# Entropy production in turbulent mixing

Joël Sommeria

► To cite this version:

Joël Sommeria. Entropy production in turbulent mixing. Kleidon A.; Lorenz R.D. Non-Equilibrium Thermodynamics and the production of Entropy: life, Earth, and Beyond, Springer, pp.79-91, 2005, 9783540224952. 10.1007/11672906_7. hal-00261125

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# Entropy Production in Turbulent Mixing

Joël Sommeria

Laboratoire des Ecoulements Géophysiques et Industriels, LEGI/Coriolis, 21
Avenue des Martyrs, 38 000 Grenoble, France.

**Summary.**We review the statement and application of a Maximum Entropy Production (MEP) principle for the modeling of turbulence. More specifically it applies to two-dimensional turbulence, for which a formalism of statistical mechanics has been proposed. In that case entropy measures the randomness of turbulent fluctuations rather than the molecular fluctuations considered in usual thermodynamics. Nevertheless the same MEP formulation also applies in usual thermodynamics, and we first show how it provides a general understanding of the classical diffusion law. In all cases the outcome of this MEP is a form of diffusion law, with additional terms taking into account long-range interactions (it provides generalized Fokker-Planck equations). This outcome is not unique, but depends on the assumed constraints. We show how the turbulence model can be improved by taking into account all the conservation laws, and by sorting out the deterministic effects of small eddies, limiting MEP to the unknown, random contribution. Finally the extension to other systems with long-range interactions is briefly discussed. This includes applications to gravitational systems. Remarkably, the same transport equations apply in biology, to the chemotactic aggregation of bacterial population.

## 7.1 Introduction

The “principle” of maximum entropy production (MEP) is a guideline to guess the evolution of a complex system. Although this idea is appealing, its scientific status is not clear. Different principles of maximum entropy production have been proposed in different contexts, as reviewed in this book, and it is not clear whether a unique physical law can emerge.

We discuss here how this idea of maximum entropy production can be specified and applied to the modeling of some macroscopic systems, ranging from turbulence to the dynamics of galaxies. Entropy measures the number of available microscopic states corresponding to some macroscopic state. The microscopic states generally represent thermal molecular motion. In our context, it will rather represent the turbulent fluctuations, while the macroscopic state represents some mean flow or coarse-grained fluid motion.

Turbulence involves random behavior, like molecular motion, so the application of similar statistical mechanics procedures seems appropriate. Navier (1823) first derived the usual Laplacian expression for viscosity by statistical modeling of “atomic” motion. At that time the concept of atoms was not
established, and not clearly distinguished from dust particles or small fluid
parcels.

In the twentieth century, it became clear that turbulence has specific prop-
erties, different from the thermal motion of molecules. There is a gap between
the smallest scale of fluid motion and the microscopic scale of thermal fluc-
tuations. While molecules interact only at short range, fluid parcels interact
by pressure forces, which are long ranged. Turbulent motion thus involves
a wide range of eddy sizes, unlike thermal motion. Therefore the separation
between the large scale motion, considered as macroscopic, and the turbulent
fluctuations is not clear. Another major difficulty is the irreversible behavior
of fluid turbulence: the energy of motion is transferred toward small scales by
cascade phenomena and dissipated by viscosity at the smallest scale of eddy
motion, the so-called Kolmogoroff (Kolmogorov) scale. As the usual proce-
dures of statistical mechanics apply close to equilibrium, their application to
turbulent motion remains problematic.

These difficulties are partly overcome in two-dimensional turbulence,
which is a good model for large scale atmospheric or oceanic motion. Two-
dimensional turbulence is also relevant for conducting fluids or plasmas in a
magnetic field. Such two dimensional fluid motion can be described in terms of
interacting vortices which conserve their circulation, like a charge. The long-
range interaction between vortices is indeed formally analogous to electro-
static interactions between charges. An application of equilibrium statistical
mechanics to a set of singular vortices was first proposed by Onsager (1949),
and further developed by Joyce and Montgomery (1973). An extension to a
non-singular distribution of vorticity was proposed by Robert and Sommeria
(1991) and independently by Miller (1990). The motivation was to explain
the observed self-organization of the turbulent motion into steady coherent
vortices. The most striking example is the Great Red Spot of Jupiter, a huge
vortex persisting for more than 300 years in a very turbulent surrounding.

