Homogenization Of Potential Vorticity Research Articles

The Role of Surface Potential Vorticity in the Vertical Structure of Mesoscale Eddies in Wind-Driven Ocean Circulations

Abstract The vertical structure of ocean eddies is generally surface-intensified, commonly attributed to the dominant baroclinic modes arising from the boundary conditions (BCs). Conventional BC considerations mostly focus on either flat- or rough-bottom conditions. The impact of surface buoyancy anomalies—often represented by surface potential vorticity (PV) anomalies—has not been fully explored. Here, we study the role of the surface PV in setting the vertical distribution of eddy kinetic energy (EKE) in an idealized adiabatic ocean model driven by wind stress. The simulated EKE profile in the extratropical ocean tends to peak at the surface and have an e-folding depth typically smaller than half of the ocean depth. This vertical structure can be reasonably represented by a single surface quasigeostrophic (SQG) mode at the energy-containing scale resulting from the large-scale PV structure. Due to isopycnal outcropping and interior PV homogenization, the surface meridional PV gradient is substantially stronger than the interior PV gradient, yielding surface-trapped baroclinically unstable modes with horizontal scales comparable to or smaller than the deformation radius. These surface-trapped eddies then grow in size both horizontally and vertically through an inverse energy cascade up to the energy-containing scale, which dominates the vertical distribution of EKE. As for smaller horizontal scales, the EKE distribution decays faster with depth. Guided by this interpretation, an SQG-based scale-aware parameterization of the EKE profile is proposed. Preliminary offline diagnosis of a high-resolution simulation shows the proposed scheme successfully reproducing the dependence of the vertical structure of EKE on the horizontal grid resolution.

Two-dimensional turbulence above topography: Vortices and potential vorticity homogenization.

The evolution of unforced and weakly damped two-dimensional turbulence over random rough topography presents two extreme states. If the initial kinetic energy [Formula: see text] 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 [Formula: see text] 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 [Formula: see text]. [Formula: see text] is the separator between high-energy evolution and low-energy evolution.

Upper-Bound General Circulation of the Ocean: A Theoretical Exposition

This paper considers the general ocean circulation (GOC) within the thermodynamical closure of our climate theory, which aims to deduce the generic climate state from first principles. The preceding papers of this theory have reduced planetary fluids to warm/cold masses and determined their bulk properties, which provide prior constraints for the derivation of the upper-bound circulation when the potential vorticity (PV) is homogenized in moving masses. In a companion paper on the general atmosphere circulation (GAC), this upper bound is seen to reproduce the observed prevailing wind, therefore forsaking discordant explanations of the easterly trade winds and the polar jet stream. In this paper on the ocean, we again show that this upper bound may replicate broad features of the observed circulation, including a western-intensified subtropical gyre and a counter-rotating tropical gyre feeding the equatorial undercurrent. Since PV homogenization has short-circuited the wind curl, the Sverdrup dynamics does not need to be the sole progenitor of the western intensification, as commonly perceived. Together with GAC, we posit that PV homogenization provides a unifying dynamical principle of the large-scale planetary circulation, which may be interpreted as the maximum macroscopic motion extractable by microscopic stirring, within the confines of thermal differentiation.

Formation of Anticyclones above Topographic Depressions

AbstractLong-lived anticyclonic eddies (ACs) have been repeatedly observed over several North Atlantic basins characterized by bowl-like topographic depressions. Motivated by these previous findings, the authors conduct numerical simulations of the spindown of eddies initialized in idealized topographic bowls. In experiments with one or two isopycnal layers, it is found that a bowl-trapped AC is an emergent circulation pattern under a wide range of parameters. The trapped AC, often formed by repeated mergers of ACs over the bowl interior, is characterized by anomalously low potential vorticity (PV). Several PV segregation mechanisms that can contribute to the AC formation are examined. In one-layer experiments, the dynamics of the AC are largely determined by a nonlinearity parameter ϵ that quantifies the vorticity of the AC relative to the bowl’s topographic PV gradient. The AC is trapped in the bowl for low , but for moderate values () partial PV segregation allows the AC to reside at finite distances from the center of the bowl. For higher , eddies freely cross the topography and the AC is not confined to the bowl. These regimes are characterized across a suite of model experiments using ϵ and a PV homogenization parameter. Two-layer experiments show that the trapped AC can be top or bottom intensified, as determined by the domain-mean initial vertical energy distribution. These findings contrast with previous theories of mesoscale turbulence over topography that predict the formation of a prograde slope current, but do not predict a trapped AC.

