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Mathematics
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Vortex Dynamics: Theory and Application to Geophysical Flows
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Vortex Dynamics: Theory and Application to Geophysical Flows

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "C2: Dynamical Systems".

Deadline for manuscript submissions: closed (20 February 2021) | Viewed by 19934

Special Issue Editors

Prof. Dr. Xavier Carton

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Guest Editor
Laboratoire d’Océanographie Physique et Spatiale, Institut Universitaire Européen de la Mer, Universite de Bretagne Occidentale, 29280 Plouzané, France
Interests: ocean dynamics; mesoscale vortex stability and interactions; continental slope currents; outflows from marginal seas
Special Issues, Collections and Topics in MDPI journals
Dr. Mikhail A. Sokolovskiy

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Guest Editor
Institute of Water Problems, Russian Academy of Science, 3 Gubkina Street, 119333 Moscow, Russia
Interests: vortex dynamics in stratified/homogeneous rotating fluid; application to the geophysical environs
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Dr. Jean Reinaud

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Guest Editor
School of Mathematics and Statistics, University of St. Andrews, North Haugh, St. Andrews KY169SS, UK
Interests: geophysical fluid dynamics; vortex equilibria; vortex stability and interactions
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Vortices are key features of fluid flows. It is long since known that they are central to flight dynamics and to ship motion. In the oceans and planetary atmospheres, they carry momentum, heat, energy, and tracers over long distances. Their role in atmospheric chemistry and ocean biology is amply demonstrated. In geophysical fluids, vortices play a central role in the spectral transfers of energy and of enstrophy between scales. In ocean dynamics, recent progress of theory and a major increase in computer performance have allowed the investigation of dynamical relations between vortices and smaller-scale features.

This Special Issue is dedicated to the publication of novel results on the three-dimensional structure and dynamics of vortices in rotating and/or stratified flows. Papers on layer-wise models of vortex dynamics are also invited. Papers focusing on their generation mechanism, stability, evolution, and interactions; on their relation with smaller-scale flows; and on their effects on tracer transport are solicited. Papers should preferably provide elements of mathematical theories in these contexts, but can also rely on extensive numerical modelling or data analysis.

The aim of this issue is to provide readers with an overview of recent progress in this field, with application to the dynamics of planetary oceans and atmospheres.

Prof. Dr. Xavier Carton
Dr. Mikhail Sokolovskiy
Dr. Jean Reinaud
Guest Editors

Manuscript Submission Information

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Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • theoretical and numerical studies of vortex dynamics
  • role of potential vorticity concentrations in rotating and stratified flow dynamics
  • vortex stability and/or evolution under external forcing
  • nonlinear interaction between vortices

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Published Papers (4 papers)

