High vertical shear and dissipation in equatorial topographic wakes

Submesoscale coherent vortices (SCVs) are a ubiquitous feature of topographic wakes in the extratropical oceans. Recent studies demonstrate a mechanism wherein high vorticity bottom boundary layers (BBLs) on the slopes of the topography separate (forming shear layers), undergo instabilities, and subsequently merge in the horizontal and align in the vertical to form vertically coherent, columnar, SCVs (i.e. with low vertical shear). Background rotation is critical to the vertical alignment of unstable vortical filaments into coherent SCVs. In the tropics, however, the weakening of rotation prevents this alignment. Employing an idealized framework of steady barotropic flow past an isolated seamount in a background of constant stratification N and rotation rate f, we examine the wake structure for a range of f values spanning values from the poles to the tropics. We find a systematic increase in the interior vertical shear with decreasing f that manifests as a highly layered wake structure consisting of vertically thin, ‘pancake’ SCVs possessing a high vertical shear. A monotonic increase in the wake energy dissipation rate is concomitantly observed with decreasing f. By examining the evolution equations for the vertical shear and vertical enstrophy, we find that the interior shear generation is an advective process, with the location of peak shear generation approximately colocated with maximum energy dissipation. This leads to the inference that high wake dissipation in tropical tropographic wakes is caused by parameterized shear instabilities induced by interior advective generation of vertical shear in the near wake region.

The Drag on the Barotropic Tide due to the Generation of Baroclinic Motion

The interaction of a barotropic flow with topography generates baroclinic motion that exerts a stress on the barotropic flow. Here, explicit solutions are calculated for the spatial-mean flow (i.e., the barotropic tide) resulting from a spatially uniform but time-varying body force (i.e., astronomical forcing) acting over rough topography. This approach of prescribing the force contrasts with that of previous authors who have prescribed the barotropic flow. It is found that the topographic stress, and thus the impact on the spatial-mean flow, depend on the nature of the baroclinic motion that is generated. Two types of stress are identified: (i) a “wave drag” force associated with propagating wave motion, which extracts energy from the spatial-mean flow, and (ii) a topographic “spring” force associated with standing motion at the seafloor, including bottom-trapped internal tides and propagating low-mode internal tides, which significantly damps the time-mean kinetic energy of the spatial-mean flow but extracts no energy in the time-mean. The topographic spring force is shown to be analogous to the force exerted by a mechanical spring in a forced-dissipative harmonic oscillator. Expressions for the topographic stresses appropriate for implementation as baroclinic drag parameterizations in global models are presented.

Energetics of Seamount Wakes. Part I: Energy Exchange

Seamounts have been theorized to act as “stirring rods,” converting barotropic flow into an unsteady wake, turbulence, and diapycnal mixing. The energetics of these processes are not well understood, but they may have implications for basin-scale mixing calculations. This study presents the results of a series of simulations for idealized seamounts in steady barotropic flow, with varying degrees of stratification and rotation. The kinetic energy within each simulation domain is decomposed into the mean kinetic energy, unsteady eddy energy, and turbulent kinetic energy; evolution equations are derived for each. Within the evolution equations, energy exchange terms arise, which relate the various forms of kinetic energy and potential energy. Key exchange terms, such as the rate at which the mean flow is converted into eddy energy, are compared across the Froude–Rossby parameter space. It is shown that the conversion terms associated with mesoscale motions are a function of the Burger number, which is consistent with a quasigeostrophic flow regime. Conversely, conversion terms associated with turbulent processes scale with the product of the Froude and Rossby number. The amount of energy extracted from the mean flows suggests that wake effects may be significant for the parameter range and model assumptions studied. These results suggest that some seamounts may indeed act as oceanic stirring rods.

Energetics of Seamount Wakes. Part II: Wave Fluxes

Seamounts are thought to facilitate ocean mixing through unsteady wake processes, and through the generation of internal waves, which propagate away from the seamount and later break. The relative importance of these processes is examined for idealized, isolated seamounts (with characteristic width D and height H) in uniform barotropic flow U. A range of Coriolis parameters f and buoyancy frequencies N are used such that a broad parameter space of low Froude numbers (U/NH) and low Rossby numbers (U/fD) is considered. Results indicate that eddy processes energetically dominate the internal wave energy flux in this range of parameter space. The internal wave field is specifically examined and partitioned into steady lee waves and unsteady, wake-generated waves. It is found that the lee wave energy flux cannot be explained by existing analytical theories. A lee wave model by Smith is then extended into the low-Froude-number regime and the effect of rotation is included. While strongly stratified experiments have previously indicated that only the top U/N of an obstacle generates internal waves, the effect of rotation appears to modify this wavemaking height. Once the U/N height is revised to account for rotation, the lee wave energy flux can be reasonably accurately reproduced by the extended Smith model.

