Point vortex dynamics for coupled surface/interior QG and propagating heton clusters in models for ocean convection

Abstract

The dynamic behavior of baroclinic point vortices in two-layer quasi-geostrophic flow provides a compact model for studying the transport of heat in a variety of geophysical flows including recent heton models for open ocean convection as a response to spatially localized intense surface cooling. In such heton models, the exchange of heat with the region external to the compact cooling region reaches a statistical equilibrium through the propagation of tilted heton clusters. Such tilted heton clusters are aggregates of cyclonic vortices in the upper layer and anti-cyclonic vortices in the lower layer which collectively propagate almost as an elementary tilted heton pair even though the individual vortices undergo shifts in their relative locations. One main result in this paper is a mathematical theorem demonstrating the existence of large families of long-lived propagating heton clusters for the two-layer model in a fashion compatible to a remarkable degree with the earlier numerical simulations. Two-layer quasi-geostrophic flow is an idealization of coupled surface/interior quasi-geostrophic flow. The second family of results in this paper involves the systematic development of Hamiltonian point vortex dynamics for coupled surface/interior QG with an emphasis on propagating solutions that transport heat. These are novel vortex systems of mixed species where surface heat particles interact with quasi-geostrophic point vortices. The variety of elementary two-vortex exact solutions that transport heat include two surface heat particles of opposite strength, tilted pairs of a surface heat particle coupled to an interior vortex of opposite strength and two interior tilted vortices of opposite strength at different depths. The propagation speeds of the tilted elementary hetons in the coupled surface/interior QG model are compared and contrasted with those in the simpler two-layer heton models. Finally, mathematical theorems are presented for the existence of large families of propagating long-lived tilted heton clusters for point vortex solutions in coupled surface/interior QG flow.     

Experiments on instability of columnar vortex pairs in rotating fluid

We present results from a new series of experiments on the geophysically important issue of the instability of anticyclonic columnar vortices in a rotating fluid in circumstances such that the Rossby number exceeds unity. The vortex pair consisting of a cyclonic and an anticyclonic vortex is induced by a rotating flap in a fluid which is itself initially in a state of solid-body rotation. The anticyclonic vortex is then subject to either centrifugal or elliptical instability, depending on whether its initial ellipticity is small or large, while the cyclone always remains stable. The experimental results demonstrate that the perturbations due to centrifugal instability have a typical form of toroidal vortices of alternating sign (rib vortices). The perturbations due to elliptical instability are of the form of sinuous deformation of the vortex filament in the plane of maximal stretching which corresponds to the plane of symmetry for the vortex pair. The initial perturbations in both cases are characterized by a definite wave number in the vertical direction. The characteristics of the unstable anticyclone are determined by the main nondimensional parameter of the flow - the Rossby number. The appearance of both centrifugal and elliptical instabilities are in accord with the predictions of theoretical criteria for these cases.     

Spontaneous Emission of Gravity Waves by Interacting Vortex Dipoles in a Stratified Fluid: Laboratory Experiments

Results from a new series of experiments on the geophysically important issue of spontaneous emission of internal gravity waves during unsteady interactions of vortical structures are presented. Vortex dipoles are a common element of a quasi-two-dimensional turbulent flow. Vortex dipoles perform translational motion and can collide with other vortices. During collision events the flow is unsteady and unbalanced and a further adjustment process associated with these events can therefore result in the spontaneous emission of gravity waves. Our laboratory experiments demonstrate that gravity waves are emitted when two translating vortex dipoles interact (collide) in a layered fluid, in accord with the current theoretical results. The emission was evident both in a two-layer system and in a fluid with a linear distribution of density with depth. The waves were generated during the period of deceleration of the secondary dipoles which constitute a vortex quadrupole emerging immediately after the collision of the primary dipoles.     

Linear theory of the response of a two layer ocean to a moving hurricane‡

Abstract

This paper describes the linear response of an inviscid two‐layer model of a deep ocean on an f‐plane to a hurricane translating across the surface at constant speed. The forcing is a localized, radially‐symmetric pattern of positive wind stress curl and negative pressure anomaly. Only the steady state response is considered. The principal result is the identification of an internal wake in the lee of the storm, present when the translation speed of the storm exceeds the baroclinic long wave speed. The amplitude of the wake depends on the length of time over which the stress is experienced at a given point. The angle of the wedge filled by the wake is small, an effect due to the fact that the scale of a hurricane is typically larger than the baroclinic radius of deformation. After the wake disperses, a geostrophically balanced baroclinic ridge remains along the storm track.     

