Topographic Vortex Research Articles

Modon solutions in an N-layer quasi-geostrophic model

Modons, or dipolar vortices, are common and long-lived features of the upper ocean, consisting of a pair of counter-rotating monopolar vortices moving through self-advection. Such structures remain stable over long times and may be important for fluid transport over large distances. Here, we present a semi-analytical method for finding fully nonlinear modon solutions in a multi-layer quasi-geostrophic model with arbitrarily many layers. Our approach is to reduce the problem to a multi-parameter linear eigenvalue problem which can be solved using numerical techniques from linear algebra. The method is shown to replicate previous results for one- and two-layer models and is applied to a three-layer model to find a solution describing a mid-depth propagating, topographic vortex.

Erratum to: Canyon Vortices: Application of the Theory of Topographic Vortices to the Phenomenon of Ice Rings in Baikal

Erratum to: Canyon Vortices: Application of the Theory of Topographic Vortices to the Phenomenon of Ice Rings in Baikal

Fractal structure of chaotic scattering in a simple hydrodynamic model with a point vortex embedded in a time-(quasi)periodic background flow

This paper is a study of a nonlinear dynamical system with 3/2 degrees of freedom consisting of a point vortex and an unsteady background flow. This background flow consists of two parts, a steady and a (quasi)periodic with two sinusoidal components with different frequencies and amplitudes. The considered system with a separatrix loop in the phase portrait is the simplest model of chaotic scattering of passive tracers on a topographic vortex, captured by a seamount, in the ocean. We have found that depending on variation of amplitudes of two sinusoidal components, the passive tracers enters and leaves the vortex region in portions. The frequency of escape and the size of such portions depend on the parameters of the background flow. It is shown that the structure of the fractal-like scattering function changes at low levels through the formation (disappearance) of segments according to two mechanisms, the action of which is determined by the relation between two frequencies of the perturbation.

Canyon Vortices: Application of the Theory of Topographic Vortices to the Phenomenon of Ice Rings in Baikal

A phenomenon of ice rings in Lake Baikal is considered. A hydrodynamic theory is proposed according to which geostrophic vortices form under ice in the area near the ice rings. These vortices are due to bed relief, namely, underwater canyons; therefore, they are referred to as canyon vortices. It is shown that a divergent ring vortex forms under ice in the ring domain at the crossing of a vertical cylindrical Stewartson layer on the lateral surface of the canyon geostrophic vortex and under-ice horizontal Ekman layer in the form of a horizontal vortex torus, and the melting of ice below in the form of a ring is due to the formation of this vortex torus.

Theoretical and Laboratory Modeling of Topographic Vortex Bifurcation on Vortex Tori over Two-Stage Axisymmetric Elevation

Theoretical and Laboratory Modeling of Topographic Vortex Bifurcation on Vortex Tori over Two-Stage Axisymmetric Elevation

A modification of the invariant imbedding method for a singular boundary value problem

We consider a dynamically-consistent analytical model of a 3D topographic vortex. The model is governed by equations derived from the classical problem of the axisymmetric Taylor–Couette flow. Using linear expansions, these equations can be reduced to a differential sixth-order equation with variable coefficients. For this differential equation, we formulate a boundary value problem, which has a number of issues for numerical solving. To avoid these issues and find the eigenvalues and eigenfunctions of the boundary value problem, we suggest a modification of the invariant imbedding method (the Riccati equation method). In this paper, we show that such a modification is necessary since the boundary conditions possess singular matrices, which sufficiently complicate the derivation of the Riccati equation. We suggest algebraic manipulations, which permit the initial problem to be reduced to a problem with regular boundary conditions. Also, we propose a method for obtaining a numerical solution of the matrix Riccati equation by means of recurrence relations, which allow us to obtain a matrizer converging to the required eigenfunction. The suggested method is tested by calculating the corresponding eigenvalues and eigenfunctions, and then, by constructing fluid particle trajectories on the basis of the eigenfunctions.

