The equilibrium statistical mechanics of simple quasi-geostrophic models

Salmon, Rick; Holloway, Greg; Hendershott, Myrl C.

doi:10.1017/s0022112076000463

Cited by 250 publications

(205 citation statements)

References 12 publications

(2 reference statements)

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“…In the limit as the skewness parameter 3 0,⌸ converges to the standard Gaussian distribution, and the corresponding statistical equilibrium theory reduces to the standard energy-enstrophy theory (6,9,10). In applying the equilibrium statistical theory formulated above, it is useful to adopt the viewpoint that the prior distribution ⌸ encodes the statistics of the potential vorticity fluctuations induced by small-scale forcing, and that the solution Q to Eq.…”

Section: The Equilibrium Statistical Theorymentioning

confidence: 99%

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: 99%

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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“…For the case of single-layer flow over topography, this variational solution yields a steady flow locked to the topography. The solution is like the solution of SALMON et at., (1976) and is qualitatively like simulated flows which are decaying toward such an asymptotic state.…”

Section: Process Models In Eddy Dynamicsmentioning

confidence: 99%

“…Why do eddies lock to topography? SALMON, HOLLOWAY and HENDERSHOTT (1976) suggest that statistical tendencies arise on account of the many interacting degrees of freedom in much the way that thermodynamic laws arise from many molecular collisions. Constructing the analogue of classical statistical mechanics, one obtains qualitatively many of the observed statistical tendencies: ratios of barotropic baroclinic energy, correlations with topography and cascade directions as disequilibrium tendencies combine toward classical equilibrium solutions.…”

Section: Process Models In Eddy Dynamicsmentioning

confidence: 99%

The Mid-Ocean Dynamics Experiment

1978

Deep Sea Research

150

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Abstract--The Mid-Ocean Dynamics Experiment (MODE-1) was designed to investigate mid-ocean mesoscale eddies. An intensive and extensive program of measurements in three spatial dimensions and time was undertaken in an area southwest of Bermuda from March through mid-July 1973. Principal components of the experiment were an array of moored current meters and temperature-pressure recorders, hydrographic stations, drifting neutrally buoyant floats at 1500 m tracked by SOFAR, and acoustic and electromagnetic profilers. During MODE-1 a smaller scale survey relying on ship-tracked neutrally buoyant floats, a conductivity-temperature (CTD) survey, and a moored current meter array, MINIMODE, was carried out. The experiment was preceded by MODE-0, consisting of measurements by a series of moored current meters and other instruments in the general area selected for MODE-1.MODE-I observations were generally within a 300-km radius circle centered at 26°N, 69°40'W, with a greater concentration of observations in the interior ofthe circle. The region covers varied topography, wit h a flat abyssal plain sloping upward to t he continental rise in t he western half and rough topography in the eastern half.Descriptive, dynamical, numerical results of the experiment are presented. It is concluded that midocean eddies are part of an energetic and structured variability field superimposed on the weaker gyrescale mean circulation. In the western North Atlantic there is a band of eddy variability of around 100-day period and 70-km scale in which currents are horizontally nearly isotropic; vertical scales are of the order of the depth. The experiment provided conclusive evidence of the existence of mid-ocean eddies and serves as the basis for future experiments, such as POLYMODE, to extend our knowledge of these systems.DURING the past decade, physical oceanographers have directed a considerable effort to exploring and modeling mid-ocean mesoscale 'eddies', which are fluctuating current cells extending throughout the water column. Eddies occur irregularly throughout the World's oceans. They have swirl speeds of 5 to 50 cm s-1, and space-time scales of tens to hundreds of kilometers and weeks to months. Associated with the horizontal currents are fluctuating vertical displacements of up to several hundred meters in the main thermocline. Eddies, where they exist, usually dominate the mid-ocean flow.Understanding the eddies and their role in the general circulation has emerged as a central problem in ocean circulation dynamics. Moreover, understanding eddy dynamics is a prerequisite for both valid modeling of large-scale exchanges of momentum and energy, heat and salt, geochemicals, nutrients and other passive solutes, and for realistic coupled ocean-atmospheric models necessary for longer range weather prediction and climate modeling.

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“…As a result various statistical theories have been proposed, which di er from each other mainly by t h e i r individual choice of conserved quantities and the \discretization" procedure used to represent the continuum by a countable set of \particles" (spectral representation, point-vortices, lattice models, etc.) Examples are the energy-enstrophy theory ( 3,4]), the point-vortex theory ( 5,6,7,8]), the Miller-Robert theory ( 9,10,11]), and a very recent few constraint statistical theory based on extrema of the potential vorticity ( 12,13]). Each statistical theory yields a di erent relationship between the vorticity a n d the stream function, which de nes a coarse-grained (macroscopic) description of the steady ow at statistical equilibrium, superimposed with local random uctuations.…”

Section: Introductionmentioning

confidence: 99%

“…Barotropic ows over topography are the simplest prototype geophysical ows where even the simplest energy-enstropy statistical theory yields striking predictions for a \most probable state" mean ow ( 4,18]). The emergence of such states as averages over large time intervals in damped and driven numerical simulations has been studied extensively with many interesting results ( 1,19,20,21,22]).…”

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confidence: 99%

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

Grote

Majda

2000

Nonlinearity

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Crude closure algorithms based on equilibrium statistical theories are developed here for prototypical geophysical ows involving barotropic ow o ver topography. These crude closure algorithms are developed utilizing the simplest energy-enstrophy statistical theory for ow with topography in these algorithms, only a single parameter, the energy, is tracked by the algorithm and the entire ow structure is predicted through the equilibrium statistical state. In particular, no explicit parametrization of a sub-grid scale energy spectrum is utilized in the algorithm. The predictions of the crude closure algorithm are compared with direct pseudo-spectral numerical simulations of the barotropic ow equations with random small scale forcing and dissipation for a variety of random topographies in basin, channel, and periodic geometries. In most situations studied here, the energy is tracked within small errors by the crude closure while the velocity errors rarely exceed 10% provided that the enstrophy/energy ratio is not large or growing signi cantly in time. Examples are also introduced where the crude closure algorithm based on the energy enstrophy theory fails in these circumstances, a crude closure algorithm based on more sophisticated equilibrium statistical theories is introduced as a possible remedy.

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The equilibrium statistical mechanics of simple quasi-geostrophic models

Cited by 250 publications

References 12 publications

Statistical equilibrium predictions of jets and spots on Jupiter

Statistical equilibrium predictions of jets and spots on Jupiter

The Mid-Ocean Dynamics Experiment

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

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