Inhibiting Shear Instability Induced by Large Amplitude Internal Solitary Waves in Two‐Layer Flows with a Free Surface

Abstract

We consider a strongly nonlinear long wave model for large amplitude internal waves in two-layer flows with the top free surface. It is shown that the model suffers from the Kelvin-Helmholtz (KH) instability so that any given shear (even if arbitrarily small) between the layers makes short waves unstable. Because a jump in tangential velocity is induced when the interface is deformed, the applicability of the model to describe the dynamics of internal waves is expected to remain rather limited. To overcome this major difficulty, the model is written in terms of the horizontal velocities at the bottom and the interface, instead of the depth-averaged velocities, which makes the system linearly stable for perturbations of arbitrary wavelengths as long as the shear does not exceed a certain critical value.

The Kelvin-Helmholtz Instabilities in Two-Fluids Shallow Water Models

“…The problem is however much more technical than for the rigid lid case, and we resort to numerical computations. As noticed in [35,1,11,40], the free surface case is also marked by a peculiar phenomenon: low frequency modes are stable for small shears as in the rigid lid case, but also for large enough shears.…”

“…We also consider here the free surface case where the upper boundary is now, like the interface, a free surface. This framework is of course relevant for many applications and has been considered for instance in [19,3,20,21,1,24,26,11,40]. We follow the same approach as above: we exhibit a condition for the stability of low frequency modes, and a stronger condition for the stability of all modes.…”

“…The investigation of the next order models (GN/GN) is not done in this paper because it is highly technical, and because it can be done following the same approach as in the rigid lid case. Note that GN/GN type models have been studied in the free surface case in [1], and that it is possible to generalize this study by deriving regularized models with the techniques used here in the rigid lid case in order to obtain a better description of the formation of Kelvin-Helmholtz instabilities.…”

Effect of variation in density on the stability of bilinear shear currents with a free surface

“…We now show that G 0, and G = 0 only if λ = 2ρ − 1. Following the same notation as Fuller [30,34] we consider the inner determinants ∆ 7 , ∆ 5 , and ∆ 3 for this quartic G in (D.4):…”

The square root depth wave equations

“…Grid refinement only makes the problem worse. Regularization by keeping higher order expansion terms is possible (Barros & Choi 2009), but such methods tend to destroy the Hamiltonian property of the system and thus may degrade its travelling wave structure. Nevertheless, if one is to consider the wave generation problem, one must consider the effects of both topography and shear.…”