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We study stimulated generation – the transfer of energy from balanced flows to existing internal waves – using an asymptotic model that couples barotropic quasi-geostrophic flow and near-inertial waves with $\\text{e}^{\\text{i}mz}$ vertical structure, where $m$ is the vertical wavenumber and $z$ is the vertical coordinate. A detailed description of the conservation laws of this vertical-plane-wave model illuminates the mechanism of stimulated generation associated with vertical vorticity and lateral strain. There are two sources of wave potential energy, and corresponding sinks of balanced kinetic energy: the refractive convergence of wave action density into anti-cyclones (and divergence from cyclones); and the enhancement of wave-field gradients by geostrophic straining. We quantify these energy transfers and describe the phenomenology of stimulated generation using numerical solutions of an initially uniform inertial oscillation interacting with mature freely evolving two-dimensional turbulence. In all solutions, stimulated generation co-exists with a transfer of balanced kinetic energy to large scales via vortex merging. Also, geostrophic straining accounts for most of the generation of wave potential energy, representing a sink of 10 %–20 % of the initial balanced kinetic energy. However, refraction is fundamental because it creates the initial eddy-scale lateral gradients in the near-inertial field that are then enhanced by advection. In these quasi-inviscid solutions, wave dispersion is the only mechanism that upsets stimulated generation: with a barotropic balanced flow, lateral straining enhances the wave group velocity, so that waves accelerate and rapidly escape from straining regions. This wave escape prevents wave energy from cascading to dissipative scales.

We derive an asymptotic model that describes the nonlinear coupled evolution of (i) near-inertial waves (NIWs), (ii) balanced quasi-geostrophic flow and (iii) near-inertial second harmonic waves with frequency near $2f_{0}$ , where $f_{0}$ is the local inertial frequency. This ‘three-component’ model extends the two-component model derived by Xie & Vanneste (J. Fluid Mech., vol. 774, 2015, pp. 143–169) to include interactions between near-inertial and $2f_{0}$ waves. Both models possess two conservation laws which together imply that oceanic NIWs forced by winds, tides or flow over bathymetry can extract energy from quasi-geostrophic flows. A second and separate implication of the three-component model is that quasi-geostrophic flow catalyses a loss of NIW energy to freely propagating waves with near- $2f_{0}$ frequency that propagate rapidly to depth and transfer energy back to the NIW field at very small vertical scales. The upshot of near- $2f_{0}$ generation is a two-step mechanism whereby quasi-geostrophic flow catalyses a nonlinear transfer of near-inertial energy to the small scales of wave breaking and diapycnal mixing. A comparison of numerical solutions with both Boussinesq and three-component models for a two-dimensional initial value problem reveals strengths and weaknesses of the model while demonstrating the extraction of quasi-geostrophic energy and production of small vertical scales.

The electrical conductivity of the Earth’s core is an important physical parameter that controls the core dynamics and the thermal evolution of the Earth. In this study, the effect of core electrical conductivity on core surface flow models is investigated. Core surface flow is derived from a geomagnetic field model on the presumption that a viscous boundary layer forms at the core–mantle boundary. Inside the boundary layer, where the viscous force plays an important role in force balance, temporal variations of the magnetic field are caused by magnetic diffusion as well as motional induction. Below the boundary layer, where core flow is assumed to be in tangentially geostrophic balance or tangentially magnetostrophic balance, contributions of magnetic diffusion to temporal variation of the magnetic field are neglected. Under the constraint that the core flow is tangentially geostrophic beneath the boundary layer, the core electrical conductivity in the range from 105Sm-1 to 107Sm-1 has less significant effect on the core flow. Under the constraint that the core flow is tangentially magnetostrophic beneath the boundary layer, the influence of electrical conductivity on the core flow models can be clearly recognized; the magnitude of the mean toroidal flow does not increase or decrease, but that of the mean poloidal flow increases with an increase in core electrical conductivity. This difference arises from the Lorentz force, which can be stronger than the Coriolis force, for higher electrical conductivity, since the Lorentz force is proportional to the electrical conductivity. In other words, the Elsasser number, which represents the ratio of the Lorentz force to the Coriolis force, has an influence on the difference. The result implies that the ratio of toroidal to poloidal flow magnitudes has been changing in accordance with secular changes of rotation rate of the Earth and of core electrical conductivity due to a decrease in core temperature throughout the thermal evolution of the Earth.

