The instability of baroclinic Rossby waves over two-dimensional topography is examined using a nonlinear model, a linear stability calculation and wave triads. In all cases, the Rossby waves are unstable, as seen previously over a flat bottom. But topography decreases the growth rates and changes the structure of the unstable waves. When the topographic height (or slope) exceeds a critical value, the instability is ‘locked’ to topography, in that the most unstable mode, particularly in the lower layer, resembles the bathymetry. In this limit, the growth rate becomes independent of topographic height. A triad calculation suggests that the growth rates in the locked state should depend on the lateral scale of the bathymetry but not its height, and that locking does not occur for topographic scales smaller than the surface deformation radius. The results suggest an alternate way that topographically locked flow can be generated, and indicate that baroclinic instability can be much different over steep bathymetry.

We numerically study weakly decaying two-dimensional turbulence over topography of varying roughness by comparing long-term flow states from simulations to the minimum enstrophy state (MES) proposed by Bretherton & Haidvogel (J. Fluid Mech., vol. 78, 1976, issue 1, pp. 129–154). The presence of isolated vortices in the numerical simulations leads to significant differences from the theory. These vortices are either roaming or topographically locked, with this distinction depending on the topographic roughness, the initial length scale and the initial non-dimensional energy ($E/E_{\#}$), where $E_{\#}$ denotes the critical energy proposed by Siegelman & Young (Proc. Natl. Acad. Sci. USA, vol. 120, 2023, issue 44, p. e2308018120). For low-energy and weakly rough topography, scatter plots show that the numerical results deviate from the MES due to the presence of vortices. As topographic roughness increases, the number, size and mobility of vortices decrease, leading to closer agreement between the numerical results and the MES. High energy and weakly rough topography lead to roaming vortices on a field of homogenised background potential vorticity. However, we observe topographically trapped eddies on highly rough topographies, even when initial length scales are much smaller than the domain size. The degree to which numerical results deviate from MES scaling is due to the presence of vortices, which, in turn, depend on topographic roughness and initial conditions in complex ways, suggesting a rich range of ways in which turbulence organises the long-term flow.

Dissipation mechanisms of low-mode internal tides, which travel far from their topographic generation sites, are an important consideration for the large-scale circulation and energy budget in the ocean. Modelling studies often decouple scattering and generation, i.e. study scattering in the absence of a local barotropic tide, or study generation in the absence of an incident internal tide. In this two-dimensional study using a semi-analytical Green function approach, we model the combined effects of internal wave generation by a barotropic forcing and scattering of an incident mode-1 internal wave, at an isolated Gaussian bottom topography in uniform stratification. Four different parameters govern the energetics – the non-dimensional topographic height (height ratio $h^*$) and slope (criticality $\epsilon$), and the normalised amplitude ($U_0$) and phase ($\varTheta$) of the barotropic forcing with respect to the incident mode-1 internal wave. The theory is first quantitatively validated by comparisons with numerical simulations for three different combinations of $(h^*,\epsilon )$, followed by a detailed parametric sweep. For a given topography and $U_0$, on an average across $\varTheta$, the total internal wave energy flux is the sum of the energy fluxes associated with generation in the presence of the barotropic forcing alone and the incident mode-1. For a given $ \varTheta$, however, the total energy flux can deviate significantly from its mean value due to constructive/destructive interferences of the individual modes; this occurs over a surprisingly wide range of $U_0$, $h^*$ and $\epsilon$. Depending on $U_0$, these deviations can be interpreted as either the extent to which a background barotropic forcing affects internal wave scattering, or the extent to which an incident mode-1 internal wave influences internal wave generation by barotropic forcing. The presence of barotropic forcing can significantly modify the scattering characteristics, including the possibility of losing a non-negligible fraction of the incident internal wave energy to another form. Similarly, internal wave generation characteristics can be sensitively dependent on the presence of an incident internal wave. These energy flux loss or gain effects are typically found for short subcritical ($h^*\lesssim 0.4$, $\epsilon \lt 1$) and sufficiently steep, tall ($h^*\gtrsim 0.4$) topographies.

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.

