Inertia–gravity waves are scattered by background flows as a result of Doppler shift by a non-uniform velocity. In the Wentzel–Kramers–Brillouin regime, the scattering process reduces to a diffusion in spectral space. Other inhomogeneities that the waves encounter, such as density variations, also cause scattering and spectral diffusion. We generalise the spectral diffusion equation to account for these inhomogeneities. We apply the result to a rotating shallow-water system, for which height inhomogeneities arise from velocity inhomogeneities through geostrophy, and to the Boussinesq system for which buoyancy inhomogeneities arise similarly. We compare the contributions that height and buoyancy variations make to the spectral diffusion with the contribution of the Doppler shift. In both systems, we find regimes where all contributions are significant. We support our findings with exact solutions of the diffusion equation and with ray tracing simulations in the shallow-water case.

Ocean turbulence at meso- and submesocales affects the propagation of surface waves through refraction and scattering, inducing spatial modulations in significant wave height (SWH). We develop a theoretical framework that relates these modulations to the current that induces them. We exploit the asymptotic smallness of the ratio of typical current speed to wave group speed to derive a linear map – the U2H map – between surface current velocity and SWH anomaly. The U2H map is a convolution, non-local in space, expressible as a product in Fourier space by a factor independent of the magnitude of the wavenumber vector. Analytic expressions of the U2H map show how the SWH responds differently to the vortical and divergent parts of the current, and how the anisotropy of the wave spectrum is key to large current-induced SWH anomalies. We implement the U2H map numerically and test its predictions against WAVEWATCH III numerical simulations for both idealised and realistic current configurations.

The refraction of surface gravity waves by currents leads to spatial modulations in the wave field and, in particular, in the significant wave height. We examine this phenomenon in the case of waves scattered by a localised current feature, assuming (i) the smallness of the ratio between current velocity and wave group speed, and (ii) a swell-like, highly directional wave spectrum. We apply matched asymptotics to the equation governing the conservation of wave action in the four-dimensional position–wavenumber space. The resulting explicit formulas show that the modulations in wave action and significant wave height past the localised current are controlled by the vorticity of the current integrated along the primary direction of the swell. We assess the asymptotic predictions against numerical simulations using WAVEWATCH III for a Gaussian vortex. We also consider vortex dipoles to demonstrate the possibility of ‘vortex cloaking’ whereby certain currents have (asymptotically) no impact on the significant wave height. We discuss the role of the ratio of the two small parameters characterising assumptions (i) and (ii) above, and show that caustics are significant only for unrealistically large values of this ratio, corresponding to unrealistically narrow directional spectra.

Lagrangian averaging plays an important role in the analysis of wave–mean-flow interactions and other multiscale fluid phenomena. The numerical computation of Lagrangian means, e.g. from simulation data, is, however, challenging. Typical implementations require tracking a large number of particles to construct Lagrangian time series, which are then averaged using a low-pass filter. This has drawbacks that include large memory demands, particle clustering and complications of parallelisation. We develop a novel approach in which the Lagrangian means of various fields (including particle positions) are computed by solving partial differential equations (PDEs) that are integrated over successive averaging time intervals. We propose two strategies, distinguished by their spatial independent variables. The first, which generalises the algorithm of Kafiabad (J. Fluid Mech., vol. 940, 2022, A2), uses end-of-interval particle positions; the second uses directly the Lagrangian mean positions. The PDEs can be discretised in a variety of ways, e.g. using the same discretisation as that employed for the governing dynamical equations, and solved on-the-fly to minimise the memory footprint. We illustrate the new approach with a pseudo-spectral implementation for the rotating shallow-water model. Two applications to flows that combine vortical turbulence and Poincaré waves demonstrate the superiority of Lagrangian averaging over Eulerian averaging for wave–vortex separation.