Evidence of persistent layering, with a vertical stacking of sharp variations in temperature, has been presented recently at the vertical and lateral periphery of energetic oceanic vortices through seismic imaging of the water column. The stacking has vertical scales ranging from a few metres up to 100 m and a lateral spatial coherence of several tens of kilometres comparable with the vortex horizontal size. Inside this layering, in situ data display a $[{ k}_{h}^{- 5/ 3} { k}_{h}^{- 2} ] $ scaling law of horizontal scales for two different quantities, temperature and a proxy for its vertical derivative, but for two different ranges of wavelengths, between 5 and 50 km for temperature and between 500 m and 5 km for its vertical gradient. In this study, we explore the dynamics underlying the layering formation mechanism, through the slow dynamics captured by quasi-geostrophic equations. Three-dimensional high-resolution numerical simulations of the destabilization of a lens-shaped vortex confirm that the vertical stacking of sharp jumps in density at its periphery is the three-dimensional analogue of the preferential wind-up of potential vorticity near a critical radius, a phenomenon which has been documented for barotropic vortices. For a small-Burger (flat) lens vortex, baroclinic instability ensures a sustained growth rate of sharp jumps in temperature near the critical levels of the leading unstable modes. Such results can be obtained for a background stratification which is due to temperature only and does not require the existence of salt anomalies. Aloft and beneath the vortex core, numerical simulations well reproduce the $[{ k}_{h}^{- 5/ 3} { k}_{h}^{- 2} ] $ scaling law of horizontal scales for the vertical derivative of temperature that is observed in situ inside the layering, whatever the background stratification. Such a result stems from the tracer-like behaviour of the vortex stretching component and previous studies have shown that spectra of tracer fields can be steeper than $- 1$ , namely in $- 5/ 3$ or $- 2$ , if the advection field is very compact spatially, with a $- 5/ 3$ slope corresponding to a spiral advection of the tracer. Such a scaling law could thus be of geometric origin. As for the kinetic and potential energy, the ${ k}_{h}^{- 5/ 3} $ scaling law can be reproduced numerically and is enhanced when the background stratification profile is strongly variable, involving sharp jumps in potential vorticity such as those observed in situ. This raises the possibility of another plausible mechanism leading to a $- 5/ 3$ scaling law, namely surface-quasi-geostrophic (SQG)-like dynamics, although our set-up is more complex than the idealized SQG framework. Energy and enstrophy fluxes have been diagnosed in the numerical quasi-geostrophic simulations. The results emphasize a strong production of energy in the oceanic submesoscales range and a kinetic and potential energy flux from mesoscale to submesoscales range near the critical levels. Such horizontal submesoscale production, which is correlated to the accumulation of thin vertical scales inside the layering, thus has a significant slow dynamical component, well-captured by quasi-geostrophy.

Galilean non-invariance of the shallow-water equations describing the motion of a rotating fluid implies that a homogeneous background flow modifies the dynamics of localized vortices even without the$\\beta $-effect. In particular, in a divergent quasi-geostrophic model on a$\\beta $-plane, which originates from the shallow-water model, the equation of motion in the reference frame attached to a uniform zonal background flow has the same form as in the absence of this flow, but with a modified$\\beta $-parameter depending linearly on the flow velocity$\\bar{U}$. The evolution of a singular vortex (SV) embedded in such a flow consists of two stages. In the first, quasi-linear stage, the SV motion is induced by the secondary dipole ($\\beta $-gyres) generated in the neighbourhood of the SV. During the next, nonlinear stage, the SV merges with the$\\beta $-gyre of opposite sign to form a compact vortex pair interacting with far-field Rossby waves radiated previously by the SV, while the other$\\beta $-gyre loses connection with the SV and disappears. In the absolute reference frame and with$\\beta = 0$, the SV drifts downstream and at an angle to the background flow. The SV always lags behind the background flow, with the strongest resistance during the quasi-linear stage and weakening resistance at the nonlinear stage of SV evolution. In the general case where$\\beta \\gt 0$, the SV can move both upstream (for small-to-moderate$\\bar{U} \\gt 0$) and downstream (for$\\bar{U} \\lt 0$or sufficiently large$\\bar{U} \\gt 0$). Under weak-to-moderate westward and all eastward flows the SV cyclone (anticyclone) also moves northward (southward), its meridional drift increasing with$\\bar{U}$.

In this investigation, a data-driven turbulence closure framework is introduced and deployed for the subgrid modelling of Kraichnan turbulence. The novelty of the proposed method lies in the fact that snapshots from high-fidelity numerical data are used to inform artificial neural networks for predicting the turbulence source term through localized grid-resolved information. In particular, our proposed methodology successfully establishes a map between inputs given by stencils of the vorticity and the streamfunction along with information from two well-known eddy-viscosity kernels. Through this we predict the subgrid vorticity forcing in a temporally and spatially dynamic fashion. Our study is both a priori and a posteriori in nature. In the former, we present an extensive hyper-parameter optimization analysis in addition to learning quantification through probability-density-function-based validation of subgrid predictions. In the latter, we analyse the performance of our framework for flow evolution in a classical decaying two-dimensional turbulence test case in the presence of errors related to temporal and spatial discretization. Statistical assessments in the form of angle-averaged kinetic energy spectra demonstrate the promise of the proposed methodology for subgrid quantity inference. In addition, it is also observed that some measure of a posteriori error must be considered during optimal model selection for greater accuracy. The results in this article thus represent a promising development in the formalization of a framework for generation of heuristic-free turbulence closures from data.

