The Ellipsoidal Vortex: A Novel Approach to Geophysical Turbulence

McKiver, William J.

doi:10.1155/2015/613683

Cited by 15 publications

(22 citation statements)

References 35 publications

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“…This implies that the distributions of PVA and vertical displacement of isopycnals are not steady but deform and rotate, anticyclonically if

ϕ_{0} > 0

, with some phase speed. Rotation of three‐dimensional ellipsoidal vortices is a well‐known result in geophysical flows (McKiver, ; Meacham, ). This deformation and rotation is not accounted for in this synoptic vortex model, which assumes that the geopotential (1) is valid only at a fixed time.…”

Section: The Octupolar W Distribution Of the Ellipsoidal Upright Vortexmentioning

confidence: 99%

“…Thus, one of the simplest baroclinic vortex models with vertical velocity results from removing the circular symmetry constraint and replacing it with an elliptical distribution, while preserving the rotation axis vertically oriented condition. This leads to the unsteady elliptical vortex with horizontal semiaxes

A_{normalx} \neq A_{normaly}

(see Meacham () for quasigeostrophic (QG) dynamics, and McKiver () and McKiver and Dritschel () for results beyond QG dynamics, and references therein). The vertical velocity field of this vortex archetype, herein referred to as the elliptical upright vortex, has an octupolar three‐dimensional distribution (Viúdez & Dritschel, ) and vanishes at the vortex equator plane z = 0.…”

Section: Introductionmentioning

confidence: 99%

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Two Modes of Vertical Velocity in Subsurface Mesoscale Eddies

Viúdez

2018

JGR Oceans

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Two fundamental modes of vertical velocity (w) in mesoscale subsurface eddies are described using the quasigeostrophic (QG) approximation and nonhydrostatic numerical modeling. The first mode of w (the spheroidal mode) arises when a spheroidal upright subsurface eddy acquires horizontal eccentricity and becomes an ellipsoid, still upright, vertically symmetric vortex. In this case, the vertical displacement of isopycnals vanishes at the middepth z 5 0. Conservation of potential vorticity anomaly (PVA) on elliptical concave/convex isopycnals entails a three-dimensional octupolar pattern of w which also vanishes at z 5 0. The second mode of w (the tilted mode) arises when the eddy remains spheroidal but its vertical axis tilts relative to the vertical direction. In this case, the displacement of isopycnals is largest at the middepth z 5 0 and has a dipolar distribution. The associated w is largest at the middepth and develops also a dipolar pattern. In both spheroidal and tilted modes, the vertical velocity pattern may be inferred from the fast advection of PVA conserving fluid particles on slower translating concave/convex or tilted isopycnals. This implies that the vertical velocity of both modes is approximately QG and may be correctly inferred from the QG omega equation as long as the Rossby number remains small. Under more general circumstances, the vortex is both spheroidal and tilted. In this case, both spheroidal and tilted modes coexist but remain, to a large extent, uncoupled, rotating with different and, at least at a first order of approximation, constant phase speeds.Plain Language Summary The vertical velocity in ocean eddies depends on how the fluid particles depart from plane horizontal circular trajectories. We describe two basic ways by which this departure may take place. In the first way, the fluid particles at different depths rotate around the same vertical axis, but their trajectories are no longer circles but almost ellipses on concave or convex ellipsoidal surfaces of constant density. In the second way, the fluid particles at different depths describe almost circles on concave or convex spheroidal surfaces of constant density, but their axis of rotation is no longer vertical but tilted. These two different ways of vertical motion may coexist in the same vortex so that the vertical velocity of the fluid particles is the sum of these two ways of motion.

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“…This implies that the distributions of PVA and vertical displacement of isopycnals are not steady but deform and rotate, anticyclonically if

ϕ_{0} > 0

Section: The Octupolar W Distribution Of the Ellipsoidal Upright Vortexmentioning

confidence: 99%

A_{normalx} \neq A_{normaly}

Section: Introductionmentioning

confidence: 99%

Two Modes of Vertical Velocity in Subsurface Mesoscale Eddies

Viúdez

2018

JGR Oceans

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“…This question was answered in the affirmative by Meacham Meacham (1992) and by Zhmur and Shchepetkin (1991) who discovered a class of three dimensional solutions to the QG equations in which the PV takes a uniform value inside an ellipsoidal domain. The discovery of these exact single-ellipsoid solutions originated a large literature concerned with using these solutions to study two or more interacting vortices (see Miyazaki et al 2001, Dritschel et al 2004, McKiver 2015. A particularly important research topic has been to determine the conditions under which ellipsoidal vortices can merge, as discussed in Reinaud and Dritschel (2005).…”