The scattering of three-dimensional inertia-gravity waves by a turbulent geostrophic flow leads to the redistribution of their action through what is approximately a diffusion process in wavevector space. The corresponding diffusivity tensor was obtained by Kafiabad et al. (J. Fluid Mech., vol. 869, 2019, R7) under the assumption of a time-independent geostrophic flow. We relax this assumption to examine how the weak diffusion of wave action across constant-frequency cones that results from the slow time dependence of the geostrophic flow affects the distribution of wave energy. We find that the stationary wave-energy spectrum that arises from a single-frequency wave forcing is localised within a thin boundary layer around the constant-frequency cone, with a thickness controlled by the acceleration spectrum of the geostrophic flow. We obtain an explicit analytic formula for the wave-energy spectrum which shows good agreement with the results of a high-resolution simulation of the Boussinesq equations.

Stochastic models of varying complexity have been proposed to describe the dispersion of particles in turbulent flows, from simple Brownian motion to complex temporally and spatially correlated models. A method is needed to compare competing models, accounting for the difficulty in estimating the additional parameters that more complex models typically introduce. We employ a data-driven method, Bayesian model comparison, which assigns probabilities to competing models based on their ability to explain observed data. We focus on the comparison between the Brownian and Langevin dynamics for particles in two-dimensional isotropic turbulence, with data that consist of sequences of particle positions obtained from simulated Lagrangian trajectories. We show that, while on sufficiently large time scales the models are indistinguishable, there is a range of time scales on which the Langevin model outperforms the Brownian model. While our set-up is highly idealised, the methodology developed is applicable to more complex flows and models of particle dynamics.

The response of a semi-infinite ocean to a slowly travelling atmospheric perturbation crossing the coast provides a simple example of the breakdown of nearly geostrophic balance induced by a boundary. We examine this response in the linear shallow-water model at small Rossby number $\varepsilon \ll 1$. Using matched asymptotics, we show that a long Kelvin wave, with $O(\varepsilon ^{-1})$ length scale and $O(\varepsilon )$ amplitude relative to quasi-geostrophic response, is generated as the perturbation crosses the coast. Accounting for this Kelvin wave restores the conservation of mass that is violated in the quasi-geostrophic approximation.

We consider the dynamics of a two-dimensional incompressible perfect fluid on a Möbius strip embedded in $\mathbb {R}^{3}$. The vorticity–stream function formulation of the Euler equations is derived from an exterior-calculus form of the momentum equation. The non-orientability of the Möbius strip and the distinction between forms and pseudo-forms this introduces lead to unusual properties: a boundary condition is provided by the conservation of circulation along the single boundary of the strip, and there is no integral conservation for the vorticity density or for any odd function thereof. A finite-difference numerical implementation is used to illustrate the Möbius-strip realisation of familiar phenomena: translation of vortices along boundaries, shear instability and decaying turbulence.

In the presence of inertia-gravity waves, the geostrophic and hydrostatic balance that characterises the slow dynamics of rapidly rotating, strongly stratified flows holds in a time-averaged sense and applies to the Lagrangian-mean velocity and buoyancy. We give an elementary derivation of this wave-averaged balance and illustrate its accuracy in numerical solutions of the three-dimensional Boussinesq equations, using a simple configuration in which vertically planar near-inertial waves interact with a barotropic anticylonic vortex. We further use the conservation of the wave-averaged potential vorticity to predict the change in the barotropic vortex induced by the waves.

We investigate the transport of inertial particles by cellular flows when advection dominates over inertia and diffusion, that is, for Stokes and Péclet numbers satisfying $St \ll 1$ and $Pe \gg 1$. Starting from the Maxey–Riley model, we consider the distinguished scaling $St \, Pe = O(1)$ and derive an effective Brownian dynamics approximating the full Langevin dynamics. We then apply homogenisation and matched-asymptotics techniques to obtain an explicit expression for the effective diffusivity $\bar {D}$ characterising long-time dispersion. This expression quantifies how $\bar {D}$, proportional to $Pe^{-1/2}$ when inertia is neglected, increases for particles heavier than the fluid and decreases for lighter particles. In particular, when $St \gg Pe^{-1}$, we find that $\bar {D}$ is proportional to $St^{1/2}/(\log ( St \, Pe))^{1/2}$ for heavy particles and exponentially small in $St \, Pe$ for light particles. We verify our asymptotic predictions against numerical simulations of the particle dynamics.

