A boundary integral representation is derived for the translational oscillations of a triaxial ellipsoid in a uniformly stratified fluid. The representation is of single-layer type, a distribution of sources and sinks over the surface of the ellipsoid. The added mass tensor of the ellipsoid is deduced from it and, from this tensor, the impulse response function together with the energy radiated away as internal waves. Horizontal oscillations correspond to the generation of an internal or baroclinic tide by the oscillation of the barotropic tide over ellipsoidal topography at the bottom of the stratified ocean. Such topography is unconditionally supercritical, namely of slope larger than the slope of the wave rays, irrespective of the frequency of oscillation. So far, analytical work on supercritical topographies has been limited, for the most part, to two-dimensional set-ups. Here, for the ellipsoidal seamount, the orientation of the barotropic tide and the anisotropy of the topography have their effects analysed in detail. As the height of the seamount increases, the rate of conversion of barotropic energy into baroclinic form is seen to first increase according to the square law expected for a topography of small slope, then saturate and eventually decrease.

As wind farms continue to grow in size, mesoscale effects such as blockage and gravity waves become increasingly important. Allaerts & Meyers (J. Fluid Mech., vol. 862, 2019, pp. 990–1028) proposed an atmospheric perturbation model (APM) that can simulate the interaction of wind farms and the atmospheric boundary layer while keeping computational costs low. The model resolves the mesoscale flow, and couples to a wake model to estimate the turbine inflow velocities at the microscale. This study presents a new way of coupling the mesoscale APM to a wake model, based on matching the velocity between the models throughout the farm. This method performs well, but requires good estimates of the turbine-level velocity fields by the wake model. Additionally, we investigate the mesoscale effects of a large wind farm, and find that aside from the turbine forces and increased turbulence levels, the dispersive stresses due to subgrid flow heterogeneity also play an important role at the entrance of the farm, and contribute to the global blockage effect. By using the wake model coupling, we can explicitly incorporate these stresses in the model. The resulting APM is validated using 27 prior large-eddy simulations of a large wind farm under different atmospheric conditions. The APM and large-eddy simulation results are compared on both mesoscale and turbine scale, and on turbine power output. The APM captures the overall effects that gravity waves have on wind farm power production, and significantly outperforms standard wake models.

We examine the fluid flow forced by precession of a rotating cylindrical container using numerical simulations and experimental flow measurements with ultrasonic Doppler velocimetry. The analysis is based on the decomposition of the flow field into contributions with distinct azimuthal symmetry or analytically known inertial modes and the corresponding calculation of their amplitudes. We show that the predominant fraction of the kinetic energy of the precession-driven fluid flow is contained only within a few large-scale modes. The most striking observation shown by simulations and experiments is the transition from a flow dominated by large-scale structures to a more turbulent behaviour with the small-scale fluctuations becoming increasingly important. At a fixed rotation frequency (parametrized by the Reynolds number, $Re$) this transition occurs when a critical precession ratio is exceeded and consists of a two-stage collapse of the directly driven flow going along with a massive modification of the azimuthal circulation (the zonal flow) and the appearance of an axisymmetric double-roll mode limited to a narrow range of precession ratios. A similar behaviour is found in experiments which make it possible to follow the transition up to Reynolds numbers of $Re\approx 2\times 10^6$. We find that the critical precession ratio decreases with rotation, initially showing a particular scaling ${\propto }Re^{-({1}/{5})}$ but developing an asymptotic behaviour for $Re\gtrsim 10^5$ which might be explained by the onset of turbulence in boundary layers.

Evolutions of internal waves of different modes, particularly mode 1 and mode 2, passing over variable bathymetry are investigated based on a new numerical scheme. The problem is idealized as interfacial waves propagating on two interfaces of a three-layer density stratified fluid system with large-amplitude bottom topography. The Dirichlet-to-Neumann operators are introduced to reduce the spatial dimension by one and to adapt the three-layer system and significant topographic effects. However, for simplicity, nonlinear interactions between interfaces are neglected. Numerical techniques such as the Galerkin approximation, proven effective in previous works, are applied to save computational costs. Shoaling of linear waves on an uneven bottom is first studied to validate the proposed formulation and the corresponding numerical scheme. Then, for two-dimensional numerical experiments, mode transition phenomena excited by locally confined bottom obstacles and quickly varying topographies, including the Bragg resonance, mode-2 excitation, wave homogenization, etc., are investigated. In three-dimensional simulations, internal wave refraction by a Luneberg lens is considered, and good agreement is found in comparison with the ray theory. Finally, in the limiting case, when the top layer can be negligible (for example, a gas layer of extremely small density), the problem is reduced to a two-and-a-half-layer fluid system, where an interface and a surface are unknown free boundaries. In this situation, the surface signature of an internal wave is simulated and verified by introducing the realistic bathymetry of the Strait of Gibraltar and qualitatively compared with the satellite image.

