No abstract
Statistical equilibrium models of coherent structures in two-dimensional and barotropic quasi-geostrophic turbulence are formulated using canonical and microcanonical ensembles, and the equivalence or nonequivalence of ensembles is investigated for these models. The main results show that models in which the global invariants are treated microcanonically give richer families of equilibria than models in which they are treated canonically. Such global invariants are those conserved quantities for ideal dynamics which depend on the large scales of the motion; they include the total energy and circulation. For each model a variational principle that characterizes its equilibrium states is derived by invoking large deviations techniques to evaluate the continuum limit of the probabilistic lattice model. An analysis of the two different variational principles resulting from the canonical and microcanonical ensembles reveals that their equilibrium states coincide only when the microcanonical entropy function is concave. These variational principles also furnish Lyapunov functionals from which the nonlinear stability of the mean flows can be deduced. While in the canonical model the well-known Arnold stability theorems are reproduced, in the microcanonical model more refined theorems are obtained which extend known stability criteria when the microcanonical and canonical ensembles are not equivalent. A numerical example pertaining to geostrophic turbulence over topography in a zonal channel is included to illustrate the general results.
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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