The scattering of inertia-gravity waves by large-scale geostrophic turbulence in a rapidly rotating, strongly stratified fluid leads to the diffusion of wave energy on the constant-frequency cone in wavenumber space. We derive the corresponding diffusion equation and relate its diffusivity to the wave characteristics and the energy spectrum of the turbulent flow. We check the predictions of this equation against numerical simulations of the three-dimensional Boussinesq equations in initial-value and forced scenarios with horizontally isotropic wave and flow fields. In the forced case, wavenumber diffusion results in a $k^{-2}$ wave energy spectrum consistent with as-yet-unexplained features of observed atmospheric and oceanic spectra.

Oceanic internal tides and other inertia–gravity waves propagate in an energetic turbulent flow whose length scales are similar to the wavelengths. Advection and refraction by this flow cause the scattering of the waves, redistributing their energy in wavevector space. As a result, initially plane waves radiated from a source such as a topographic ridge become spatially incoherent away from the source. To examine this process, we derive a kinetic equation which describes the statistics of the scattering under the assumptions that the flow is quasigeostrophic, barotropic and well represented by a stationary homogeneous random field. Energy transfers are quantified by computing a scattering cross-section and shown to be restricted to waves with the same frequency and identical vertical structure, hence the same horizontal wavelength. For isotropic flows, scattering leads to an isotropic wave field. We estimate the characteristic time and length scales of this isotropisation, and study their dependence on parameters including the energy spectrum of the flow. Simulations of internal tides generated by a planar wavemaker carried out for the linearised shallow-water model confirm the pertinence of these scales. A comparison with the numerical solution of the kinetic equation demonstrates the validity of the latter and illustrates how the interplay between wave scattering and transport shapes the wave statistics.

Many fluctuation-driven phenomena in fluids can be analysed effectively using the generalised Lagrangian-mean (GLM) theory of Andrews & McIntyre (J. Fluid Mech., vol. 89, 1978, pp. 609–646) This finite-amplitude theory relies on particle-following averaging to incorporate the constraints imposed by the material conservation of certain quantities in inviscid regimes. Its original formulation, in terms of Cartesian coordinates, relies implicitly on an assumed Euclidean structure; as a result, it does not have a geometrically intrinsic, coordinate-free interpretation on curved manifolds, and suffers from undesirable features. Motivated by this, we develop a geometric generalisation of GLM that we formulate intrinsically using coordinate-free notation. One benefit is that the theory applies to arbitrary Riemannian manifolds; another is that it establishes a clear distinction between results that stem directly from geometric consistency and those that depend on particular choices. Starting from a decomposition of an ensemble of flow maps into mean and perturbation, we define the Lagrangian-mean momentum as the average of the pull-back of the momentum one-form by the perturbation flow maps. We show that it obeys a simple equation which guarantees the conservation of Kelvin’s circulation, irrespective of the specific definition of the mean flow map. The Lagrangian-mean momentum is the integrand in Kelvin’s circulation and distinct from the mean velocity (the time derivative of the mean flow map) which advects the contour of integration. A pseudomomentum consistent with that in GLM can then be defined by subtracting the Lagrangian-mean momentum from the one-form obtained from the mean velocity using the manifold’s metric. The definition of the mean flow map is based on choices made for reasons of convenience or aesthetics. We discuss four possible definitions: a direct extension of standard GLM, a definition based on optimal transportation, a definition based on a geodesic distance in the group of volume-preserving diffeomorphisms, and the ‘glm’ definition proposed by Soward & Roberts (J. Fluid Mech., vol. 661, 2010, pp. 45–72). Assuming small-amplitude perturbations, we carry out order-by-order calculations to obtain explicit expressions for the mean velocity and Lagrangian-mean momentum at leading order. We also show how the wave-action conservation of GLM extends to the geometric setting. To make the paper self-contained, we introduce in some detail the tools of differential geometry and main ideas of geometric fluid dynamics on which we rely. These include variational formulations which we use for alternative derivations of some key results. We mostly focus on the Euler equations for incompressible inviscid fluids but sketch out extensions to the rotating–stratified Boussinesq, compressible Euler, and magnetohydrodynamic equations. We illustrate our results with an application to the interaction of inertia-gravity waves with balanced mean flows in rotating–stratified fluids.