Internal solitary waves are a widely observed phenomenon in natural waters. Mathematically, they are fundamentally a nonlinear phenomenon that differs from the paradigm of turbulence, in that energy does not move across scales. Internal solitary waves may be computed from the Dubreil–Jacotin Long equation, which is a scalar partial differential equation that is equivalent to the stratified Euler equations. When a background shear current is present the algebraic complexity of the problem increases substantially. We present an alternative point of view for characterizing the situation with a shear current using Lagrangian (particle-like) models analysed with graph theoretic methods. We find that this yields a novel, data-centric framework for analysis that could prove useful well beyond the study of internal solitary waves.

The stimulation of instability and transport in the bottom boundary layer by internal solitary waves has been documented for over twenty years. However, the challenge of shallow slopes and a disparity of scales between the large-scale wave and the small-scale boundary layer has proven challenging for simulations. We present laboratory scale simulations that resolve the three-dimensionalisation in the boundary layer during the entire shoaling process. We find that the late stage, in which the incoming wave fissions into boluses, provides the most consistent source of three-dimensionalisation. In the early stage of shoaling, three-dimensionalisation occurs not so much due to separation bubble instability, but due to the interaction of vortices shed from the separation bubble with the overlying pycnocline. This interaction overturns the pycnocline, and creates bursts in kinetic energy and viscous dissipation, suggesting that the shed vortices induce turbulent motion and sediment resuspension in the water column above and behind the separation bubble.

We investigated the stability of the bottom boundary layer (BBL) beneath periodic internal solitary waves (ISWs) of depression over a flat bottom through two-dimensional direct numerical simulations. We explored the convective versus absolute/global nature of the BBL instability in response to changes in Reynolds number, and the sensitivity of the instability to seeding noise in the front of the ISW – spanning laboratory to geophysical scales. The BBL was laminar at $Re_{ISW}=90$ and convectively unstable at $Re_{ISW}=300$. At laboratory-scale $Re_{ISW}=300$, the convective wave packet was periodically amplified by each successive ISW, until vortex shedding occurred. The associated noise-amplification behaviour potentially explains the discrepancies of the critical $Re_{ISW}$ between the lock–release laboratory experiments and our Dubreil–Jacotin–Long-initialized numerical simulations as the result of the difference in background noise. Instability energy decreased under the front shoulder of the ISW, analogous to flow relaminarization under a favourable pressure gradient. At geophysical-scale $Re_{ISW}=900$, the BBL was initially convectively unstable, and then the instability tracked with the ISW, appearing phenomenologically similar to a global instability. The simulated initial convective instability at both $Re_{ISW}=300$ and $Re_{ISW}=900$ is in agreement with local linear stability analysis which predicts that the instability group speed is always lower than the ISW celerity. Increased free stream perturbations in front of the ISW and larger $Re_{ISW}$ shift the location of vortex shedding (and enhanced bed shear stress) beneath the wave, closer to the ISW trough, thereby potentially changing the location of maximum sediment resuspension, in agreement with field observations at higher $Re_{ISW}$.

In this paper, a high-level Green–Naghdi (HLGN) model for large-amplitude internal waves in a two-layer fluid system, where the upper-fluid layer is of finite depth and the lower-fluid layer is of infinite depth, is developed under the rigid-lid free-surface approximation. The equations of the present HLGN model follow Euler's equations under the sole assumption that the horizontal and vertical velocity distributions along the vertical column are presented by known shape functions for each layer. The linear dispersion relations of the HLGN model for different levels are presented and compared with those obtained by other strongly nonlinear models for deep water, including the fully nonlinear models that include the dispersion effects $O(\mu )$ (where $\mu$ is the ratio of the upper-fluid layer depth to a typical wavelength) derived by Choi & Camassa (Phys. Rev. Lett., vol. 77, 1996, pp. 1759–1762) and $O(\mu ^2)$ derived by Debsarma et al. (J. Fluid Mech., vol. 654, 2010, pp. 281–303). It is shown that the HLGN model has a wider application range than other models. Solutions of travelling large-amplitude internal solitary waves in the absence and presence of background shear-current are then investigated by using the HLGN model. For the no-current cases, results obtained by the HLGN model show better agreement with Euler's solution on wave profile, velocity profile at the maximum interface displacement and wave speed compared with those obtained by other models. For the background shear-current cases, results obtained by the HLGN model also show good agreement with those obtained by solving the Dubreil-Jacotin–Long equation.

