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20 pages, 12180 KiB

Open AccessArticle

Clustering of Floating Tracers in a Random Velocity Field Modulated by an Ellipsoidal Vortex Flow

by Konstantin Koshel, Dmitry Stepanov, Nata Kuznetsova and Evgeny Ryzhov

Symmetry 2023, 15(12), 2211; https://0-doi-org.brum.beds.ac.uk/10.3390/sym15122211 - 18 Dec 2023

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The influence of a background vortex flow on the clustering of floating tracers is addressed. The vortex flow considered is induced by an ellipsoidal vortex evolving in a deformation. The system exhibits various vortex motion regimes: (1) a steady state, (2) oscillation and [...] Read more.

The influence of a background vortex flow on the clustering of floating tracers is addressed. The vortex flow considered is induced by an ellipsoidal vortex evolving in a deformation. The system exhibits various vortex motion regimes: (1) a steady state, (2) oscillation and (3) rotation of the ellipsoidal vortex core. The latter two induce an unsteady velocity field for the tracer, thus leading to irregular (chaotic) tracer motion. Superimposing a stochastic divergent velocity field onto the deterministic vortex flow allows us to observe significantly different tracer evolution. An ellipsoidal vortex has ellipsoidal symmetry, and the tracer’s trajectories exhibit the same symmetry inside the vortex. Outside the vortex, the external deformation flow symmetry dominates. Diffusion scattering and chaotic advection give tracers the opportunity to leave the region of ellipsoidal symmetry and form a picture of shear flow symmetry. We use the method of characteristics to integrate the floating tracer density evolution equation and the Euler Ito scheme for obtaining the floating tracer trajectories with a random velocity field. The cluster area and cluster mass from the statistical topography are used as the quantitative diagnostics of a floating tracer’s clustering. For the case of a steady ellipsoidal vortex embedded into the deformation flow with a random velocity field component, we found that the clustering characteristics were weakened by the steady vortex. For the cases of an unsteady ellipsoidal vortex, we observed clustering in the floating tracer density field if the contribution of the divergent component was greater than or equal to that of the rotational (nondivergent) component. Even when the initial floating tracer patch was set on the boundary of the oscillating ellipsoidal vortex, we observed the formation of clusters. In the case of a rotating ellipsoidal vortex, we also observed pronounced clustering. Thus, we argue that unsteady ellipsoidal vortex regimes (oscillation and rotation), which induce chaotic motion of the nearby passive tracer’s trajectories, are still conducive to clustering of floating tracers observed in the density field, despite the intense deformation introduced by strain and shear. Full article

(This article belongs to the Special Issue Geophysical Fluid Dynamics and Symmetry)

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11 pages, 1502 KiB

Open AccessArticle

The Importance of Being Asymmetric for Geophysical Vortices

by Georgi G. Sutyrin

Symmetry 2023, 15(12), 2204; https://0-doi-org.brum.beds.ac.uk/10.3390/sym15122204 - 15 Dec 2023

Viewed by 682

Abstract

Several types of spatial symmetry in vortex structures within rotating stratified fluids are examined by looking at self-propagating configurations in the quasigeostrophic model. The role of symmetry breaking in the dynamics of geophysical waves, vortices and instabilities is highlighted. In particular, the energy [...] Read more.

Several types of spatial symmetry in vortex structures within rotating stratified fluids are examined by looking at self-propagating configurations in the quasigeostrophic model. The role of symmetry breaking in the dynamics of geophysical waves, vortices and instabilities is highlighted. In particular, the energy exchange of the large-scale vertical shear with monopolar and dipolar vortices is analyzed. Various coupled vortex-wave structures are described in terms of wavy and evanescent modes. The Rossby wave radiation is shown to induce a zonal asymmetry, which is needed for the energy support and self-amplification of vortices in large-scale flow. The consequences for the evolution of the most long-lived vortices in the subtropical westward flows are discussed. Full article

(This article belongs to the Special Issue Geophysical Fluid Dynamics and Symmetry)

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14 pages, 2003 KiB

Open AccessFeature PaperArticle

Dynamics of a Circular Foil and Two Pairs of Point Vortices: New Relative Equilibria and a Generalization of Helmholtz Leapfrogging

by Ivan A. Bizyaev and Ivan S. Mamaev

Symmetry 2023, 15(3), 698; https://0-doi-org.brum.beds.ac.uk/10.3390/sym15030698 - 10 Mar 2023

Cited by 2 | Viewed by 968

Abstract

In this paper, we study the plane-parallel motion of a circular foil interacting with two vortex pairs in an infinite volume of an ideal fluid. We assumed that the circulation of the velocity of the fluid around the foil was zero. We showed [...] Read more.

