Session E1: Geophysical Flows: Oceanography I

Shear Instabilities, Internal Waves and Turbulence in an Equatorial Undercurrent Model

Direct Numerical Simulation is used to investigate the role of both shear instabilities and internal waves in turbulent mixing in an Equatorial Undercurrent model. The flow condition corresponds to a surface mixed layer moving in an opposite direction of an underlying linearly-stratified stream where the gradient Richardson number is larger than 0.25. Holmboe shear instability emerges at the base of the mixed layer, moves at the speed of the local velocity, and ejects wisps of fluid from the bottom stream upward. At the crests of the primary Holmboe instability, a secondary Kelvin-Helmholtz shear instability causes isopycnal overturns resulting in significant turbulent mixing. Vortices formed by the Kelvin-Helmholtz instability are occasionally ejected downward and stretched by the local shear into horseshoe vortices creating intermittent bursts of turbulence in the bottom stream. Horizontally propagating internal waves with wavelength and frequency equal to that of the Holmboe instability are trapped in the bottom stream where they persist longer than the turbulent mixing caused by either the two shear instabilities or the horseshoe vortices. Internal waves and turbulence in the bottom stream of our model have characteristics similar to the near-N oscillations and the deep-cycle turbulence observed in the Pacific Equatorial Undercurrent.   

Excitation of harmonic modes by an internal wave beam incident on a simulated ocean pycnocline

Laboratory experiments have been performed to investigate the reflection of an internal wave beam with a ``pycnocline'' layer situated below an unstratified layer in order to simulate observed oceanic processes. An oscillating cylinder was used to generate internal wave beams in the well-known ``St. Andrew's Cross'' pattern. Interactions with the pycnocline were observed using the synthetic schlieren technique. As the beam refracted into the pycnocline, a discrete spectrum of harmonic modes was excited. For modest pycnocline stratifications (relative to the stratified layer below) only a few modes were observed, containing only a few percent of the kinetic energy of the primary beam. However, strongly stratified pycnoclines resulted in many excited modes and a more significant transfer of energy. The observed nonlinear energy transfer may be indicative of the validity of a postulated mechanism for the formation of oceanic internal solitary waves by internal waves incident on the pycnocline.   

Nonlinear Harmonic Generation During the Interaction of an Internal Wave Beam with a Sharp Oceanic Pycnocline

The phenomenon of nonlinear harmonic generation upon the encounter of an internal wave beam with a sharp oceanic pycnocline underlying a well-mixed layer is examined using 2-D direct numerical simulation. Use of a spectral multidomain penalty scheme enables the detailed resolution of phenomena within the pycnocline. Values of the ratio, r, of pycnocline to lower layer Brunt-Vaisala (BV) frequency and pycnocline thickness, h/$\lambda _{x}$, normalized by the incident horizontal wavelength are considered in the ranges [1,10] and [0.1,1], respectively. Nonlinear harmonic generation is enhanced with increasing values of r/h, i.e. the gradient of the BV frequency within the pycnocline. We provide scaling laws for the content of the first two harmonics as a function of environmental parameters and explore the possibility of a non-monotonic dependence which indicates particular regimes for resonance.   

Interaction of an internal tide beam with an upper ocean pycnocline

Direct numerical simulation and linear inviscid theory are used to study the interaction of an internal wave beam with an upper ocean pycnocline at different values of Froude number, $Fr$. At low $Fr$, both theory and numerics agree well as to the behavior of the internal wave beam after reflection. At moderate values of $Fr$, nonlinear response initiates the formation of higher harmonics. The harmonics that have frequency higher than the buoyancy frequency of the lower medium, are unable to propagate into the lower medium. These harmonics, affected by multiple reflections inside the pycnocline, are trapped. At high value of $Fr$, the IW beam undergoes shear instabilities during its tunneling through the transitional layer of the pycnocline. The role of turbulence in modifying the internal wave beam structure will be discussed.   

