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In a paper that appeared 70 years ago in the Journal of the Dutch Physical Society, Burgers 1 addressed the question whether the statistics of turbulent flows can be investigated by the methods of statistical mechanics. In the past the methods of statistical mechanics had been very successfully applied to the microscopic motion of molecules. In extending their application to the realm of turbulent motion, one encounters the difficulty that the fluid dynamical equations are continuous. By using a representation of the flow fields in terms of Fourier components this problem can be solved to the extent that a phase space can still be defined although it is infinite dimensional.

Another difficulty is that, in turbulent motion, energy is not conserved but flows through the system. Burgers therefore proposed that the statistics of a turbulent system is controlled by an average balance between input and output of energy and not, as is appropriate to assume in the realm of molecular motion, by the conservation of energy. Taking the dissipation to be quadratic, when expressed in terms of the Fourier coefficients, and constraining the statistics to respect an average balance between forcing and dissipation, he applied the techniques of statistical mechanics and concluded that the dissipation is equally partitioned among the Fourier components.

This conclusion was both interesting and problematic. Equipartition of dissipation leads to an unphysical infinite total dissipation if the phase space of the system is infinite dimensional. Quantum mechanics does not come to the rescue here as it had done earlier when an analogous problem arose in the statistical mechanics of electromagnetic radiation. Despite a series of publications , many of which are reprinted in the memorial volume by Nieuwstadt and Steketee 2 , a completely satisfying solution did not emerge and Burgers finally abandoned the subject.

Several years later Onsager 3 took it up again but decided to pursue a course that is more in line with equilibrium statistical mechanics, as detailed in the review article by Eyink and Sreenivasan 4. It will thus be investigated whether statistical mechanics can be used to deal with forced-dissipative turbulent systems, using as a basic assumption that the statistics is controlled by an average balance between forcing and dissipation. The problem of the infinite dissipation is not resolved but moderated by limiting ourselves to finitely truncated spectral representations of fluid flows.

We will phrase the theory in the language of probability theory and the principle of maximum entropy, as advocated by Jaynes 6. The method will be applied to a simple one-layer model of the large-scale atmospheric circulation. The model to be considered describes the motion of a single layer of incompressible fluid on the surface of a rotating sphere. Orography is taken into account and the flow is assumed to be geostrophically balanced and thus approximately governed by the horizontal advection of quasigeostrophic potential vorticity.

The equation that is used, is a somewhat simplified version of an equation discussed by the author 7. The system is forced by relaxation towards a zonally symmetric circulation that consists of jet-streams in both hemispheres, and is damped by a term that has the same structure as the viscosity term in fluid dynamics. By projecting the advection equation of potential vorticity onto the finite set of spherical harmonics, one obtains a dynamical system of quadratically non-linear equations in the Fourier coefficients. When integrated numerically, this finite-dimensional dynamical system displays chaotic turbulent motion, not unlike what is seen in large-scale atmospheric flow.

To demonstrate this, we show in Figure 1 two snapshots of the vorticity and the zonally averaged zonal velocity, separated by 10 days in time, at the end of an integration of days. This has been shown 5 to work rather well if the statistics is controlled by conservation of energy and enstrophy, i. Fortunately, the mathematics is similar in both cases because all constraints are quadratic and leads to a probability density function that is a product of normal distributions.

Once the probability density function is known, all relevant statistics can be calculated, such as spectra of energy and enstropy and average vorticity fields. A cubic stochastic model 5C. Comparisons with other methods 5C. An application to CBL turbulence simulations The unclosed LES equations 6. The stochastic model considered 6. The closure of LES equations 6. Hybrid methods 6.

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The closure of the equation for filtered velocities 6. The transport equation for the SGS stress tensor 6. The general algebraic expression for the SGS stress tensor 6. Linear and quadratic algebraic SGS stress tensor models 6. Scaling analysis 6. The theoretical calculation of parameters 6. The closure of the scalar FDF transport equation 6.

The scalar-conditioned convective flux 6. The diffusion coefficient 6. The scalar mixing frequency 6. The need for the unification of turbulence models 7. Industrial applications of turbulence models 7. Basic studies by DNS 7. Unified turbulence models 7. A unified stochastic model 7. A unified model for filtered variables 7. Some unsolved questions 7. The structure of unified turbulence models 7.

The parameters of unified turbulence models ST source term in temperature equation see 4. E thermal expansion coefficient see 4. The understanding of turbulence is highly relevant to the protection of our natural living conditions and many technological developments.