Vortex tori above bottom perturbations in a rotating fluid

In this study, a model that describes one of the mechanisms of diapycnal mixing in topographic vortices above seamounts in the Ocean is presented. It is known that topographic eddies are described in the quasi-geostrophic approximation; i.e., vertical velocities are considered to be negligibly small (of about the Rossby number). However, the data of observations indicate the presence of a well-mixed fluid column in the vicinity of seamounts. Description of such vertical mixing is possible only due to account of the vertical velocity component. The spectral problem for the stability of a topographic vortex over the perturbation of the bottom relief in the form of a simple cylinder and for a more complex configuration consisting of two coaxial cylinders is solved. It is shown that the eigenvalues of the spectrum are the bifurcation points of the flow, due to which secondary toroidal vortices are generated. These toroidal vortices perform intense vertical mixing of waters above seamounts in the Ocean. The method of invariant imbedding is used to solve the spectral problem.

Interaction of a monopole vortex with an isolated topographic feature in a three-layer geophysical flow

Abstract. In the frame of a three-layer, quasi-geostrophic analytical model of an f-plane geophysical flow, the Lagrangian advection induced by the interaction of a monopole vortex with an isolated topographic feature is addressed. Two different cases when the monopole is located either within the upper or the middle layer are of our interest. In the bottom layer, there is a delta-function topographic feature, which generates a closed recirculation region in its vicinity due to the background flow. This recirculation region extends to the middle and upper layers, and it plays the role of a topographic vortex. The interaction between the monopole and the topographic vortex causes a complex, including chaotic, advection of fluid particles. We show that the model's parameters, namely the monopole and topographic vortices' strengths and initial positions, and the layers' depths and densities, are responsible for the diverse advection patterns. While the patterns are rather complicated, one can single out two major processes, which mostly govern the fluid particle advection. The first one is the variation in time of the system's phase space structure, so that within the closed region of the topographic vortex, there appear periodically unclosed particle pathways by which the particles leave the topographic vortex. The second one is chaotic advection that arises from the nonstationarity of the monopole–topography interaction.

Secular variations in the main geomagnetic field within the scope of the dynamic model of field sources

The effect of different-level sources on the spatial structure of the secular variations has been considered based on the dynamic model of sources of the main geomagnetic field developed by us. It has been obtained that the development of 13 most powerful dipoles only roughly characterizes global anomalies of the secular variations, and each anomaly results from the superposition of the dynamics of several sources. The model secular variations have been compared with the data from the observatories. It has been obtained that it is impossible to describe local anomalies of the observed secular variations ignoring sources of the third order of smallness as compared to the main dipole. It has been assumed that topographic vortices, originating around inhomogeneities of the core-mantle boundary, can be physical sources responsible for dipoles of the third order.

Evaluating the stochastic layer thickness in a two-layer topographic vortex model

The thickness of a stochastic layer is evaluated within the framework of the model of a two-layer inviscid fluid in unidirectional pulsating background flow over a submerged delta-shaped obstacle. Estimates of the width of a boundary layer, in which passive impurity exchange takes place between an external flow and a vortexed domain, are obtained and the limits of their applicability are analyzed. It is established that the sto- chastic layer thickness exhibits a jumplike change with increasing perturbation amplitude.

Chaotic advection and nonlinear resonances in an oceanic flow above submerged obstacle

