A regularized model for strongly nonlinear internal solitary waves

Abstract

The strongly nonlinear long-wave model for large amplitude internal waves in a two-layer system is regularized to eliminate shear instability due to the wave-induced velocity jump across the interface. The model is written in terms of the horizontal velocities evaluated at the top and bottom boundaries instead of the depth-averaged velocities, and it is shown through local stability analysis that internal solitary waves are locally stable to perturbations of arbitrary wavelengths if the wave amplitudes are smaller than a critical value. For a wide range of depth and density ratios pertinent to oceanic conditions, the critical wave amplitude is close to the maximum wave amplitude and the regularized model is therefore expected to be applicable to the strongly nonlinear regime. The regularized model is solved numerically using a finite-difference method and its numerical solutions support the results of our linear stability analysis. It is also shown that the solitary wave solution of the regularized model, found numerically using a time-dependent numerical model, is close to the solitary wave solution of the original model, confirming that the two models are asymptotically equivalent.

A Model for Strongly Nonlinear Long Interfacial Waves with Background Shear

“…In this case, the calculation above shows that a quadratic truncation of the equations is stable for w 2 0 < 2/3. This calculation confirms the result of [19] whereby, using wall variables, the MCC equations were stabilized.…”

“…Thus, the nonhydrostatic system is linearly unstable at high wavenumbers and mathematically ill-posed. This has been noted before and some possible resolutions to the problem have been proposed [13,19]. Note that even when w = 0 initially, it is clear that in (16) and (17), w will not remain zero and thus high-frequency oscillations will become unstable due to the linearized behavior about the new nonzero state.…”

“…Seeking nontrivial solutions of (18) and (19) proportional to e i(kx−ωt) yields a dispersion relation…”

“…One may use a numerical filter for high wavenumbers, thus damping the unstable frequencies, or regularize the equations consistently with the underlying physics. In [19], for example, a stable model is formulated in terms of values of u 1 and u 2 at the top and bottom walls, respectively, as opposed to their vertical averagesū 1 andū 2 used here. Here, an alternative is proposed, based on exchanges between time and space derivatives used to regularize surface wave equations (see [16]).…”

Dispersive dam-break and lock-exchange flows in a two-layer fluid

“…With regards to the oceanographic application discussed above, the surface tension cannot be justified physically, and is perhaps best considered as a numerical filter that is required for regularisation. Note that an alternative approach to regularisation has been described recently by Choi, Barros & Jo (2009). Nevertheless, τ is retained throughout our analysis since there is no obstacle in principle to setting up experiments in a two-fluid system with finite τ , with which the results below might be compared.…”