Three-dimensional small-scale instabilities of plane internal gravity waves

Abstract

We study the evolution of three-dimensional (3-D), small-scale, small-amplitude perturbations on a plane internal gravity wave using the local stability approach. The plane internal wave is characterised by its non-dimensional amplitude, $A$, and the angle the group velocity vector makes with gravity, $\unicode[STIX]{x1D6F7}$. For a given $(A,\unicode[STIX]{x1D6F7})$, the local stability equations are solved on the periodic fluid particle trajectories to obtain growth rates for all two-dimensional (2-D) and 3-D perturbation wave vectors. For small $A$, the local stability approach recovers previous results of 2-D parametric subharmonic instability (PSI) while offering new insights into 3-D PSI. Higher-order triadic resonances, and associated deviations from them, are also observed at small $A$. Moreover, for small $A$, purely transverse instabilities resulting from parametric resonance are shown to occur at select values of $\unicode[STIX]{x1D6F7}$. The possibility of a non-resonant instability mechanism for transverse perturbations at finite $A$ allows us to derive a heuristic, modified gravitational instability criterion. We then study the extension of small $A$ to finite $A$ internal wave instabilities, where we recover and build upon existing knowledge of small-scale, small-amplitude internal wave instabilities. Four distinct regions of the $(A,\unicode[STIX]{x1D6F7})$-plane based on the dominant instability modes are identified: 2-D PSI, 3-D oblique, quasi-2-D shear-aligned, and 3-D transverse. Our study demonstrates the local stability approach as a physically insightful and computationally efficient tool, with potentially broad utility for studies that are based on other theoretical approaches and numerical simulations of small-scale instabilities of internal waves in various settings.

Experimental study on superharmonic wave generation by resonant interaction between internal wave modes

“…Specifically, three plane internal waves that satisfy ω 1 + ω 2 + ω 3 = 0 (positive or negative frequencies ω 1,2,3 ) and k 1 + k 2 + k 3 = 0 spontaneously exchange energy between each other, and are termed together as a resonant triad. In fact, triadic resonance is the underlying mechanism for almost all instabilities in finite-amplitude plane internal waves [20,21]. Triadic resonance is possible in internal wave modes, too [22], and this paper presents an experimental investigation of the same.…”

“…We proceed to compare the observed spatial growth of the steady-state superharmonic modal amplitude in the resonant forcing frequency experiment with the theoretical predictions based on the amplitude evolution Eqs. (20) and (21). Figure 5(a) shows the spatial evolution of the steady-state primary wave modal amplitudes in the case-1 resonant triad experiment with (A 3 , A 4 ) = (5, 5) mm and the resonant forcing frequency of ω 0 /N = 0.4841.…”

“…The spatial growth of the steady-state superharmonic wave at small distances from the wave generator is captured accurately by the amplitude evolution equations, (20) and (21), wherein it was assumed that the steady-state modal amplitudes evolve over relatively large spatial scales. During the transient period in the experiment, however, the wave amplitudes at a fixed spatial location are temporally evolving, thus rendering the amplitude evolution equations, (20) and (21), not well suited. We proceed to explore how well amplitude evolution equations based on a slow temporal evolution could describe the observed wave fields before steady state is reached.…”

“…Following a procedure that is very similar to how Eqs. (20) and (21) were derived, the amplitude evolution equations can now be shown to be…”

“…The coefficients α j appeared earlier in the amplitude evolution Eqs. (20) and (21), and c g, j is the group speed of mode j at frequency ω j . We investigate the usefulness of Eqs.…”