The effect of an isolated submarine obstacle on the motion of fluid particles in a periodic external flow is studied within the framework of the barotropic, quasi-geostrophic approximation on f -plane. The concept of background currents advanced by Kozlov [1995. Background currents in geophysical hydrodynamics. Izvestia, Atmos. Oceanic Phys. 31 (2), 245–250] is used to construct a dynamically consistent stream function satisfying the potential vorticity conservation law. It is shown that a system of two topographic vortices revolving about a rotation center can form in a circular external flow. Unsteady periodic perturbations, associated with either variations in the background current or deviations of the external flow from circulation, are analyzed. Unsteadiness in the external flow essentially complicates the pattern of the motion of fluid particles. Vortex-type quasi-periodic structures, identified with nonlinear resonances that form in Lagrangian equations of fluid particle advection, are examined. They either surround the stationary configuration by a vortex chain—a ringlet-like structure [ Kennelly, M.A., Evans, R.H., Joyce, T.M., 1985. Small-scale cyclones on the periphery of Gulf Stream warm-core rings. J. Geophys. Res. 90(5), 8845–8857], or they form a complex-structure multivortex domain. Asymptotic estimates and numerical modeling are used to study the distribution and widths of the nonlinear resonance domains that appear under unsteady perturbations of different types. The onset of chaotic regimes owing to the overlapping of nonlinear resonance domains is analyzed. Transport fluxes determined by chaotic advection and barriers for transport (KAM-tori) and the conditions of their existence are studied. The relation of the rotation frequency of fluid particles on their initial position (when the dependence is calculated in the undisturbed system) is shown to completely determine the main features of the pattern of Lagrangian trajectories and chaotization effects. Because of nonlinear effects, the domain involved in quasi-periodic and chaotic motions can be much larger than the domain occupied by steady topographic vortices. The results of study by Sokolovskiy et al. [1988. On the influence of an isolated submerged obstacle on a barotropic tidal flow. Geophys. Astrophys. Fluid Dyn. 88(1), 1–30] concerning the due regard on the irrotational background component as the necessary factor for the transportation of fluid particles from the vortex domain to infinity are confirmed.

Chaotic advection induced by a topographic vortex in baroclinic ocean

Chaotic advection induced by a topographic vortex in baroclinic ocean

Erosion of a Surface Vortex by a Seamount on the β Plane

Abstract This paper investigates the behavior of a surface-intensified anticyclone encountering a seamount on the β plane in a stratified ocean. The eddy may be strongly eroded, and sometimes subdivided, provided that it gets close enough to the seamount. In case of subdivision, the detached part has a vertical structure different from that of the initial eddy, and a subsurface vortex may result. The basic erosion mechanism previously observed with f-plane experiments is still active on the β plane. Deep fluid motions induced by the initial vortex across the isobaths generate topographic vortices whose upper parts exert a shear/strain on the initial eddy, causing its filamentation. On the β plane, this process is further complicated by the presence of additional eddies created by fluid motion across the planetary vorticity gradient. Experiments without any topography show that these eddies by themselves can erode the initial vortex. In particular, a deep positive potential vorticity pole influences the near-bottom signature of the original vortex with a strong temporal variability. This reflects on the manner in which the surface eddy feels an underlying seamount. Sensitivity experiments show that the eddy erosion rate after encountering a seamount can no longer be related to basic parameters such as the minimum eddy–seamount distance, as it was on the f plane. The additional vorticity poles influencing the eddy on the β plane make the result of the eddy–seamount encounter very sensitive to small variations of the initial conditions, and impossible to predict.

Boundary Effect on the Mixing and Transport of Passive Impurities in a Nonstationary Flow

The simplest model of a point topographic vortex situated near a linear boundary in a nonstationary incident flow is considered. The results of numerical experiments reveal a new feature in the behavior of trajec- tories that is not observed in absence of the boundary. A criterion for distinguishing such trajectories is pro- posed. © 2005 Pleiades Publishing, Inc.

Hamiltonian fractals and chaotic scattering of passive particles by a topographical vortex and an alternating current

We investigate the dynamics of passive particles in a two-dimensional incompressible open flow composed of a fixed topographical point vortex and a background current with a periodic component, the model inspired by the phenomenon of topographic vortices over mountains in the ocean and atmosphere. The tracer dynamics is found to be typically chaotic in a mixing region and regular in far upstream and downstream regions of the flow. Chaotic advection of tracers is proven to be of a homoclinic nature with transversal intersections of stable and unstable manifolds of the saddle point. In spite of simplicity of the flow, chaotic trajectories are very complicated alternatively sticking nearby boundaries of the vortex core and islands of regular motion and wandering in the mixing region. The boundaries act as dynamical traps for advected particles with a broad distribution of trapping times. This implies the appearance of fractal-like scattering function: dependence of the trapping time on initial positions of the tracers. It is confirmed numerically by computing a trapping map and trapping time distribution which is found to be initially Poissonian with a crossover to a power-law at the PDF tail. The mechanism of generating the fractal is shown to resemble that of the Cantor set with the Hausdorff fractal dimension of the scattering function to be equal to d≃1.84.

