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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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We investigate quasi-geostrophic waves in a rotating spherical shell permeated by an imposed magnetic field. A projection of the momentum equation onto a well chosen class of velocity fields results in a quasi-geostrophic reduced model where, in contrast with previous such models, the evolution of the velocity is not prescribed by an equation for the axial vorticity. We consider fieldsthat may be longitudinally dependent. Increasing the angular rotation frequency, we find that the non-axisymmetric Alfvén waves that are present at low rotation morph into inertial waves, torsional Alfvén waves and low frequency magnetostrophic waves that satisfy Taylor’s constraint (i.e. vanishing acceleration of the geostrophic cylinders by the magnetic forces). [ABSTRACT FROM PUBLISHER]

The evolution of localised jets and periodic nonlinear waves in rotating shallow water magnetohydrodynamics (rotating SWMHD) and standard rotating shallow water model (RSW) is compared within the framework of translationally-invariant 1.5-dimensional configurations, which are traditionally used in geophysical fluid dynamics for studying geostrophic adjustment and frontogenesis. Such configurations also allow for exact nonlinear wave solutions in both models. A theory of the magneto-geostrophic adjustment, i.e. adjustment of an arbitrary initial configuration to a state of magneto-geostrophic equilibrium in RSWMHD, is developed and confirmed by numerical simulations with a finite-volume well-balanced code. The code is resolving all kinds of waves in the model and corresponding weak solutions equally well. It is benchmarked by reproducing exact solutions – steady essentially nonlinear Alfvèn and mixed magneto-inertia-gravity waves – and used to demonstrate robustness of these solutions with respect to localised along-wave perturbations. It is also shown how the results on adjustment can be extended to the fully 2-dimensional case. [ABSTRACT FROM PUBLISHER]

In this paper, we derive and study approximate balance models for nearly geostrophic shallow water flow where the Coriolis parameter is permitted to vary across the domain as long as it remains nondegenerate. This situation includes, for example, the β-plane approximation to the shallow water equations at mid-latitudes. Our approach is based on changing configuration space coordinates in the underlying variational principle in such a way that consistent asymptotics in the transformed Lagrangian leads to a degenerate Lagrangian structure. In this paper, we restrict our attention to first-order models. We show that the resulting models can be formulated in terms of an advected potential vorticity with a nonlinear vorticity inversion relation. We study the associated solvability conditions and identify a subfamily of models for which these conditions are satisfied without additional restrictions on the data. Finally, we provide the link between our framework and the theory of constrained Hamiltonian systems. [ABSTRACT FROM AUTHOR]

We examine the equilibrium form, properties, stability and nonlinear evolution of steadily-rotating simply-connected vortex patches in the single-layer quasi-geostrophic model of geophysical fluid dynamics. This model, valid for rotating shallow-water flow in the limit of small Rossby and Froude numbers, has an intrinsic length scale L D called the “Rossby deformation length” relating the strength of the stratification to that of the background rotation. Here, we generate steadily-rotating vortex equilibria for a wide range of γ = L/L D , where L is the typical horizontal length scale of the vortex. We vary both γ (over the range 0.02 ≤ γ ≤ 10) and the vortex aspect ratio λ (over the range 0 < λ < 1). We find two modes of instability arising at sufficiently small aspect ratio λ < λ c (γ): an asymmetric (dominantly wave 3) mode at small γ (or large L D ) and a symmetric (dominantly wave 4) mode at large γ (or small L D ). At marginal stability, the asymmetric mode dominates for γ ≲ 3, while the symmetric mode dominates for γ ≳ 3. The nonlinear evolution of weakly-perturbed unstable equilibria results in major structural changes, in most cases producing two dominant vortex patches and thin, quasi-passive filaments. Overall, the nonlinear evolution can be classified into three principal types: (1) vacillations for a limited range of aspect ratios λ when 5 ≤ γ ≤ 6, (2) filamentation and a single-dominant vortex for γ ≲ 1, and (3) vortex splitting – asymmetric for 1 ≲ γ ≲ 4 and symmetric for γ ≳ 4. [ABSTRACT FROM AUTHOR]

This work addresses the linear dynamics underlying the formation of density interfaces at the periphery of energetic vortices, well outside the vortex core, both in the radial and axial directions. We compute numerically the unstable modes of an anticyclonic Gaussian vortex lens in a continuously stratified rotating fluid. The most unstable mode is a slow mode, associated with a critical layer instability located at the vortex periphery. Although the most unstable disturbance has a characteristic vertical scale which is comparable to the vortex height, interestingly, the critical levels of the successively fastest growing modes are closely spaced at intervals along the axial direction that are much smaller than the vortex height. [ABSTRACT FROM AUTHOR]

