The quasi-biennial oscillation (QBO) of Earth’s stratosphere is a slowly reversing, large-scale mean flow that is generated by fast, small-scale waves. The variability of QBO reversals in recent years has triggered significant interests in the intrinsic variability of wave-driven mean flows. In this paper, we show a direct connection between the statistical properties of gravity waves randomly emitted at the bottom of a stably stratified fluid and the statistics of mean-flow reversals. We perform wave-resolved, direct numerical simulations of the two-dimensional Navier–Stokes equations under the Boussinesq approximation. We generate waves monochromatic in space at the bottom of the layer using three different types of temporal forcing: a constant-amplitude monochromatic forcing, a finite band polychromatic forcing and a stochastic forcing. We show that the stochastic forcing scheme consistently generates a mean flow with variable reversals and investigate the dependence of the reversal statistics on the wave Reynolds number and forcing correlation time. In particular, we demonstrate that the mean-flow reversals become increasingly variable as the forcing correlation time approaches the characteristic time scale of the reversals. The monochromatic and polychromatic forcing schemes trigger QBO-type flows that are highly regular for most values of the control parameters considered. Thus, the mean-flow variability under stochastic forcing is not linked to secondary mean-flow instabilities in our simulations, but rather evidence that small-amplitude waves can alter large-scale oscillations when their generation is chaotic. Finally, we demonstrate that the first-order statistics of the mean flow are relatively insensitive to the forcing type.

Internal waves are an important feature of stratified fluids, both in oceanic and lake basins and in other settings. Many works have been published on the generic feature of internal wave trapping onto planar wave attractors and super-attractors in two and three dimensions and the exceptional class of standing global internal wave modes. However, most of these works did not deal with waves that escape trapping. By using continuous symmetries, we analytically prove the existence of internal wave whispering gallery modes (WGMs), internal waves that propagate continuously without getting trapped by attractors. The WGM’s neutral stability with respect to different perturbations enables whispering gallery beams, a continuum of rays propagating together coherently. The systems’ continuous symmetries also enable projection onto two-dimensional planes that yield effective two-dimensional billiards preserving the original dynamics. By examining rays deviating from these WGMs in parabolic channels, we discover a new type of wave attractor that is located along the channel’s critical depth – the depth at which the bottom slope is identical to the ray slope, instead of cross-channel, as in previous works. This new critical-slope wave attractor leads to a new understanding of WGMs as sitting at the border between the two basins of attraction. Finally, both critical-slope wave attractors and whispering gallery beams are used to propose explanations for along-channel energy fluxes in submarine canyons and tidal energy intensification near critical slopes.

The behaviour of near-inertial waves (NIWs) in baroclinic currents is investigated, with a focus on wave trapping and critical layer dynamics. We present theoretical analysis, supported by numerical simulations, of the Sawyer–Eliassen equation, which describes waves in the plane perpendicular to an arbitrary balanced background flow. Gradients of the background velocity and buoyancy field modulate the wave properties, in particular defining the range of frequencies for which the Sawyer–Eliassen equation is hyperbolic, i.e. wave-like. Variations in the lower limit of this range, the local minimum frequency, lead to the trapping of low-frequency, typically sub-inertial, waves through either total internal reflection or the formation of critical layers. Both mechanisms are studied. We introduce a local coordinate rotation that not only elucidates these dynamics by simplifying the governing equation, but also allows us, through direct analogy, to draw upon theory and intuition developed for barotropic problems. In the majority of physically relevant cases, the transformed coordinates are aligned with and perpendicular to isopycnals, and are thus easily utilised. Employing the coordinate transformation, we consider along-isopycnal modes, and study the behaviour of waves approaching a critical level without the need for a full ray-tracing approximation. Finally, we find qualitative differences in the critical layers that form in strongly and weakly baroclinic flows, most notably in their location. In the weakly baroclinic case, NIW energy accumulates about the minimum in the relative vorticity of the background flow, whereas in the strongly baroclinic case, we find slantwise critical layers concentrated in fronts.

