Quasi-geostrophic Vortices Research Articles

The effect of ageostrophy on the stability of thin oceanic vortices

This paper examines the stability of vortices in a two-layer ocean on the f -plane. The mean depth h ¯ 1 of the upper layer is assumed to be much smaller than the depth h ¯ 2 of the lower layer. Using the primitive equations, we derive an asymptotic criterion for baroclinic instability of compensated (i.e. confined to the upper layer) vortices. Surprisingly, it coincides exactly with a similar criterion derived from the quasigeostrophic equations [Benilov, E.S., 2003. Instability of quasigeostrophic vortices in a two-layer ocean with thin upper layer. J. Fluid Mech. 475, 303–331]. Thus, to leading order in h ¯ 1 / h ¯ 2 , ageostrophy does not affect the stability properties of thin compensated vortices. As a result, whether a vortex is stable or not, depends on its shape, not amplitude (although the growth rate of an unstable vortex does depend on its amplitude).

The critical merger distance between two co-rotating quasi-geostrophic vortices

This paper examines the critical merger or strong interaction distance between two equal-potential-vorticity quasi-geostrophic vortices. The interaction between the two vortices depends on five parameters: their volume ratio, their height-to-width aspect ratios and their vertical and horizontal offsets. Due to the size of the parameter space, a direct investigation solving the full quasi-geostrophic equations is impossible. We instead determine the critical merger distance approximately using an asymptotic approach. We associate the merger distance with the margin of stability for a family of equilibrium states having prescribed aspect and volume ratios, and vertical offset. The equilibrium states are obtained using an asymptotic solution method which models vortices by ellipsoids. The margin itself is determined by a linear stability analysis. We focus on the interaction between oblate to moderately prolate vortices, the shapes most commonly found in turbulence. Here, a new unexpected instability is found and discussed for prolate vortices which is manifested by the tilting of vortices toward each other. It implies than tall vortices may merge starting from greater separation distances than previously thought.

Stability of a Two-Layer Quasigeostrophic Vortex over Axisymmetric Localized Topography

Abstract The stability of a quasigeostrophic vortex over a radially symmetric topographic feature (elevation or depression) in a two-layer ocean on the f plane is examined. This article’s concern is with compensated vortices, that is, those in which the lower layer is at rest (the disturbances, however, are present in both layers). Through numerical solution of the linear normal-mode problem, it is demonstrated that a bottom elevation is a stabilizing influence for a cyclone and a destabilizing influence for an anticyclone, whereas a bottom depression acts in the opposite way. These conclusions are interpreted using an asymptotic theory developed for the case of a thin upper layer. It is demonstrated that an elevation moves the critical level of an unstable mode toward the periphery of the cyclone, which leads to its stabilization. Estimates based on realistic oceanic parameters show that stabilization occurs for relatively small topography (5%–15% of the lower layer’s depth).

The theory of quasi-geostrophic von Kármán vortex streets in two-layer fluids on a beta-plane

In this study self-organized periodic coherent vortex structures arising in geophysical turbulent flows at low Rossby number are investigated by developing a conceptual model based on an analytical theory of von Kármán vortex streets affected by stratification and differential rotation. In the framework of a quasi-geostrophic (QG) two-layer beta-plane model vortex streets with three different types of vertical structures (barotropic, upper layer and hetonic) are analysed using the point vortex approximation. The streets are found to be exact solutions of the potential vorticity equation and to be characterized by four non-dimensional parameters. Von Kármán streets are semi-localized solutions which form a bridge between vortex pairs (limit of symmetric dilute streets) and two parallel vortex sheets (limit of dense streets). On the beta-plane QG von Kármán streets can only move to the east, i.e. with a speed outside the range of speeds of Rossby waves, so that a dynamical asymmetry in the zonal direction is introduced. A complete classification on a diagram of states shows that critical bounds exist in the parameter space, prescribing for example a maximum distance between vortex rows beyond which no QG vortex streets can be found. Typically a fast and a slow vortex street with different flow structures are found in the region of existence. As a function of distance between vortex rows baroclinic QG vortex streets show a characteristic non-monotonic speed behaviour at scales of the order of the baroclinic Rossby radius. A wide region of possible existence of QG von Kármán streets is found in atmospheric, oceanic and planetary conditions as well as in rotating tank experiments. The theory can be applied to describe the coherent part of turbulent baroclinic intermittent zonal jet-like and frontal flows and provides a scaling for such flows.