Upgradient and Downgradient Potential Vorticity Fluxes Produced by Forced Rossby Waves. Part I: Basic Experiments

AbstractThe mean flow and potential vorticity (PV) flux produced by Rossby waves are examined, particularly by focusing on the effects of stratification and nonlinearity on upgradient and downgradient PV fluxes. Rossby waves are excited by an external forcing confined near the surface and produce a northward (upgradient) PV flux in the surface layer. While the meridional PV flux is considerably weak in the deep layer in the weakly nonlinear case, the southward (downgradient) PV flux is produced as nonlinearity increases. In both the surface and deep layers, the distribution of the PV flux and mean flow is qualitatively similar to that in recirculation gyres obtained in an eddy-resolving model of the wind-driven circulation. A perturbation analysis shows that the primary and harmonic waves are excited by external forcing and wave–wave interaction between the primary waves, respectively. The meridional PV fluxes in the surface and deep layers are mostly produced by the primary and harmonic waves, respectively. The southward PV flux in the deep layer is produced by the interaction between the barotropic harmonic wave and the first baroclinic component of nonlinear forcing by the primary waves. The irreversible processes that are implicitly assumed in the PV homogenization theory (Rhines and Young) do not substantially affect the southward PV flux. The qualitative features of the PV flux remain unchanged even when nonlinearity is increased beyond the range in which the perturbation theory is exactly applicable.

Formation and evolution of target patterns in Cahn-Hilliard flows

We study the evolution of the concentration field in a single eddy in the two-dimensional (2D) Cahn-Hilliard system to better understand scalar mixing processes in that system. This study extends investigations of the classic studies of flux expulsion in 2D magnetohydrodynamics and homogenization of potential vorticity in 2D fluids. Simulation results show that there are three stages in the evolution: (A) formation of a "jelly roll" pattern, for which the concentration field is constant along spirals; (B) a change in isoconcentration contour topology; and (C) formation of a target pattern, for which the isoconcentration contours follow concentric annuli. In the final target pattern stage, the isoconcentration bands align with stream lines. The results indicate that the target pattern is a metastable state. The band merger process continues on a time scale exponentially long relative to the eddy turnover time. The band merger process resembles step merger in drift-ZF staircases; this is characteristic of the long-time evolution of phase-separated patterns described by the Cahn-Hilliard equation.

Nonlinear baroclinic equilibration at finite supercriticality

The nonlinear equilibration of finite amplitude baroclinic waves in Phillips two-layer model is investigated at finite supercriticality. The aims are to quantify the robustness and relevance of the nonlinear theory of Warn, Gauthier and Pedlosky (WGP) for the evolution of the developing baroclinic wave, and to assess the tightness of pseudomomentum and improved pseudoenergy bounds for disturbance amplitude and energy. A high-resolution numerical model is used to perform a parameter sweep in (β, W)-space, where β is the inverse criticality of the initial flow, and W is the ratio of the channel width to the (internal) Rossby radius. At low supercriticalities, the main predictions of WGP are found to be accurate at short times, but at long times the fully nonlinear results are found to diverge from WGP's solution. The mechanism for equilibration involves the elimination of the lower layer potential vorticity (PV) gradient, but as the supercriticality increases this is achieved by the roll-up of a train of opposite-signed vortices, rather than by coarse-grain PV homogenization as in WGP. Peak wave amplitudes are typically ≈90% of the maximum attainable under the pseudomomentum bound. New formulae are given for the pseudoenergy bound on disturbance energy which, unlike the WGP solution and the pseudomomentum bound, have non-trivial dependence on W. A detailed assessment is made of the extent to which these bounds are attained.

