Session E27: Waves: Internal and Interfacial Waves

Triadic instability, vortex cluster, and weak inertial wave turbulence: observations on a wave attractor in an axisymmetric setup

Numerous studies have delved into the properties of inertia-gravity wave reflection, showing in particular that in a confined trapezoidal domain the wave beam experiences a focusing effect, and eventually ends on a limit trajectory called attractor. Evidence of this particular structure have been found in 2D experimental and numerical surveys, and it has been more recently the focus of 3D investigations. Wave attractors are of primordial importance since nonlinear triadic cascades may occur in the branches of the attractor, due to the local energy focusing, leading to energy transfer between scales.

We present experimental and numerical observations of the fate of an axisymmetric inertial wave attractor excited in a fully confined trapezoidal cylindrical domain. Due to focusing and defocusing effects caused by wave reflections and by the radial geometry itself, wave instability occurs while forcing the attractor, eventually leading to an azimuthal symmetry breakdown characterized by the emergence of a regular polygonal vortex cluster. A systematic study of the role of the forcing amplitude, performed using DNS, shows different behaviors: a linear regime, a 3D triadic resonant instability, and a spectral enrichment reminiscent of weak inertial wave turbulence.   

Turbulent aspects of a subsurface-recirculating core in a shoaling internal solitary wave of depression through high-resolution/accuracy three-dimensional simulations

The three-dimensional convective breaking of an internal solitary wave (ISW) of depression, shoaling over a realistic gentle bathymetric slope and complex background stratification/current profiles sampled in the South China Sea, is examined via high-resolution/accuracy simulations. These massively parallel simulations are based on a hybrid high-order spectral-element-method/Fourier Galerkin flow solver which enables the reliable representation of the complex shoaling bathymetry, under the constraint of the ISW propagating in the normal-to-the-isobath direction. As the ISW shoals into shallower waters it undergoes a distinct convective instability: as the ISW continues to maintain its symmetric waveform, the plunging isopycnal originating from the rear of the wave produces a heavy-over-light configuration in the wave interior, effectively leading to the formation of a recirculating subsurface core. A subsequent secondary transverse instability follows, leading to turbulence transition and the resulting formation of finer-scale turbulence therein. Beyond outlining the structural features of the three-dimensional breaking process, the presentation also investigates the ISW-driven turbulent dissipation and its connection to the persistence of the recirculating core.   

Aspects of the spectral-element-based simulation of a model internal swash zone

We examine the design and implementation of a numerically simulated internal swash zone in a two-layer continuous stratification, in two-dimensions. The domain is chosen sufficiently long compared to the wavelength to allow the wave to develop before impacting the downstream sloping boundary. A high-accuracy/resolution spectral-element-method incompressible flow solver is used. A first topic of focus is the implementation of wave-generating boundary conditions at the deep-water boundary. We also investigate subtleties of the implementation the zero-flux boundary condition for the density on the downstream sloping boundary. In particular, challenges that may emerge due to straining of the near-bed isopycnal field by incident finite amplitude periodic internal waves are considered. We will study the interaction of the incident created wave with the sloping boundary, namely the interactions with the reflected wave and any subsequent waves resulting from the periodic deep-water forcing. We will also discuss the implications of our findings for future three-dimensional simulations of internal-wave-induced transition and turbulence on the deformed wall.   

Solitary wave fission in a viscous fluid conduit

This talk presents a theoretical and experimental study of the long-standing fluid mechanics problem involving the temporal resolution of a large localised initial disturbance into a sequence of solitary waves. Of fundamental importance in a range of applications including tsunami and internal ocean wave modelling, this problem is studied in the context of the viscous fluid conduit system–the driven, cylindrical, free interface between two miscible Stokes fluids with high viscosity contrast. Owing to buoyancy-induced nonlinear self-steepening balanced by stress-induced interfacial dispersion, the disturbance evolves into a slowly modulated wavetrain and further into a sequence of solitary waves. An extension of Whitham modulation theory is used to resolve the fission of an initial disturbance into solitary waves. The developed theory predicts the relationship between the initial disturbance's profile, the number of emergent solitary waves and their amplitude distribution. The theoretical predictions for the fluid conduit system are confirmed both numerically and experimentally.  The number of observed solitary waves is consistently within one to two waves of the prediction, and the amplitude distribution shows remarkable agreement.   

