal waves have been under investigation for some time, with stratified flow over topography a primary example. [13][14][15] One particular case is the resonant generation of internal waves. 14,16 Some of these large waves are found in the lee of the topography, others advance slowly upstream and a third class remains trapped over the topography.…”