Baroclinic Adjustment in an Atmosphere–Ocean Thermally Coupled Model: The Role of the Boundary Layer Processes

Abstract Baroclinic eddy equilibration and the roles of different boundary layer processes in limiting the baroclinic adjustment are studied using an atmosphere–ocean thermally coupled model. Boundary layer processes not only affect the dynamical constraint of the midlatitude baroclinic eddy equilibration but also are important components in the underlying surface energy budget. The authors' study shows that baroclinic eddies, with the strong mixing of the surface air temperature, compete against the fast boundary layer thermal damping and enhance the meridional variation of surface sensible heat flux, acting to reduce the meridional gradient of the surface temperature. Nevertheless, the requirement of the surface energy balance indicates that strong surface baroclinicity is always maintained in response to the meridionally varying solar radiation. With the strong surface baroclinicity and the boundary layer processes, the homogenized potential vorticity (PV) suggested in the baroclinic adjustment are never observed near the surface or in the boundary layer. Although different boundary layer processes affect baroclinic eddy equilibration differently with more dynamical feedbacks and time scales included in the coupled system, their influence in limiting the PV homogenization is more uniform compared with the previous uncoupled runs. The boundary layer PV structure is more determined by the strength of the boundary layer damping than the surface baroclinicity. Stronger boundary layer processes always prevent the lower-level PV homogenization more efficiently. Above the boundary layer, a relatively robust PV structure with homogenized PV around 600–800 hPa is obtained in all of the simulations. The detailed mechanisms through which different boundary layer processes affect the equilibration of the coupled system are discussed in this study.

Equatorial Zonal Jet Formation through the Barotropic Instability of Low-Frequency Mixed Rossby–Gravity Waves, Equilibration by Inertial Instability, and Transition to Superrotation

Abstract Depth-dependent barotropic instability of short mixed Rossby–gravity (MRG) waves is proposed as a mechanism for the formation of equatorial zonal jets. High-resolution primitive equation simulations show that a single MRG wave of very short zonal wavelength and small to moderate amplitude is unstable and leads to the development of a largely barotropic, zonally symmetric flow, featuring a westward jet at the equator and extra-equatorial jets alternating in direction with latitude. At higher but still moderate amplitude, westward flow still prevails at the equator at depths of maximum horizontal velocity amplitude in the initial wave, but the long-term equilibrated state can also feature eastward “superrotating” jets at the equator near the depths of zero horizontal velocity in the initial wave. The formation of the superrotating jets in the simulations is found to be sensitive to the inclusion of the nontraditional Coriolis force in the equations of motion. A linear theory is used to demonstrate the existence of exponentially growing horizontally nondivergent perturbations with a dominant zonally symmetric zonal velocity component. An argument for the sense and positioning of the jets relative to the equator is given in terms of inertial instability and the meridional mixing of planetary vorticity by the small zonal-scale components of the linearly unstable modes. In the long time evolution of the flow, if the amplitude of the westward equatorial jet becomes too great, zonally symmetric inertial instability limits the growth of the jets, and inertial adjustment leads to the homogenization of potential vorticity in latitude and depth around the equator.