A New Look at the Problem of Tropical Cyclones in Vertical Shear Flow: Vortex Resiliency

A new paradigm for the resiliency of tropical cyclone (TC) vortices in vertical shear flow is presented. To elucidate the basic dynamics, the authors follow previous work and consider initially barotropic vortices on an f plane. It is argued that the diabatically driven secondary circulation of the TC is not directly responsible for maintaining the vertical alignment of the vortex. Rather, an inviscid damping mechanism intrinsic to the dry adiabatic dynamics of the TC vortex suppresses departures from the upright state. Recent work has demonstrated that tilted quasigeostrophic vortices consisting of a core of positive vorticity surrounded by a skirt of lesser positive vorticity align through projection of the tilt asymmetry onto vortex Rossby waves (VRWs) and their subsequent damping (VRW damping). This work is extended here to the finite Rossby number (Ro) regime characteristic of real TCs. It is shown that the VRW damping mechanism provides a direct means of reducing the tilt of intense cyclonic vortices (Ro > 1) in unidirectional vertical shear. Moreover, intense TC-like, but initially barotropic, vortices are shown to be much more resilient to vertical shearing than previously believed. For initially upright, observationally based TC-like vortices in vertical shear, the existence of a “downshear-left” tilt equilibrium is demonstrated when the VRW damping is nonnegligible. On the basis of these findings, the axisymmetric component of the diabatically driven secondary circulation is argued to contribute indirectly to vortex resiliency against shear by increasing Ro and enhancing the radial gradient of azimuthal-mean potential vorticity. This, in addition to the reduction of static stability in moist ascent regions, increases the efficiency of the VRW damping mechanism.

Instability of quasi-geostrophic vortices in a two-layer ocean with a thin upper layer

We examine the stability of a quasi-geostrophic vortex in a two-layer ocean with a thin upper layer on the f-plane. It is assumed that the vortex has a sign-definite swirl velocity and is localized in the upper layer, whereas the disturbance is present in both layers. The stability boundary-value problem admits three types of normal modes: fast (upper-layer-dominated) modes, responsible for equivalent-barotropic instability, and two slow baroclinic types (mixed- and lower-layer-dominated modes). Fast modes exist only for unrealistically small vortices (with a radius smaller than half of the deformation radius), and this paper is mainly focused on the slow modes. They are examined by expanding the stability boundary-value problem in powers of the ratio of the upper-layer depth to the lower-layer depth. It is demonstrated that the instability of slow modes, if any, is associated with critical levels, which are located at the periphery of the vortex. The complete (sufficient and necessary) stability criterion with respect to slow modes is derived: the vortex is stable if and only if the potential-vorticity gradient at the critical level and swirl velocity are of the same sign. Several vortex profiles are examined, and it is shown that vortices with a slowly decaying periphery are more unstable baroclinically and less barotropically than those with a fast-decaying periphery, with the Gaussian profile being the most stable overall. The asymptotic results are verified by numerical integration of the exact boundary-value problem, and interpreted using oceanic observations.