Characterization of non-linear internal waves using PIV/PLIF techniques

Internal waves are a fascinating physical phenomenon that play an important role in the mixing and dynamics of both atmospheric and oceanographic flows. This experimental study addresses non-linear internal waves due to their importance in shaping the circulation and distributions of heat and carbon within density stratified systems. We aim to fully understand the dynamics of internal waves by measuring the density and velocity fields using combined PLIF/PIV measurements and comparing the experimental results with the theoretical non-linear wave solution. Non-linear theory is required due to the non-negligible amplitude of the wave compared to the wavelength. A laboratory-scale apparatus was created to replicate the flow characteristics of standing internal waves in a two-layer stratified system. Experimental results are presented for configurations with a density jump of 1.1 and 1.5 σ_t (separately). The interface location, density gradient, wave amplitude and period, velocity and vorticity fields, kinetic energy, and shear strain rate are quantified at several phases in one wave cycle. The experimental results are compared with the corresponding predictions based on third-order Stokes internal-wave theory. The results showed that the 3^rd-order non-linear theory does an outstanding job of describing the internal wave flow.   

Laboratory Observations and Consequences of Parametric Subharmonic Instability in an Internal Wave Field

Diapycnal mixing in the ocean interior modulates the meridional overturning circulation. Since energy is typically injected at scales that are too large to mix effectively, mixing requires nonlinear processes to transfer energy toward small scales where 3D turbulence occurs. Here we consider internal waves generated by the flow of the barotropic tide over ocean ridges. The dynamics of a barotropic tide over an ocean are akin to a stationary fluid forced by an oscillating topographic feature. Therefore, we were able to simulate the energy transfer in a laboratory experiment by oscillating an artificial ridge in a linearly stratified tank. We observed the emergence of the iconic internal wave beams, and after a sufficiently long time, the beams engendered Parametric Subharmonic Instability (PSI) which created vertically periodic mixed regions that spread as thin intrusions. Our work shows that under appropriate conditions internal wave beams efficiently transfer energy from large forcing scales into small mixing scales through PSI, an effective agent of mixing in the ocean.   

Observations of Breathers in Soliton-Cnoidal Wave Interaction in a Viscous Fluid Conduit - Part II

Conduits generated by the buoyant injection of a miscible, Stokes fluid into another Stokes fluid with high viscosity contrast have been studied due to their remarkable nonlinear wave behavior. Applications in geological and geophysical contexts include the dynamics of channelized flows in magmatic and glacial systems. Breathers, localized disturbances to nonlinear periodic wavetrains, have been theoretically studied in the strongly nonlinear regime of a viscous fluid conduit using the so-called conduit equation. The present work utilizes a laboratory setup for fluid interfacial waves that can reliably generate solitons and nonlinear periodic waves by varying the flow rate of dyed, diluted glycerin injected into the bottom of a pure glycerin reservoir. The generated cnoidal-type waves quantitatively agree with exact conduit equation periodic traveling wave solutions. Breathers of the elevation and depression types are observed to be the result of the nonlinear superposition of solitons and cnoidal waves. In this study, the first experimental results exhibiting the complex, coherent, nonlinear structures of breathers in a fluid conduit are presented, motivating future investigations in other fluid dynamic contexts such as shallow water waves and internal waves.    

Theoretical characterization of viscous conduit breathers (envelope solitary waves) - Part I

 The spatio-temporal evolution of the circular interface between two miscible fluids of high viscosity contrast, a viscous conduit has proven to be an ideal platform for studying nonlinear dispersive hydrodynamic (DH) excitations. Despite the bulk, two-fluid dynamics at low Reynolds numbers, the fluid conduit interface is effectively non-dissipative, thanks to extremely slow rates of mass diffusion. The present study investigates the existence and characterization of breather solutions of the conduit equation, a long wavelength, fully nonlinear PDE model of conduit interfacial dynamics. Bright and dark breathers represent a class of fundamental multi-scale, propagating DH excitations. Bright breathers have been obtained numerically and investigated across the entire range of nonlinearity. Crucially, the study has suggested a three-parameter characterization of these solutions, with a counterintuitive, continuous deformation into the dark breathers across the zero-dispersion line. The talk will highlight the novelties of the numerical scheme used to compute conduit breathers, the identification of universal frameworks to study such solutions, and future applications in internal oceanic waves.   

Not Participating

We describe and model experimental results on the dynamics of a "ludion" - a neutrally buoyant body - immersed in a layer of stably stratified water. The ludion or Cartesian diver, initially positioned at its equilibrium height and free to move horizontally, can oscillate vertically when forced by pressure oscillations caused by the motion of a piston. Depending on the ratio of the forcing frequency to the buoyancy frequency, the ludion can emit its own internal gravity waves that we measure by PIV. Our experimental results describe first the resonance of the vertical motions of the ludion when excited at different frequencies. A theoretical oscillator model is then derived taking into account added mass and added friction coefficients and its predictions are compared to the experimental data. Then, for the larger oscillation amplitudes, we observe a bifurcation towards free horizontal motions. Although the internal gravity wave frequencies are affected by the Doppler shift induced by the horizontal displacement velocities, they are not the cause of the horizontal swimming. This does not however, exclude possible interactions between the ludion and his internal gravity waves and possible hydrodynamic quantum analogies as already observed with the bouncing drops of Couder.