The Role of Boundary Layer Processes in Limiting PV Homogenization

Abstract A β-plane multilevel quasigeostrophic channel model with interactive static stability and a simplified parameterization of atmospheric boundary layer physics is used to study the role of different boundary layer processes in eddy equilibration and their relative effect in maintaining the strong boundary layer potential vorticity (PV) gradient. The model results show that vertical thermal diffusion, along with the surface heat exchange, is primarily responsible for limiting PV homogenization by baroclinic eddies in the boundary layer. Under fixed SST boundary conditions, these two processes act as the source of the mean flow baroclinicity in the lower levels and result in stronger eddy heat fluxes. Reducing surface friction alone does not result in efficient elimination of the boundary layer PV gradient, but the equilibrium state temperature gradient is still largely influenced by surface friction and its response to changes in surface friction is not monotonic. In the regime of strong surface friction, with reduced poleward eddy heat flux, a strong temperature gradient is still retained. When the surface friction is sufficiently weak along with the stronger zonal wind, the critical level at the center of the jet drops below the surface. As a result, in the lower levels, the eddy heat flux forcing on the mean flow moves away from the center of the jet and the equilibrium state varies only slightly with the strength of the vertical momentum diffusion in the boundary layer.

Deep western boundary current and southern frontal systems of the Antarctic Circumpolar Current southeast of the Kerguelen Plateau

The structures of the deep western boundary current and southern frontal systems of the Antarctic Circumpolar Current (ACC) southeast of the Kerguelen Plateau were investigated. Top‐to‐bottom hydrography was conducted around 58°S, and two underway observations were made around 56°S and 61°S. Middepth floats were also used to describe the subsurface flow and temperature fields. Along the 58°S section, a cold dome was located just off the plateau, and northwestward and southeastward flows were found in its western and eastern sides. The observed flow structure was highly barotropic. The maximum volume transport of the bottom water was 22 Sv equatorward, including a possible cyclonic recirculation of 10 Sv. The transport of the deep recirculation can be explained by the potential vorticity homogenization in the closed isopachs formed by the bottom slope off the plateau and the overlying ACC flowing southeastward through the Fawn Trough. The spatial scale of the cold dome, which showed a signature of the southern ACC front, was around 100 km at 58°S, and the similar cold domes were also found around 56°S and 61°S, along with a signature of the southern boundary of ACC around 61°S. The sharp excursion of the southern ACC front was also confirmed by the middepth float trajectories and the distribution of temperature maximum. This indicates the equatorward excursion up to 54°S, with the warmer ACC water just east of the southern frontal systems.

A Barotropic Model of the Angular Momentum–Conserving Potential Vorticity Staircase in Spherical Geometry

Abstract An idealized analytical model of the barotropic potential vorticity (PV) staircase is constructed, constrained by global conservation of absolute angular momentum, perfect homogenization of PV in mixing zones between (prograde) westerly jets, and the requirement of barotropic stability. An imposed functional relationship is also assumed between jet speed and latitudinal separation using a multiple of the “dynamical Rossby wave” Rhines scale inferred from the strength of westerly jets. The relative simplicity of the barotropic system provides a simple relation between absolute angular momentum and PV (or absolute vorticity). A family of solutions comprising an arbitrary number of jets is constructed and is used to illustrate the restriction of jet spacing and strength imposed by the constraints of global conservation of angular momentum and barotropic stability. Asymptotic analysis of the theoretical solution indicates a limiting ratio of jet spacing to the dynamical Rhines scale equal to the square root of 6, meaning that westerly jets are spaced farther apart than predicted by the dynamical Rhines scale. It is inferred that an alternative “geometrical” Rhines scale for jet spacing can be obtained from conservation of absolute angular momentum on the sphere if the strength of zonal jets is known from other considerations. Numerical simulations of the full (nonaxisymmetric) equations reveal a pattern of zonal jet evolution that is consistent with our construction of ideal PV staircases in spherical geometry (which can be considered as limiting cases), as well as with the asymptotic analysis of a geometrical Rhines scale. The evolution of the PV staircase originating from an upscale cascade of energy in the barotropic model is therefore seen to depend on conservation of energy (for the strength of jets) and conservation of absolute angular momentum (for the spacing and number of jets). Further analysis of the numerical results confirms a “Taylor identity” relating the flux of eddy potential vorticity to mean-flow acceleration. Eddy fluxes are responsible for the occasional transitions between mode number as well as for maintaining the sharp westerly jets against small-scale dissipation. Suggestions are made for extending the theoretical model to PV staircases that are asymmetric between hemispheres or with latitudinal variation of amplitude, as modeled in the shallow-water system.

