Quasigeostrophic Ellipsoidal Vortices in a Two-Dimensional Strain Field

Hashimoto, Hideki; Shimonishi, Tomoyuki; Miyazaki, Takeshi

doi:10.1143/jpsj.68.3863

Cited by 17 publications

(17 citation statements)

References 12 publications

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“…The stability of QG ellipsoidal vortices has been studied for some special background flows, e.g. a pure horizontal strain flow (Meacham et al 1994) and a pure shear flow for which the strain rate equals the background rotation (Hashimoto et al 1999). McKiver & Dritschel (2006) studied the combined effects of horizontal strain and vertical shear on QG ellipsoidal vortices, analysing their linear stability to both ellipsoidal and non-ellipsoidal disturbances.…”

Section: Introductionmentioning

confidence: 99%

Ellipsoidal vortices in rotating stratified fluids: beyond the quasi-geostrophic approximation

Tsang

Dritschel

2014

J. Fluid Mech.

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We examine the basic properties and stability of isolated vortices having uniform potential vorticity in a non-hydrostatic rotating stratified fluid, under the Boussinesq approximation. For simplicity, we consider a uniform background rotation and a linear basic-state stratification for which both the Coriolis and buoyancy frequencies, f and N , are constant. Moreover, we take f /N ≪ 1, as typically observed in the Earth's atmosphere and oceans. In the small Rossby number 'quasigeostrophic' limit, when the flow is weak compared to the background rotation, there exists exact solutions for steadily-rotating ellipsoidal volumes of uniform potential vorticity in an unbounded flow (Zhmur & Shchepetkin 1991;Meacham 1992). Furthermore, a wide range of these solutions are stable so long as the horizontal and vertical aspect ratios λ and µ do not depart greatly from unity (Dritschel et al. 2005). In the present study, we examine the behaviour of ellipsoidal vortices at Rossby numbers up to near unity in magnitude. We find that there is a monotonic increase in stability as one varies the Rossby number from nearly −1 (anticyclone) to nearly +1 (cyclone). That is, quasi-geostrophic vortices are more stable than anticyclones at finite (negative) Rossby number, and generally less stable than cyclones at finite (positive) Rossby number. Ageostrophic effects strengthen both the rotation and the stratification within a cyclone, enhancing its stability. The converse is true for an anticyclone. For all Rossby numbers, stability is reinforced by increasing λ towards unity or decreasing µ. An unstable vortex often restabilises by developing a near-circular cross section, typically resulting in a roughly ellipsoidal vortex, but occasionally a binary system is formed. Throughout the nonlinear evolution of a vortex, the emission of inertiagravity waves is negligible across the entire parameter space investigated. Thus, vortices at small to moderate Rossby numbers, and any associated instabilities are (ageostrophically) balanced. A manifestation of this balance is that, at finite Rossby number, an anticyclone rotates faster than a cyclone.

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Section: Introductionmentioning

confidence: 99%

Ellipsoidal vortices in rotating stratified fluids: beyond the quasi-geostrophic approximation

Tsang

Dritschel

2014

J. Fluid Mech.

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“…McKiver & Dritschel (2003) revisited the topic, proposing a new set of prognostic variables which are used in the ELM. Note that in an independent effort Hashimoto, Shimonishi & Miyazaki (1999), Miyazaki, Ueno & Shimonishi (1999) and Miyazaki, Furuichi & Takahashi (2001) proposed a 'moment expansion ellipsoidal model' to study the interaction between ellipsoids. Our own dynamical approach differs from theirs.…”

Section: Introductionmentioning

confidence: 99%

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

Reinaud

Dritschel

2005

J. Fluid Mech.

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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.

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“…Recently, these ellipsoidal solutions have been studied by numerous authors, cf. Meacham et al (1994), Meacham, Morrison & Flierl (1997), Miyazaki, Ueno & Shimonishi (1999), Hashimoto, Shimonishi &McKiver &.…”

Section: Introductionmentioning

confidence: 99%

The quasi-geostrophic ellipsoidal vortex model

2004

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We present a simple approximate model for studying general aspects of vortex interactions in a rotating stably-stratified fluid. The model idealizes vortices by ellipsoidal volumes of uniform potential vorticity, a materially conserved quantity in an inviscid, adiabatic fluid. Each vortex thus possesses 9 degrees of freedom, 3 for the centroid and 6 for the shape and orientation. Here, we develop equations for the time evolution of these quantities for a general system of interacting vortices. An isolated ellipsoidal vortex is well known to remain ellipsoidal in a fluid with constant background rotation and uniform stratification, as considered here. However, the interaction between any two ellipsoids in general induces weak non-ellipsoidal perturbations. We develop a unique projection method, which follows directly from the Hamiltonian structure of the system, that effectively retains just the part of the interaction which preserves ellipsoidal shapes. This method does not use a moment expansion, e.g. local expansions of the flow in a Taylor series. It is in fact more general, and consequently more accurate. Comparisons of the new model with the full equations of motion prove remarkably close.

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Quasigeostrophic Ellipsoidal Vortices in a Two-Dimensional Strain Field

Cited by 17 publications

References 12 publications

Ellipsoidal vortices in rotating stratified fluids: beyond the quasi-geostrophic approximation

Ellipsoidal vortices in rotating stratified fluids: beyond the quasi-geostrophic approximation

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

The quasi-geostrophic ellipsoidal vortex model

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