In this paper, we study the plane-parallel motion of a circular foil interacting with two vortex pairs in an infinite volume of an ideal fluid. We assumed that the circulation of the velocity of the fluid around the foil was zero. We showed that the equations of motion possess an invariant submanifold such that the foil performed translational motion and the vortices were symmetric relative to the foil’s direction of motion. A qualitative analysis of the motion on this invariant submanifold was made. New relative equilibria were found, a bifurcation diagram was constructed, and a stability analysis is given. In addition, trajectories generalizing Helmholtz leapfrogging were found where the vortices passed alternately through each other, while remaining at a finite distance from the foil. Full article

(This article belongs to the Special Issue Geophysical Fluid Dynamics and Symmetry)

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18 pages, 1113 KiB

Open AccessArticle

Eady Baroclinic Instability of a Circular Vortex

by Armand Vic, Xavier Carton and Jonathan Gula

Symmetry 2022, 14(7), 1438; https://0-doi-org.brum.beds.ac.uk/10.3390/sym14071438 - 13 Jul 2022

Cited by 1 | Viewed by 1175

Abstract

The stability of two superposed buoyancy vortices is studied linearly in a two-level Surface Quasi-Geostrophic (SQG) model. The basic flow is chosen as two circular vortices with uniform buoyancy, coaxial, and the same radius. A perturbation with a single angular mode is added [...] Read more.

The stability of two superposed buoyancy vortices is studied linearly in a two-level Surface Quasi-Geostrophic (SQG) model. The basic flow is chosen as two circular vortices with uniform buoyancy, coaxial, and the same radius. A perturbation with a single angular mode is added to this mean flow. The SQG equations linearized in perturbation around this basic flow form a two-dimensional ODE for which the normal and singular mode solutions are numerically computed. The instability of these two vortices depends on several parameters. The parameters varied here are: the vertical distance between the two levels and the two values of the vortex buoyancies (called vortex intensity hereafter); the other parameters remain fixed. For normal modes, the system is stable if the levels are sufficiently far from each other vertically, to prevent vertical interactions of the buoyancy patches. Stability is also reached if the layers are close to each other, but if the vortices have very different intensities, again preventing the resonance of Rossby waves around their contours. The system is unstable if the vortex intensities are similar and if the two levels are close to each other. The growth rates of the normal modes increase with the angular wave-number, also corresponding to shorter vertical distances. The growth rates of the singular modes depend more on the distance between the levels than on the ratio of the vortex intensities, at a short time; as expected, they converge towards the growth rates of the normal modes. This study remaining linear does not predict the final evolution of such unstable vortices. This nonlinear evolution will be studied in a sequel of this work. Full article

(This article belongs to the Special Issue Geophysical Fluid Dynamics and Symmetry)

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11 pages, 2544 KiB

Open AccessArticle

On (n,1) Wave Attractors: Coordinates and Saturation Time

by Ilias Sibgatullin, Alexandr Petrov, Xiulin Xu and Leo Maas

Symmetry 2022, 14(2), 319; https://0-doi-org.brum.beds.ac.uk/10.3390/sym14020319 - 04 Feb 2022

Cited by 2 | Viewed by 1567

Abstract

The simplest geometry of the domain, for which internal wave attractors were for the first time investigated both experimentally and numerically, has the shape of a trapezium with one vertical wall and one inclined lateral wall, characterized by two parameters. Using the symmetries [...] Read more.

The simplest geometry of the domain, for which internal wave attractors were for the first time investigated both experimentally and numerically, has the shape of a trapezium with one vertical wall and one inclined lateral wall, characterized by two parameters. Using the symmetries of such a geometry we give an exact solution for the coordinates of the wave attractors with one reflection from each of the lateral boundaries and an integer amount n of reflections from each of the horizontal boundaries. The area of existence for each (n,1) attractor has the form of a triangle in the (d,

τ

) parameter plane, and the shape of this triangle is explicitly given with the help of inequalities or vertices. The expression for the Lyapunov exponents and their connection to the focusing parameters is given analytically. The corresponding direct numerical simulations with low viscosity fully support the analytical results and demonstrate that in bounded domains (n,1) wave attractors can be effective transformers of the global forcing into traveling waves. The saturation time from the state of rest to the final wave regime depends almost linearly on the number of cells, n. Full article

(This article belongs to the Special Issue Geophysical Fluid Dynamics and Symmetry)

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29 pages, 34699 KiB

Open AccessArticle

Vertical Shear Processes in River Plumes: Instabilities and Turbulent Mixing

by Adam Ayouche, Xavier Carton and Guillaume Charria

Symmetry 2022, 14(2), 217; https://0-doi-org.brum.beds.ac.uk/10.3390/sym14020217 - 23 Jan 2022

Cited by 4 | Viewed by 2264

Abstract

In this paper, the problem of vertical shear flow instabilities at the base of a river plume and their consequences in terms of turbulent energy production and mixing is addressed. This study was carried out using 2D non-hydrostatic simulations and a linear stability [...] Read more.