The turbulent equilibration of an unstable baroclinic jet

The evolution of an unstable baroclinic jet, subject to a small perturbation, is examined numerically in a quasi-geostrophic two-layer β-channel model. After a period of initial wave growth, wave breaking leads to turbulence within each layer, and to the eventual equilibration of the flow. The equilibrated flow must satisfy certain dynamical constraints: total momentum is conserved, the total energy is bounded and the flow must be realizable via some area-preserving (diffusive) rearrangement of the initial potential vorticity field in each layer. A theory is introduced that predicts the equilibrated flow in terms of the initial flow parameters. The idea is that the equilibrated state minimizes available potential energy, subject to the constraints on total momentum and total energy, and the further ‘kinematic’ constraint that the potential vorticity changes through a process of complete homogenization within well-delineated regions in each layer. Within a large region of parameter space, the theory accurately predicts the cross-channel structure and strength of the equilibrated jet, the regions where potential vorticity mixing takes place, and total eddy mass (temperature) fluxes. Results are compared with predictions from a maximum-entropy theory that allows for more general rearrangements of the initial potential vorticity field, subject to the known dynamical constraints. The maximum-entropy theory predicts that significantly more available potential energy is released than is observed in the simulations, and that an unphysical ‘exchange’ of bands of fluid will occur across the channel in the lower layer. The kinematic constraint of piecewise potential vorticity homogenization is therefore important in limiting the ‘efficiency’ of release of available potential energy in unstable baroclinic flows. For a typical initial flow, it is demonstrated that if the dynamical constraints alone are considered, then over twice as much potential energy is available for release compared to that actually released in the simulations.

Temperature Phase Tilt in Unstable Baroclinic Waves

Abstract A general property on the phase relation in linear baroclinic instability is proved analytically: in a potential vorticity homogenization regime, the complex geometry of the quasigeostrophic equations determines that the phase lines of temperature and pressure disturbances tilt with height in opposite directions.

The role of westward intensification in potential vorticity homogenization

The role of westward intensification in potential vorticity homogenization

Transformed Eulerian-Mean Theory. Part II: Potential Vorticity Homogenization and the Equilibrium of a Wind- and Buoyancy-Driven Zonal Flow

Abstract The equilibrium of a modeled wind- and buoyancy-driven, baroclinically unstable, flow is analyzed using the transformed Eulerian-mean (TEM) approach described in Part I. Within the near-adiabatic interior of the flow, Ertel potential vorticity is homogenized along mean isopycnals—a finding readily explained using TEM theory, given the geometry of the domain. The equilibrium, zonal-mean buoyancy structure at the surface is determined entirely by a balance between imposed surface fluxes and residual mean and eddy buoyancy transport within a “surface diabatic layer.” Balance between these same processes and the wind stress determines the stratification, and hence potential vorticity, immediately below this layer. Ertel potential vorticity homogenization below then determines the mean buoyancy structure everywhere. Accordingly, the equilibrium structure of this flow can be described—and quantitatively reproduced—from knowledge of the eddy mixing rates within the surface diabatic zone and the depth of this zone, together with potential vorticity homogenization beneath. These results emphasize the need to include near-surface buoyancy transport, as well as interior PV transport, in eddy parameterization schemes. They also imply that, in more realistic models, the surface buoyancy balances may be impacted by processes in remote locations that allow diapycnal flow.