We present direct numerical simulations of planar intrusions from a constant source into a linearly stratified ambient fluid for Reynolds numbers between $200$ and $5000$, inlet widths $W\geqslant 2.2 \sqrt {Q/N}$ and ambient layer thicknesses $H_a$ between $W$ and $20\sqrt {Q/N}$, where $Q$ is the supply rate (area per unit time) and $N$ is the buoyancy frequency. Across this broad parameter space, the intrusions form a universal self-similar shape with a constant thickness of approximately $2.2\sqrt {Q/N}$ at the source tapering towards a tip that propagates at a constant speed of approximately $0.7\sqrt {NQ}$. This broad-scale structure does not change regardless of whether the intrusions are subcritical or supercritical relative to internal waves. The perturbations to the ambient resulting from the intrusive flow appear as a near-universal uplift/depression of isopycnals immediately above/below the intrusion, upstream blocking ahead of the intrusion and, for subcritical intrusions, columnar disturbances. A moderate-amplitude wave train is also formed on the surface of subcritical intrusions. This appears at approximately the mid-length of the intrusion, with the waves propagating towards the tip. We also compare our results with the solution of the Mei shallow-water model. The comparison is poor and we rederive the model, carefully examining the underlying assumptions against the simulation data. The only assumption that is violated is that the ambient density is unperturbed. We present extensions to the model allowing for a density perturbation based on (i) simple data fits and (ii) a solution to the Dubreil-Jacotin–Long equation for shallow ambients. These significantly improve the predictive ability of the model for this geometry.

We explore the bifurcation structure of mode-1 solitary waves in a three-layer fluid confined between two rigid boundaries. A recent study Lamb (2023 J. Fluid Mech., vol. 962, A17) proposed a method to predict the coexistence of solitary waves with opposite polarity in a continuously stratified fluid with a double pycnocline by examining the conjugate states for the Euler equations. We extend this line of inquiry to a piecewise-constant three-layer stratification, taking advantage of the fact that the conjugate states for the Euler equations are exactly preserved by the strongly nonlinear model that we will refer to as the three-layer Miyata-Maltseva-Choi-Camassa (MMCC3) equations. In this reduced setting, solitary waves are governed by a Hamiltonian system with two degrees of freedom, whose critical points are used to explain the bifurcation structure. Through this analysis, we also discover families of solutions that have not been previously reported for a three-fluid system. Using the shared conjugate state structure between the MMCC3 model and the full Euler equations, we propose criteria for distinguishing the full range of solution behaviours. This alignment between the reduced and full models provides strong evidence that partitioning the parameter space into regions associated with distinct solution types is valid within both theories. This classification is further substantiated by numerical solutions to both models, which show excellent agreement.

We revisit the interaction of an initially uniform near-inertial wave (NIW) field with a steady background flow, with the goal of predicting the subsequent organisation of the wave field. To wit, we introduce an exact analogy between the Young–Ben Jelloul (YBJ) equation and the quantum dynamics of a charged particle in a steady electromagnetic field, whose potentials are expressed in terms of the background flow. We derive the time-averaged spatial distributions of wave kinetic energy, potential energy and Stokes drift in two asymptotic limits. In the ‘strongly quantum’ limit, where the background flow is weak compared with wave dispersion, we compute the wave statistics by extending a strong-dispersion expansion initially introduced by YBJ. In the ‘quasi-classical’ limit, where the background flow is strong compared with wave dispersion, we compute the wave statistics by leveraging the equilibrium statistical mechanics of classical systems. We compare our predictions with numerical simulations of the YBJ equation, using an instantaneous snapshot from a two-dimensional turbulent flow as the steady background flow. The agreement is very good in both limits. In particular, we quantitatively describe the preferential concentration of NIW energy in anticyclones. We predict weak NIW concentration in both asymptotic limits of weak and strong background flow, and maximal anticyclonic concentration for background flows of intermediate strength, providing theoretical underpinning to observations reported by Danioux, Vanneste and Bühler (2015 J. Fluid Mech., vol. 773, R1).