While two-dimensional turbulence tends to self-organize into steady flows,
it is often maintained in unsteady regimes by forcing. This is the case for
instance for oceanic currents. Then it is desirable to use Large Eddy Sim-
ulations: an explicit numerical model for large scales, involving a statistical
model of the sub-grid, unresolved scales. A MEP principle has been proposed
for this purpose by Robert and Sommeria (1992). The idea is that the un-
resolved turbulent motion should increase the entropy at the highest rate
consistent with known constraints.

This MEP is intended as a method of research rather than an intangible
law of physics. In the spirit of Jaynes (1985), the idea is to make the most
objective guess of the transport by the unresolved turbulent motion, taking
into account the known constraints. This guess can be progressively improved
by taking into account new constraints. The opposite approach would be to
sort out the chain of elementary instabilities and other elementary processes.
Turbulence is not pure disorder: well defined flow phenomena can be distin-
guished.
However elementary they may seem, all processes rapidly become quite intricate, and some interactions remain unknown or neglected. The cumulative effects of small errors may lead to erroneous predictions if the general trends are not well represented. Any approximation to the dynamical equations is only valid for a finite time, and for long time predictions, the respect of the system properties is more important than the precision of the model. These properties involve the conservation of global quantities, like energy, but also the general trends to irreversibility, corresponding to entropy increase. The MEP approach allows to progressively improve our modeling capability, always keeping into account these global properties.

This formulation of MEP for turbulence modeling will be summarized in Sect. 7.3. This formulation provides also a new way to understand diffusion process in usual thermodynamics, so it will be introduced first in this context (next section). Finally, extensions to gravitational systems will be discussed in Sect. 7.4. Gravitational interactions, which are also long ranged, indeed possess interesting similarities with vortices. Likewise, the thermodynamics of gravitational systems is fascinating as entropy increase is an apparent source of order. This is of broad relevance for the organization of the universe and consequently for the evolution of life. Note finally that a connection has been recently pointed out between gravitational systems and the chemical interactions which tend to organize bacterial populations in clusters, the chemotactic aggregation (Chavanis 2003).

<figure><img src="image_2.png" /><figcaption>**Fig. 7.1.** The Great Red Spot of Jupiter (top) and White Oval (bottom) are large atmospheric vortices remaining coherent amidst turbulence. Photo: NASA</figcaption></figure>
## 7.2 MEP in Classical Thermodynamics

The definition of entropy is subtle. It begins with a definition of elementary events, or a probability density $f$ for continuous variables. In classical statistical mechanics, these elementary events are defined as the set of positions $q$ and velocity $\dot{q}$ of each particle. More precisely, an event corresponds to the system being in a small interval $dq\dot{q}$. The choice of $q, \dot{q}$ as coordinates is important. For instance, taking $q' = q^2/2$ instead of $q$, would yield a different probability $f'$, such that $f(q)dq = f'(q')dq'$, so that $f'(q') = f(q)/q$. If the probability density $f$ of $q$ is uniform, the probability $f'$ of $q'$ is not uniform due to the different weights of the volume elements.

Therefore the probability density seems to be an arbitrary choice of our state of knowledge. It is, however, natural to choose $q$ instead of $q^2$ as we expect all the positions to be equivalent, so that all the intervals $dq$ should have the same statistical weight. Similarly we expect a priori that the two sides of a tossed coin have the same probability. Concerning velocity, the choice of a uniform probability for each component is not so obvious.

In this respect a key ingredient is the Liouville theorem, which states that the volume element in phase space, that is, the product of $dq\dot{q}$, for each particle is conserved in time. This applies if the chosen coordinates $q, \dot{q}$ are a set of canonical variables of the Hamiltonian dynamics. The probability density in phase space is therefore locally conserved, like a dye concentration transported and stirred in an incompressible fluid. We then expect for an isolated system to reach a uniform probability among the available microscopic states. All the states of given energy tend to be reached with equal probability if there is no other constraint. This choice of density probability is justified by its consistence with the dynamical evolution. When applying the concept of entropy to more general systems, for instance turbulence, the absence of a Liouville's theorem is a fundamental difficulty, and consistency with the dynamics must be checked.