Mountain-generated inertia–gravity waves (IGWs) affect the dynamics of both the atmosphere and the ocean through the mean force they exert as they interact with the flow. A key to this interaction is the presence of critical-level singularities or, when planetary rotation is taken into account, inertial-level singularities, where the Doppler-shifted wave frequency matches the local Coriolis frequency. We examine the role of the latter singularities by studying the steady wavepacket generated by a multiscale mountain in a rotating linear shear flow at low Rossby number. Using a combination of Wentzel–Kramers–Brillouin (WKB) and saddle-point approximations, we provide an explicit description of the form of the wavepacket, of the mean forcing it induces and of the mean-flow response. We identify two distinguished regimes of wave propagation: Regime I applies far enough from a dominant inertial level for the standard ray-tracing approximation to be valid; Regime II applies to a thin region where the wavepacket structure is controlled by the inertial-level singularities. The wave–mean-flow interaction is governed by the change in Eliassen–Palm (or pseudomomentum) flux. This change is localised in a thin inertial layer where the wavepacket takes a limiting form of that found in Regime II. We solve a quasi-geostrophic potential-vorticity equation forced by the divergence of the Eliassen–Palm flux to compute the wave-induced mean flow. Our results, obtained in an inviscid limit, show that the wavepacket reaches a large-but-finite distance downstream of the mountain (specifically, a distance of order $(k_{\ast }\unicode[STIX]{x1D6E5})^{1/2}\unicode[STIX]{x1D6E5}$, where $k_{\ast }^{-1}$ and $\unicode[STIX]{x1D6E5}$ measure the wave and envelope scales of the mountain) and extends horizontally over a similar scale.

An inertia–gravity wave (IGW) propagating in a vertically sheared, rotating stratified fluid interacts with the pair of inertial levels that surround the critical level. An exact expression for the form of the IGW is derived here in the case of a linear shear and used to examine this interaction in detail. This expression recovers the classical values of the transmission and reflection coefficients $|T|=\text{e}^{-{\rm\pi}{\it\mu}}$ and $|R|=0$, where ${\it\mu}^{2}=J(1+{\it\nu}^{2})-1/4$, $J$ is the Richardson number and ${\it\nu}$ the ratio between the horizontal transverse and along-shear wavenumbers. For large $J$, a WKB analysis provides an interpretation of this result in term of tunnelling: an IGW incident on the lower inertial level becomes evanescent between the inertial levels, returning to an oscillatory behaviour above the upper inertial level. The amplitude of the transmitted wave is directly related to the decay of the evanescent solution between the inertial levels. In the immediate vicinity of the critical level, the evanescent IGW is well represented by the quasi-geostrophic approximation, so that the process can be interpreted as resulting from the coupling between balanced and unbalanced motion. The exact and WKB solutions describe the so-called valve effect, a dependence of the behaviour in the region between the inertial levels on the direction of wave propagation. For $J<1$ this is shown to lead to an amplification of the wave between the inertial levels. Since the flow is inertially unstable for $J<1$, this establishes a correspondence between the inertial-level interaction and the condition for inertial instability.

An overlooked conservation law for near-inertial waves (NIWs) propagating in a steady background flow provides a new perspective on the concentration of these waves in regions of anticyclonic vorticity. The conservation law implies that this concentration is a direct consequence of the decrease in spatial scales experienced by an initially homogeneous wave field. Scaling arguments and numerical simulations of a reduced-gravity model of mixed-layer NIWs confirm this interpretation and elucidate the influence of the strength of the background flow relative to the dispersion.