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16 pages, 6379 KB  
Article
Mathematical Modeling of Vortex Interaction Using a Three-Layer Quasigeostrophic Model. Part 2: Finite-Core-Vortex Approach and Oceanographic Application
by Mikhail A. Sokolovskiy, Xavier J. Carton and Boris N. Filyushkin
Mathematics 2020, 8(8), 1267; https://doi.org/10.3390/math8081267 - 2 Aug 2020
Cited by 5 | Viewed by 3261
Abstract
The three-layer version of the contour dynamics/surgery method is used to study the interaction mechanisms of a large-scale surface vortex with a smaller vortex/vortices of the middle layer (prototypes of intrathermocline vortices in the ocean) belonging to the middle layer of a three-layer [...] Read more.
The three-layer version of the contour dynamics/surgery method is used to study the interaction mechanisms of a large-scale surface vortex with a smaller vortex/vortices of the middle layer (prototypes of intrathermocline vortices in the ocean) belonging to the middle layer of a three-layer rotating fluid. The lower layer is assumed to be dynamically passive. The piecewise constant vertical density distribution approximates the average long-term profile for the North Atlantic, where intrathermocline eddies are observed most often at depths of 300–1600 m. Numerical experiments were carried out with different initial configurations of vortices, to evaluate several effects. Firstly, the stability of the vortex compound was evaluated. Most often, it remains compact, but when unstable, it can break as vertically coupled vortex dipoles (called hetons). Secondly, we studied the interaction between a vertically tilted cyclone and lenses. Then, the lenses first undergo anticlockwise rotation determined by the surface cyclone. The lenses can induce alignment or coupling with cyclonic vorticity above them. Only very weak lenses are destroyed by the shear stress exerted by the surface cyclone. Thirdly, under the influence of lens dipoles, the surface cyclone can be torn apart. In particular, the shedding of rapidly moving vortex pairs at the surface reflects the presence of lens dipoles below. More slowly moving small eddies can also be torn away from the main surface cyclone. In this case, they do not appear to be coupled with middle layer vortices. They are the result of large shear-induced deformation. Common and differing features of the vortex interaction, modeled in the framework of the theory of point and finite-core vortices, are noted. Full article
(This article belongs to the Special Issue Vortex Dynamics: Theory and Application to Geophysical Flows)
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13 pages, 4893 KB  
Article
Mathematical Modeling of Vortex Interaction Using a Three-Layer Quasigeostrophic Model. Part 1: Point-Vortex Approach
by Mikhail A. Sokolovskiy, Xavier J. Carton and Boris N. Filyushkin
Mathematics 2020, 8(8), 1228; https://doi.org/10.3390/math8081228 - 26 Jul 2020
Cited by 5 | Viewed by 8347
Abstract
The theory of point vortices is used to explain the interaction of a surface vortex with subsurface vortices in the framework of a three-layer quasigeostrophic model. Theory and numerical experiments are used to calculate the interaction between one surface and one subsurface vortex. [...] Read more.
The theory of point vortices is used to explain the interaction of a surface vortex with subsurface vortices in the framework of a three-layer quasigeostrophic model. Theory and numerical experiments are used to calculate the interaction between one surface and one subsurface vortex. Then, the configuration with one surface vortex and two subsurface vortices of equal and opposite vorticities (a subsurface vortex dipole) is considered. Numerical experiments show that the self-propelling dipole can either be captured by the surface vortex, move in its vicinity, or finally be completely ejected on an unbounded trajectory. Asymmetric dipoles make loop-like motions and remain in the vicinity of the surface vortex. This model can help interpret the motions of Lagrangian floats at various depths in the ocean. Full article
(This article belongs to the Special Issue Vortex Dynamics: Theory and Application to Geophysical Flows)
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19 pages, 751 KB  
Article
On the Effects of Circulation around a Circle on the Stability of a Thomson Vortex N-gon
by Leonid Kurakin and Irina Ostrovskaya
Mathematics 2020, 8(6), 1033; https://doi.org/10.3390/math8061033 - 24 Jun 2020
Cited by 4 | Viewed by 3378
Abstract
The stability problem of the stationary rotation of N identical point vortices is considered. The vortices are located on a circle of radius R 0 at the vertices of a regular N-gon outside a circle of radius R. The circulation [...] Read more.
The stability problem of the stationary rotation of N identical point vortices is considered. The vortices are located on a circle of radius R 0 at the vertices of a regular N-gon outside a circle of radius R. The circulation Γ around the circle is arbitrary. The problem has three parameters N, q, Γ , where q = R 2 / R 0 2 . This old problem of vortex dynamics is posed by Havelock (1931) and is a generalization of the Kelvin problem (1878) on the stability of a regular vortex polygon (Thomson N-gon) on the plane. In the case of Γ = 0 , the problem has already been solved: in the linear setting by Havelock, and in the nonlinear setting in the series of our papers. The contribution of this work to the solution of the problem consists in the analysis of the case of non-zero circulation Γ ≠ 0 . The linearization matrix and the quadratic part of the Hamiltonian are studied for all possible parameter values. Conditions for orbital stability and instability in the nonlinear setting are found. The parameter areas are specified where linear stability occurs and nonlinear analysis is required. The nonlinear stability theory of equilibria of Hamiltonian systems in resonant cases is applied. Two resonances that lead to instability in the nonlinear setting are found and investigated, although stability occurs in the linear approximation. All the results obtained are consistent with those known for Γ = 0 . This research is a necessary step in solving similar problems for the case of a moving circular cylinder, a model of vortices inside an annulus, and others. Full article
(This article belongs to the Special Issue Vortex Dynamics: Theory and Application to Geophysical Flows)
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15 pages, 22078 KB  
Article
Sub-Mesoscale Frontal Instabilities in the Omani Coastal Current
by Mathieu Morvan and Xavier Carton
Mathematics 2020, 8(4), 562; https://doi.org/10.3390/math8040562 - 11 Apr 2020
Cited by 2 | Viewed by 3055
Abstract
The Omani Coastal Current (OCC) flowing northward along the southern coast of Oman during the summer monsoon is associated with an upwelling system. The mesoscale circulation of the western Arabian Sea is dominated by energetic mesoscale eddies down to about 1000 m depth. [...] Read more.
The Omani Coastal Current (OCC) flowing northward along the southern coast of Oman during the summer monsoon is associated with an upwelling system. The mesoscale circulation of the western Arabian Sea is dominated by energetic mesoscale eddies down to about 1000 m depth. They drive the pathways of the upwelling water masses and the Persian Gulf Outflow water. This paper focuses on the sub-mesoscale frontal dynamics in the OCC by analyzing the results from a regional realistic numerical simulation performed with a primitive equation model. Off the Omani coast, the interaction between the upwelling fronts and the mesoscale eddies triggers the frontogenesis at play in the surface mixed layer during the summer monsoon. In spring, sub-mesoscale eddies are generated at the Cape of Ra’s al Hadd due to the horizontal shear instabilities undergone by the OCC. The OCC also drives and elongates Peddies formed during the Summer monsoon and located below the thermocline. Finally, the interaction between mesoscale eddies and the upwelling system leads to the formation of sub-mesoscale eddies at depth through baroclinic instabilities. Full article
(This article belongs to the Special Issue Vortex Dynamics: Theory and Application to Geophysical Flows)
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