The evolution of an initially circular vortex near an escarpment. Part II: numerical results

Dunn, McDonald and Johnson [6] have obtained leading-order analytical results for the motion of an initally circular, quasigeostrophic vortex, near an infinitely long escarpment using f-plane dynamics. In the limit that the vortex is weak (i.e., topographic vortex stretching dominates advection by the vortex), advection of fluid across the escarpment is inhibited and at short times the escarpment acts like a wall. On the other hand, if advection by the vortex dominates over topographic vortex stretching (i.e., an intense vortex), expressions for the trajectory of the vortex centre may be obtained. In this paper the contour advection scheme of Dritschel [2] is used to investigate the full range of vortex strengths. It is found that the pseudoimage description of the motion of a weak vortex is accurate beyond the time for which linear theory is formally valid. The analytic prediction of the drift of intense vortices is also shown to be accurate. The physical mechanisms for the evolution of an intense vortex are identified. In particular there is a ‘trapped region’ near the vortex periphery, which protects the vortex from deformation, and a ‘trailing eddy’, which contributes to motion perpendicular to the escarpment. These features are similar to those found in recent analytical and numerical studies of vortex motion on the β-plane. For the case of a moderate intensity vortex, advection by the vortex and topographic vortex stretching are equally important, and in this highly nonlinear case no theory is available. Contour advection simulations reveal that anticyclones located on the shallow side of the escarpment are able to cross the escarpment, while cyclones located on the shallow side are ‘back-reflected’ and move away from the escarpment. Moderate vortices undergo enhanced motion in the direction perpendicular to the escarpment, compared with the weak and intense vortices. These characteristics of the motion are associated with dipole formation, and are consistent with recent experimental and numerical studies of barotropic vortices near escarpment topography.

The merger of vertically offset quasi-geostrophic vortices

We examine the critical merging distance between two equal-volume, equal-potential- vorticity quasi-geostrophic vortices. We focus on how this distance depends on the vertical offset between the two vortices, each having a unit mean height-to-width aspect ratio. The vertical direction is special in the quasi-geostrophic model (used to capture the leading-order dynamical features of stably stratified and rapidly rotating geophysical flows) since vertical advection is absent. Nevertheless vortex merger may still occur by horizontal advection.In this paper, we first investigate the equilibrium states for the two vortices as a function of their vertical and horizontal separation. We examine their basic properties together with their linear stability. These findings are next compared to numerical simulations of the nonlinear evolution of two spheres of potential vorticity. Three different regimes of interaction are identified, depending on the vertical offset. For a small offset, the interaction differs little from the case when the two vortices are horizontally aligned. On the other hand, when the vertical offset is comparable to the mean vortex radius, strong interaction occurs for greater horizontal gaps than in the horizontally aligned case, and therefore at significantly greater full separation distances. This perhaps surprising result is consistent with the linear stability analysis and appears to be a consequence of the anisotropy of the quasi-geostrophic equations. Finally, for large vertical offsets, vortex merger results in the formation of a metastable tilted dumbbell vortex.

A Theory for the Vertical Alignment of a Quasigeostrophic Vortex

This article presents a new theory for the rate at which a quasigeostrophic vortex realigns, under conservative dynamics, after being tilted by an episode of external vertical shear. The initial tilt is viewed as the excitation of a three-dimensional “vortex Rossby mode.” This mode, that is, the tilt, decays exponentially with time during its early evolution. The decay rate γ is proportional to the potential vorticity gradient at a critical radius, where the fluid rotation is resonant with the mode. The decay rate γ also depends on the internal Rossby deformation radius lR, which is proportional to the stratification strength of the atmospheric or oceanic layer containing the vortex. The change of γ with lR is sensitive to the form of the vortex. For the case of a “Rankine-with-skirt” vortex, the magnitude of γ increases (initially) with increasing lR. On the other hand, for the case of a “Gaussian” vortex, the magnitude of γ decreases with increasing lR. The relevance of this theory to tropical cyclogenesis is discussed.

