Hamiltonian moment reduction for describing vortices in shear

Meacham, S. P.; Morrison, Philip; Flierl, Glenn R.

doi:10.1063/1.869352

Cited by 54 publications

(70 citation statements)

References 22 publications

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“…II) is somewhat different in flavor from the others. Imagine that you have succeeded in obtaining a finite Hamiltonian system out of some fluid model, the Kida vortex being a good example (see, for example, Meacham et al, 1997). What should you expect of the dynamics?…”

Section: Introductionmentioning

confidence: 99%

Hamiltonian description of the ideal fluid

1998

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The Hamiltonian viewpoint of fluid mechanical systems with few and infinite number of degrees of freedom is described. Rudimentary concepts of finite-degree-of-freedom Hamiltonian dynamics are reviewed, in the context of the passive advection of a scalar or tracer field by a fluid. The notions of integrability, invariant-tori, chaos, overlap criteria, and invariant-tori breakup are described in this context. Preparatory to the introduction of field theories, systems with an infinite number of degrees of freedom, elements of functional calculus and action principles of mechanics are reviewed. The action principle for the ideal compressible fluid is described in terms of Lagrangian or material variables. Hamiltonian systems in terms of noncanonical variables are presented, including several examples of Eulerian or inviscid fluid dynamics. Lie group theory sufficient for the treatment of reduction is reviewed. The reduction from Lagrangian to Eulerian variables is treated along with Clebsch variable decompositions. Stability in the canonical and noncanonical Hamiltonian contexts is described. Sufficient conditions for stability, such as Rayleigh-like criteria, are seen to be only sufficient in the general case because of the existence of negative-energy modes, which are possessed by interesting fluid equilibria. Linearly stable equilibria with negative energy modes are argued to be unstable when nonlinearity or dissipation is added. The energy-Casimir method is discussed and a variant of it that depends upon the notion of dynamical accessibility is described. The energy content of a perturbation about a general fluid equilibrium is calculated using three methods.[S0034-6861(98)00102-0] CONTENTS

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Section: Introductionmentioning

confidence: 99%

Hamiltonian description of the ideal fluid

1998

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“…Meacham et al (1994) further generalised the ellipsoidal model to include a uniform background horizontal and vertical shear. Meacham et al (1997) showed that the time-dependent vortex motion (which is potentially chaotic) can be described exactly in terms of a finite degreeof-freedom model obtained from the method of 'Hamiltonian moment reduction'. The stability of QG ellipsoidal vortices has been studied for some special background flows, e.g.…”

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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“…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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Hamiltonian moment reduction for describing vortices in shear

Cited by 54 publications

References 22 publications

Hamiltonian description of the ideal fluid

Hamiltonian description of the ideal fluid

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

The quasi-geostrophic ellipsoidal vortex model

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