Recent studies have shown that, in coastal waters where water depth decreases significantly due to rapid bathymetric changes, the non-equilibrium dynamics (NED) substantially increases the occurrence probability of extreme (rogue) waves. Nevertheless, research on depth-induced NED has been predominantly confined to unidirectional irregular waves, while the role of directionality remains largely unexplored. The scarce studies on multidirectional waves mainly rely on numerical simulations and have yielded conflicting results. In this work, we report on an experimental investigation of wave directionality on the depth-induced non-equilibrium wave statistics. High-order statistical moments, skewness and kurtosis, are used as proxies for the non-equilibrium wave response. Our results indicate that the directional spreading has a minor effect on decreasing the maximum values of these statistical moments. In contrast, the incidence direction plays a significant role in the non-equilibrium wave response, which is attributed to the effective bottom slope.

Wavy topography can exert a significant influence on gravity-driven flows in porous media. Building on the low-dimensional theoretical framework for a wavy topography of height $f(x) = A[1 - \cos (\lambda x)]$, where $A$ is the amplitude and $\lambda$ is the wavenumber of the topography, under small-slope conditions ($A\lambda \ll 1$) Di et al. (2025 J. Fluid Mech., vol. 1016, A16), we extend the framework to constant-flux injection while incorporating uniform drainage and localised leakage through low-permeability substrates. A key dimensionless topographic intensity, emerges as the ratio of the pressure gradient required to overcome topographic slopes to the characteristic viscous gradient driving the flow, thereby quantifying topographic resistance. Our results show that a larger topographic intensity retards current advancement, while drainage, governed by the drainage intensity, imposes an upper bound on propagation distance. Leakage proves highly sensitive to the along-slope position of fissured zones. Comparisons with a macroscopic sharp-interface flow model indicate that the low-dimensional model simplifies the two-phase dynamics in substrates via a Darcy’s sink term, yielding underestimates of propagation during drainage and leakage. Applied to the field of carbon dioxide sequestration, our low-dimensional model reveals how injection flux modulates the early-stage flow dynamics over wavy cap rocks, offering theoretical insights into sequestration performance.

A combined experimental and numerical investigation of equilibrium states arising from quasi-two-dimensional turbulent flows in a rotating quadrangular basin with a central flat region and steep slopes adjacent to the sidewalls is presented. The study examines freely decaying and continuously forced regimes. Laboratory experiments show that decaying turbulence consistently evolves into a robust equilibrium state characterised by: (i) a boundary current around the basin along the topographic contours, and (ii) a central anticyclone – features accurately reproduced by shallow-water numerical simulations at laboratory scale. Additional simulations using a mesoscale basin suggest the relevance of these findings to oceanic regimes for different initial conditions and topographic parameters. In the case of continuously forced flows, time-averaged fields reveal qualitatively similar structures, despite the randomness of the applied forcing and the consequent absence of a strict equilibrium. These results demonstrate the emergence of robust flow patterns with implications for the modelling and understanding of semi-permanent flows that are often found in statistical theories of geophysical turbulence.

This paper describes a high-order strongly nonlinear (SNL) model for long waves in the presence of a variable bottom, which is a generalisation of the model for a flat bottom (Choi 2022a, J. Fluid Mech. vol. 945, A15). This asymptotic model written in terms of the bottom velocity is obtained using systematic expansion with a single small parameter measuring the ratio of the water depth to the characteristic wavelength and is found linearly stable at any order of approximation. To test the high-order SNL model with a variable bottom, we solve numerically the first- and second-order models using a pseudo-spectral method to study the deformation or generation of long waves over a variable bottom. Specifically, we consider two examples: (i) the propagation of cnoidal waves over a fixed bottom topography, and (ii) the forced generation of solitary waves by a submerged topography moving steadily with a transcritical speed. The computed results are then compared with the fully nonlinear computation using a boundary integral method as well as the numerical solutions of the weakly nonlinear long wave model. It is found that the second-order SNL model for the bottom velocity is suitable for stable numerical computations and produces accurate solutions even for a relatively large-amplitude initial wave or submerged topography.