An elliptical region having a particular distribution of anomalous buoyancy or temperature at the surface of an otherwise unbounded rotating stratified fluid is shown to steadily rotate under the quasi-geostrophic approximation. The particular distribution is proportional to the vertical thickness of an ellipsoid, divided by its mean thickness, in the limit of vanishing thickness. The steady rotation of this structure or vortex is assured by the known steady rotation of any ellipsoid, and can be obtained by an appropriate limit. It is found by numerical experimentation that this vortex is stable if its minor to major aspect ratio λ exceeds 0.427, approximately. Notably, a two-dimensional elliptical vortex (having uniform vorticity) is stable for λ > 1/3. Instabilities of surface vortices are characterised by the ejection of a weak tongue of buoyancy, which subsequently rolls up into a street of weak vortices. The main vortex is thereby reduced in aspect ratio and remains robust for long times. [ABSTRACT FROM AUTHOR]

The linear baroclinic and parametric instabilities of boundary currents with piecewise-constant potential vorticity are studied in a two-layer quasi-geostrophic model. The growth rates of both the exponential modes and of the optimal perturbations are calculated for the baroclinic instability of steady coastal currents. We show that the growth rates of the exponential modes are maximal for a vertically symmetric flow. Furthermore, the vertical asymmetries induced by different layer thicknesses, the presence of a barotropic potential vorticity or bottom topography, all act to dampen the growth rates and favor growth at shorter wavelengths. It is shown that this behavior can be predicted from the conditions for vertical resonance of Rossby waves on the two potential vorticity fronts. Also, the baroclinic instability of the optimal perturbations has larger growth rates at shorter wavelengths and shorter time scales. As well, the presence of a sloping bottom of moderate amplitude favors the growth of these optimal perturbations. Finally, we compute the growth rates of parametric instability of oscillatory coastal flows. We show that subharmonic resonance is the most unstable mode of growth. In addition, a second region of parametric instability is found (for the first time) away from marginality of exponential-mode baroclinic instability. It is shown that the functional dependency of the growth rates of parametric instability, for optimal excitation, are similar to that of the optimal perturbations of baroclinic instability. To explain this a mechanism for parametric instability, involving the rapid growth of short-wave optimal perturbations, is proposed. [ABSTRACT FROM AUTHOR]

An analytical dispersion relation is derived for linear perturbations to a Rankine vortex governed by surface quasi-geostrophic dynamics. Such a Rankine vortex is a circular region of uniform anomalous surface temperature evolving under quasi-geostrophic dynamics with uniform interior potential vorticity. The dispersion relation is analysed in detail and compared to the more familiar dispersion relation for a perturbed Rankine vortex governed by the Euler equations. The results are successfully verified against numerical simulations of the full equations. The dispersion relation is relevant to problems including wave propagation on surface temperature fronts and the stability of vortices in quasi-geostrophic turbulence. [ABSTRACT FROM AUTHOR]

The influence of external flow field on the statistical equilibrium state of quasi-geostrophic point vortices (vortex cloud) is investigated numerically. The numerical computations are performed using the fast special-purpose computer for molecular dynamics simulations, MDGRAPE-3. The equilibrium state in otherwise quiescent fluid is axisymmetric, whose radial distribution depends on both the vertical distribution of vortices P(z) and the total energy of the vortex system E. At a certain critical energy value Ec, the number of microscopic state with a given angular momentum attains its maximum (zero-inverse temperature state), where the radial distribution is Gaussian at any vertical height. When the energy is smaller (E < Ec: positive temperature), the radial distribution becomes flatter than the Gaussian. In contrast, if the energy is higher (E > Ec: negative temperature), the radial distribution becomes sharper showing tighter concentration near the axis of symmetry. If an equilibrium vortex cloud of positive temperature is immersed in the horizontal strain field Ue = ey, Ve = ex, the vortex distribution is stretched in the y-direction, and the azimuthally averaged radial distribution becomes Gaussian-like. Similarly, when the equilibrium state of positive temperature is immersed in the vertical shear field Uτ = τz, Vτ = 0, the vortex cloud is tilted in the y-direction, and the radial distribution becomes Gaussian-like. These findings explain how the internal vorticity distributions inside interacting vortex clouds of positive temperature change to be nearly zero-inverse temperature state. [ABSTRACT FROM AUTHOR]