Successive formation of planetary lenses in an intermediate layer

Abstract

We study the formation of lenses of the ocean's intermediate water using a 2.5-layerβ-plane primitive equation model with localized injection of water mass. For the injecting rate of 1.0 Sv, we have observed that strong vortices are shed regularly. These vortices propagate westward much faster than the second baroclinic long Rossby wave. They are totally isolated from each other and show strong baroclinicity as well. Moreover, they remain stable over a sufficiently long period of time. Regular formation of such strong vortices in the intermediate layer has not been reported previously. The translation speed is explained using the Euler's momentum integral theorem for the nonlinear baroclinic vortex on the β-plane. We have demonstrated that coupling between the primary motion in the intermediate layer and the secondary motion in the upper layer with a meridional shift is crucial to the fast westward translation of the intense vortices. A simple dispersion formula relating the zonal translation speed with the vortex radius is also derived under the assumption of quasi-geostrophy. It has turned out that the analytical relation explains the numerical results surprisingly well despite the limitation of its derivation.     

Baroclinic nonlinear exchanges of energy and potential enstrophy in the quasi-geostrophic two-layer model

Abstract

Merilees and Warn's (1975) nonlinear interaction analysis of two-dimensional nondivergent flow is extended to examine the quasi-geostrophic two-layer model. Two sets of triads exist in this model (Salmon, 1978). The purely barotropic triads are the same as the triads examined by Merilees and Warn. Baroclinic-barotropic triads are found to exchange more energy or potential enstrophy with smaller or larger scales depending on the scale of motion as compared with the internal Rossby deformation radius and the relative wavenumber position of baroclinic and barotropic components.     

Influence of condensation and latent heat release upon barotropic and baroclinic instabilities of vortices in a rotating shallow water f-plane model

Analysis of the influence of condensation and related latent heat release upon developing barotropic and baroclinic instabilities of large-scale low Rossby-number shielded vortices on the f-plane is performed within the moist-convective rotating shallow water model, in its barotropic (one-layer) and baroclinic (two-layer) versions. Numerical simulations with a high-resolution well-balanced finite-volume code, using a relaxation parameterisation for condensation, are made. Evolution of the instability in four different environments, with humidity (i) behaving as passive scalar, (ii) subject to condensation beyond a saturation threshold, (iii) subject to condensation and evaporation, with three different parameterisations of the latter, are inter-compared. The simulations are initialised with unstable modes determined from the detailed linear stability analysis in the “dry” version of the model. In a configuration corresponding to low-level mid-latitude atmospheric vortices, it is shown that the known scenario of evolution of barotropically unstable vortices, consisting in formation of a pair of dipoles (dipolar breakdown) is substantially modified by condensation and related moist convection, especially in the presence of surface evaporation. No enhancement of the instability due to precipitation was detected in this case. Cyclone-anticyclone asymmetry with respect to sensitivity to the moist effects is evidenced. It is shown that inertia-gravity wave emission during the vortex evolution is enhanced by the moist effects. In the baroclinic configuration corresponding to idealised cut-off lows in the atmosphere, it is shown that the azimuthal structure of the leading unstable mode is sensitive to the details of stratification. Scenarios of evolution are completely different for different azimuthal structures, one leading to dipolar breaking, and another to tripole formation. The effects of moisture considerably enhance the perturbations in the lower layer, especially in the tripole formation scenario.     

Baroclinic and topographic influences on the transport in western boundary currents

Abstract

One of the central unsolved theoretical problems of the large scale ocean circulation is concerned with explaining the very large transports measured in western boundary currents such as the Gulf Stream and the Kuroshio. The only theory up to now that can explain the size of these transports is that of non-linear recirculation in which the advective terms in the momentum equations became important near the western boundary. In this paper an alternative explanation is suggested. When bottom topography and baroclinic effects are included in a wind-driven ocean model it is shown that the western boundary current can have a transport larger than that predicted from the wind stress distribution even when the nonlinear advective terms are ignored. The explanation lies in the presence of pressure torques associated with bottom topography which can contribute to the vorticity balance in the same sense as the wind stress curl.

Three numerical experiments have been carried out to explore the nature of this process using a three dimensional numerical model. The first calculation is done for a baroclinic ocean of constant depth, the second for a homogeneous ocean with an idealized continental slope topography, and the third for a baroclinic ocean with the same continental slope topography. The nature of the vorticity balance and of the circulation around closed paths is examined in each case, and it is shown that bottom pressure torques lead to enhanced transport in the western boundary current only for the baroclinic case with variable depth.     