To advance understanding of the influence hill-slope and hill-shape have on neutrally stratified turbulent air flow over isolated forested hills, we interrogate four turbulence-resolving simulations. A spectrally friendly fringe technique enables the use of periodic boundary conditions to simulate flow over isolated two-dimensional (2D) and three-dimensional (3D) hills of cosine shape. The simulations target recently conducted wind tunnel (WT) experiments that are configured to fall outside the regimes for which current theory applies. Simulation skill for flow over isolated 3D hills is demonstrated through matching the canopy and hill configuration with the recently conducted WT experiments and comparing results. The response of the mean and turbulent flow components to 2D versus 3D hills along the hill-centreline are discussed. The phase and amplitude of spatially varying flow perturbations over forested hills are evaluated for flows outside the regime valid for current theory. Flow over isolated 2D forested hills produces larger amplitude vertical motions on a hill’s windward and leeward faces and the speed-up of the mean wind compared with that over isolated 3D forested hills at the hill-centreline. The 3D hills generate surface pressure minima over hill-crests that are only half the magnitude of those over 2D hills. The spatial region over which hill-induced negative pressure drag acts increases with increasing hill steepness. Assumptions in partitioning the flow into an upper layer with an inviscid response to the hill’s pressure field are robust and lead to solid predictions of hill-induced perturbations to the mean flow; however, applying those assumptions to predict the evolution of the turbulent moments only provides approximate explanations at best.

We investigate interactions between two like-signed vortices over either an isolated seamount or a basin (a depression in the bathymetry), using a quasi-geostrophic, two-layer model on the $f$-plane. When the vortex pair is centred over the seamount, the vortices are pushed together by the secondary flow generated in the bottom layer, facilitating their merger. Over a basin, the deep anomalies are much stronger and their interaction strains out the surface vortices. The results are supported by an analytical estimation of the initial potential vorticity anomalies in the lower layer and by analysis of the linear stability of a single vortex over the bathymetry. Similar phenomena are observed when the vortex pair is displaced from the bathymetric centre and when the initial vortices are initially compensated. Sub-deformation-scale vortices are less influenced by bathymetry than larger vortices. The results help explain asymmetries noted previously in turbulence simulations over bathymetry.

We present a theoretical framework for porous media gravity currents propagating over rigid curvilinear surfaces. By reducing the flow dynamics to low-dimensional models applicable on surfaces where curvature effects are negligible, we demonstrate that, for finite-volume releases, the flow behaviour in both two-dimensional and axisymmetric configurations is primarily governed by the ratio of the released viscous fluid volume to the characteristic volume of the curvilinear surface. Our theoretical predictions are validated using computational fluid dynamics simulations based on a sharp-interface model for macroscopic flow in porous media. In the context of carbon dioxide geological sequestration, our findings suggest that wavy cap rock geometries can enhance trapping capacity compared with traditional flat-surface assumptions, highlighting the importance of incorporating realistic topographic features into subsurface flow models.

The modification of a gravity current past a thin two-dimensional barrier is studied experimentally, focusing on propagation characteristics as well as turbulence and mixing at the gravity-current head near the obstacle. The broader aim is to develop an eddy-diffusivity parametrisation based on local governing variables to represent gravity-current/obstacle interactions in numerical weather prediction models. A gravity current is produced in a rectangular tank by releasing a salt solution via a lock-exchange mechanism into an aqueous ethanol solution with matched refractive index, and it is allowed to interact with the barrier. A combined particle image velocimetry and planar laser-induced fluorescence system is used to obtain instantaneous velocity and density fields. The experiments span two Reynolds numbers and four obstacle heights, with each case replicated ten times for conducting phase-aligned ensemble averaging. Four evolutionary stages of the front are identified: approach, vertical deflection, collapse and reattachment. Particular focus is placed on the vertical deflection and collapse stages (dubbed collision phase), which includes flow (hydraulic) adjustment, flow modulation over the obstacle, instabilities, turbulence and mixing, and relaxation to a gravity current downstream. The time scales for various flow stages were identified. The results demonstrate that the normalised eddy diffusivity changes significantly throughout these stages and with the dimensionless height of the obstacle.