Drift Velocity of Radiating Quasigeostrophic Vortices

Abstract Because of the beta effect, quasigeostrophic monopole vortices propagate westward and excite Rossby waves. This wave radiation depletes the vortex energy, and causes cyclones to drift northward and anticylones southward (in the Northern Hemishpere). In the present work explicit solutions describing such radiating vortices are found by perturbation analysis, assuming the vortex amplitude to be large, and consequently the radiation to be a small perturbation. From these solutions the radiated energy is calculated and then used to obtain a simple expression for the meridional drift. The zonal drift is also modified by the wave radiation, but to calculate this component the complete explicit solution is not necessary; it is enough to consider the ratio of the loss of pseudoenergy to the loss of pseudomomentum.

Quasigeostrophic Ellipsoidal Vortex Model

An ellipsoidal vortex model is developed in order to investigate the dynamics of vortex interactions in a quasigeostrophic turbulence. Each coherent vortex is modeled by an ellipsoid of uniform potential vorticity. It is assumed that each vortex remains to be an ellipsoid, although its principal axes and inclination angles change under the influence of the strain field induced by other vortices. The dynamics of ellipsoidal vortices are shown to be a Hamiltonian system of finite degree of freedom extracted from the partial differential equations that describe quasigeostrophic vortex dynamics. The center of vorticity and the angular momentum are conserved, besides the total energy and Casimirs of the system, such as the vortex height and the vortex volume. Chaotic motions are observed even in a two-body system.

Equilibrium Statistical Predictions for Baroclinic Vortices: The Role of Angular Momentum

We develop a point-vortex equilibrium statistical model for baroclinic quasigeostrophic vortices within the context of a two-layer quasigeostrophic fluid that evolves in all of space. Angular momentum, which follows from the rotational symmetry of the unbounded domain, is the key conserved quantity, introducing a length scale that confines the most probable states of the statistical theory. We apply the theory as a model of localized convection in a preconditioned gyre. To illustrate this application, the preconditioned cyclonic, largely barotropic gyres are modeled as „zero inverse temperature” states, which are explicit solutions to the mean-field equations with a Gaussian probability distribution of vortices. Convection is modeled by a cloud of point-vortex hetons – purely baroclinic arrangements of point vortices, cyclonic above and anticyclonic below – which capture the short-term, geostrophically balanced response to strong surface cooling. Numerical heton studies (Legg and Marshall, 1993, 1998) have shown that a preexisting barotropic rim current can suppress baroclinic instability and confine anomalies of potential vorticity and temperature introduced by the cold-air outbreak. Here, we demonstrate that the lateral extent of the most probable states of the statistical theory are constrained by the angular momentum. Without resolution of the detailed dynamics, the equilibrium statistical theory predicts that baroclinic instability is suppressed for preconditioned flows with potential vorticity of the same sign in each layer provided that the strength of convective overturning does not change the sign of potential vorticity in one of the layers. This result agrees with detailed simulations (Legg and Marshall, 1998) and supports the potential use of these statistical theories as parametrizations for crude closure.

The effect of changing the Coriolis force gradient parameter on the escape probability and mean residence time

We investigate a quasigeostrophic vortex flow with random perturbations. The system we will look at is rotating and has a gradient in the Coriolis force (the force due to the earth's rotation). When these two characteristics are present, the conserved quantity is the potential vorticity q=∇ 2 ψ+ βy. Thus, the dynamics are determined by the quasigeostrophic equation ( ∂ t+ v → ·∇)q=0 . In this, the parameter β is proportional to the Coriolis force gradient. Whenever we have a system with random perturbations, we may talk of the escape probability of fluid particles crossing a portion of the boundary for a fluid domain, and the mean residence time of fluid particles inside a fluid domain. The goal of this paper is to determine the relationship between the parameter β and the escape probability and the mean residence time. We will look at a vortex of the flow and determine the escape probability crossing the upper and lower boundaries of the vortex of a particle (we assume that the particles are uniformly distributed in the vortex). We will also calculate the mean residence time for a particle inside a vortex. We find that as β increases, the escape probability for a particle crossing the lower boundary decreases. We also find that as β increases the mean residence time of a particle inside a vortex decreases.