On the mean atmospheric circulation over Antarctica

Abstract

The south-easterly surface flow down the slopes of Antarctica induces a transfer of westerly angular momentum to the atmosphere, which must be removed from the Antarctic domain by atmospheric transports. It is suggested that synoptic eddies protruding from the northern baroclinic zone into the polar regions are modified by the topography such that they are able to perform these meridional transports. A simple linear two-layer model of the axisymmetric circulation of Antarctica is presented where the eddy effects are incorporated via a K-ansatz. It is shown that qualitatively realistic mean flow patterns can be obtained with this model. The limitations of this approach are exposed.     

On the nonlinear evolution of columnar vortices in a rotating environment

We examine the three-dimensional, nonlinear evolution of columnar vortices in a rotating environment. As the initial vorticity distribution, a wavetrain of finite amplitude Kelvin-Helmholtz vortices in shear is employed. Through direct numerical simulation of the Navier-Stokes equations we seek to better understand the process of maturation of the various three-dimensional modes of instability to which such vortical flows are subject, especially those which exist as a consequence of the action of the Coriolis force. In the absence of rotational influence, we thereby demonstrate that the nonlinear evolution of columnar vortices is most strongly controlled by one or the other of two mechanisms. One mechanism of instability is identifiable as a so-called elliptical instability, which promotes the initial bending of vortex tubes in a sinusoidal fashion, while the other is a hyperbolic mode, which is responsible for the development of streamwise vortex streaks in the "braids" between adjacent vortex cores. In the rotating case, anticyclonic vortices are strongly destabilized by weak background rotation, while rapid rotation stabilizes both the cyclones and anticyclones. The strong anticyclones are subject to two distinct forms of instability, namely a Coriolis force modified elliptical instability and an inertial (centrifugal) instability. The former instability is very similar to the nonrotating form of the elliptical instability as it promotes bending of vortex tubes, while the latter instability grows on the edge of the vortex core and generates streaks of vorticity, which surround the vortex core itself. These results of direct numerical simulation fully verify the results of previous linear stability analyses. Taken together, they provide a simple explanation for the broken symmetry that is often observed to be characteristic of the von Karman vortex streets that develop in the atmospheric lee of oceanic islands.     

The Theory of the Beta-Plane Baroclinic Topographic Modons

Form-preserving, uniformly translating, horizontally localized solutions (modons) are considered within the framework of nondissipative quasi-geostrophic dynamics for a two-layer model with meridionally sloping bottom. A general classification of the beta-plane baroclinic topographic modons ( g -BTMs) is given, and three distinct domains are shown to exist in the plane of the parameters. The first domain corresponds to the regular modons with the translation speed outside the range of the phase speeds of linear waves. In the second domain, modons cannot exist: only non-localized solutions are permissible here. The third domain contains both linear periodic waves and the so-called anomalous modons traveling without resonant radiation. Exact modon solutions with piecewise linear relation between the potential vorticity and streamfunction are found and analyzed. Special attention is given to the smooth regular dipole-plus-rider solutions (anomalous modons cannot carry a smooth axisymmetric rider). As distinct from their flat-bottom analogs, g -BTMs may have nonzero total angular momentum. This feature combined with the ability of g -BTMs to bear smooth riders of arbitrary amplitude provides the existence of almost monopolar (in both layers) stationary vortices.     

Vortex merger in surface quasi-geostrophy

The merger of two identical surface temperature vortices is studied in the surface quasi-geostrophic model. The motivation for this study is the observation of the merger of submesoscale vortices in the ocean. Firstly, the interaction between two point vortices, in the absence or in the presence of an external deformation field, is investigated. The rotation rate of the vortices, their stationary positions and the stability of these positions are determined. Then, a numerical model provides the steady states of two finite-area, constant-temperature, vortices. Such states are less deformed than their counterparts in two-dimensional incompressible flows. Finally, numerical simulations of the nonlinear surface quasi-geostrophic equations are used to investigate the finite-time evolution of initially identical and symmetric, constant temperature vortices. The critical merger distance is obtained and the deformation of the vortices before or after merger is determined. The addition of external deformation is shown to favor or to oppose merger depending on the orientation of the vortex pair with respect to the strain axes. An explanation for this observation is proposed. Conclusions are drawn towards an application of this study to oceanic vortices.     

