Crude closure for flow with topography through large-scale statistical theory

Grote, Marcus J.; Majda, Andrew J.

doi:10.1088/0951-7715/13/3/304

Cited by 7 publications

(11 citation statements)

References 26 publications

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“…11 represents the large-scale structure resulting from an inverse cascade process. Such a perspective is confirmed in a variety of flow regimes for various idealized models by comparing crude closure schemes based on equilibrium statistical theory to direct numerical simulations with random small-scale forcing (16)(17)(18). This viewpoint contrasts with other numerical studies of equilibrium statistical theories that emphasize freely decaying turbulence (12,19).…”

Section: The Equilibrium Statistical Theorymentioning

confidence: 89%

Statistical equilibrium predictions of jets and spots on Jupiter

Turkington

Majda

Haven

et al. 2001

Proc. Natl. Acad. Sci. U.S.A.

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An equilibrium statistical theory of coherent structures is applied to midlatitude bands in the northern and southern hemispheres of Jupiter. The theory imposes energy and circulation constraints on the large-scale motion and uses a prior distribution on potential vorticity fluctuations to parameterize the small-scale turbulent eddies. Nonlinearly stable coherent structures are computed by solving the constrained maximum entropy principle governing the equilibrium states of the statistical theory. The theoretical predictions are consistent with the observed large-scale features of the weather layer if and only if the prior distribution has anticyclonic skewness, meaning that intense anticyclones predominate at small scales. Then the computations show that anticyclonic vortices emerge at the latitudes of the Great Red Spot and the White Ovals in the southern band, whereas in the northern band no vortices form within the zonal jets. Recent observational data from the Galileo mission support the occurrence of intense small-scale anticyclonic forcing. The results suggest the possibility of using equilibrium statistical theory for inverse modeling of the smallscale characteristics of the Jovian atmosphere from observed features.P rominent examples of long-lived large-scale vortices in geophysical flows are those observed on the Jovian planets, such as the Great Red Spot (GRS) on Jupiter (1-3). The emergence and persistence of such coherent structures at specific latitudes, such as 22.4°S for the GRS, in a background zonal shear flow that seems to violate all of the standard stability criteria (1) are a genuine puzzle needing a theoretical explanation. The present article contributes to such an explanation by using a recent equilibrium statistical theory (4-6) to predict the coherent structures in the weather layer of Jupiter. This statistical theory is based on a few judiciously chosen dynamical invariants and does not involve any detailed resolution of the fluid dynamics. An important input to the theory is therefore a prior probability distribution for the one-point statistics of the potential vorticity, which parameterizes the unresolved small-scale turbulent eddies that produce the large-scale coherent structures. Below it is demonstrated that the equilibrium states of the statistical theory simultaneously have three key properties:(i) Coherent monopolar vortices, such as the GRS, emerge at the appropriate latitudes within the zonal mean velocity profile, such as the Limaye profile derived from Voyager data.(ii) The coherent vortices are anticyclones if and only if the prior distribution on potential vorticity fluctuations has anticyclonic skewness.(iii) All steady flows realized as equilibrium states are nonlinearly stable, including zonal shear flows that contain prograde and retrograde jets and embedded vortices.Here the theory is developed for a one and one-half layer quasigeostrophic model, which yields the limiting behavior of a corresponding shallow-water model in a standard fashion (7,8). The one and one-...

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Section: The Equilibrium Statistical Theorymentioning

confidence: 89%

Statistical equilibrium predictions of jets and spots on Jupiter

Turkington

Majda

Haven

et al. 2001

Proc. Natl. Acad. Sci. U.S.A.