A straight front separating two semi-infinite regions of uniform potential vorticity (PV) in a rotating shallow-water fluid gives rise to a localized fluid jet and a geostrophically balanced shelf in the free surface. The linear stability of this configuration, consisting of the simplest non-trivial PV distribution, has been studied previously, with ambiguous results. We revisit the problem and show that the flow is weakly unstable when the maximum Rossby number ${\textsfi R}\,{>}\,1$. The instability is surprisingly weak, indeed exponentially so, scaling like $\exp[-4.3/({\textsfi R} - 1)]$ as ${\textsfi R}\,{\to}\,1$. Even when ${\textsfi R}\,{=}\,\sqrt{2}$ (when the maximum Froude number ${\textsfi F}\,{=}\,1$), the maximum growth rate is only $7.76\,{\times}\,10^{-6}$ times the Coriolis frequency. Its existence nonetheless sheds light on the concept of ‘balance’ in geophysical flows, i.e. the degree to which the PV controls the dynamical evolution of these flows.

The amplitude equations governing the nonlinear interaction among normal modes are derived for a multilayer quasi-geostrophic channel. The set of normal modes can represent any wavy disturbance to a parallel shear flow, which may be stable or unstable. Orthogonality in the sense of pseudomomentum or pseudoenergy is used to obtain the amplitude equations in a direct fashion, and pseudoenergy and pseudomomentum conservation laws permit the properties of the interaction coefficients to be deduced. Particular attention is paid to triads exhibiting explosive resonant interaction, as they lead to nonlinear instability of the basic flow. The relationship between this mechanism and the most recently discovered nonlinear stability conditions is discussed.

Situations in which the basic velocity is constant in each layer are treated in detail. A particular formulation of the stability condition is given that emphasizes the close connection between linear and nonlinear stability. It is established that this stability condition is also a necessary condition: when it is not satisfied, and when the flow is linearly stable, explosive resonant interaction of baroclinic Rossby waves acts as a destabilizing mechanism. Two- and three-layer models are specifically considered; their stability features are presented in the form of stability diagrams, and interaction coefficients are calculated in particular cases.

An approach is presented for studying Rossby wave interaction in a shear flow with both regular and singular modes (i.e. those possessing a critical level). The approach relies on a truncated normal mode expansion of the equations of motion. Such an expansion remains valid in the presence of singular modes, provided that these modes are not considered individually, but that complete packets are taken into account in the truncated system. Mathematically, this means that the interaction equations need to be integrated with respect to the phase velocity (or, equivalently, the critical level position) of the singular modes.

The action of two regular modes on a packet of singular modes is treated in detail; in particular, asymptotic results are deduced for the long-term behaviour of the packet. The case of a linear shear is considered as an illustration: analytical expressions are derived for the normal modes and their pseudomomentum, and they are used to present explicit results for the evolution of the packet of singular modes.

The weakly nonlinear dynamics of quasi-geostrophic flows over a one-dimensional, periodic or random, small-scale topography is investigated using an asymptotic approach. Averaged (or homogenized) evolution equations which account for the flow–topography interaction are derived for both homogeneous and continuously stratified quasi-geostrophic fluids. The scaling assumptions are detailed in each case; for stratified fluids, they imply that the direct influence of the topography is confined within a thin bottom boundary layer, so that it is through a new bottom boundary condition that the topography affects the large-scale flow. For both homogeneous and stratified fluids, a single scalar function entirely encapsulates the properties of the topography that are relevant to the large-scale flow: it is the correlation function of the topographic height in the homogeneous case, and a linear transform thereof in the continuously stratified case.

Some properties of the averaged equations are discussed. Explicit nonlinear solutions in the form of one-dimensional travelling waves can be found. In the homogeneous case, previously studied by Volosov, they obey a second-order differential equation; in the stratified case on which we focus they obey a nonlinear pseudodifferential equation, which reduces to the Peierls–Nabarro equation for sinusoidal topography. The known solutions to this equation provide examples of nonlinear periodic and solitary waves in continuously stratified fluid over topography.

The influence of bottom topography on large-scale baroclinic instability is also examined using the averaged equations: they allow a straightforward extension of Eady's model which demonstrates the stabilizing effect of topography on baroclinic instability.