The quasi-geostrophic two-layer model is a widely used tool to study baroclinic instability in the ocean. One instability criterion for the inviscid two-layer model is that the potential vorticity (PV) gradient must change sign between the layers. This has a well-known implication if the model includes a linear bottom slope: for sufficiently steep retrograde slopes, instability is suppressed for a flow parallel to the isobaths. This changes in the presence of bottom friction as well as when the PV gradients in the layers are not aligned. We derive the generalised instability condition for the two-layer model with non-zero friction and arbitrary mean flow orientation. This condition involves neither the friction coefficient nor the bottom slope; even infinitesimally weak bottom friction destabilises the system regardless of the bottom slope. We then examine the instability characteristics as a function of varying slope orientation and magnitude. The system is stable across all wavenumbers only if friction is absent and if the planetary, topographic and stretching PV gradients are aligned. Strong bottom friction decreases the growth rates but also alters the dependence on bottom slope. In conclusion, the often mentioned stabilisation by steep bottom slopes in the two-layer model holds only in very specific circumstances, thus probably plays only a limited role in the ocean.

We investigate the effects of bottom roughness on bottom boundary-layer (BBL) instability beneath internal solitary waves (ISWs) of depression. Applying both two-dimensional (2-D) numerical simulations and linear stability theory, an extensive parametric study explores the effect of the Reynolds number, pressure gradient, roughness (periodic bump) height $h_b$ and roughness wavelength $\lambda _b$ on BBL instability. The simulations show that small $h_b$, comparable to that of laboratory-flume materials ($\sim$100 times less than the thickness of the viscous sublayer $\delta _v$), can destabilize the BBL and trigger vortex shedding at critical Reynolds numbers much lower than what occurs for numerically smooth surfaces. We identify two mechanisms of vortex shedding, depending on $h_b/\delta _v$. For $h_b/\delta _v \gtrapprox 1$, vortices are forced directly by local flow separation in the lee of each bump. Conversely, for $h_b/\delta _v \lessapprox 10^{-1}$ the roughness seeds perturbations in the BBL, which are amplified by the BBL flow. Roughness wavelengths close to those associated with the most unstable BBL mode, as predicted by linear instability theory, are preferentially amplified. This resonant amplification nature of the BBL flow, beneath ISWs, is consistent with what occurs in a BBL driven by surface solitary waves and by periodic monochromatic waves. Using the $N$-factor method for Tollmien–Schlichting waves, we propose an analogy between the roughness height and seed noise required to trigger instability. Including surface roughness, or more generally an appropriate level of seed noise, reconciles the discrepancies between the vortex-shedding threshold observed in the laboratory versus that predicted by otherwise smooth-bottomed 2-D spectral simulations.

The transformation of internal waves on a stepwise underwater obstacle is studied in the linear approximation. The transmission and reflection coefficients are derived for a two-layer fluid. The results are obtained and presented as functions of incident wave wavenumber, density ratio of layers, pycnocline position, and height of the bottom step. Excitation coefficients of evanescent modes are also calculated, and their importance is demonstrated. This allows one to estimate the number of evanescent modes necessary to take into account to attain the required accuracy for the transformation coefficients.

The breaking and energy distribution of mode-1 depression internal solitary wave interactions with Gaussian ridges are examined through laboratory experiments. A series of processes, such as shoaling, breaking, transmission and reflection, are captured completely by measuring the velocity field in a large region. It is found that the maximum interface descent ($a_{max}$) during wave shoaling is an important parameter for diagnosing the type of wave–ridge interaction and energy distribution. The wave breaking on the ridge depends on the modified blockage parameter $\zeta _m$, the ratio of the sum of the upper layer depth and $a_{max}$ to the water depth at the top of the ridge. As $\zeta _m$ increases, the interaction type transitions from no breaking to plunging and mixed plunging–collapsing breaking. Within the scope of this experiment, the energy distribution can be characterized solely by $\zeta _m$. The transmission energy decreases monotonically with increasing $\zeta _m$, and there is a linear relationship between $\zeta _m^2$ and the reflection coefficient. The value of $a_{max}$ can be determined from the basic initial parameters of the experiment. Based on the incident wave parameters, the depth of the upper and lower layers, and the topographic parameters, two new simple methods for predicting $a_{max}$ on the ridge are proposed.

