Energy-conserving and Hamiltonian low-order models in geophysical fluid dynamics

Gluhovsky, Alexander

doi:10.5194/npg-13-125-2006

Cited by 34 publications

(30 citation statements)

References 75 publications

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“…Like the former, the latter problem may also be difficult since linear models are inappropriate, whereas the multitude of nonlinear models is overwhelming. As way to handle the problem, one might consider physically sound low-order models that possess fundamental properties of fluid dynamical equations (in the spirit of Lorenz, 1963, 1982and Obukhov, 1973, see also Gluhovsky, 2006), thereby infusing more physics into time series analysis. From the perspective of complexity theory, the development of appropriate statistical methods for atmospheric and climate data analysis should be based on time series spawned by the underlying dynamics rather than on traditional time series models (cf., Nicolis and Nicolis, 2007).…”

Section: Discussionmentioning

confidence: 99%

Statistical inference from atmospheric time series: detecting trends and coherent structures

Gluhovsky

2011

Nonlin. Processes Geophys.

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Abstract. Standard statistical methods involve strong assumptions that are rarely met in real data, whereas resampling methods permit obtaining valid inference without making questionable assumptions about the data generating mechanism. Among these methods, subsampling works under the weakest assumptions, which makes it particularly applicable for atmospheric and climate data analyses. In the paper, two problems are addressed using subsampling: (1) the construction of simultaneous confidence bands for the unknown trend in a time series that can be modeled as a sum of two components: deterministic (trend) and stochastic (stationary process, not necessarily an i.i.d. noise or a linear process), and (2) the construction of confidence intervals for the skewness of a nonlinear time series. Non-zero skewness is attributed to the occurrence of coherent structures in turbulent flows, whereas commonly employed linear time series models imply zero skewness.

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Section: Discussionmentioning

confidence: 99%

Statistical inference from atmospheric time series: detecting trends and coherent structures

Gluhovsky

2011

Nonlin. Processes Geophys.

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“…Finally, a LOM for the magnetohydrodynamics of a toroidal confinement device (Chen et al, 1990) can easily be shown to have the form of a gyrostat with three nonlinear terms and one pair of linear terms. It is also notable that Hamiltonian LOMs based on gyrostats inherit the Hamiltonian structure (hence all conservation properties) of the original equations (Gluhovsky, 2006). These examples provide further evidence that coupled gyrostats can be employed in a physically motivated, modular approach to designing LOMs based on spectral Galerkin approximations.…”

Section: Discussionmentioning

confidence: 84%

“…A physicallymotivated, modular approach to designing LOMs using the Galerkin method has been introduced (Gluhovsky, 1982(Gluhovsky, , 1986Gluhovsky and Tong, 1999;Gluhovsky et al, 2002;Gluhovsky, 2006). This approach is based on designing LOMs so that their mathematical equations have a structure isomorphic to those for systems known in mechanics as coupled gyrostats, with friction and forcing.…”

Section: Introductionmentioning

confidence: 99%

Gyrostatic extensions of the Howard-Krishnamurti model of thermal convection with shear

Tong

Gluhovsky

2008

Nonlin. Processes Geophys.

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Abstract. The Howard & Krishnamurti (1986) low-order model (LOM) of Rayleigh-Bénard convection with spontaneous vertical shear can be extended to incorporate various additional physical effects, such as externally forced vertical shear and magnetic field. Designing such extended LOMs so that their mathematical structure is isomorphic to those of systems of coupled gyrostats, with damping and forcing, allows for a modular approach while respecting conservation laws. Energy conservation (in the limit of no damping and forcing) prevents solutions that diverge to infinity, which are present in the original Howard & Krishnamurti LOM. The first LOM developed here (as a candidate model of transverse rolls) involves adding a new Couette mode to represent externally forced vertical shear. The second LOM is a modification of the Lantz (1995) model for magnetoconvection with shear. The modification eliminates an invariant manifold in the original model that leads to potentially unphysical behavior, namely solutions that diverge to infinity, in violation of energy conservation. This paper reports the first extension of the coupled gyrostats modeling framework to incorporate externally forced vertical shear and magnetoconvection with shear. Its aim is to demonstrate better model building techniques that avoid pathologies present in earlier models; consequently we do not focus here on analysis of dynamics or model validation.

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“…The problem of energy conservation was solved in [50] where a universal criterion for the truncation to energy-conserving finite-mode models was established. Moreover, truncation to systems in coupled gyrostats form [18,19] may also lead to models that retain the conservation properties of the original equations. We also note, that a single gyrostat is a Nambu system and hence using such a truncation, conservation of the underlying geometry may be implemented at least in some minimal form.…”

Section: Maximum Simplificationmentioning

confidence: 99%

Minimal atmospheric finite-mode models preserving symmetry and generalized Hamiltonian structures

Bihlo

Staufer

2011

Physica D: Nonlinear Phenomena

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A typical problem with the conventional Galerkin approach for the construction of finitemode models is to keep structural properties unaffected in the process of discretization. We present two examples of finite-mode approximations that in some respect preserve the geometric attributes inherited from their continuous models: a three-component model of the barotropic vorticity equation known as Lorenz' maximum simplification equations [Tellus, 12, 243-254 (1960)] and a six-component model of the two-dimensional Rayleigh-Bénard convection problem. It is reviewed that the Lorenz-1960 model respects both the maximal set of admitted point symmetries and an extension of the noncanonical Hamiltonian form (Nambu form). In a similar fashion, it is proved that the famous Lorenz-1963 model violates the structural properties of the Saltzman equations and hence cannot be considered as the maximum simplification of the Rayleigh-Bénard convection problem. Using a six-component truncation, we show that it is again possible retaining both symmetries and the Nambu representation in the course of discretization. The conservative part of this six-component reduction is related to the Lagrange top equations. Dissipation is incorporated using a metric tensor.

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Energy-conserving and Hamiltonian low-order models in geophysical fluid dynamics

Cited by 34 publications

References 75 publications

Statistical inference from atmospheric time series: detecting trends and coherent structures

Statistical inference from atmospheric time series: detecting trends and coherent structures

Gyrostatic extensions of the Howard-Krishnamurti model of thermal convection with shear

Minimal atmospheric finite-mode models preserving symmetry and generalized Hamiltonian structures

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