Marek Stastna

Large fully nonlinear internal solitary waves: The effect of background current

In this paper we consider what effect the presence of a nonconstant background current has on the properties of large, fully nonlinear solitary internal waves in a shallow, stratified ocean. In particular, we discuss how the amplitude of the largest nonbreaking wave that it is possible to calculate depends on the background current as well as the nature of the upper bound. We find that the maximum wave amplitude is given by one of three possibilities: The onset of wave breaking, the conjugate flow amplitude or a failure of the wave calculating algorithm to converge (associated with shear instability). We also discuss how wave properties such as propagation speed, half-width, etc. vary with background current amplitude.

On the resonant generation of large-amplitude internal solitary and solitary-like waves

In this paper we discuss numerical simulations of the generation of large-amplitude solitary waves in a continuously stratified fluid by flow over isolated topography. We employ the fully nonlinear theory for internal solitary waves to classify the numerical results for mode-1 waves and compare with two classes of approximate theories, weakly nonlinear theory leading to the Korteweg–deVries and Gardner equations and conjugate flow theory which makes no approximation with respect to nonlinearity, but neglects dispersion entirely. We find that both weakly nonlinear theories have a limited range of applicability. In contrast, the conjugate flow theory predicts the nature of the limiting upstream propagating response (a dissipationless bore), successfully describes the bores vertical structure, and gives a value of the inflow speed,

Short note: A short note on the discontinuous Galerkin discretization of the pressure projection operator in incompressible flow

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Poroelastic acoustic wave trains excited by harmonic line tractions

, above which no upstream propagating response is possible. The numerical experiments demonstrate the existence of a class of large-amplitude response structures that are generated and trapped over the topography when the inflow speed exceeds

Internal wave boundary layer interaction: A novel instability over broad topography

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Numerical simulation of supercritical trapped internal waves over topography

. While similar in structure to fully nonlinear solitary waves, these trapped disturbances can induce isopycnal displacements more than 100% larger than those induced by the limiting solitary wave while remaining laminar. We develop a theory to describe the vertical structure at the crest of these trapped disturbances and describe its range of validity. Finally, we turn to the generation of mode-2 solitary-like waves. Mode-2 waves cannot be truly solitary owing to the existence of a small mode-1 tail that radiates energy downstream from the wave. We demonstrate that, for stratifications dominated by a single pycnocline, mode-2 wave dissipation is dominated by wave breaking as opposed to mode-1 wave radiation. We propose a phenomenological criterion based on weakly nonlinear theory to test whether mode-2 wave generation is to be expected for a given stratification.

Intrinsic Breaking of Internal Solitary Waves in a Deep Lake

This note reports on the issue of spurious compressibility artifacts that can arise when the popular pressure projection (PP) method is used for unsteady simulations of incompressible flow using the symmetric interior penalty discontinuous Galerkin (SIP-DG) method. Through a spectral analysis of the projection operators SIP-DG discretization, we demonstrate that the eigenfunctions of the operator do not form a basis that allows for the correct enforcement of the incompressibility constraint. This short-coming can cause numerical instabilities for inviscid, advection-dominated, and density stratified flow simulations, especially for long-time integrations and/or under-resolved situations. To remedy this problem, we propose a local post-processing projection that enforces incompressibility exactly to allow for stable and robust long-time integrations.

On fully nonlinear, vertically trapped wave packets in a stratified fluid on the f-plane

A two-dimensional boundary-value problem for a porous half-space with an open boundary, described by the widely recognized Biots equations of poroelasticity, is considered. Using complex analysis techniques, a general solution is represented as a superposition of contributions from the four different types of motion corresponding to P1, P2, S and Rayleigh waves. Far-field asymptotic solutions for the bulk modes, as well as near-field numerical results, are investigated. Most notably, this analysis reveals the following: (i) a line traction generates three wave trains corresponding to the bulk modes, so that P1, P2 and S modes emerge from corresponding wave trains at a certain distance from the source, (ii) bulk modes propagating along the plane boundary are subjected to geometric attenuation, which is found quantitatively to be x−3/2, similar to the classical results in perfect elasticity theory, (iii) the Rayleigh wave is found to be predominant at the surface in both the near (due to the negation of the P1 and S wave trains) and the far field (due to geometric attenuation of the bulk modes), and (iv) the recovery of the transition to the classical perfect elasticity asymptotic results validates the asymptotics established herein.

Mass transport by mode-2 internal solitary-like waves

It has been known for some time that internal wave-induced currents can drive near bed instabilities in the bottom boundary layer over a flat bottom. When the bottom is not flat, the situation can become quite complicated, with a diverse set of mechanisms responsible for instability and the subsequent transition to turbulence. Using numerical simulations, we demonstrate the existence of a mode of instability due to internal solitary wave propagation over broad topography that is fundamentally different from the two dominant paradigms of flow separation over sharp topography and global instability in the wave footprint that occurs over a flat bottom observed at high Reynolds number. We discuss both the two and three-dimensional evolution of the instability on experimental scales. The instability takes the form of a roll up of vorticity near the crest of the topography. As this region is unstratified in our simulations, little three-dimensionalization is observed. However, the instability-induced currents p...

Transcritical generation of nonlinear internal waves in the presence of background shear flow

We consider the steady flow of a stratified fluid over topography in a fluid of finite vertical extent, as typified by experimental flumes with a rigid lid or the ocean under the rigid lid approximation. We do not specify a functional form of the upstream stratification or background current and derive a general version of the Dubreil–Jacotin–Long equation appropriate for the problem. This elliptic equation is strongly nonlinear and we develop an efficient, pseudospectral, iterative method for its numerical solution. The method allows us to compute laminar, trapped waves with amplitudes more than 50% of the depth of the fluid. We find that when either a background shear current is present or the topography is narrow enough, multiple steady states are possible and we confirm this finding by using integrations of the full time-dependent Euler equations. We discuss instances of waves with closed streamlines, finding that the presence of shear allows for waves with vortex cores that persist for long times in ...