Concentrated wave beams are analysed both theoretically and numerically in a general rotating and stratified axisymmetric medium, where both the rotation rate and the Brunt–Väisälä frequency vary with position. The fluid is assumed to be incompressible, weakly diffusive and weakly viscous. The analysis is conducted within the Boussinesq approximation and a linear framework, with a prescribed frequency. An asymptotic solution is derived in the limit of weak viscosity and diffusivity, describing a harmonic beam of inertia gravity waves localised around a characteristic (or ray path), similar to those generated by boundary singularities or critical points. This solution is shown to break down when the characteristic reaches a turning point which corresponds to the transition from oscillatory to evanescent behaviour. A local asymptotic analysis near the turning point demonstrates that the wave beam reflects, preserving its transverse structure while acquiring a phase shift of $\pm \pi /2$. These theoretical predictions are validated through numerical simulations, which show that the wave beam structure, both near and far from the turning point, is accurately reproduced.

The Earth’s quasi-biennial oscillation (QBO) is a natural example of wave–mean flow interaction and corresponds to the alternating directions of winds in the equatorial stratosphere. It is due to internal gravity waves (IGWs) generated in the underlying convective troposphere. In stars, a similar situation is predicted to occur, with the interaction of a stably stratified radiative zone and a convective zone. In this context, we investigate the dynamics of this reversing mean flow by modelling a stably stratified envelope and a convectively unstable core in polar geometry. Here, the coupling between the two zones is achieved self-consistently, and IGWs generated through convection lead to the formation of a reversing azimuthal mean flow in the upper layer. We characterise the mean flow oscillations by their periods, velocity amplitudes and regularity. Despite a continuous broad spectrum of IGWs, our work shows good qualitative agreement with the monochromatic model of Plumb & McEwan (1978, J. Atmos. Sci. vol. 35, no. 10, pp. 1827–1839). While the latter was originally developed in the context of the Earth’s QBO, then our study could prove relevant for its stellar counterpart in massive stars, which host convective cores and radiative envelopes.

We derive equations for three-dimensional internal wave beams propagating over a uniform slope in a uniformly stratified fluid. Using small-amplitude expansions, linear solutions for internal waves are obtained under weakly viscous conditions. Furthermore, a set of equations is constructed for the Lagrangian mean flow induced by the weakly nonlinear internal waves, providing the corresponding Lagrangian mean flow solutions within the boundary layer. The momentum equations of the Lagrangian mean velocity show that the Lagrangian mean flow is driven by the internal wave-induced body force, with its barotropic component related to the pressure gradient force, and its baroclinic component influenced by both viscosity and buoyancy. The Lagrangian-averaged buoyancy equation demonstrates that only the horizontal velocity for the Lagrangian mean flow exists throughout a vertically stable stratification region. This study emphasises the potential role of the Lagrangian mean flow in transporting time-averaged potential vorticity and solves for the Lagrangian mean flow in the inviscid region via mean potential vorticity conservation. The main results of the internal waves and the Lagrangian mean flow are visualised, revealing that the range of the boundary layer is related to the Reynolds number and that the intensity of the Lagrangian mean flow within the boundary layer is affected by the incidence angle and the reflection obliqueness. Theoretical analysis is provided to explain these phenomena.

Internal waves in a two-layer fluid with rotation are considered within the framework of Helfrich’s $f$-plane extension of the Miyata–Maltseva–Choi–Camassa model. We develop simultaneous asymptotic expansions for the evolving mean fields and deviations from them to describe a large class of uni-directional waves via the Ostrovsky equation, which fully decouples from mean-field variations. The latter generate additive inertial oscillations in the shear and in the phase of both the interfacial displacement and shear. Unlike conventional derivations leading to the Ostrovsky equation, our formulation does not impose the zero-mean constraints on the initial conditions of any variable. Using the constructed solutions, we model the evolution of quasi-periodic initial conditions close to the cnoidal wave solutions of the Korteweg–de Vries (KdV) equation but with local defects, both with and without rotation. We show that rotation leads to the emergence of bursts of internal waves and shear currents, qualitatively similar to the wavepackets generated from solitons and modulated cnoidal waves in earlier studies, but emerging much faster. We also show that cnoidal waves with expansion defects discussed in this work are generalised travelling waves of the KdV equation: they satisfy all conservation laws of the KdV equation (appropriately understood), as well as the Weirstrass–Erdmann corner condition for broken extremals of the associated variational problem and a natural weak formulation. Being smoothed in numerical simulations, they behave, in the absence of rotation, as long-lived states with no visible evolution, while rotation leads to the emergence of strong bursts.

