Finite Amplitude Free Convection as an Initial Value Problem—I

Saltzman, Barry

doi:10.1175/1520-0469(1962)019<0329:fafcaa>2.0.co;2

Cited by 473 publications

(272 citation statements)

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“…This model is derived through projection of the governing equations onto an orthonormal basis and subsequent low-order truncation, following the approach of Platzman (1960), Saltzman (1962), Lorenz (1963), and many others since, such as Lorenz (1982), Legras and Ghil (1985), Ghil and Childress (1987), Olbers (2001), andCrucifix (2012). S. Pierini and colleagues had already investigated this decadal variability with a relatively realistic primitive equation ocean model (Pierini 2006;Pierini et al 2009;Pierini 2010Pierini , 2014b.…”

Section: Model Description a Model Formulationmentioning

confidence: 99%

Exploring the Pullback Attractors of a Low-Order Quasigeostrophic Ocean Model: The Deterministic Case

2016

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A low-order quasigeostrophic double-gyre ocean model is subjected to an aperiodic forcing that mimics time dependence dominated by interdecadal variability. This model is used as a prototype of an unstable and nonlinear dynamical system with time-dependent forcing to explore basic features of climate change in the presence of natural variability. The study relies on the theoretical framework of nonautonomous dynamical systems and of their pullback attractors (PBAs), that is, of the time-dependent invariant sets attracting all trajectories initialized in the remote past. The existence of a global PBA is rigorously demonstrated for this weakly dissipative nonlinear model. Ensemble simulations are carried out and the convergence to PBAs is assessed by computing the probability density function (PDF) of localization of the trajectories. A sensitivity analysis with respect to forcing amplitude shows that the PBAs experience large modifications if the underlying autonomous system is dominated by small-amplitude limit cycles, while less dramatic changes occur in a regime characterized by large-amplitude relaxation oscillations. The dependence of the attracting sets on the choice of the ensemble of initial states is then analyzed. Two types of basins of attraction coexist for certain parameter ranges; they contain chaotic and nonchaotic trajectories, respectively. The statistics of the former does not depend on the initial states whereas the trajectories in the latter converge to small portions of the global PBA. This complex scenario requires separate PDFs for chaotic and nonchaotic trajectories. General implications for climate predictability are finally discussed.

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Section: Model Description a Model Formulationmentioning

confidence: 99%

Exploring the Pullback Attractors of a Low-Order Quasigeostrophic Ocean Model: The Deterministic Case

2016

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“…These equations are simplified by introducing the stream function in place of the two velocity components. Saltzman (1961) and Lorenz (1963) …”

Section: The Lorenz Modelmentioning

confidence: 99%

Statistical Mechanics of Nonequilbrium Liquids

Morriss¹,

Evans²

2007

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“…In particular, we demonstrate that for a simple system (three modes) the equations governing these instabilities are exactly the same as those that govern the Rayleigh-Benard instability [Saltzman, 1962;Lorenz, 1963J. The analogy between these two instabilities is the following.…”

mentioning

confidence: 59%

“…show that the equations correspond exactly to the Lorenz equations which approximately describe the Rayleigh-Benard instability (Saltzman, 1962;Lorenz, 1963]. Following the analysis of Lorenz (1963), it is shown that the three mode system can exhibit a strange attractor with chaotic behavior [Ruelle and Takens, 1971].…”

Section: Introductionmentioning

confidence: 66%

Ionospheric turbulence: Interchange instabilities and chaotic fluid behavior

Huba

Hassam

Schwartz

et al. 1985

Geophysical Research Letters

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We develop a set of mode coupling equations which describe the nonlinear evolution of the Rayleigh‐Taylor and gradient drift instabilities which are relevant to the ionosphere. We show that for a three mode system, the nonlinear equations describing these instabilities correspond exactly to the Lorenz equations which approximately describe the Rayleigh‐Benard instability. It is shown that the three mode system can exhibit a strange attractor with chaotic behavior. Ion inertia plays a critical role in this phenomenon in that if it is neglected, as in the collisional limit, the three mode system does not exhibit chaos and a stable convection pattern results.

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Finite Amplitude Free Convection as an Initial Value Problem—I

Cited by 473 publications

References 0 publications

Exploring the Pullback Attractors of a Low-Order Quasigeostrophic Ocean Model: The Deterministic Case

Exploring the Pullback Attractors of a Low-Order Quasigeostrophic Ocean Model: The Deterministic Case

Statistical Mechanics of Nonequilbrium Liquids

Ionospheric turbulence: Interchange instabilities and chaotic fluid behavior

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