In this paper, the problem of vertical shear flow instabilities at the base of a river plume and their consequences in terms of turbulent energy production and mixing is addressed. This study was carried out using 2D non-hydrostatic simulations and a linear stability analysis. The initial conditions used in these simulations were similar to those observed in river plumes near estuaries. Unstable stratified sheared flows follow three stages of evolution: (i) the generation of billows induced by vertical shear instabilities, (ii) intensification, and (iii) elongation. The elongation of the generated billows is related to the strain intensity, which depends on the physical setting involved (velocity shear, stratification thickness, and bottom slope). Two vertical shear instabilities were found in our study: the Holmboe and Kelvin–Helmholtz instabilities. The Kelvin–Helmholtz instability has a smaller growth time and longer wavelengths; the Holmboe instability is characterized by a longer growth time and shorter wavelengths. The Kelvin–Helmholtz instability is intensified when the bottom is sloped and for large shears. The Holmboe instability is stronger when the stratification thickness is reduced compared to the shear thickness and when the bottom is sloped. For mixing, the flow can be: (i) pre-turbulent, (ii) quasi-turbulent, or (iii) turbulent. The pre-turbulent flow corresponds to more mass mixing than momentum mixing and to more Eddy Kinetic Energy dissipation than Eddy Available Potential Energy dissipation. Such a flow is encountered over a flat bottom whatever the initial shear is. The quasi-turbulent and turbulent flows are reached when the bottom is sloped and when the stratification thickness is reduced. Using turbulent mixing statistics (mixing coefficients, mixing efficiency, Eddy Kinetic Energy, and Eddy Available Potential Energy dissipation rates), we showed that, despite their slow growth, Holmboe instabilities contribute more efficiently to turbulent mixing than Kelvin–Helmholtz instabilities. Holmboe instabilities are the only source of turbulent mixing when sharp density gradients are observed (small buoyancy thickness experiment). Our simulations highlight the contribution of the Holmboe instability to turbulent mixing. Full article

(This article belongs to the Special Issue Geophysical Fluid Dynamics and Symmetry)

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14 pages, 4644 KiB

Open AccessArticle

The Symmetry and Stability of the Flow Separation around a Sphere at Low and Moderate Reynolds Numbers

by Junwei Li and Benmou Zhou

Symmetry 2021, 13(12), 2286; https://0-doi-org.brum.beds.ac.uk/10.3390/sym13122286 - 01 Dec 2021

Cited by 7 | Viewed by 2685

Abstract

The flow separation state reflects the symmetry and stability of flow around spheres. The three-dimensional structures of flow around a rigid sphere at moderate Reynolds number (Re) between 20 and 400 by using finite volume method with adaptive mesh refinement are [...] Read more.

The flow separation state reflects the symmetry and stability of flow around spheres. The three-dimensional structures of flow around a rigid sphere at moderate Reynolds number (Re) between 20 and 400 by using finite volume method with adaptive mesh refinement are presented, and the process of separation angles changing from stable to oscillating state with increasing of Re is analyzed. The results show that the flow is steady, and the separation angles are stable and axisymmetric at Re in less than 200. The flow is unsteady and time-periodic, and the flow separation becomes regular fluctuations and asymmetric at Re = 300, which leads to the nonzero value of lateral force and the phase difference between lift and lateral force. At Re = 400, the flow is unsteady, non-periodic, and asymmetric, as is the flow separation. It’s concluded that the flow separation angle increases when Re increases within a range between 40 and 200. With Re continues to increase, the flow separation state changes from stable to periodically regular until quasi-periodically irregular. The vortex structure changes from no shedding to asymmetric periodic shedding, and finally to asymmetric and intermittently periodic vortex shedding. These results have important implications for the stability of flow around spheres. Full article

(This article belongs to the Special Issue Geophysical Fluid Dynamics and Symmetry)

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Review

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24 pages, 8210 KiB

Open AccessReview

Transport Barriers in Geophysical Flows: A Review

by Sergey Prants

Symmetry 2023, 15(10), 1942; https://0-doi-org.brum.beds.ac.uk/10.3390/sym15101942 - 20 Oct 2023

Cited by 1 | Viewed by 966

Abstract

In the Lagrangian approach, the transport processes in the ocean and atmosphere are studied by tracking water or air parcels, each of which may carry different tracers. In the ocean, they are salt, nutrients, heat, and particulate matter, such as plankters, oil, radionuclides, [...] Read more.

In the Lagrangian approach, the transport processes in the ocean and atmosphere are studied by tracking water or air parcels, each of which may carry different tracers. In the ocean, they are salt, nutrients, heat, and particulate matter, such as plankters, oil, radionuclides, and microplastics. In the atmosphere, the tracers are water vapor, ozone, and various chemicals. The observation and simulation reveal highly complex patterns of advection of tracers in turbulent-like geophysical flows. Transport barriers are material surfaces across which the transport is minimal. They can be classified into elliptic, parabolic, and hyperbolic barriers. Different diagnostics in detecting transport barriers and the analysis of their role in the dynamics of oceanic and atmospheric flows are reviewed. We discuss the mathematical tools, borrowed from dynamical systems theory, for detecting transport barriers in simple kinematic and dynamic models of vortical and jet-like flows. We show how the ideas and methods, developed for simple model flows, can be successfully applied for studying the role of barriers in oceanic and atmospheric flows. Special attention is placed on the significance of transport barriers in important practical issues: anthropogenic and natural pollution, advection of plankton, cross-shelf exchange, and propagation of upwelling fronts in coastal zones. Full article

(This article belongs to the Special Issue Geophysical Fluid Dynamics and Symmetry)

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