In problems of dynamical evolution, the system is assumed to be in a particular state, described by a probability distribution $f(q, \dot{q})$. Some a priori information is therefore available, quantified by the information entropy $\int f \ln f \, dq \, d\dot{q}$. The entropy measures the number of possible microscopic configurations consistent with the given probability density. Minimizing this entropy with the normalization constraint $\int f \ln f \, dq \, d\dot{q} = N$ (total number of particles) yields the uniform probability distribution. This is the state of equilibrium expected to be reached at the final stage, when all the available information on the initial state has been lost.

Usual thermodynamic systems tend to relax very quickly to a local equilibrium by molecular interactions, while the global equilibrium is reached after the much longer time scale of diffusion. Maximizing the physical entropy $S = -\int f \ln f \, d\dot{q}t$ (i.e., minimizing the information entropy $-S$) for a given local density $\rho(\mathbf{r}) = \int f \, d\dot{q}$ and local energy density $\rho(\mathbf{r})=(1/2)\int f \, \dot{q}^2$
$d\dot{q}$ yields the usual Gaussian distribution for velocity, characterizing a local thermodynamic equilibrium.

The slower evolution is described by the Navier-Stokes equation, which expresses the conservation of momentum. It describes the global uniformisation of density (and pressure) through acoustic wave propagation. To present MEP ideas in the absence of such complications, we shall consider that a state of uniform pressure and temperature has been reached, but a chemical has a non-uniform concentration $c(\mathbf{r})$. Then the further evolution of the system by diffusion will be associated with an increase of the sole compositional entropy

$$S = - \int c(\mathbf{r}) \ln(c(\mathbf{r})) d^3 r. \qquad (7.1)$$

The conservation of the total mass $\int c(r) d^3 r$ of the chemical is equivalent to the local expression:

$$\frac{\partial c}{\partial t} = - \nabla \cdot \mathbf{J}, \qquad (7.2)$$

where the flux vector $\mathbf{J}$ must have a zero normal component at the edge of the system, assumed to be isolated.

We know that entropy must increase, leading to uniformisation of $c$. By combining (7.1) and (7.2), the rate of entropy change is

$$\dot{S} = - \int J \cdot \nabla (\ln c) d^3 r. \qquad (7.3)$$

In the traditional thermodynamic approach for non-equilibrium, a linear relationship between the flux $\mathbf{J}$ and the concentration gradient is assumed

$$\mathbf{J} = -\kappa \nabla c. \qquad (7.4)$$

This is usually justified as the first term in an expansion in terms of $\nabla c$, which expresses the distance to equilibrium. The diffusion coefficient $\kappa$ must be positive to assure the entropy increase. Indeed (7.3) leads to $\dot{S} = \int \kappa(\nabla c)^2 d^3 r$.

The same result can be obtained by a MEP principle, expressed as follows for each macroscopic state of the system, characterized by $c(\mathbf{r})$, the flux $\mathbf{J}$ maximizes the entropy production (7.3) with the constraint $|\mathbf{J}| < A(\mathbf{r})$. We do not specify the bound $A(\mathbf{r})$, it must only exist.

It can be shown that the bound on $|\mathbf{J}|$ is reached: it is more advantageous in terms of entropy production to increase $|\mathbf{J}|$, as it is obvious from the expression (7.3) of entropy production. It is also obvious that for a given modulus, $\mathbf{J}$ optimizes $\dot{S}$ when it is aligned with $\nabla c$, so that the classical expression (7.4) is recovered.

