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The evolution of unforced and weakly damped two-dimensional turbulence over random rough topography presents two extreme states. If the initial kinetic energy$E$is sufficiently high, then the topography is a weak perturbation, and evolution is determined by the spontaneous formation and mutual interaction of coherent axisymmetric vortices. High-energy vortices roam throughout the domain and mix the background potential vorticity (PV) to homogeneity, i.e., in the region between vortices, which is most of the domain, the relative vorticity largely cancels the topographic PV. If$E$is low, then vortices still form but they soon become locked to topographic features: Anticyclones sit above topographic depressions and cyclones above elevated regions. In the low-energy case, with topographically locked vortices, the background PV retains some spatial variation. We develop a unified framework of topographic turbulence spanning these two extreme states of low and high energy. A main organizing concept is that PV homogenization demands a particular kinetic energy level$E_{♯}$.$E_{♯}$is the separator between high-energy evolution and low-energy evolution.

 

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.

 

The Stokes velocity u S , defined approximately by Stokes (1847, Trans. Camb. Philos. Soc. , 8 , 441–455.), and exactly via the Generalized Lagrangian Mean, is divergent even in an incompressible fluid. We show that the Stokes velocity can be naturally decomposed into a solenoidal component, u sol S , and a remainder that is small for waves with slowly varying amplitudes. We further show that u sol S arises as the sole Stokes velocity when the Lagrangian mean flow is suitably redefined to ensure its exact incompressibility. The construction is an application of Soward & Roberts’s glm theory (2010, J. Fluid Mech. , 661 , 45–72. ( doi:10.1017/S0022112010002867 )) which we specialize to surface gravity waves and implement effectively using a Lie series expansion. We further show that the corresponding Lagrangian-mean momentum equation is formally identical to the Craik–Leibovich (CL) equation with u sol S replacing u S , and we discuss the form of the Stokes pumping associated with both u S and u sol S . This article is part of the theme issue ‘Mathematical problems in physical fluid dynamics (part 1)’. 

Vortex crystals are quasiregular arrays of like-signed vortices in solid-body rotation embedded within a uniform background of weaker vorticity. Vortex crystals are observed at the poles of Jupiter and in laboratory experiments with magnetized electron plasmas in axisymmetric geometries. We show that vortex crystals form from the free evolution of randomly excited two-dimensional turbulence on an idealized polar cap. Once formed, the crystals are long lived and survive until the end of the simulations (300 crystal-rotation periods). We identify a fundamental length scale, L γ = ( U / γ ) 1 / 3 , characterizing the size of the crystal in terms of the mean-square velocity U of the fluid and the polar parameter γ = f p / a p 2 , with f p the Coriolis parameter at the pole and a p the polar radius of the planet. 

Inertia-gravity waves in the atmosphere and ocean are transported and refracted by geostrophic turbulent currents. Provided that the wave group velocity is much greater than the speed of geostrophic turbulent currents, kinetic theory can be used to obtain a comprehensive statistical description of the resulting interaction (Savva et al. , J. Fluid Mech. , vol. 916, 2021, A6). The leading-order process is scattering of wave energy along a surface of constant frequency, $\omega$ , in wavenumber space. The constant- $\omega$ surface corresponding to the linear dispersion relation of inertia-gravity waves is a cone extending to arbitrarily high wavenumbers. Thus, wave scattering by geostrophic turbulence results in a cascade of wave energy to high wavenumbers on the surface of the constant- $\omega$ cone. Solution of the kinetic equations shows establishment of a wave kinetic energy spectrum $\sim k_h^{-2}$ , where $k_h$ is the horizontal wavenumber. 

Abstract Anticyclonic vortices focus and trap near-inertial waves so that near-inertial energy levels are elevated within the vortex core. Some aspects of this process, including the nonlinear modification of the vortex by the wave, are explained by the existence of trapped near-inertial eigenmodes. These vortex eigenmodes are easily excited by an initialwave with horizontal scale much larger than that of the vortex radius. We study this process using a wave-averaged model of near-inertial dynamics and compare its theoretical predictions with numerical solutions of the three-dimensional Boussinesq equations. In the linear approximation, the model predicts the eigenmode frequencies and spatial structures, and a near-inertial wave energy signature that is characterized by an approximately time-periodic, azimuthally invariant pattern. The wave-averaged model represents the nonlinear feedback of the waves on the vortex via a wave-induced contribution to the potential vorticity that is proportional to the Laplacian of the kinetic energy density of the waves. When this is taken into account, the modal frequency is predicted to increase linearly with the energy of the initial excitation. Both linear and nonlinear predictions agree convincingly with the Boussinesq results. 