The interaction of two co-rotating quasi-geostrophic vortices in the vicinity of a surface buoyancy filament

In this paper, we investigate the interaction between two like-signed quasi-geostrophic uniform potential vorticity internal vortices in the vicinity of a surface buoyancy anomaly filament in a three dimensional, stably stratified and rapidly rotating fluid. The surface buoyancy distribution locally modifies the pressure fields and generates a shear flow. We start the study by first considering the effects of a uniform linear horizontal shear on the binary vortex interaction. We confirm that a cooperative shear facilitates the merger of a pair of vortices while an adverse shear has the opposite effect. We next investigate the binary vortex interaction in the vicinity of the surface buoyancy filament explicitly. Here, not only the filament generates a shear flow, but it also responds dynamically to the forcing by the vortex pair. The filament destabilises and forms buoyancy billows at the surface. These billows interact with the internal vortices. In particular, a surface billow may pair with one of the internal vortices. In such cases, the like-signed internal vortex pair may separate if they are initially moderately distant from each other.     

Formation of strong stationary vortex turbulence in the terrestrial magnetosheath

The microscopic consequences of the presence of nonlinear vortex structures in the near-Earth plasma dispersive medium are studied in this work. In dispersive media, strongly localized vortex structures contain trapped particles, cause pronounced density fluctuations, and intensify transfer processes, mixing in a medium; i.e., they can form strong vortex turbulence. Turbulence is represented as a gas in the ensemble of strongly localized (therefore, weakly interacting) identical vortices composing the ground state. Vortices with different amplitudes are randomly located in space (since they interact with one another) and are described statistically. It is assumed that the steady turbulent state is formed through a balance of mutually competing effects: spontaneous generation of vortices due to nonlinear steepening of the disturbance front, ^noise transfer to small scales, and collisional or collisionless damping of disturbances in the HF region. Noise scaling in the inertial interval takes place since structures merge during their collision. A magnetized plasma medium in the magnetosheath is considered. A new type of turbulent fluctuation spectra with respect to wavenumbers k ^−8/3, which is in satisfactory agreement with satellite observations in space plasma, has been determined. The medium particle diffusion on an ensemble of vortices has also been studied. It has been established that the interaction between structures themselves and between structures and medium particles causes anomalous diffusion in the medium. The effective diffusion coefficient square roothly depends on the noise stationary level.     

Transport through Prince William Sound: numerical study in a nowcast/forecast system

Using 1-year simulated data from extended Prince William Sound (PWS) nowcast/forecast system, both barotropic and baroclinic transports through two-strait, semi-enclosed PWS are examined. With major tidal constituents removed, hourly time series of volume transports through two straits are significantly correlated with net transport well balanced by the time rate of change of the PWS spatial-mean sea level. A transition frequency band occurs within the coherence function of hourly volume transports, which is characterized by a nearly 180° phase shift between low-frequency (>30 h) and high-frequency (<6 h) bands. The transition band is implicitly related to the horizontally divergent and horizontally non-divergent flows inside the Sound. Further investigation of monthly and annual mean volume transports indicates strong seasonal variability of flows through two straits. On the other hand, baroclinic transport through PWS demonstrates the transition between a two-layered flow structure during the wintertime and a well-defined three-layered structure, i.e., inflow in both the surface and bottom layer with outflow in the intermediate layer, in the remainder of the year. This three-layer exchange flow is determined to be mainly buoyancy-driven, geostrophic flow, and thus largely affected by seasonal variability of buoyancy over the shelf and PWS.     

Vortex pairs and power station cooling water

Cooling water discharged from power stations in the U.K. is frequently released from an outlet in an estuary or the sea. The warm water forms a thermal plume which is slightly buoyant and which spreads horizontally over the water surface while mixing vertically downwards with the cooler ambient water. In this paper, the possibility of vortex pair production at the cooling water outlet is considered as a mechanism contributing to this spreading of the warm water.The motion of a vortex pair contained between two rigid plane boundaries is an idealization of the flow between the water surface and the sea bed. The resulting motion is calculated from potential theory and viscous effects are neglected. The problem of deciding what strength to assign the vortices is discussed and specific consideration of shear and buoyancy at the outlet is detailed. It is observed that bifurcation of the vortex pair is determined by the initial position of the vortices and is unlikely to occur in conditions relevant to U.K. power station discharges.It is calculated that, in the absence of turbulence, the motion of such vortex pairs would result in horizontal spreading of the warm water which is greater than that observed at site surveys. It is concluded that turbulence in the ambient receiving water is sufficient to destroy vortices produced by the discharge during the early stages of the plume development.     

