Statistical mechanics of Fofonoff flows in an oceanic basin

We study the minimization of potential enstrophy at fixed circulation and energy in an oceanic basin with arbitrary topography. For illustration, we consider a rectangular basin and a linear topography h=by which represents either a real bottom topography or the beta-effect appropriate to oceanic situations. Our minimum enstrophy principle is motivated by different arguments of statistical mechanics reviewed in the article. It leads to steady states of the quasigeostrophic (QG) equations characterized by a linear relationship between potential vorticity q and stream function psi. For low values of the energy, we recover Fofonoff flows [J. Mar. Res. 13, 254 (1954)] that display a strong westward jet. For large values of the energy, we obtain geometry induced phase transitions between monopoles and dipoles similar to those found by Chavanis and Sommeria [J. Fluid Mech. 314, 267 (1996)] in the absence of topography. In the presence of topography, we recover and confirm the results obtained by Venaille and Bouchet [Phys. Rev. Lett. 102, 104501 (2009)] using a different formalism. In addition, we introduce relaxation equations towards minimum potential enstrophy states and perform numerical simulations to illustrate the phase transitions in a rectangular oceanic basin with linear topography (or beta-effect).Comment: 26 pages, 28 figure

Generalized thermodynamics and Fokker-Planck equations. Applications to stellar dynamics, two-dimensional turbulence and Jupiter's great red spot

We introduce a new set of generalized Fokker-Planck equations that conserve energy and mass and increase a generalized entropy until a maximum entropy state is reached. The concept of generalized entropies is rigorously justified for continuous Hamiltonian systems undergoing violent relaxation. Tsallis entropies are just a special case of this generalized thermodynamics. Application of these results to stellar dynamics, vortex dynamics and Jupiter's great red spot are proposed. Our prime result is a novel relaxation equation that should offer an easily implementable parametrization of geophysical turbulence. This relaxation equation depends on a single key parameter related to the skewness of the fine-grained vorticity distribution. Usual parametrizations (including a single turbulent viscosity) correspond to the infinite temperature limit of our model. They forget a fundamental systematic drift that acts against diffusion as in Brownian theory. Our generalized Fokker-Planck equations may have applications in other fields of physics such as chemotaxis for bacterial populations. We propose the idea of a classification of generalized entropies in classes of equivalence and provide an aesthetic connexion between topics (vortices, stars, bacteries,...) which were previously disconnected.Comment: Submitted to Phys. Rev.

Relaxation equations for two-dimensional turbulent flows with a prior vorticity distribution

Using a Maximum Entropy Production Principle (MEPP), we derive a new type of relaxation equations for two-dimensional turbulent flows in the case where a prior vorticity distribution is prescribed instead of the Casimir constraints [Ellis, Haven, Turkington, Nonlin., 15, 239 (2002)]. The particular case of a Gaussian prior is specifically treated in connection to minimum enstrophy states and Fofonoff flows. These relaxation equations are compared with other relaxation equations proposed by Robert and Sommeria [Phys. Rev. Lett. 69, 2776 (1992)] and Chavanis [Physica D, 237, 1998 (2008)]. They can provide a small-scale parametrization of 2D turbulence or serve as numerical algorithms to compute maximum entropy states with appropriate constraints. We perform numerical simulations of these relaxation equations in order to illustrate geometry induced phase transitions in geophysical flows.Comment: 21 pages, 9 figure

Extending Electromagnetic Methods to Map Coastal Pore Water Salinities

The feasibility of mapping pore water salinity based on surface electromagnetic (EM) methods over land and shallow marine water is examined in a coastal wetland on Tampa Bay, Florida. Forward models predict that useful information on seabed conductivity can be obtained through \u3c1.5 m of saline water, using floating EM-31 and EM-34 instruments from Geonics Ltd. The EM-31 functioned as predicted when compared against resistivity soundings and pore water samples and proved valuable for profiling in otherwise inaccessible terrain due to its relatively small size. Experiments with the EM-34 in marine water, however, did not reproduce the theoretical instrument response. The most effective technique for predicting pore water conductivities based on EM data entailed (1) computing formation factors from resistivity surveys and pore water samples at representative sites and (2) combining these formation factors with onshore and offshore EM-31 readings for broader spatial coverage. This method proved successful for imaging zones of elevated pore water conductivities/salinities associated with mangrove forests, presumably caused by salt water exclusion by mangrove roots. These zones extend 5 to 10 m seaward from mangrove trunks fringing Tampa Bay. Modeling indicates that EM-31 measurements lack the resolution necessary to image the subtle pore water conductivity variations expected in association with diffuse submarine ground water discharge of fresher water in the marine water of Tampa Bay. The technique has potential for locating high-contrast zones and other pore water salinity anomalies in areas not accessible to conventional marine- or land-based resistivity arrays and hence may be useful for studies of coastal-wetland ecosystems