We can derive this relationship in a more formal way, by introducing a Lagrange multiplier $\lambda(\mathbf{r})$ associated with the constraint on $|\mathbf{J}|$ at each point. Then the first variations must satisfy the relationship
$$ \delta \dot{S} - \int \lambda(\mathbf{r}) \delta \mathbf{J}^2 d^3 r = 0. \qquad (7.5) $$

Using (7.3) this relationship writes $\int [\nabla(\ln c) + 2\lambda\mathbf{J}] \delta\mathbf{J} = 0$, which must be satisfied for any variation $\delta J$. This is only possible if the integrand vanishes, leading again to (7.4) (with a diffusion coefficient $\kappa=(2 \lambda c)^{-1}$).

This seems a quite natural derivation. However we must keep in mind that some assumptions are hidden. First the entropy definition relies on the choice of the particle positions $q$ as the elementary variable. Choosing $q^2$ for example would lead to a uniform concentration per unit of $r^2$. This is justified by Liouville's theorem for a Hamiltonian system, but generalization is not obvious. Secondly we need to distinguish the fast process, and the slow process of diffusion which controls the global relaxation to equilibrium. We also neglect memory effects, assuming that the flux $\mathbf{J}$ at any time depends only on the present macroscopic state.

## 7.3 MEP in Two-Dimensional Turbulence

As stated in the introduction, our MEP approach was introduced in the context of 2D turbulence. This is a complex flow confined to a surface, for instance a planetary atmosphere at large scale, or a flow in a volume in the presence of constraining external force, resulting from the Coriolis or electromagnetic effects (see e.g., Sommeria (2001) for a recent review). We consider the fluid as inviscid, so the relevant equation is Euler's equation in two dimensions, and not Navier-Stokes. This is quite justified, as the solutions have been proved to remain regular for any time in two dimensions. Euler's equation is best written in terms of the vorticity field $\omega=(\text{curl } \mathbf{u})_z$, where $z$ is the normal to the plane of motion:

$$ \begin{gathered} \frac{\partial \omega}{\partial t} + \mathbf{u} \cdot \nabla \omega = 0 \qquad (7.6) \\ \nabla \cdot \mathbf{u} = 0. \end{gathered} $$

It expresses the material conservation of the vorticity $\omega$ by the divergenceless flow $\mathbf{u}$. The condition $\nabla \cdot \mathbf{u} = 0$ is equivalent to the statement that $\mathbf{u}$ is derived from a stream function $\psi$ by $u = e_z \times \nabla \psi$ where $e_z$ is the vertical unit vector. Each fluid parcel preserves its local rotation like a small gyroscope, but it is transported and stirred by the velocity field induced by all the other vorticity parcels.

The total energy of the flow is conserved. It can be expressed in terms of the vorticity $\omega$ and the stream function $\psi$, by using integration by parts:

$$ E = \frac{1}{2} \int u^2 d^2 r = \frac{1}{2} \int \psi \omega d^2 r. \qquad (7.7) $$
Note that the expression of energy is formally analogous to the potential energy for charge interactions with a potential $\psi$. This potential similarly satisfies the Poisson equation $-\Delta\psi = \omega$.

<figure><img src="image_3.png" /><figcaption>Fig. 7.2. Numerical simulation of two-dimensional turbulence, represented by vorticity maps (from Bouchet 2003). The initial state (left) is made of uniform vorticity patches. These patches deform into complex filaments (right) which tend to wrap up into large coherent vortices</figcaption></figure>

It is known that vorticity is stirred at small scale, as shown in numerical simulations (e.g., Fig. 7.2). We are not interested in these fine scales but rather in the velocity field, which is much smoother, as it depends on some locally averaged vorticity. The idea developed by Robert and Sommeria (1991) and Miller (1990) is to describe the system in a macroscopic way, as the local probability density $\rho(\sigma, \mathbf{r})$ of finding the vorticity level $\sigma$ in a neighborhood of the position $\mathbf{r}$. We are mostly interested in the local vorticity average $\bar{\omega}$, and the associated “coarse-grained” velocity field defined by curl $\mathbf{u} = \bar{\omega}$. As each vorticity parcel conserves its vorticity, we have the global conservation of $\gamma(\sigma) = \int \rho(\sigma, \mathbf{r}) d^2\mathbf{r}$ for each vorticity value $\sigma$. This conservation law can be written in a local form, by introducing a diffusion flux $\mathbf{J}(\sigma, \mathbf{r})$.

$$ \frac{\partial \rho}{\partial t} + \mathbf{u} \cdot \nabla \rho = - \nabla \cdot \mathbf{J}. \qquad (7.8) $$

Each vorticity level behaves like a chemical species: it is globally conserved but it is transported and mixed. In comparison with the diffusion (7.2), we have sorted out the transport $\mathbf{u} \cdot \nabla \rho$ by the explicitly resolved (coarse-grained) velocity $\mathbf{u}$. We limit the MEP application to the transport by the local, unknown fluctuations.

