Experimental study of parametric subharmonic instability for internal plane waves

Abstract

Internal waves are believed to be of primary importance as they affect ocean mixing and energy transport. Several processes can lead to the breaking of internal waves and they usually involve non linear interactions between waves. In this work, we study experimentally the parametric subharmonic instability (PSI), which provides an efficient mechanism to transfer energy from large to smaller scales. It corresponds to the destabilization of a primary plane wave and the spontaneous emission of two secondary waves, of lower frequencies and different wave vectors. Using a time-frequency analysis, we observe the time evolution of the secondary waves, thus measuring the growth rate of the instability. In addition, a Hilbert transform method allows the measurement of the different wave vectors. We compare these measurements with theoretical predictions, and study the dependence of the instability with primary wave frequency and amplitude, revealing a possible effect of the confinement due to the finite size of the beam, on the selection of the unstable mode.

Nonlinear internal wave penetration via parametric subharmonic instability

“…having prominent peaks at the forcing frequency, ω 0 ≈ 1.03 s −1 , as well as at ω 1 ≈ 0.73 s −1 and ω 2 ≈ 0.31 s −1 , corresponding to the two daughter waves. These frequency peaks satisfy the resonance condition ω 0 = ω 1 + ω 2 to within 1%, thus confirming the formation of a resonant wave triad; the spatial resonance condition, which is a well established experimental feature, 13,14 was also checked and confirmed. The relatively high forcing frequency results in steeply oriented constant phase lines for the forced wave field, which in turn yields strong and weak vertical and horizontal velocities, respectively.…”

“…Complementary laboratory studies in uniform stratifications have demonstrated conditions under which internal wave fields become unstable to minute perturbations via PSI. 13,14 In this letter, we present the results of a laboratory experiment in which internal waves are excited in a relatively shallow, upper layer of high buoyancy frequency, N 1 , sitting atop a deeper layer of low buoyancy frequency, N 2 . The internal wave field is excited at a relatively high frequency ω 0 (i.e., N 2 < ω 0 < N 1 ) via boundary forcing imposed at the top of the upper layer; as such, the wave field excited at the forcing frequency is confined to the upper layer because it is evanescent in the lower layer.…”

The wave instability pathway to turbulence

“…One breakthrough of Bourget et al (2013) is the use of a new wave generation mechanism that creates a beam with three across-beam wavelengths spanning its width. Thus they are able to observe the onset of instability in the beam, as shown in figure 1(e,f ).…”

“…The other breakthrough of Bourget et al (2013) is the application of novel analysis methods that separate the signal of the primary beam and the subharmonic waves it excites. For example, at an arbitrary point in the tank measurements of the perturbation vertical density gradient versus time can be Fourier transformed.…”

“…The development of PSI in the x-z plane observed at two times in three circumstances: (a,b) the displacement of dye-lines in experiments with a vertically oscillating rectangular tank at t = 1200 and 1220 s (Benielli & Sommeria (1998); (c,d)) perturbations of the spanwise vorticity field computed in simulations of a wave beam in a doubly periodic domain, shown initially and after 40 buoyancy periods (Clark & Sutherland (2010); (e,f ) perturbations of the vertical density gradient in experiments on a wave beam emanating from a camshaft wave generator, shown after 10 and 50 wave periods (Bourget et al 2013). …”