Bottom-trapped currents as statistical equilibrium states above topographic anomalies

Oceanic geostrophic turbulence is mostly forced at the surface, yet strong bottom-trapped flows are commonly observed along topographic anomalies. Here we consider the case of a freely evolving, initially surface-intensified velocity field above a topographic bump, and show that the self-organization into a bottom-trapped current can result from its turbulent dynamics. Using equilibrium statistical mechanics, we explain this phenomenon as the most probable outcome of turbulent stirring. We compute explicitly a class of solutions characterized by a linear relation between potential vorticity and streamfunction, and predict when the bottom intensification is expected. Using direct numerical simulations, we provide an illustration of this phenomenon that agrees qualitatively with theory, although the ergodicity hypothesis is not strictly fulfilled

Equilibrium statistical mechanics and energy partition for the shallow water model

The aim of this paper is to use large deviation theory in order to compute the entropy of macrostates for the microcanonical measure of the shallow water system. The main prediction of this full statistical mechanics computation is the energy partition between a large scale vortical flow and small scale fluctuations related to inertia-gravity waves. We introduce for that purpose a discretized model of the continuous shallow water system, and compute the corresponding statistical equilibria. We argue that microcanonical equilibrium states of the discretized model in the continuous limit are equilibrium states of the actual shallow water system. We show that the presence of small scale fluctuations selects a subclass of equilibria among the states that were previously computed by phenomenological approaches that were neglecting such fluctuations. In the limit of weak height fluctuations, the equilibrium state can be interpreted as two subsystems in thermal contact: one subsystem corresponds to the large scale vortical flow, the other subsystem corresponds to small scale height and velocity fluctuations. It is shown that either a non-zero circulation or rotation and bottom topography are required to sustain a non-zero large scale flow at equilibrium. Explicit computation of the equilibria and their energy partition is presented in the quasi-geostrophic limit for the energy-enstrophy ensemble. The possible role of small scale dissipation and shocks is discussed. A geophysical application to the Zapiola anticyclone is presented.Comment: Journal of Statistical Physics, Springer Verlag, 201

The role of fluctuations across a density interface

A statistical mechanics theory for a fluid stratified in density is presented. The predicted statistical equilibrium state is the most probable outcome of turbulent stirring. The slow temporal evolution of the vertical density profile is related to the presence of irreversible mixing, which alters the global distribution of density levels. We propose a model in which the vertical density profile evolves through a sequence of statistical equilibrium states. The theory is then tested with laboratory experiments in a two-layer stably stratified fluid forced from below by an oscillating grid. Quantitative measurements of density fluctuations across the interface are made by planar laser induced fluorescence. These fluctuations are splitted in a "wave" part and a "turbulent" part. The wave part of the density field is well described by a previous theory due to Phillips. We argue that statistical mechanics predictions apply for the turbulent part of the density field sufficiently close to the interface. However inside the mixed layer density fluctuations are instead controlled by a balance between eddy flux downward and dissipation by cascade to small scales. We report exponential tails for the density pdf in this region

A statistical mechanics approach to mixing in stratified fluids

Predicting how much mixing occurs when a given amount of energy is injected into a Boussinesq fluid is a longstanding problem in stratified turbulence. The huge number of degrees of freedom involved in those processes renders extremely difficult a deterministic approach to the problem. Here we present a statistical mechanics approach yielding prediction for a cumulative, global mixing efficiency as a function of a global Richardson number and the background buoyancy profile.Comment: Accepted in Journal of Fluid Mechanic

Synchronization and coordination of sequences in two neural ensembles

There are many types of neural networks involved in the sequential motor behavior of animals. For high species, the control and coordination of the network dynamics is a function of the higher levels of the central nervous system, in particular the cerebellum. However, in many cases, especially for invertebrates, such coordination is the result of direct synaptic connections between small circuits. We show here that even the chaotic sequential activity of small model networks can be coordinated by electrotonic synapses connecting one or several pairs of neurons that belong to two different networks. As an example, we analyzed the coordination and synchronization of the sequential activity of two statocyst model networks of the marine mollusk Clione. The statocysts are gravity sensory organs that play a key role in postural control of the animal and the generation of a complex hunting motor program. Each statocyst network was modeled by a small ensemble of neurons with Lotka-Volterra type dynamics and nonsymmetric inhibitory interactions. We studied how two such networks were synchronized by electrical coupling in the presence of an external signal which lead to winnerless competition among the neurons. We found that as a function of the number and the strength of connections between the two networks, it is possible to coordinate and synchronize the sequences that each network generates with its own chaotic dynamics. In spite of the chaoticity, the coordination of the signals is established through an activation sequence lock for those neurons that are active at a particular instant of time.This work was supported by National Institute of Neurological Disorders and Stroke Grant No. 7R01-NS-38022, National Science Foundation Grant No. EIA-0130708, Fundación BBVA and Spanish MCyT Grant No. BFI2003-07276

Ribbon Turbulence

We investigate the non-linear equilibration of a two-layer quasi-geostrophic flow in a channel forced by an imposed unstable zonal mean flow, paying particular attention to the role of bottom friction. In the limit of low bottom friction, classical theory of geostrophic turbulence predicts an inverse cascade of kinetic energy in the horizontal with condensation at the domain scale and barotropization on the vertical. By contrast, in the limit of large bottom friction, the flow is dominated by ribbons of high kinetic energy in the upper layer. These ribbons correspond to meandering jets separating regions of homogenized potential vorticity. We interpret these result by taking advantage of the peculiar conservation laws satisfied by this system: the dynamics can be recast in such a way that the imposed mean flow appears as an initial source of potential vorticity levels in the upper layer. The initial baroclinic instability leads to a turbulent flow that stirs this potential vorticity field while conserving the global distribution of potential vorticity levels. Statistical mechanical theory of the 1-1/2 layer quasi-geostrophic model predict the formation of two regions of homogenized potential vorticity separated by a minimal interface. We show that the dynamics of the ribbons results from a competition between a tendency to reach this equilibrium state, and baroclinic instability that induces meanders of the interface. These meanders intermittently break and induce potential vorticity mixing, but the interface remains sharp throughout the flow evolution. We show that for some parameter regimes, the ribbons act as a mixing barrier which prevent relaxation toward equilibrium, favouring the emergence of multiple zonal jets

The catalytic role of beta effect in barotropization processes

The vertical structure of freely evolving, continuously stratified, quasi-geostrophic flow is investigated. We predict the final state organization, and in particular its vertical structure, using statistical mechanics and these predictions are tested against numerical simulations. The key role played by conservation laws in each layer, including the fine-grained enstrophy, is discussed. In general, the conservation laws, and in particular that enstrophy is conserved layer-wise, prevent complete barotropization, i.e., the tendency to reach the gravest vertical mode. The peculiar role of the $\beta$-effect, i.e. of the existence of planetary vorticity gradients, is discussed. In particular, it is shown that increasing $\beta$ increases the tendency toward barotropization through turbulent stirring. The effectiveness of barotropisation may be partly parameterized using the Rhines scale $2\pi E_{0}^{1/4}/\beta^{1/2}$. As this parameter decreases (beta increases) then barotropization can progress further, because the beta term provides enstrophy to each layer

Statistical mechanics of two-dimensional and geophysical flows

International audienceThe 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. The equilibrium microcanonical measure is built from the Liouville theorem. 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 two-dimensional turbulence, the Great Red Spot and other Jovian vortices, ocean jets like the Gulf-Stream, and ocean vortices. We also present recent results for non-equilibrium situations, for the studies of either the relaxation towards equilibrium or non-equilibrium steady states