The expression of entropy is now obtained by replacing the concentration $c$ by $\rho$ in (7.1), and integrating over the vorticity levels $\sigma$. A weak form of a Liouville’s theorem can be invoked to justify this entropy (Robert 2000). We
have now two additional constraints: first the fluid parcels locally exclude each other, resulting in the local normalization $∫ρ(σ, r)dσ = 1$. The other constraint is brought by energy conservation. The energy (7.7) can be written in terms of the densities by replacing $ω$ by its local average $ω̄$.

Introducing these additional constraints, MEP yields the diffusion current

$$ \mathbf{J} = -\kappa [\nabla \rho + \beta \rho(\sigma - \bar{\omega})\nabla \psi] . \quad (7.9) $$

It contains a diffusion flux in $\nabla \rho$ and a second term in which $\nabla \psi$ acts at large scale. It is formally similar to a drift term, like for a charged particle submitted to an electric field $-\nabla \psi$. The coefficient $\beta$ is determined by the condition of global energy conservation.

<figure><img src="image_4.png" /><figcaption>Fig. 7.3. Evolution of a two-dimensional vortex ring using the MEP model (top) and a high resolution numerical simulation (bottom). The initial ring (left) develops sheat instabilities (middle), and eventually self-organizes into a unique coherent vortex (right). Voriticity filaments are visible in the high resolution simulations, but the MEP diffusive flux smooth them out while preserving the organization into the final steady state. From Sommeria (2001)</figcaption></figure>

At equilibrium, the two terms balance each other, and we get a steady-state solution characterized by a given relationship between the density $ρ$ and the stream function. This represents some mean flow, for instance a large vortex, which is predicted to emerge from turbulence. It can be directly obtained as a statistical equilibrium by maximizing the entropy with given energy. This explanation of self-organization by entropy maximization may
seem paradoxical, but the organization is in reality due to the constraint on energy: Full mixing of vorticity would be inconsistent with energy conservation, and the optimum state appears to be a large coherent vortex surrounded by well mixed vorticity.

In this model, we have replaced the initial vorticity equation by a set of equations for each vorticity level $\sigma$. It is still computationally advantageous as only a coarse spatial resolution is needed, thanks to the smoothing effect of the diffusive flux $J$. By contrast the resolution of the initial Euler's equation requires a very high resolution to resolve the fine scale filaments. It turns out that taking into account a few vorticity levels $\sigma$ is generally sufficient. Furthermore, if the initial condition is made of a patch with a single non-zero vorticity level $\sigma_0$, then the probability density $\rho(\sigma, r)$ is a Dirac distribution in $\sigma$, and we can identify $\rho$ with the coarse-grained vorticity $\bar{\omega}$.

An application of this model to the organization into a single coherent vortex, like in the case of the Great Red Spot of Jupiter, is shown in Fig. 7.3. The initially developed vortices merge into a single one, which remains indefinitely as an equilibrium state. Usual turbulence models lead instead to an eventual decay of the vortex by diffusion. A specific application to the Great Red Spot is discussed by Bouchet and Sommeria (2002).

Traditional turbulence modeling relies on closure approximations: hypothesis on the probability laws are assumed, consisting in neglecting some correlations. Kinetic models of molecular motion, like the Boltzmann or the Fokker-Planck equations, also rely on a similar closure approximation. A derivation of a kinetic transport equation for two-dimensional turbulence has been proposed by Chavanis (2000). This approach confirms the general form of the MEP result given by equation (7.9): transport by velocity fluctuations involves a usual diffusion term, plus a long range “drift term”. Its physical origin is interpreted as a long range “polarization” of vorticity by the influence of local vorticity fluctuations. The expression of this drift term is, however, more complex, and memory effects cannot be neglected: the polarization results from the history of flow straining.