This study introduces a boundary element method to solve the three-dimensional problem of internal tide generation over arbitrary isolated seamounts in a uniformly stratified finite-depth fluid with background rotation, without assumptions on the size or slope of the topography. Focusing on linearly propagating waves with small tidal excursions, the approach employs a vertical mode decomposition to describe the wavefield and the wave energy flux. We apply the model to the generation of internal tides by a unidirectional barotropic tide interacting with an axisymmetric Gaussian seamount. We study the conversion rate and flow field for various topographic configurations. We qualitatively recover some of the two-dimensional results of Papoutsellis et al. (2023 J. Fluid Mech. 964, A20), and find topographies with weak conversion rates, as discussed by Maas (2011 J. Fluid Mech. 684, 5–24). Furthermore, our results reveal the previously underestimated influence of the Coriolis frequency on the wavefield and on the spatial distribution of radiated energy flux. Due to Coriolis effects, the energy fluxes are shifted slightly counter-clockwise in the northern hemisphere. We explain in detail how this shift increases with the magnitude of the Coriolis frequency and the topographic features and why such effects are absent in models based on the weak topography assumption.

Induced diffusion (ID), an important mechanism of spectral energy transfer due to interacting internal gravity waves (IGWs), plays a significant role in driving turbulent dissipation in the ocean interior. In this study, we revisit the ID mechanism to elucidate its directionality and role in ocean mixing under varying IGW spectral forms, with particular attention to deviations from the standard Garrett–Munk spectrum. The original interpretation of ID as an action diffusion process, as proposed by McComas et al., suggests that ID is inherently bidirectional, with its direction governed by the vertical-wavenumber spectral slope $\sigma$ of the IGW action spectrum, $n \propto m^\sigma$. However, through the direct evaluation of the wave kinetic equation, we reveal a more complete depiction of ID, comprising both a diffusive and a scale-separated transfer rooted in the energy conservation within wave triads. Although the action diffusion may reverse direction depending on the sign of $\sigma$ (i.e. red or blue spectra), the net transfer by ID consistently leads to a forward energy cascade at the dissipation scale, contributing positively to turbulent dissipation. This supports the viewpoint of ID as a dissipative mechanism in physical oceanography. This study presents a physically grounded overview of ID, and offers insights into the specific types of wave–wave interactions responsible for turbulent dissipation.

This study presents a modified intermediate long wave (mILW) equation derived from the Navier–Stokes equations via multi-scale analysis and perturbation expansion, aimed at describing internal solitary waves (ISWs) in finite-depth, stratified oceans. Compared to the classical ILW model, the proposed mILW equation incorporates cubic nonlinearities and captures the dynamical behaviour of large-amplitude ISWs more accurately. The equation reduces to the modified Korteweg–de Vries equation and modified Benjamin–Ono equations in the shallow- and deep-water limits, respectively, thus providing a unified framework across varying depth regimes. Soliton solutions are constructed analytically using Hirota’s bilinear method, and numerical simulations investigate wave–wave interactions, including rogue waves and Mach reflection. Furthermore, a smooth tanh-type density profile is adopted to reflect realistic stratification. Associated vertical modal structures and vertical velocity fields are analysed, and higher-order statistics (skewness and kurtosis) are introduced to reveal the density dependence of wave asymmetry. The results offer new insights into the nonlinear dynamics of ISWs, with implications for ocean mixing, energy transport and submarine acoustics.

The interaction of near-inertial waves (NIWs) with submesoscale vorticity filaments is explored using theory and simulations. We study three idealised set-ups representative of submesoscale flows allowing for $O(1)$ or greater Rossby numbers. First, we consider the radiation of NIWs away from a cyclonic filament and develop scalings for the decay of wave energy in the filament. Second, we introduce broad anticyclonic regions that separate the cyclonic filaments mimicking submesoscale eddy fields and analyse the normal modes of this system. Third, we extend this set-up to consider the vertical propagation and the radiation of NIW energy. We identify a key length scale $L_m$, dependent on the strength of the filament, stratification and vertical scale of the waves, that when compared with the horizontal scales of the background flow determines the NIW behaviour. A generic expression for the vertical group velocity is derived that highlights the importance of horizontal gradients for vertical wave propagation. An overarching theme of the results is that NIW radiation, both horizontally and vertically, is most efficient when $L_m$ is comparable to the length scales of the background flow.