The Holton–Lindzen–Plumb (HLP) model of the quasi-biennial oscillation (QBO) is investigated in order to assess the impact of introducing intermittency in the wave forcing. Intermittency is introduced to HLP by allowing the amplitude of the waves which force the QBO to evolve according to a stationary random process, driven by a stochastic differential equation (SDE) with an associated time scale $\tau$. Provided that $\tau$ is much shorter than the QBO period, it is shown that the impact on the QBO of the intermittent forcing is captured by a single intermittency parameter $\lambda$, and the value of $\lambda$ is proportional to $\tau$ and otherwise depends upon the details of the SDE. Numerical simulations, using a family of mean-reverting Ornstein–Uhlenbeck processes as the choice of SDE, show that the effect of increasing the intermittency parameter is invariably to decrease the QBO amplitude and increase its period. Changes to the QBO amplitude and period are indeed found to collapse onto a single curve controlled by $\lambda$, as predicted by the theory, provided that $\tau$ is small enough for the approximations used to be valid. The extension to broadband forcing is discussed in the context of stochastic gravity wave parameterisation, with the eventual goal of developing a representation of source intermittency in the most general situation with close fidelity to the physics.

Wave turbulence provides a conceptual framework for weakly nonlinear interactions in dispersive media. Dating from five decades ago, applications of wave turbulence theory to oceanic internal waves assigned a leading-order role to interactions characterized by a near equivalence between the group velocity of high-frequency internal waves with the phase velocity of near-inertial waves. This scale-separated interaction leads to a Fokker–Planck (generalized diffusion) equation. More recently, starting four decades ago, this scale-separated paradigm has been investigated using ray tracing methods. These ray methods characterize spectral transport of energy by counting the amplitude and net velocity of wave packets in phase space past a high-wavenumber gate prior to ‘breaking’. This explicitly advective characterization is based on an intuitive assignment and lacks theoretical underpinning. When one takes an estimate of the net spectral drift from the wave turbulence derivation and makes the corresponding assessment, one obtains a prediction of spectral transport that is an order of magnitude larger than either observations or reported ray tracing estimates. Motivated by this contradiction, we report two parallel derivations for transport equations describing the refraction of high-frequency internal waves in a sea of random inertial waves. The first uses standard wave turbulence techniques and the second is an ensemble-averaged packet transport equation characterized by the dispersion of wave packets about a mean drift in the spectral domain. The ensemble-averaged transport equation for ray tracing differs in that it contains the intuitively motivated advective term. We conclude that the aforementioned contradiction between theory, numerics and observations needs to be taken at face value and present a pathway for resolving this contradiction.

The concept of added mass is generalized to stratified fluids, accounting for the presence of internal waves. Once the added mass of a moving body is known, so is the hydrodynamic force exerted on it by the fluid, and the energy imparted by it to the fluid. As a function of frequency, added mass is complex. Its real part is associated with inertia and its imaginary part, only present in the frequency range of propagating waves, with wave damping. Owing to causality, these two parts satisfy Kramers–Kronig relations. The added masses of an elliptic cylinder of horizontal axis, typical of two-dimensional bodies, and a spheroid of vertical axis, typical of three-dimensional bodies, are deduced from their dipole strengths, themselves deduced from their representations as single layers. The wave power is shown to be a maximum, for fixed oscillation amplitude, at approximately $0.8$ times the buoyancy frequency. In the temporal domain, added mass appears as a new memory force taking the form of a convolution integral. The kernel of this integral combines algebraically decaying oscillations at the buoyancy frequency on the one hand; and an exponentially damped oscillation for the horizontal motion of the spheroid, implying short-term memory, an aperiodic algebraic decay for its vertical motion, implying long-term memory, and a constant for the motion of the cylinder, implying everlasting memory, on the other hand. A limitation of the study is its restriction to translational motion.