Closure models and the MEP approach are complementary. MEP provides models consistent with the long time trends of the dynamics, but relying on some guessed constraints. Furthermore the values of the diffusion coefficients are not given by such thermodynamic approach. By contrast, closure yields these coefficients, and relies on systematic approximations, which can be at least justified for short time scales. Ideally, a good model should be a closure fully consistent with the MEP. This has not been really achieved for two-dimensional turbulence. For instance the above mentioned closure model of Chavanis (2000) does not conserve energy on long time scales. Some improvements have been also proposed for MEP, introducing the constraint of energy conservation in a local way (Chavanis and Sommeria 1997). Another improvement, by Bouchet (2003), is to consider that part of the fluctuations is not random but results from the systematic straining by the coarse-grained motion. Then MEP is used only for a remaining random component. There
is probably not a unique answer to turbulence modeling, but rather a set of models suitable to respect some properties of the system, with a tradeoff between accuracy and complexity.

## 7.4 Application to Stellar Systems

We have already discussed above analogies between vortices and electrostatic charge interactions. Gravitational systems are similarly controlled by long-range interactions. We distinguish two kinds of gravitational systems, whether they are collisional or not. The first case is an ordinary self-gravitating gas, for instance during star formation. The second case corresponds to stellar dynamics in a galaxy. Then the individual “molecules” are stars.

The general trend of such systems is to form a dense core by gravitational collapse, while the released energy heats the envelope. This collapse globally increases the entropy of the system. Thus the whole process of stellar evolution can be qualitatively understood in terms of entropy increase: the initial star formation from a dilute cloud, its further evolution into a denser and denser body, associated with the expulsion of a hot gas, the solar wind or the explosion of a supernova for massive stars. These processes are well described by the fluid dynamics of compressible gas, and MEP does not seem to be of much use in this stellar case.

<figure><img src="image_5.png" /><figcaption>**Fig. 7.4.** Elliptical galaxies (here NGC3379) contain stars in a state of statistical equilibrium, which results from a global, fluid like, mixing in phase space. From http://www.licha.de/AstroWeb</figcaption></figure>
For galaxies, the dynamics are quite different, because of the very low probability of binary interactions between stars ("collisions"). Nevertheless a strong tendency to reach statistical equilibrium is observed. In particular, for elliptical galaxies (see Fig. 7.4) the probability distribution of velocity and radial density profile fit very well the prediction of statistical equilibrium (except at the periphery). According to estimates on binary star interactions such equilibrium would be reached in most cases on a time scale greater than the age of the universe.

An explanation of this paradox has been given by Lynden-Bell (1967) by introducing the notion of violent relaxation, a kind of turbulent behavior for a fluid in the six dimensional phase space of position and velocity. This fluid satisfies the Vlasov equation which has some similarities with the Euler equation for two-dimensional turbulence. Lynden-Bell (1967) proposed a similar statistical approach as described in previous section for two-dimensional turbulence.

The application of MEP to this problem was proposed by Chavanis et al. (1996); see also Chavanis (2002, 2003). The turbulent mixing for density is similarly described by a diffusion term and a drift term proportional to the gravity field. The degree of validity of this model for actual gravitational systems is still unclear.

Remarkably similar equations arise for the chemotactic aggregation of bacterial populations (Chavanis 2003). In that case the potential corresponds to the concentration of chemicals emitted by the bacteria. The balance between emission and diffusion yields a Poisson equation, like for the potential of gravity.