Vertically bounded, horizontally propagating internal waves may become unstable through triad resonant instability, in which two sibling waves in background noise draw energy from a parent internal tide. If the background stratification is uniform, then the condition for pure resonance between the parent and sibling wave frequencies and horizontal and vertical wavenumbers can be found semi-analytically from the roots of a polynomial expression. In non-uniform stratification, determining the frequencies and horizontal wavenumbers for which resonance occurs is less straightforward. We develop a theory for near-resonant excitation of a pair of sibling waves from a low-mode internal wave in which the proximity to pure resonance is characterised by the discrepancy between the forced sibling wave frequencies and the natural frequency of these modes. Knowing this discrepancy can be used methodically to determine pure resonance conditions. This inviscid theory is compared with numerical simulations of effectively inviscid waves. For comparison with laboratory experiments, the theory is adapted to include viscous effects both in the bulk of the fluid and at the side walls of the tank. We find that our theoretical predictions for frequencies and wavenumbers of the fastest growing sibling waves are generally consistent between theory, simulations and experiments, though theory overpredicts the growth rate observed in experiments. In all cases, the growth rate of sibling waves decreases with decreasing parent wave frequency, becoming negligibly small in experiments if the parent wave has frequency less than $\approx 0.7$ of the buoyancy frequency at the surface.

We consider the conceptual two-layered oscillating tank of Inoue & Smyth (2009 J. Phys. Oceanogr. vol. 39, no. 5, pp. 1150–1166), which mimics the time-periodic parallel shear flow generated by low-frequency (e.g. semi-diurnal tides) and small-angle oscillations of the density interface. Such self-induced shear of an oscillating pycnocline may provide an alternate pathway to pycnocline turbulence and diapycnal mixing in addition to the turbulence and mixing driven by wind-induced shear of the surface mixed layer. We theoretically investigate shear instabilities arising in the inviscid two-layered oscillating tank configuration and show that the equation governing the evolution of linear perturbations on the density interface is a Schrödinger-type ordinary differential equation with a periodic potential. The necessary and sufficient stability condition is governed by a non-dimensional parameter $\beta$ resembling the inverse Richardson number; for two layers of equal thickness, instability arises when $\beta \,{\gt}\,1/4$. When this condition is satisfied, the flow is initially stable but finally tunnels into the unstable region after reaching the time marking the turning point. Once unstable, perturbations grow exponentially and reveal characteristics of Kelvin–Helmholtz (KH) instability. The modified Airy function method, which is an improved variant of the Wentzel–Kramers–Brillouin theory, is implemented to obtain a uniformly valid, composite approximate solution to the interface evolution. Next, we analyse the fully nonlinear stages of interface evolution by modifying the circulation evolution equation in the standard vortex blob method, which reveals that the interface rolls up into KH billows. Finally, we undertake real case studies of Lake Geneva and Chesapeake Bay to provide a physical perspective.

In this work, we study the effect of flow curvature, or angular momentum, on the propagation and trapping characteristics of near-inertial waves (NIWs) in a curved front. The curved front is idealised as a baroclinic vortex in cyclogeostrophic balance. Motivated by ocean observations, we employ a Gaussian base flow, which by construction possesses a shield of oppositely signed vorticity surrounding its core, and we consider both cyclonic and anticyclonic representations of this flow. Following two main assumptions, i.e. that (i) the horizontal wavelength of the NIW is smaller than the length scale of the background flow (the WKBJ approximation), and (ii) the vertical wavelength of the NIW is smaller than the radial distance of interest, we derive the NIW dispersion relation and discuss the group velocity and direction of energy propagation. We show that the curvature can (i) increase the critical depth and horizontal extent of the trapping region, (ii) reduce NIW activity at the centre of the anticyclonic vortex core and enhance it in the cyclonic shield surrounding the core for high curvatures, (iii) lead to NIW trapping in the anticyclonic shield surrounding the cyclonic core, and (iv) increase the available band of NIW frequencies that are trapped. The solutions from the ray-tracing method are supported by numerical solutions of the governing equations linearised about the cyclogeostrophic base state. Finally, these methods are applied to an idealised model of oceanic mesoscale Arctic eddies showing an increase in the critical depth of trapping. Our results – while applied to polar eddies – equally apply at lower latitudes in both oceans and atmospheres, highlighting the potential importance of flow curvature in controlling the propagation of NIW energy.