Planetary geostrophic Boussinesq dynamics: barotropic flow, baroclinic instability and forced stationary waves

<p>Motions on planetary spatial scales in the atmosphere are governed by<br>the planetary geostrophic equations. However, not much attention has<br>been paid to the interaction between the baroclinic and barotropic<br>flow within the planetary geostrophic scaling. This is the focus of<br>the present study by utilizing planetary geostrophic equations for a<br>Boussinesq fluid supplemented by an asymptotically derived evolution<br>equation for the barotropic flow. The latter is effected by meridional<br>momentum flux due to baroclinic flow and drag by the surface wind. The<br>barotropic wind on the other hand affects the baroclinic flow through<br>buoyancy advection. By relaxing towards a prescribed buoyancy profile<br>the model produces realistic major features of the zonally symmetric<br>wind and temperature fields. We show that there is considerable<br>cancelation between the barotropic and the baroclinic surface zonal<br>mean zonal wind. The linear and nonlinear model response to steady<br>diabatic zonally asymmetric forcing is investigated. The arising<br>stationary waves are interpreted in terms of analytical solutions. We<br>also study the problem of baroclinic instability on the sphere within<br>the present model.</p><p>Reference: Dolaptchiev, S. I., Achatz, U. and Th. Reitz, 2019: Planetary<br>geostrophic Boussinesq dynamics: barotropic flow, baroclinic<br>instability and forced stationary waves, Quart. J. Roy. Met. Soc., 145: 3751-3765.</p>

Phase Transition to Quadrupolar Vortices in a Spherical Model of the Energy-Enstrophy Theory — Exact Solution

A new energy-enstrophy model for the equilibrium statistical mechanics of barotropic flow on a sphere is introduced and solved exactly for phase transitions to quadrupolar vortices when the kinetic energy level is high. Unlike the Kraichnan theory, which is a Gaussian model, we substitute a microcanonical enstrophy constraint for the usual canonical one, a step which is based on sound physical principles. This yields a spherical model with zero total circulation, a microcanonical enstrophy constraint and a canonical constraint on energy, with angular momentum fixed to zero. A closed-form solution of this spherical model, obtained by the Kac – Berlin method of steepest descent, provides critical temperatures and amplitudes of the symmetry-breaking quadrupolar vortices. This model and its results differ from previous solvable models for related phenomena in the sense that they are not based on a mean-field assumption.

Submesoscale Vortical Wakes in the Lee of Topography

An idealized framework of steady barotropic flow past an isolated seamount in a background of constant stratification (with frequency N) and rotation (with Coriolis parameter f) is used to examine the formation, separation, instability of the turbulent bottom boundary layers (BBLs), and ultimately, the genesis of submesoscale coherent vortices (SCVs) in the ocean interior. The BBLs generate vertical vorticity ζ and potential vorticity q on slopes; the flow separates and spawns shear layers; barotropic and centrifugal shear instabilities form submesoscale vortical filaments and induce a high rate of local energy dissipation; the filaments organize into vortices that then horizontally merge and vertically align to form SCVs. These SCVs have O(1) Rossby numbers () and horizontal and vertical scales that are much larger than those of the separated shear layers and associated vortical filaments. Although the upstream flow is barotropic, downstream baroclinicity manifests in the wake, depending on the value of the nondimensional height , which is the ratio of the seamount height to that of the Taylor height , where L is the seamount half-width. When , SCVs span the vertical extent of the seamount itself. However, for , there is greater range of variation in the sizes of the SCVs in the wake, reflecting the wake baroclinicity caused by the topographic interaction. The aspect ratio of the wake SCVs has the scaling , instead of the quasigeostrophic scaling .

Physics and Dynamics of the Oceans

This chapter focuses on the physics and dynamics of the ocean. It describes the variability of salinity and surface temperature, as well as the vertical temperature structure of the ocean, with the thermocline separating the variable top layer from the deeper ocean. It then describes the key forces in the ocean, as well as the geostrophic balance due to the Coriolis force and density differences. It derives the equations for the change of velocity with depth, the Ekman flow. Barotropic flow and baroclinic flow are elucidated and the general circulation of the ocean, with gyres and the effect of vorticity on their structure, is shown. The thermohaline circulation of the ocean with surface flow and returning deep ocean flows is described. Next, a simple model is used to show how salinity interacts with the thermohaline flow. Finally, as an example of ocean–land interaction, the El Niño phenomenon is described.

Diffusion-rotational parameterization of eddy fluxes of potential vorticity: barotropic flow in the zonal channel

The problem of parametrization of the eddy fluxes of a potential vorticity is discussed. Traditional diffusion parameterization is complemented by the inclusion of a rotational component. For the analysis of the new scheme, a quasi-geostrophic model of the dynamics of the barotropic flow in a zonal channel with a non-uniform bottom is used. An analytical solution of the problem is found and the influence of topography on the flow disturbances is discussed. It is shown that the equation for the eddy potential enstrophy allows to relate diffusion and «rotational» coefficients.