The Global Geometry of the Slow Manifold in the Lorenz–Krishnamurthy Model

Camassa, Roberto; Tin, Siu Kei

doi:10.1175/1520-0469(1996)053<3251:tggots>2.0.co;2

Cited by 18 publications

(9 citation statements)

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“…There are many indications that atmospheric and oceanic flows characterized by an O(1) Rossby number and initially in a state of 'balance' -here meaning a hypothetical state void of inertia-gravity waves (IGWs) -cannot remain that way owing to the spontaneous generation of small-amplitude IGWs (Lorenz & Krishnamurthy 1987;Bokhove & Shepherd 1996;Camassa & Tin 1996;Warn 1997;Ford, McIntyre & Norton 2000;Plougonven & Zeitlin 2002;Plougonven, Teitelbaum & Zeitlin 2003;Vanneste & Yavneh 2004;Vanneste 2004). At O(1) Rossby numbers, however, there is no consensus on the exact meaning of the terms geostrophic adjustment, Lighthill emission, and spontaneous-adjustment emission (SAE; see Ford et al 2000;Saujani & Shepherd 2002).…”

Section: Introductionmentioning

confidence: 99%

Spontaneous generation of inertia–gravity wave packets by balanced geophysical flows

Viúdez¹,

Dritschel²

2006

J. Fluid Mech.

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The generation and propagation of a packet of small-amplitude inertia-gravity waves (IGWs) in a rotating stratified balanced flow is described. The initially balanced geophysical flow is an unstable baroclinic jet which breaks up into a street of cyclonic and anticyclonic vortices. The small-amplitude unbalanced component of the flow is extracted from the large-amplitude mesoscale balanced flow using the optimal potential vorticity balance approach. This analysis reveals that during the instability the balanced flow spontaneously emits bursts of IGWs. The emission occurs along two directions, into and out of the anticyclonic vortices. The inward-waves remain trapped inside the vortices while the outward-waves propagate away from them as a packet of small-amplitude IGWs with a three-dimensional helical structure. The wave packet emission is confirmed for different spatial resolutions (128 3 , 160 3 , 192 3 and 256 3 grid points). The ratio between the balanced vertical and horizontal velocity components is of the order of 10 −3 , as is typical of mesoscale geophysical flows. The ratio between the unbalanced vertical and horizontal components is about 0.1. Since the unbalanced horizontal and the balanced vertical velocity components are of similar magnitude, the vertical velocity of the IGWs is about 10 −4 times the balanced horizontal velocity. The IGWs are dominated by frequencies close to the inertial frequency and have a clockwise-rotating horizontal velocity, similar to plane wave solutions.

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Section: Introductionmentioning

confidence: 99%

Spontaneous generation of inertia–gravity wave packets by balanced geophysical flows

Viúdez¹,

Dritschel²

2006

J. Fluid Mech.

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“…The main result of this paper, Theorem 1, and its extensions presented in Section 10, consists of a tool that, for systems of the form (2.1), is comparable in its applicability and efficiency to the celebrated Melnikov method. As such, it is useful both for solving applied problems [6,7,38], and for unifying existing perturbation-theoretic approaches to detection of homoclinic and heteroclinic orbits in near-integrable systems. In this latter vein, this paper already develops a unifying approach to the results of references [6,59,23,32].…”

Section: Discussionmentioning

confidence: 99%

“…7) and soQ l (T ) − Q u (T ) = O (ε α ), P l (T ) − P u (T ) = O (ε α ). (7.8) Consider the function N(a, b, I ) = N ε (a ε , b ε , I ε , ψ ε ) = H (x, I ) − F (I ), where F (I ) = H (X(I ), I ).…”

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confidence: 99%

A Melnikov Method for Homoclinic Orbits with Many Pulses

Camassa

Kovačič

Tin

1998

Archive for Rational Mechanics and Analysis

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We present an extension of the Melnikov method which can be used for ascertaining the existence of homoclinic and heteroclinic orbits with many pulses in a class of near-integrable systems. The Melnikov function in this situation is the sum of the usual Melnikov functions evaluated with some appropriate phase delays. We show that a nonfolding condition which involves the logarithmic derivative of the Melnikov function must be satisfied in addition to the usual transversality conditions in order for homoclinic orbits with more than one pulse to exist.

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“…This system describes coupled Rossby waves and gravity waves. It was mainly investigated from the existence of a slow manifold point of view [2][3][4][5]. Among other studies regarding 5-dimensional Lorenz system we mention Hamiltonian structure [6], chaotic behaviour [7][8][9], and analytic integrability [10].…”

Section: Introductionmentioning

confidence: 99%