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 evolution of an initially circular vortex near an escarpment. Part I: analytical results

The motion of a quasigeostrophic, equivalent-barotropic, initially circular vortex patch near an infinitely long topographic escarpment is studied using f-plane dynamics. There are two time scales in the problem: the advective time scale associated with the vortex, and the time scale for topographic vortex stretching. Analytical progress is possible when these two time scales are well-separated and results are presented here. If topographic vortex stretching dominates advection by the vortex the vortex is said to be ‘weak’. The vortex patch remains circular on the topographic time scale, and dispersive topographic waves rapidly propagate the initial disturbance away from the vicinity of the vortex. Subsequently cross-isobath motion is inhibited, and the vortex moves as though the escarpment were a plane wall. The same behaviour was observed for the motion of a weak singular vortex near an escarpment by Dunn, McDonald and Johnson [7], who named the phenomenon the ‘pseudoimage’ of the vortex. If advection dominates over topographic effects, the vortex is said to be ‘intense’. The vortex also remains circular to leading order, but the relative vorticity produced by the swirl of the vortex is less able to escape the vicinity of the vortex. The vortex follows a similar curved trajectory to those observed for intense vortices on the β-plane. The dipolar mechanism for this behaviour is described. Large time solutions are inhibited by the form of the escarpment topography, but examination of the equations leads to the conclusion that the leading order solution may be predict the motion for times beyond its formal range of validity.

On a modified arrested topographic wave in Fram Strait

[1] The bottom geostrophic flow in Fram Strait is explained in terms of a modified arrested topographic wave model. The model is based on a combination of the vorticity balance between the topographic vortex stretching and the bottom Ekman pumping due to liner friction with a relation between the bottom flow and the bottom pressure gradient over a sloping bottom. Such a combination emphasizes a dynamical relevance of along-isobath density variations at the bottom. Using an inverse solution based on the hydrographic data from the MIZEX 84 experiment, it is shown that the along-isobath bottom density variations should be dynamically important, for example, in the East Greenland Current. A relatively high correlation between an along-isobath distribution of the bottom geostrophic flow and the bottom density is reported for that current. The density variations identified in the data should be able to drive a bottom geostrophic flow with a magnitude consistent with the inverse solution and observations.

The determinant of the volume transport distribution of the Tsushima Warm Current around the Tsushima/Korea Straits

The Tsushima Warm Current (TWC) splits into, at least, two branches in the Japan Sea. One is the Coastal branch flowing along the Japanese coast, and the other is the East Korean Warm Current (EKWC) flowing northward along the Korean coast. This study proposes a mechanism that determines the volume transport distribution between these two branches around the Tsushima/Korea Straits. Results of the hydrographic observations indicate that the stratification in the western channel develops in summer because of the existence of the bottom cold water and the sea surface heating. At the same time, the volume transport of the EKWC increases. Numerical models show that the Jebar (joint effect of baroclinicity and relief) and the topographic vortex stretching (divergence) effect are predominant around the shelf slope in the Tsushima/Korea Straits, where the TWC bifurcates. The Jebar is due to the intrusion of the bottom cold water along the Korean coast. The divergence effect due to the EKWC crossing the shelf slope cancels this Jebar. It increases in summer when the Jebar is intensified.

High Rayleigh number β-convection

Abstract High resolution numerical simulations of thermal convection in a rapidly rotating channel with gravity perpendicular to the rotation vector are described. The convecting columns are subject to a β-effect resulting from topographic vortex stretching. The symmetries of the problem allow many invariant wavenumber sets, and this property is associated with the existence of stable multiple-equilibria at modest supercriticality. The transition to chaotic behavior involves the production of intermittent unstable orbits off a two-torus in energy space. At very high Rayleigh number (of order 106 to 107) the motion can be turbulent, depending on the size of β. However, the turbulence is usually characterized by an almost-periodic formation of patches of small scale convection that cause regular pulsations in the accompanying strong zonal jets. The processes maintaining these flows may be related to those responsible for the zonal currents on Jupiter and for cyclic variability on the Sun.