Energetics of mode-1 internal waves interacting with topographic ridges are investigated using high-resolution two-dimensional simulations at spatial scales of $O(100)$ m that span between classical laboratory-scale ($O(10)$ m) and field-scale simulations ($O(1000\unicode{x2013}10\,000)$ m). This paper focuses on the energetics of wave–topography interaction, with emphasis on systematically examining the partitioning of the incident wave energy as a function of wave forcing and topographic parameters. Partitioning of energy into the transmitted, reflected and dissipated components is quantified as a function of wave Froude number $Fr=U_0/c_{ph}$ ($U_0=$ velocity amplitude of forcing and $c_{ph}=$ internal wave celerity), slope criticality $=\gamma /s$, where $\gamma =$ topographic slope and $s=$ wave characteristic slope, and the ratio of topographic height $h_t$ to water depth $d$. As $Fr$ increases, dense fluid from the base of the stratified water column surges upslope with significant vertical inertia, leading to the formation of internal boluses that plunge over and onto the downstream side of the ridge, resulting in elevated dissipation. Results show that non-hydrostatic contributions to the total energy flux are significant (up to 50 %). Analysis of the energy flux budget shows that transmitted energy flux decreases monotonically as $\gamma /s$ increases for any given $Fr$ and $h_t/d$. At critical slopes ($\gamma /s=1$), the transmitted energy flux scales as a linear function of $h_t/d$, with a mild dependence on $Fr$, a key result that can be useful in energy flux parameterizations. Reflected energy flux exhibits a nonlinear dependence on the ridge height, increasing sharply when $h_t/d > 0.5$. Dissipation is enhanced at critical slopes, with a plateau evident for $\gamma /s \ge 1$ and $h_t/d = 0.5$ for all $Fr$.

Experimental studies on the sloshing of fluid layers are usually performed in rectangular tanks with fixed boundaries. In contrast, the present study uses a 4.76-m-long circular channel, a geometry with open periodic boundaries. Surface waves are excited by means of a submerged hill that, together with the tank, performs a harmonic oscillation. Laboratory measurements are made using 18 ultrasonic probes, evenly distributed over the channel to track the wave propagation. It is shown that a two-dimensional long-wave numerical model derived via the Kármán–Pohlhausen approach reproduces the experimental data as long as the forcing is monochromatic. The sloshing experiments imply a highly complex surface wave field. Different wave types such as solitary waves, undular bores and antisolitary waves are observed. For order one $\delta _{hill} = h_{hill}/h_0$, where $h_0$ is the mean water level and $h_{hill}$ the obstacle's height, the resonant reflections of solitary waves by the submerged obstacle give rise to an amplitude spectrum for which the main resonance peaks can be explained by linear theory. For smaller $\delta _{hill}$, wave transmissions lead to major differences with respect to the more common cases of sloshing with closed ducts having fully reflective ends for which wave transmission through the end walls is not possible. This ultimately results in more complex resonance diagrams and a pattern formation that changes rather abruptly with the frequency. The experiments are of interest not only for engineering applications but also for tidal flows over bottom topography.

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.

This study explores precession-driven flows in a non-axisymmetric ellipsoid spinning around its medium axis. Using an experimental approach, we focus on two aspects of the flow: the base flow of uniform vorticity and the development of fluid instabilities. In contrast to a preceding paper (J. Fluid. Mech., vol. 932, 2022, A24), where the ellipsoid rotated around its shortest axis, we do not observe bi-stability or hysteresis of the base flow, but a continuous transition from small to large differential rotation and tilt of the fluid rotation axis. We then use the model developed by Noir & Cébron (J. Fluid. Mech., vol. 737, 2013, pp. 412–439) to numerically determine regions in the parameter space of axial and equatorial deformations for which bi-stability may exist. Concerning fluid instabilities, we use three independent observations to track the onset of both boundary layer and parametric instabilities. Our results clearly show the presence of a parametric instability, yet the exact nature of the underlying mechanism (conical shear layer instability, shear instability and elliptical instability) is not unambiguously identified. A coexisting boundary layer instability, although unlikely, cannot be ruled out based on our experimental data. To make further progress on this topic, a new generation of experiments at significantly lower Ekman numbers (ratio of rotation and viscous time scales) is clearly needed.