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. 

Jupiter’s atmosphere is one of the most turbulent places in the solar system. Whereas observations of lightning and thunderstorms point to moist convection as a small-scale energy source for Jupiter’s large-scale vortices and zonal jets, this has never been demonstrated due to the coarse resolution of pre-Juno measurements. The Juno spacecraft discovered that Jovian high latitudes host a cluster of large cyclones with diameter of around 5,000 km, each associated with intermediate- (roughly between 500 and 1,600 km) and smaller-scale vortices and filaments of around 100 km. Here, we analyse infrared images from Juno with a high resolution of 10 km. We unveil a dynamical regime associated with a significant energy source of convective origin that peaks at 100 km scales and in which energy gets subsequently transferred upscale to the large circumpolar and polar cyclones. Although this energy route has never been observed on another planet, it is surprisingly consistent with idealized studies of rapidly rotating Rayleigh–Bénard convection, lending theoretical support to our analyses. This energy route is expected to enhance the heat transfer from Jupiter’s hot interior to its troposphere and may also be relevant to the Earth’s atmosphere, helping us better understand the dynamics of our own planet.

 

We use a multiple-scale expansion to average the wave action balance equation over an ensemble of sea-surface velocity fields characteristic of the ocean mesoscale and submesoscale. Assuming that the statistical properties of the flow are stationary and homogeneous, we derive an expression for a diffusivity tensor of surface-wave action density. The small parameter in this expansion is the ratio of surface current speed to gravity wave group speed. For isotropic currents, the action diffusivity is expressed in terms of the kinetic energy spectrum of the flow. A Helmholtz decomposition of the sea-surface currents into solenoidal (vortical) and potential (divergent) components shows that, to leading order, the potential component of the surface velocity field has no effect on the diffusivity of wave action: only the vortical component of the sea-surface velocity results in diffusion of surface-wave action. We validate our analytic results for the action diffusivity by Monte Carlo ray-tracing simulations through an ensemble of stochastic velocity fields. 

In the problem of horizontal convection a non-uniform buoyancy, $b_{s}(x,y)$ , is imposed on the top surface of a container and all other surfaces are insulating. Horizontal convection produces a net horizontal flux of buoyancy, $\boldsymbol{J}$ , defined by vertically and temporally averaging the interior horizontal flux of buoyancy. We show that $\overline{\boldsymbol{J}\boldsymbol{\cdot }\unicode[STIX]{x1D735}b_{s}}=-\unicode[STIX]{x1D705}\langle |\unicode[STIX]{x1D735}b|^{2}\rangle$ ; the overbar denotes a space–time average over the top surface, angle brackets denote a volume–time average and $\unicode[STIX]{x1D705}$ is the molecular diffusivity of buoyancy  $b$ . This connection between $\boldsymbol{J}$ and $\unicode[STIX]{x1D705}\langle |\unicode[STIX]{x1D735}b|^{2}\rangle$ justifies the definition of the horizontal-convective Nusselt number, $Nu$ , as the ratio of $\unicode[STIX]{x1D705}\langle |\unicode[STIX]{x1D735}b|^{2}\rangle$ to the corresponding quantity produced by molecular diffusion alone. We discuss the advantages of this definition of $Nu$ over other definitions of horizontal-convective Nusselt number. We investigate transient effects and show that $\unicode[STIX]{x1D705}\langle |\unicode[STIX]{x1D735}b|^{2}\rangle$ equilibrates more rapidly than other global averages, such as the averaged kinetic energy and bottom buoyancy. We show that $\unicode[STIX]{x1D705}\langle |\unicode[STIX]{x1D735}b|^{2}\rangle$ is the volume-averaged rate of Boussinesq entropy production within the enclosure. In statistical steady state, the interior entropy production is balanced by a flux through the top surface. This leads to an equivalent ‘surface Nusselt number’, defined as the surface average of vertical buoyancy flux through the top surface times the imposed surface buoyancy $b_{s}(x,y)$ . In experimental situations it is easier to evaluate the surface entropy flux, rather than the volume integral of $|\unicode[STIX]{x1D735}b|^{2}$ demanded by $\unicode[STIX]{x1D705}\langle |\unicode[STIX]{x1D735}b|^{2}\rangle$ .