Asymmetry of an Equilibrated Gulf Stream–Type Jet over Topographic Slope

The symmetry properties of the Gulf Stream–type jet equilibrated over topographic slope are investigated in a series of idealized numerical experiments. A baroclinically unstable zonal jet equilibrates over a sloping bottom through the process of potential vorticity (PV) homogenization underneath the main thermocline by the bottom-intensified eddy activity associated with the stream meandering. Potential vorticity homogenization underneath the main thermocline leads to formation of recirculation gyres on both sides of the jet. The magnitude of the northern recirculation gyre, as measured by its westward transport, is larger than the magnitude of the southern recirculation gyre. This asymmetry in recirculations is shown to be the result of an asymmetric PV mixing underneath the thermocline produced by an asymmetric jet. In particular, the lateral shift of the velocity maximum near the surface relative to the velocity maximum at depth is shown to be responsible for the asymmetry. The results are related to the Gulf Stream data between 73° and 65°W.

Why Are Eddy Fluxes of Potential Vorticity Difficult to Parameterize?

Abstract Eddy fluxes systematically affect the larger-scale, time-mean state, but their local behavior is difficult to parameterize. To understand how eddy fluxes of potential vorticity (PV) are controlled, the enstrophy budget is diagnosed for a five-layer, 1/16°, eddy-resolving, isopycnic model of a wind-driven, flat-bottom basin. The direction of the eddy flux across the mean PV contours is controlled by the Lagrangian evolution of enstrophy, including contributions from the temporal change and mean and eddy advection, as well as dissipation of enstrophy. During the spinup, an overall increase in enstrophy is consistent with eddy fluxes being directed downgradient on average and homogenization of PV within intermediate layers. Enstrophy becomes largest along the flanks of the gyre, where PV gradients are large, and becomes smallest in the interior. At a statistically steady state, there is a reversing pattern of up- and downgradient eddy PV fluxes, which are locally controlled by the advection of enstr...

The Role of Sloping Sidewalls in Forming Potential Vorticity Contrasts in the Ocean Interior

The role of sidewalls in determining the interior distribution of potential vorticity (PV) is investigated using eddy-resolving isopycnic experiments. The layer model is integrated at 1 /8 resolution for a wind-driven double gyre with either vertical or sloping sidewalls. If there are vertical sidewalls, eddy stirring leads to PV homogenization within unforced, interior density layers. If there are sloping sidewalls, frictional torques lead to bands of low and high PV being formed along the western boundary of the subpolar and subtropical gyres, respectively. These regions of low and high PV are transferred into the interior by a separated jet at the intergyre boundary. Over a limited domain, this injection of the PV contrast can prevent eddy homogenization from occurring. However, over a larger-scale domain, eddies provide a downgradient transfer of PV, reducing the PV contrast downstream along the jet and enabling homogenization to occur for intermediate layers within the basin interior. Diabatic mixing along the slope can introduce low PV for intermediate layers and even mask the frictional contributions.

Barotropic background currents in the Gulf of California

A field of barotropic background currents (BC) in the Gulf of California is constructed numerically. The definition of BC used here is based on suppositions about (i) the horizontal homogenization of potential vorticity in large-scale gyres, and (ii) the tendency of a current structure to evolve to a state with a minimum of system mechanical energy. In the barotropic case here, the numerical algorithm is reduced to the solution of the Dirichlet boundary problems for planetary, topographic and flowing integral streamfunction components separately. The results obtained are compared to observations and the results of other numerical models. The calculated barotropic BC realistically describe the main features of the circulation in the southern part of the Gulf of California, including its mouth. Cyclonic movement in the deep-water and anticyclonic movement near the coasts are observed. The numerical results for the central and northern parts of the gulf, where seasonal climatological differences are pronounced, coincide with observations in the upper layer only during the winter season; i.e., the circulation here is anticyclonic. Thus, there are reasons to suggest that mainly planetary-topographic forcing, especially in the southern part, forms the circulation in the lower layer. The seasonal variability of currents in the upper layer caused by non-stationary effects such as wind, heat flux and baroclinicity, is apparently in phase with these mechanisms during winter, and in antiphase during summer, particularly for the central and northern gulf.