This study investigates the interactions between flexural-gravity waves and interfacial waves in a two-layer fluid, focusing on wave blocking. Both liquid layers are of finite depth bounded on top by a viscoelastic thin plate. Both liquids are incompressible and inviscid, and their flows are two-dimensional and potential. Linear wave theory and a linear equation of a thin floating viscoelastic plate of constant thickness are used. We analyse the phenomenon of wave blocking and Kelvin–Helmholtz (KH) instability in a two-layer fluid with a discontinuous background mean flow. A quartic dispersion relation for frequency as a function of wavenumber and other parameters of the problem is derived. Two cases of uniform current and layers moving with different velocities are studied. Wave blocking occurs when roots of the dispersion relation coalesce without accounting for plate viscosity, leading to zero group velocity. Our findings indicate that wave blocking can occur for both flexural-gravity and interfacial waves under various frequency and current speed conditions, provided that plate viscosity is absent. The role of different parameters and the flow velocities of the upper and lower layers are investigated in the occurrence of wave blocking and KH instability. The loci of the roots of the dispersion relation involving plate viscosity depict that no root coalescence occurs irrespective of the values of wavenumber and frequency in the presence of plate viscosity. The amplitude ratio of the interfacial wave elevation to that of floating viscoelastic plate deflection exhibits the dead-water phenomenon as a density ratio approaches unity.

Combined theoretical and quantitative experimental study of resonant internal standing waves in a pycnocline between two miscible liquids in a narrow rectangular basin is presented. The waves are excited by a cylinder that harmonically oscillates in the vertical direction. A linear theoretical model describing the internal wave structure that accounts for pycnocline thickness, the finite wavemaker size and dissipation is developed. Separate series of measurements were performed using shadowgraphy and time-resolved particle image velocimetry. Accurate density profile measurements were carried out to monitor the variation of the pycnocline parameters in the course of the experiments; these measurements were used as the input parameters for the model simulations. The detected broadening of the pycnocline is attributed mainly to the presence of the waves and leads to the variation of the wave structure. The complex spatio-temporal structure of the observed internal wavefield was elucidated by carrying band-pass filtering in the temporal domain. The experiments demonstrate the coexistence of multiple spatial modes at the forcing frequency as well as the presence of the internal wave system at the second harmonic of the forcing frequency. The results of the theoretical model are in good agreement with the experiments.

Geophysical flows are typically composed of wave and mean motions with a wide range of overlapping temporal scales, making separation between the two types of motion in wave-resolving numerical simulations challenging. Lagrangian filtering – whereby a temporal filter is applied in the frame of the flow – is an effective way to overcome this challenge, allowing clean separation of waves from mean flow based on frequency separation in a Lagrangian frame. Previous implementations of Lagrangian filtering have used particle tracking approaches, which are subject to large memory requirements or difficulties with particle clustering. Kafiabad & Vanneste (2023, Computing Lagrangian means, J. Fluid Mech., vol. 960, A36) recently proposed a novel method for finding Lagrangian means without particle tracking by solving a set of partial differential equations alongside the governing equations of the flow. In this work, we adapt the approach of Kafiabad & Vanneste to develop a flexible, on-the-fly, partial differential equation-based method for Lagrangian filtering using arbitrary convolutional filters. We present several different wave–mean decompositions, demonstrating that our Lagrangian methods are capable of recovering a clean wave field from a nonlinear simulation of geostrophic turbulence interacting with Poincaré waves.