Hydrodynamical Modeling Of Oceanic Vortices

Mesoscale coherent vortices are numerous in the ocean.Though they possess various structures in temperature and salinity,they are all long-lived, fairly intense and mostly circular. Thephysical variable which best describes the rotation and the density anomaly associated with coherent vortices is potential vorticity. It is diagnostically related to velocity and pressure, when the vortex is stationary. Stationary vortices can be monopolar (circular or elliptical) or multipolar; their stability analysis shows thattransitions between the various stationary shapes are possible when they become unstable. But stable vortices can also undergo unsteady evolutions when perturbed by environmental effects, likelarge-scale shear or strain fields, -effect or topography. Changes in vortex shapes can also result from vortex interactions. such as the pairing, merger or vertical alignment of two vortices, which depend on their relative polarities and depths. Such interactions transfer energy and enstrophy between scales, and are essential in two-dimensional and in geostrophic turbulence. Finally, in relation with the observations, we describe a few mechanisms of vortex generation.     

Interaction between a surface jet and subsurface vortices in a three-layer quasi-geostrophic model

This study focuses on the interaction between mid depth vortices and surface jets and fronts in a three-layer quasi-geostrophic model. Such vortices may be regarded as an idealisation of meddies, eddies of Mediterranean Water in the Northeastern Atlantic Ocean, interacting with the Azores j t and front. Successively, a single vortex, a vortex doublet and a vortex pair (in the middle layer) are studied. When a single vortex is considered, the jet has a critical effect of its motion, temporarily slowing down its zonal drift and accelerating it meridionally as the vortex crosses the front. On the contrary, if the vortex does not cross the front, it can drift fairly rapidly along it. The merger of a vortex doublet (two like-signed vortices) below a surface jet is possible whatever the relative position of this doublet with respect to the jet axis. Nevertheless, doublets initially located below the front, will undergo stronger shear and merger efficiency will be diminished. The merged vortex will be circled at the surface by a large meander of the jet. Finally, eastward jet-dipole interaction experiments are performed with various orientations of the vortex dipoles. Eastward propagating dipoles below the jet follow it without deformation. Southeastward drifting dipoles finally join the previous evolution. Southward and southwestward directed dipoles cross the surface jet southeastward. The presence of meanders initially on the jet does not prevent its crossing by a single vortex. Characteristics of the surface jet meanders are also described for a possible remote detection of this process.     

Vortex evolution due to straining: a mechanism for dominance of strong,interior anticyclones

In this article we address two questions: Why do freely evolving vortices weaken on average, even when the viscosity is very small? Why, in the fluid's interior, away from vertical boundaries and under the influence of Earth's rotation and stable density stratification, do anticyclonic vortices become dominant over cyclonic ones when the Rossby number and deformation radius are finite? The context for answering these questions is a rotating, conservative, Shallow-water model with Asymmetric and Gradient-wind Balance approximations. The controlling mechanisms are vortex weakening under straining deformation (with a weakening that is substantially greater for strong cyclones than strong anticyclones) followed by a partially compensating vortex strengthening during a relaxation phase dominated by Vortex Rossby Waves (VRWs) and their eddy–mean interaction with the vortex. The outcome is a net, strain-induced vortex weakening that is greater for cyclones than anticyclones when the deformation radius is not large compared to the vortex radius and the Rossby number is not small. Furthermore, when the exterior strain flow is sustained, the vortex changes also are sustained: for small Rossby number (i.e., the quasigeostrophic limit, QG), vortices continue to weaken at a relatively modest rate, but for larger Rossby number, cyclones weaken strongly and anticyclones actually strengthen systematically when the deformation radius is comparable to the vortex radius. The sustained vortex changes are associated with strain-induced VRWs on the periphery of the mean vortex. It therefore seems likely that, in a complex flow with many vortices, anticyclonic dominance develops over a sequence of transient mutual straining events due to the greater robustness of anticyclones (and occasionally their net strengthening).     

A Formal Theory for Vortex Rossby Waves and Vortex Evolution

A formal theory is presented for the balanced evolution of a small-amplitude, small-scale wave field in the presence of an axisymmetric vortex initially in gradient-wind balance and the accompanying changes induced in the vortex by the azimuthally averaged wave fluxes. The theory is a multi-parameter, asymptotic perturbation expansion for the conservative, rotating, f-plane, shallow-water equations. It extends previous work on Rossby-wave dynamics in vortices and more generally provides a new perspective on wave/mean-flow interaction in finite Rossby-number regimes. Some illustrative solutions are presented for a perturbed vortex undergoing axisymmetrization.