Section: Introductionmentioning

confidence: 99%

Models of interacting pairs of thin, quasi-geostrophic vortices: steady-state solutions and nonlinear stability

Bersanelli

Dritschel

Lancellotti

et al. 2016

Geophysical & Astrophysical Fluid Dynamics

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We study pairwise interactions of elliptical quasi-geostrophic (QG) vortices as the limiting case of vanishingly thin uniform potential vorticity ellipsoids. In this limit, the product of the vertical extent of the ellipsoid and the potential vorticity within it is held fixed to a finite non-zero constant. Such elliptical "lenses" inherit the property that, in isolation, they steadily rotate without changing shape. Here, we use this property to extend both standard moment models and Hamiltonian ellipsoidal models to approximate the dynamical interaction of such elliptical lenses. By neglecting nonelliptical deformations, the simplified models reduce the dynamics to just four degrees of freedom per vortex. For simplicity, we focus on pairwise interactions between identical elliptical vortices initially separated in both the horizontal and vertical directions. The dynamics of the simplified models are compared with the full QG dynamics of the system, and show good agreement as expected for sufficiently distant lenses. The results reveal the existence of families of steadily rotating equilibria in the initial horizontal and vertical separation parameter space. For sufficiently large vertical separations, equilibria with varying shape exist for all horizontal separations. Below a critical vertical separation (stretched by the constant ratio of buoyancy to Coriolis frequencies N/f ), comparable to the mean radius of either vortex, a gap opens in horizontal separation where no equilibria are possible. Solutions near the edge of this gap are unstable. In the full QG system, equilibria at the edge of the gap exhibit corners (infinite curvature) along their boundaries. Comparisons of the model results with the full nonlinear QG evolution show that the early stages of the instability are captured by the Hamiltonian elliptical model but not by the moment model that inaccurately estimates shorter-range interactions. ARTICLE HISTORY

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“…This results in the appearance of exponentially diverging trajectories in the unsteady velocity field governing the fluid particle advection. Similar chaotic dynamics is present in the case of the ellipsoid vortex model (Zhmur et al, 2011;Koshel et al, 2013Koshel et al, , 2015, which is a generalization of the elliptic vortex model taking into account a linear vertical stratification (Zhmur and Pankratov, 1989;Meacham et al, 1994;Dritschel, 2011;McKiver, 2015;McKiver and Dritschel, 2016).…”

Section: Introductionmentioning

confidence: 58%

Parametric resonance in the dynamics of an elliptic vortex in a periodically strained environment

Koshel

Ryzhov

2017

Nonlin. Processes Geophys.

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Abstract. The model of an elliptic vortex evolving in a periodically strained background flow is studied in order to establish the possible unbounded regimes. Depending on the parameters of the exterior flow, there are three classical regimes of the elliptic vortex motion under constant linear deformation: (i) rotation, (ii) nutation, and (iii) infinite elongation. The phase portrait for the vortex dynamics features critical points which correspond to the stationary vortex not changing its form and orientation. We demonstrate that, given superimposed periodic oscillations to the exterior deformation, the phase space region corresponding to the elliptic critical point experiences parametric instability leading to locally unbounded dynamics of the vortex. This dynamics manifests itself as the vortex nutates along the strain axis while continuously elongating. This motion continues until nonlinear effects intervene near the region associated with the steadystate separatrix. Next, we show that, for specific values of the perturbation parameters, the parametric instability is effectively suppressed by nonlinearity in the primal parametric instability zone. The secondary zone of the parametric instability, on the contrary, produces an effective growth of the vortex's aspect ratio.

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The Ellipsoidal Vortex: A Novel Approach to Geophysical Turbulence

Cited by 15 publications

References 35 publications

Two Modes of Vertical Velocity in Subsurface Mesoscale Eddies

Two Modes of Vertical Velocity in Subsurface Mesoscale Eddies

Models of interacting pairs of thin, quasi-geostrophic vortices: steady-state solutions and nonlinear stability

Parametric resonance in the dynamics of an elliptic vortex in a periodically strained environment

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