We formulate a dynamically consistent two-layer quasigeostrophic model of geophysical flow using the concept of background currents which are characterized by a constant potential vorticity minimizing the energy. An incident current over a delta-like isolated topographic feature generates a topographically trapped vortex in the bottom layer with a singular elliptic point, and one with a regular elliptic point in the upper layer. Such vortices are finite regions of recirculation which occur in the vicinity of isolated topographic features. The corresponding Hamiltonian equations of motion for a fluid particle are known to produce chaotic advection under the presence of the periodic incident current. When a periodic incident current is superimposed, fluid is entrained and detrained from the neighbourhood of the vortex and chaotic particle motion occurs. At a small amplitude periodic incident current we have a near-integrable case of weak chaotization in narrow stochastic layer near separatrix. At a finite amplitude periodic incident current we have the case of strong chaos, which will be investigated here. For the bottom layer of the flow, there is always a regular region (regular vortex core) around the singular elliptical point even in the case of a finite amplitude periodic incident current. Particles in this regular region have regular trajectories and will remain there permanently, whereas particles with chaotic trajectories will be emanated from the vortex by the incident current. Using the Chirikov criterion of resonance overlap, we estimate the radius of this regular vortex core and a range of the optimal frequencies, i.e. those frequencies of the incident current which provide a maximal possible stochastization of fluid particle trajectories. [ABSTRACT FROM AUTHOR]

The solution for the bottom Ekman layer has a somewhat counter intuitive character, which seems to violate the maximum principle: at a certain level the velocity within the Ekman layer is higher than the velocity in the geostrophic layer above. I explain this character by looking at an analogous problem in an inertial frame of reference and show that it is the result of observing the flow from a rotating frame of reference (i.e. within a system that is not in steady state). The flow in the bottom Ekman layer is a superposition of the flow that results from the force exerted on the fluid by the rotating Earth and of the flow that results from the pressure-gradient term. Therefore, at a certain level the speed is higher than the speed of the geostrophic layer above which results from the pressure gradient alone. [ABSTRACT FROM AUTHOR]

Utilizing an eigenfunction decomposition, we study the growth and spectra of energy in the vortical (geostrophic) and wave (ageostrophic) modes of a three-dimensional (3D) rotating stratified fluid as a function of ε = f/N, where f is the Coriolis parameter and N is the Brunt-Vaisala frequency. Throughout, we employ a random large-scale forcing in a unit aspect ratio domain and set these parameters such that the Froude and Rossby numbers are roughly comparable and much less than unity. Working in regimes characterized by moderate Burger numbers, i.e. Bu = 1/ε2 < 1 or Bu ≥ 1, our results indicate profound change in the character of vortical and wave mode interactions with respect to Bu = 1. Indeed, previous analytical work concerning the qualitatively different nature of these interactions has been in limiting conditions of rotation or stratification domination (i.e. when Bu << 1 or Bu >> 1, respectively). As with the reference state of ε = 1, for ε < 1 the wave mode energy saturates quite quickly and the ensuing forward cascade continues to act as an efficient means of dissipating ageostrophic energy. Further, these saturated spectra steepen as ε decreases: we see a shift from k-1 to k-5/3 scaling for kf < k < kd (where kf and kd are the forcing and dissipation scales, respectively). On the other hand, when ε > 1 the wave mode energy never saturates and comes to dominate the total energy in the system. In fact, in a sense the wave modes behave in an asymmetric manner about ε = 1. With regard to the vortical modes, for ε ≤ 1, the signatures of 3D quasigeostrophy are clearly evident. Specifically, we see a k-3 scaling for kf < k < kd and, in accord with an inverse transfer of energy, the vortical mode energy never saturates but rather increases for all k < kf. In contrast, for ε > 1 and increasing, the vortical modes contain a progressively smaller fraction of the total energy indicating that the 3D quasigeostrophic subsystem, though always present, plays an energetically smaller role in the overall dynamics. Combining the vortical and wave modes, the total energy for k > kf and ε ≤ 1 shows a transition as k increases wherein the vortical modes contain a large portion of the energy at large scales, while the wave modes dominate at smaller scales. There is no such transition when ε > 1 and the wave modes dominate the total energy for all k > kf. [ABSTRACT FROM AUTHOR]

Geostrophic dynamics in the horizontally differential super-rotation of Venus are examined using f A (the Coriolis parameter defined by the angular velocity of a basic flow in an inertial frame) and Γ (the differential rotation parameter defined by the latitudinal gradient of the angular velocity) under the conditions that vertical shear of the basic field is not considered and the intrinsic phase velocity has a magnitude comparable to that of an eddy horizontal flow. The geostrophic and quasi-geostrophic approximations are valid in the regions of weakly differential and rigid-body super-rotations; however, they are invalid in the regions of strongly differential super-rotation even when the Rossby number R O is sufficiently smaller than unity for synoptic eddies. In a general circulation model of a Venus-like atmosphere, the horizontal divergence that results from the strong differential cannot be ignored over a wide range within latitudes ±60° and below 60&ThickSpace;km elevation because of large Γ/ f A (≥1/2). [ABSTRACT FROM AUTHOR]