On the onset of instability in self-induced shears by large amplitude internal waves

Large amplitude internal waves in stratified fluids generate a shear layer which can support Kelvin-Helmholtz instabilities when the pycnocline is sharp enough. The way perturbations propagate along such nonparallel shear flows and eventually excite the local unstable modes is not fully understood, and differences from the parallel setup can be expected. Here we consider the specific case of a solitary wave interacting with prescribed upstream baroclinic perturbations with the aim of assessing the role of the self-induced shear flow on the evolution of the perturbation. We present a few preliminary results documenting how internal waves exhibit a subtle selectivity on the nature of the perturbation itself, which affects the outcome between the two extremes from net amplification to net damping, even in locally-unstable conditions. By varying the parameters of the perturbation (wavenumber, phase speed or frequency) we find that frequency correlates best with the overall growth/decrease trend, thereby favoring an interpretation of the underlying mechanism in terms of quasi-spatial instability.   

Oblique Collisions of Internal Gravity Wave Beams

Nonlinear interactions between two colliding internal gravity wave beams in a stratified fluid are studied theoretically, making use of small-amplitude expansions. Such collisions can give rise to secondary wave beams with frequencies equal to the sum and difference of the frequencies of the primay beams, a mechanism that is believed to contribute to the energy cascade from the tides to ocean mixing. Earlier work has been confined to the special case when the propagation directions of the primary beams, and hence the induced higher-order beams, lie in the same vertical plane. Here, the general three-dimensional (3D) configuration where the colliding beams approach each other obliquely, is considered. Based on suitable radiation conditions, the wave characteristics and direction of secondary beams are deduced, thus generalizing the known selection rules for plane collisions to the 3D case. Moreover, explicit expressions for the induced-beam profiles are derived and their dependence on obliqueness of collision is examined.   

Resonant Triad Instability in Stratified Fluid

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 experimentally study the resonant triad instability (also called Parametric Subharmonic Instability), which provides an efficient way to transfer energy from large to smaller scales. It corresponds to the destabilization of a parent wave and the spontaneous emission of two daughter waves, of different frequencies and wave numbers. We observe the experimental conditions under which a monochromatic vertical mode-1 wave is unstable. Using a time-frequency analysis, we are able to follow the evolution of the instability. We quantitatively measure the growth rate of the amplitude of the two daughter waves and compare with theoretical predictions.   

Subsurface Signature of a Reflecting Internal Wave Beam

We examine the subsurface signature of an internal wave beam reflecting off a free-slip slip top surface in a linearly stratified water column using 3-D direct numerical simulations. A spectral multidomain penalty scheme in the vertical enables finer resolution of the subsurface region. The wave beam is modelled based on a recently obtained characterization of the internal wave field radiated by a stratified turbulent wake of a towed sphere. Subsurface current and strain fields are quantified as a function of beam geometry, steepness and phase-line-tilt angle with respect to the vertical. We also compute the induced surface displacements and discuss the potential of nonlinear harmonic generation.   

Reflection of internal gravity waves at a bottom topography with near-critical slope

Direct numerical simulation is performed to study plane internal wave reflection at a sloping bottom at different values of Froude number, $Fr$. The slope angle is also varied in a range of near-critical values. At low $Fr$, the numerical results agree well with linear inviscid theory of near-critical internal wave reflection. With increasing Froude number, the reflection process becomes increasingly nonlinear with the formation of higher harmonics and subsequently fine scale turbulence. At a critical value of $Fr$, turbulence is initiated via convective instability. Also, turbulent intensities are more pronounced for somewhat off-critical reflection compared to exactly critical reflection. As the Froude number increases, the near wall shear plays a dominant role in critical reflection by enhancing turbulence compared to off-critical reflection. For a fixed slope angle, as the Froude number increases the fraction of the input energy converted into the turbulent kinetic energy increases and saturates at higher Froude numbers.