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“…The crude closure numerical algorithms give a variety of interesting new behaviors when such geophysical effects are included [10,24,25]. There is a recent statistical theory that successfully predicts the Great Red Spot of Jupiter in a fashion consistent with the observations from the Galileo and Voyager missions [35,47].…”

Section: Conclusion and Remarksmentioning

confidence: 91%

“…Numerical simulation in the regime of weak forcing and weak damping (see Figures 1.2, 1.3, and 1.4, and also [24,25]) indicates the emergence and persistence of a large coherent structure. More precisely, numerical experiments demonstrate that the flow field reaches a quasi-equilibrium state in terms of energy (Figure 1.2), enstrophy, circulation, etc., and the contour plot of the vorticity field looks like a large vortex plus small (random) perturbation (see Figures 1.3 and 1.4).…”

Section: Introductionmentioning

confidence: 89%

“…But it would make sense to test if the statistical theories can be applied to the weakly forced and damped case where the situation is close to the case of an inviscid, unforced environment and the flow is in a quasi-equilibrium state. This hypothesis has been tested by comparing numerical experiments and equilibrium statistical predictions in a series of papers [10,24,25,33].…”

Section: Introductionmentioning

confidence: 99%

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The emergence of large‐scale coherent structure under small‐scale random bombardments

Majda

Wang

2005

Comm Pure Appl Math

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We provide mathematical justification of the emergence of large-scale coherent structure in a two-dimensional fluid system under small-scale random bombardments with small forcing and appropriate scaling assumptions. The analysis shows that the large-scale structure emerging out of the small-scale random forcing is not the one predicted by equilibrium statistical mechanics. But the error is very small, which explains earlier successful prediction of the large-scale structure based on equilibrium statistical mechanics.

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“…In the case of the large scales of geophysical flows, recent interesting works have used numerical simulations, for instance to study the limits of second order cumulants expansion [132], or to study numerically a self consistent closure describing the coupling of the mean flow and of the second order cumulant [77]. Another line of research, on related issues, has been to search for crude closures [88,89], or more precise mathematical results [130], when the system is subjected to random bombardments In all of the previous works, some hypothesis of a phenomenological nature are made in order to simplify the problem at some point (closure without small parameter, assumption of scale separations, Markovianization), that allows interesting studies to be pushed forward. However, there still remains a lot of work to assess either numerically or theoretically the validity or not of these hypothesis, and thus to be really able to propose a clear theory of the large scales of two dimensional and geophysical flows.…”

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confidence: 99%

Statistical mechanics of two-dimensional and geophysical flows

Bouchet¹,

Venaille²

2012

Physics Reports

256

329

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The theoretical study of the self-organization of two-dimensional and geophysical turbulent flows is addressed based on statistical mechanics methods. This review is a self-contained presentation of classical and recent works on this subject; from the statistical mechanics basis of the theory up to applications to Jupiter's troposphere and ocean vortices and jets. Emphasize has been placed on examples with available analytical treatment in order to favor better understanding of the physics and dynamics.After a brief presentation of the 2D Euler and quasi-geostrophic equations, the specificity of two-dimensional and geophysical turbulence is emphasized. The equilibrium microcanonical measure is built from the Liouville theorem. Important statistical mechanics concepts (large deviations, mean field approach) and thermodynamic concepts (ensemble inequivalence, negative heat capacity) are briefly explained and described.On this theoretical basis, we predict the output of the long time evolution of complex turbulent flows as statistical equilibria. This is applied to make quantitative models of twodimensional turbulence, the Great Red Spot and other Jovian vortices, ocean jets like the GulfStream, and ocean vortices. A detailed comparison between these statistical equilibria and real flow observations is provided.We also present recent results for non-equilibrium situations, for the studies of either the relaxation towards equilibrium or non-equilibrium steady states. In this last case, forces and dissipation are in a statistical balance; fluxes of conserved quantity characterize the system and microcanonical or other equilibrium measures no longer describe the system.

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Crude closure for flow with topography through large-scale statistical theory

Cited by 7 publications

References 26 publications

Statistical equilibrium predictions of jets and spots on Jupiter

Statistical equilibrium predictions of jets and spots on Jupiter

The emergence of large‐scale coherent structure under small‐scale random bombardments

Statistical mechanics of two-dimensional and geophysical flows

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