## 7.5 Conclusions

Diffusion equations, widely used in turbulent modeling, can be derived from a MEP principle. The diffusion of a quantity can be viewed as the process which maximizes the entropy production with the natural constraint of a bounded flux for this quantity. Note that this principle should not be confused with the principle of minimum entropy production proposed by Prigogine in 1947 (see Prigogine, 1967). The later applies to the solution of known transport equations for a system submitted to external fluxes. The MEP principle is quite different, and it is designed to guess the equations of transport. The MEP principle discussed here is also different from what has been proposed for describing the heat transport in planetary atmospheres (e.g. Paltridge, this volume; Ito and Kleidon, this volume; Lorenz, this volume). In particular we are considering a dynamical entropy describing turbulent fluctuations rather than the usual thermodynamical entropy. It is not clear whether a more unified principle can arise. Furthermore, MEP, as discussed here, cannot be viewed as a law of physics in the usual sense, with a formula that we could apply and check. It is rather a guideline for modeling complex sys-
tems, possibly learning from trial and error, in the spirit of Jaynes' ideas. It does not replace a more detailed analysis of the system, but it provides an objective way of using available information.

We can refine the model by adding new constraints as we better understand the behavior of the system. A first constraint is provided by conservation laws, in particular for energy. In the case of long-range interactions, either for vortices or gravitational systems, this results in an additional drift term, which leads to self-organisation into large structures, coherent vortices or galaxies. At a next step, we can distinguish some deterministic transport effects and restrict the statistical description to a smaller random contribution. This refinement respects the general trend of entropy increase, and the known constraints of the system. This is unlike the usual approximation procedures, turbulent closure or kinetic models, which, although improving short term predictions, are prone to progressively drift away from reality on long time scales.

## References

* Bouchet F, Sommeria J (2002) Emergence of intense jets and Jupiter Great Red Spot as maximum entropy structures. J Fluid Mech 464: 165-207.

* Bouchet F (2003) Parametrisation of two-dimensional turbulence using an anisotropic maximum entropy production principle. Phys Fluids, submitted.

* Chavanis PH, Sommeria J, Robert R (1996) Statistical Mechanics of Two-dimensional Vortices and Collisionless Stellar Systems. Astrophys J 471: 385.

* Chavanis PH, Sommeria J (1997) Thermodynamical approach for small scale parametrisation in 2D turbulence. Phys Rev Lett 78: 3302-3305.

* Chavanis PH (2000) Quasi-linear theory of the 2D Euler equation. Phys Rev Lett 84: 5512-5515.

* Chavanis PH (2002) Statistical mechanics of two-dimensional vortices and stellar systems, in Dauxois T (eds) Dynamics and thermodynamics of systems with long range interactions, Lecture Notes in Physics, Springer Verlag, Berlin.

* Chavanis PH (2003) Generalized thermodynamics and Fokker-Planck equations: applications to stellar dynamics and two-dimensional turbulence. Phys Rev E 68: 036108.

* Jaynes ET (1985) Where do we go from here? in: Ray Smith C, Grandy WT (eds) Maximum entropy and Bayesian methods in inverse problems, Reidel, Dordrecht, Holland.

* Joyce G, Montgomery D (1973) Negative temperature states for the two-dimensional guiding center plasma. J Plasma Physics 10: 107-121.

* Lynden-Bell D (1967) Statistical mechanics of violent relaxation in stellar systems. Month Notes Roy Astron Soc 136: 101-121.

* Miller J (1990) Statistical mechanics of Euler equations in two dimensions. Phys Rev Lett 65: 2137-2140.

* Navier CLMH (1823) Mémoire sur les lois du mouvement des fluides. Mém Acad Roy Sci 6: 389-440.

* Onsager L (1949) Statistical Hydrodynamics. Nuovo Cimento Suppl 6: 279-287.
Prigogine I (1967) Introduction to thermodynamics of irreversible processes. Wiley, New York.

Robert R, Sommeria J (1991) Statistical equilibrium states for two dimensional flows. J Fluid Mech 229: 291-310.

Robert R, Sommeria J (1992) Relaxation towards a statistical equilibrium state in two-dimensional perfect fluid dynamics. Phys Rev Lett 69: 2776-2779.

Robert R (2000) On the statistical mechanics of 2D Euler and Vlasov Poisson equations. Comm Math Phys 212: 245-256.

Sommeria J (2001) Two-dimensional turbulence. in: Lesieur M, Yaglom A, David F, New trends in turbulence, EDP/Springer, 387-447.