The oscillations of buoyant bodies in stratified fluids are deduced from the variations of their added mass. Three configurations are considered: a body displaced from its neutral level then released; a Cartesian diver set into oscillation by a modulation of the hydrostatic pressure, then released; and a body attached to a pendulum to which an impulse is applied. The first configuration is related to the dynamics of Lagrangian floats in the ocean. Two particular bodies are considered: an elliptic cylinder of horizontal axis, typical of two-dimensional bodies; and a spheroid of vertical axis, typical of three-dimensional bodies. The ultimate motion of the body consists of oscillations at the buoyancy frequency with an amplitude decaying algebraically with time. Before that, the resonant response of the system is observed, either aperiodic exponential decay when the system has no intrinsic dynamics, or exponentially damped oscillation otherwise. Comparison with available measurements demonstrates the need to include viscous dissipation in the analysis. At high Stokes number, dissipation comes from the Basset–Boussinesq memory force and is affected negligibly by the stratification; at low Stokes number, dissipation comes from Stokes resistance and exhibits a significant effect of the stratification.

Direct numerical simulations (DNSs) of three-dimensional cylindrical release gravity currents in a linearly stratified ambient are presented. The simulations cover a range of stratification strengths $0< S\leq 0.8$ (where $S=(\rho _b^*-\rho _0^*)/(\rho _c^*-\rho _0^*), \rho _b^*, \rho _0^*$ and $\rho _c^*$ are the dimensional density at the bottom of the domain, top of the domain and the dense fluid, respectively) at two different Reynolds numbers. A comparison between the stratified and unstratified cases illustrates the influence of stratification strength on the dynamics of cylindrical gravity currents. Specifically, the front velocity in the slumping phase decreases with increasing stratification strength whereas the duration of the slumping phase increases with increments of $S$. The Froude number calculated in this phase shows a good agreement with models proposed by Ungarish & Huppert (J. Fluid Mech., vol. 458, 2002, pp. 283–301) and Ungarish (J. Fluid Mech., vol. 548, 2006, pp. 49–68), originally developed for planar gravity currents in a stratified ambient. In the inertial phase, the front velocity across cases with different stratification strengths adheres to a power-law scaling with an exponent of $-$1/2. Higher Reynolds numbers led to more frequent lobe splitting and merging, with lobe size diminishing as stratification strength increased. Strong interactions among inner vortex rings occurred during the slumping phase, leading to the early formation of hairpin vortices in weakly stratified cases, while strongly stratified cases exhibited delayed vortex formation and less turbulence.

An experimental study of long interfacial gravity waves was conducted in a closed wave tank containing two layers of viscous immiscible fluids. The study focuses on the development in time of the mean particle drift that occurs close to the interface where the two fluids meet. From a theoretical analysis by Weber & Christensen (Eur. J. Mech.-B/Fluids, vol. 77, 2019, pp. 162–170) it is predicted that the growing drift in this region is associated with the action of the virtual wave stress. This effect has not been explored experimentally before. Interfacial waves of different amplitudes and frequencies were produced by a D-shaped paddle. Particle tracking velocimetry was used to track the time development of the Lagrangian mean drift. The finite geometry of the wave tank causes a mean return flow that is resolved by mass transport considerations. The measurements clearly demonstrate the importance of the virtual wave stress mechanism in generating wave drift currents near the interface.

We study the spectral energy transfer due to wave–triad interactions in the Garrett–Munk spectrum of internal gravity waves based on a numerical evaluation of the collision integral in the wave kinetic equation. Our numerical evaluation builds on the reduction of the collision integral on the resonant manifold for a horizontally isotropic spectrum. We evaluate directly the downscale energy flux available for ocean mixing, whose value is in close agreement with the finescale parameterization. We further decompose the energy transfer into contributions from different mechanisms, including local interactions and three types of non-local interactions, namely parametric subharmonic instability, elastic scattering (ES) and induced diffusion (ID). Through analysis on the role of each mechanism, we resolve two long-standing paradoxes regarding the mechanism for forward cascade in frequency and zero ID flux for the GM76 spectrum. In addition, our analysis estimates the contribution of each mechanism to the energy transfer in each spectral direction, and reveals new understanding of the importance of local interactions and ES in the energy transfer.