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.

Rotating Rayleigh–Bénard convection is investigated numerically with the use of an asymptotic model that captures the rapidly rotating, small Ekman number limit, $Ek \\rightarrow 0$. The Prandtl number ($Pr$) and the asymptotically scaled Rayleigh number ($\\widetilde {Ra} = Ra Ek^{4/3}$, where $Ra$ is the typical Rayleigh number) are varied systematically. For sufficiently vigorous convection, an inverse kinetic energy cascade leads to the formation of a pair of large-scale vortices of opposite polarity, in agreement with previous studies of rapidly rotating convection. With respect to the kinetic energy, we find a transition from convection dominated states to a state dominated by large-scale vortices at an asymptotically reduced (small-scale) Reynolds number of $\\widetilde {Re} \\approx 6$ ($\\widetilde {Re} = Re Ek^{1/3}$, where $Re$ is the Reynolds number associated with vertical flows) for all investigated values of $Pr$. The ratio of the depth-averaged kinetic energy to the kinetic energy of the convection reaches a maximum at $\\widetilde {Re} \\approx 24$, then decreases as $\\widetilde {Ra}$ is increased. This decrease in the relative kinetic energy of the large-scale vortices is associated with a decrease in the convective correlations with increasing Rayleigh number. The scaling behaviour of the convective flow speeds is studied; although a linear scaling of the form $\\widetilde {Re} \\sim \\widetilde {Ra}/Pr$ is observed over a limited range in Rayleigh number and Prandtl number, a clear departure from this scaling is observed at the highest accessible values of $\\widetilde {Ra}$. Calculation of the forces present in the governing equations shows that the ratio of the viscous force to the buoyancy force is an increasing function of $\\widetilde {Ra}$, that approaches unity over the investigated range of parameters.

Upper-ocean turbulence at scales smaller than the mesoscale is believed to exchange surface and thermocline waters, which plays an important role in both physical and biogeochemical budgets. But what energizes this submesoscale turbulence remains a topic of debate. Two mechanisms have been proposed: mesoscale-driven surface frontogenesis and baroclinic mixed-layer instabilities. The goal here is to understand the differences between the dynamics of these two mechanisms, using a simple quasi-geostrophic model. The essence of mesoscale-driven surface frontogenesis is captured by the well-known surface quasi-geostrophic model, which describes the sharpening of surface buoyancy gradients and the subsequent breakup in secondary roll-up instabilities. We formulate a similarly archetypical Eady-like model of submesoscale turbulence induced by mixed-layer instabilities. The model captures the scale and structure of this baroclinic instability in the mixed layer. A wide range of scales are energized through a turbulent inverse cascade of kinetic energy that is fuelled by the submesoscale mixed-layer instability. Major differences to mesoscale-driven surface frontogenesis are that mixed-layer instabilities energize the entire depth of the mixed layer and produce larger vertical velocities. The distribution of energy across scales and in the vertical produced by our simple model of mixed-layer instabilities compares favourably to observations of energetic wintertime submesoscale flows, suggesting that it captures the leading-order balanced dynamics of these flows. The dynamics described here in an oceanographic context have potential applications to other geophysical fluids with layers of different stratifications.

In rotating stratified flows including in the atmosphere and ocean, inertia-gravity waves (IGWs) often coexist with geostrophically balanced turbulent flows. Advection and refraction by such flows lead to wave scattering, redistributing IGW energy in the position–wavenumber phase space. We give a detailed description of this process by deriving a kinetic equation governing the evolution of the IGW phase-space energy density. The derivation relies on the smallness of the Rossby number characterising the geostrophic flow, which is treated as a random field with known statistics, makes no assumption of spatial scale separation, and neglects wave–wave interactions. It extends previous work restricted to near-inertial waves, barotropic flows or waves much shorter than the flow scales. The kinetic equation describes energy transfers that are restricted to IGWs with the same frequency, as a result of the time scale separation between waves and flow. We formulate the kinetic equation on the constant-frequency surface – a double cone in wavenumber space – using polar spherical coordinates, and we examine the form of the two scattering cross-sections involved, which quantify energy transfers between IGWs with, respectively, the same and opposite directions of vertical propagation. The kinetic equation captures both the horizontal isotropisation and the cascade of energy across scales that result from scattering. We focus our attention on the latter to assess the predictions of the kinetic equation against direct simulations of the three-dimensional Boussinesq equations, finding good agreement.