Quasigeostrophic Wire-Vortex Model

A wire-vortex model is developed in order to investigate the dynamics of vortex interactions in a quasigeostrophic turbulence. Each coherent vortex is modeled by a straight wire of finite length. A tilted wire-vortex rotates around the vertical axis steadily by its self-induction in an otherwise quiescent fluid, whereas its inclination angle changes if it is embedded in a local strain field induced by other vortices. The dynamics of wire-vortices are shown to be a Hamiltonian system of finite degree of freedom extracted from the partial differential equations that describe quasigeostrophic vortex dynamics. The center of vorticity and the angular momentum are conserved, besides the total energy and Casimirs of the system, such as the vortex height and the vortex volume. Chaotic motions are observed even in a two-body system.

Linear Instability of Barotropic Submesoscale Coherent Vortices Observed in the Ocean

The linear instability of circular quasigeostrophic vortices with horizontal cross sections of angular velocity identical to those that fit the observations of submesoscale eddies in the ocean is investigated analytically and numerically. The theoretical necessary conditions for instability are formulated as a single condition on the mean potential vorticity or mean angular velocity rather than the streamfunction, which is more readily applicable to oceanic lenses, eddies, and meddies. It is shown that the suggested cross sections that best fit the observed angular velocity of several long-lived vortices are all unstable to small, wavelike perturbations, and that the e-folding time for perturbation growth at all wavenumbers and cross sections is on the order of 1 day. For all cross sections considered, azimuthal wavenumber 1 is stable while all higher azimuthal wavenumbers, as well as all vertical wavenumbers, are unstable. The main contribution to the instability comes from the jump in potential vorticity at the radius of maximum angular velocity. When the potential vorticity is continuous at this radius, the growth rates become smaller and the details of potential vorticity distribution become important. The fast growth rates obtained by the numerical calculations clearly emphasize the insufficient spatial resolution of existing observations for deciphering the exact velocity cross sections of submesoscale oceanic vortices, especially near the radius of maximum angular velocity.

The Evolution of Frontal-Geostrophic Vortices in a Two-Layer Ocean

Abstract The properties of large amplitude vortices are numerically investigated using a set of two-layer primitive equations. Numerical experiments are systematically conducted for the f and β plants, quasigeostrophic (small amplitude) and frontal-geostrophic (large amplitude) dynamics, cyclone and anticyclone, and vortex sizes comparable to the radius of deformation and larger. In particular, the evolution and migration of frontal-geostrophic vortices are discussed with reference to the evolution equations in a two-layer ocean. It is shown that the anticyclonic vortex with large amplitude displacement evolves differently from the quasigeostrophic vortex and that it is extremely long-lived. The evolution of anticyclonic vortices, in which interfacial displacement is large and the size is greater than the radius of deformation, is different from that of cyclonic vortices. While the shapes of anticyclonic vortices change from circle to ellipse because of instability, the large amplitude effect of interfaci...

Chaotic oscillations and breakup of quasigeostrophic vortices in the N-layer approximation

This paper examines the onset of chaotic oscillations in the N-layer model of stratified quasigeostrophic flow due to interaction of N vortex sections in different layers, and then considers the possible role of chaotic self-induced motion in the breakup of quasigeostrophic vortices. In order to investigate the nature of the long-time evolution of the vortex using a low-dimensional dynamical system, attention is restricted to vortices with uniform potential vorticity within the core and sufficiently large values of the stratification parameter that the vortex cross section remains nearly circular with time. Computations using contour dynamics in many-layered systems are performed to check that the deformation of the vortex cross section is small. It is shown that for finite-amplitude perturbations of the vortex centerline, the barotropic interaction between sections of the vortex in different layers can cause the vortex to oscillate chaotically. The approach to chaotic motion is initially investigated using a system with only four layers, in which the vortex is approximated by equal-strength segments of either point vortices or circular patches. The onset of chaos is shown to be sensitive to the stratification parameter, the vortex core radius, and the initial configuration. The consequences of chaos on the vortex evolution and breakup are then demonstrated by numerical computations with a large number of layers for vortices with finite-core area.