We compare a higher-order asymptotic construction for balance in geophysical flows with the method of ‘optimal balance’, a purely numerical approach to separating inertia–gravity waves from vortical modes. Both methods augment the linear geostrophic mode with dependent inertia–gravity wave mode contributions, the so-called slaved modes, such that the resulting approximately balanced states are characterized by very small residual wave emission during subsequent time evolution. In our benchmark setting – the single-layer rotating shallow water equations in the quasi-geostrophic regime – the performance of both methods is comparable across a range of Rossby numbers and for different initial conditions. Cross-balancing, i.e. balancing the model with one method and diagnosing the imbalance with the other, suggests that both methods find approximately the same balanced state. Our results also reinforce results from previous studies suggesting that spontaneous wave emission from balanced flow is very small. We further compare two numerical implementations of each of the methods: one pseudospectral, and the other a finite difference scheme on the standard C-grid. We find that a state that is balanced relative to one numerical scheme is poorly balanced for the other, independent of the method that was used for balancing. This shows that the notion of balance in the discrete case is fundamentally tied to a particular scheme.

The response to harmonic horizontal oscillations of a stably stratified fluid-filled two-dimensional square container is examined as the forcing amplitude is increased. For the studied forcing frequency, the response flow at very small forcing amplitudes is a synchronous periodic flow with piecewise-constant vorticity in regions delineated by the characteristics emanating from the corners of the container, regularized by viscosity. The second temporal harmonic of the forced response flow resonantly excites an intrinsic mode of the stratified container, whose magnitude grows as the square of the forcing amplitude. Above a critical forcing amplitude, a sequence of pairs of other container modes are excited via triadic resonances with the second-harmonic-driven mode. The flows are computed from the Navier–Stokes–Boussinesq equations and the ensuing dynamics is analysed using Fourier techniques, providing a comprehensive picture of the transition to internal wave turbulence.

The instability and vortex shedding in the bottom boundary layer caused by internal solitary waves of depression propagating along a shallow pycnocline of a fluid are computed by finite-volume code in two dimensions. The calculated transition to instability agrees very well with laboratory experiments (Carr et al., Phys. Fluids, vol. 20, issue 6, 2008, 06603) but disagrees with existing computations that give a very conservative instability threshold. The instability boundary expressed by the amplitude depends on the depth $d$ of the pycnocline divided by the water depth $H$, and decays by a factor of 2.2 when $d/H$ is 0.21, and by a factor of 1.6 when $d/H$ is 0.16, and the stratification Reynolds number increases by a factor of 32. The instability occurs at moderate amplitude at large scale. The calculated oscillatory bed shear stress is strong in the wave phase and increases with the scale. Its non-dimensional magnitude at stratification Reynolds number 650 000 is comparable to the turbulent stress that can be extracted from field measurements of internal solitary waves of similar nonlinearity, moving along a pycnocline of similar relative depth.

We model linear, inviscid, internal tides generated by the interaction of a barotropic tide with one-dimensional topography. Starting from the body-forcing formulation of the hydrodynamic problem, we derive a coupled-mode system (CMS) using a local eigenfunction expansion of the stream function. For infinitesimal topography, we solve this CMS analytically, recovering the classical weak topography approximation (WTA) formula for the barotropic-to-baroclinic energy conversion rate. For arbitrary topographies, we solve this CMS numerically. The CMS enjoys faster convergence with respect to existing modal solutions and can be applied in the subcritical and supercritical regimes for both ridges and shelf profiles. We show that the non-uniform barotropic tide affects the baroclinic field locally over topographies with large slopes and we study the dependence of the radiated energy conversion rate on the criticality. We show that non-radiating or weakly radiating topographies are common in the subcritical regime. We also assess the region of validity of the WTA approximation for the commonly used Gaussian ridge and a compactly supported bump ridge studied here for the first time. Finally, we provide numerical evidence showing that in the strongly supercritical regime, the energy conversion rate for a ridge (respectively shelf) approaches the value obtained by the knife-edge (respectively step) topography.

The reflection and transmission of internal waves at a horizontal interface between layers of constant buoyancy frequency are treated. This interface is a simple model of the atmospheric tropopause. The waves are weakly nonlinear and obey the nonlinear Schrödinger equation away from the interface. The waves are horizontally periodic and vertically confined to a long packet. The interfacial conditions are formally taken to third order, and include higher-order linear as well as nonlinear effects. The higher-order linear affects show an instability at the interface for steep waves. Numerical results for shorter wave packets show that the higher-order linear terms result in non-physical wave generation, and are ultimately neglected for such cases. Nonlinear effects result in a decrease in the reflected wave amplitudes and an increase in the transmitted wave amplitudes when compared to linear theory. Overall, the nonlinear effects make the interface more transparent to upwardly propagating internal waves.