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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.

 

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 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 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.