On the stability of baroclinic vortex streets composed of quasi-geostrophic point eddies

The stability of quasi-geostrophic vortex streets is investigated for a single, a symmetric double, and a staggered double row, where component baroclinic point vortices are discriminated from ordinary ones by concentrated potential vorticity instead of concentrated vorticity. The former two types of rows always turn out unstable, though the growth rate itself decreases monotonically with increasing F ∗ , an index of horizontal divergence, or the inverse of the deformation radius nondimensionalized in terms of the longitudinal spacing of the vortex street. As regards a staggered double row, a critical value F c ∗ = 3.04 divides F ∗ into two regimes. Below F c ∗ , only one value of b ∗ = b s ∗ is neutrally stable, where b = b a is the aspect ratio of the vortex street with a and b the separations between eddies along and perpendicular to the street, respectively. The stable configuration b s ∗ slowly increases with F ∗ from the Kármán ratio 0.281 at F ∗ = 0 to 0.322 at F ∗ = F c ∗ . For F ∗ above F c ∗ , the staggered double row is stable for a band of b ∗ . The appearance of the stable region is explained by a near-field approximation, where an eddy is assumed to exert its vortex force only on adjacent eddies. The near-field approximation yields a stability diagram which agrees well with that obtained from rigorous calculation for large F ∗ . In particular, the configuration b ∗ ≥ √ 3 2 is shown to be unstable irrespective of F ∗ .

Jupiter's Great Red Spot and zonal winds as a self-consistent, one-layer, quasigeostrophic flow.

We present the point of view that both the vortices and the east-west zonal winds of Jupiter are confined to the planet's shallow weather layer and that their dynamics is completely described by the weakly dissipated, weakly forced quasigeostrophic (QG) equation. The weather layer is the region just below the tropopause and contains the visible clouds. The forcing mimics the overshoot of fluid from an underlying convection zone. The late-time solutions of the weakly forced and dissipated QG equations appear to be a small subset of the unforced and undissipated equations and are robust attractors. We illustrate QG vortex dynamics and attempt to explain the important features of Jupiter's Great Red Spot and other vortices: their shapes, locations with respect to the extrema of the east-west winds, stagnation points, numbers as a function of latitude, mergers, break-ups, cloud morphologies, internal distributions of vorticity, and signs of rotation with respect to both the planet's rotation and the shear of their surrounding east-west winds. Initial-value calculations in which the weather layer starts at rest produce oscillatory east-west winds. Like the Jovian winds, the winds are east-west asymmetric and have Karman vortex streets located only at the west-going jets. From numerical calculations we present an empirically derived energy criterion that determines whether QG vortices survive in oscillatory zonal flows with nonzero potential vorticity gradients. We show that a recent proof that claims that all QG vortices decay when embedded in oscillatory zonal flows is too restrictive in its assumptions. We show that the asymmetries in the cloud morphologies and numbers of cyclones and anticyclones can be accounted for by a QG model of the Jovian atmosphere, and we compare the QG model with competing models.

Tearing of an aligned vortex by a current difference in two-layer quasi-geostrophic flow

A study of two-layer quasi-geostrophic vortex flow is performed to determine the effect of a current difference between the layers on a vortex initially extending through both layers. In particular, the conditions under which the vortex can resist being torn by the current difference are examined. The vortex evolution is determined using versions of the contour dynamics and discrete vortex methods which are modified for two-layer quasi-geostrophic flows. The vortex response is found to depend upon the way in which the current difference between the layers is maintained. In the first set of flows studied, the current difference is generated by a (stronger) third vortex in the upper layer located at a large distance from the (weaker) vortex under study.