To develop environmental protection control strategies, we need knowledge about the mechanisms of the global warming up of our earth's atmosphere, the development and path of hurricanes and the transport of radioactive substances in the turbulent atmosphere. With regard to technological developments, we are interested to see how the efficiency of the mixing of species in chemical reactors or propulsion systems of high-speed aircraft can be enhanced.

Studies of turbulent phenomena by means of measurements are usually very expensive. In addition to this, one obtains in this way only limited and often quite inaccurate information about some of the quantities of interest. To understand turbulent phenomena in their complexity, we need models that enable the simulation of turbulent flows. Basic ways to construct such models, and characteristic advantages and disadvantages of different approaches, will be described in this book, as illustrated in Fig.

An overview of essential features of computational methods for turbulent flows will be given first in order to explain the motivation for the discussions performed in the following chapters. In section 1. These equations become inapplicable to flows with a well-developed turbulence as usually occurs in nature or technological problems. The question how suitable equations for such flows can be constructed will be addressed in section 1.

In agreement with the course of their development, this discussion will be split into the consideration of stochastic models for large-scale turbulence and stochastic models for small-scale turbulence which are currently under development. This leads then to the important question of the unification of turbulence models. There are different ways to derive the basic equations for turbulent flows.

## Turbulence

One way is the use of empirical assumptions in combination with symmetry constraints. Another way is to develop a model for the underlying molecular motion and to derive then the basic equations of fluid and thermodynamics as a consequence of. The equations of fluid 5. Stochastic models for 6. Stochastic models for and thermodynamics large-scale turbulence small-scale turbulence. The unification of turbulence models.

### Submission history

An illustration of the organization of the book. Chapter 1 provides an overview on the questions addressed. Chapters 2 and 3 deal with fundamentals regarding the use of stochastic variables and processes. The basic equations of fluid and thermodynamics are then introduced in chapter 4. Environmental and technological flows are usually characterized by a well-developed turbulence.

The problem related to the basic equations is that they are practically inapplicable to calculate such turbulent flows. Therefore, one has to develop models for turbulent motions on the fluid dynamic scale. The question how such stochastic turbulence models can be constructed is the main object of this book. This question will be differentiated into the consideration of models for large- and small-scale turbulence, which are treated in chapters 5 and 6.

The unification of these approaches is the concern of chapter 7. This approach will be applied here in order to demonstrate the analogy to the construction of stochastic turbulence models and to explain the limitations of the applicability of the basic equations. The basic equations 3.

We consider a macroscopic volume 1 cm3 of a gas that consists of a very large number of molecules. The molecules may be seen as particles mass points that move in an irregular way. This presentation of evolution equations is called the Lagrangian approach. One considers the variables of interest as properties of particles.

The mathematical expression of this idea is equation 1. By providing initial and boundary conditions for all the particles considered, the dynamics of the fluid would be completely described by 1. However, because of the huge number of particles, the equations 1. On the other hand, one is in general not so much interested in having information on the state of each molecule, but in characteristic properties of a large number of molecules, that is, in averages or means, both terms will be applied throughout this book as having the same meaning.

The value of 1. The simplest way to construct equations for means is the use of ensemble means, which are defined in Appendix 1A. The fluid and thermodynamic equations can be obtained then on the basis of a stochastic model for the molecular motion, as shown in chapter 4. This can be done by rewriting stochastic equations that have the structure of 1. This corresponds to the transition into the Eulerian approach where fluid properties are considered at fixed positions. The molecular stress tensor pik is defined in terms of molecular quantities.

Suitable approximations for it can be found by means of 1. The application of 1. This technique is a unique tool to investigate basic mechanisms of turbulence but its use is restricted to weakly turbulent flows because the computational costs grow rapidly with the Reynolds number, which represents a measure for the intensity of turbulence. Unfortunately, it is obvious today that DNS cannot be used at least for the foreseeable future for the calculation of flows of technological or environmental relevance.

To overcome this problem of the inapplicability of the basic equations of fluid and thermodynamics to most of the industrial problems one used these equations to construct equations for coarser quantities, this means for variables that represent fluid properties on a coarser scale. The first way that was applied to obtain such equations is to average the basic equations by adopting ensemble means. The problem related to use of 1. Suitable approximations for the Reynolds stress tensor can often be found by adopting approximations in a transport equation for it, which can be obtained in analogy to 1.

Nevertheless, such a closure strategy becomes unfeasible with regard to the simulation of reacting flows. To model them, one needs closed transport equations for thermochemical variables as mass fractions or temperature. Turbulence models 5. Apart from some cases where one can successfully assess the deviations from limiting cases of very fast or slow chemistry, suitable closure of such equations for mean quantities can hardly be obtained on the basis of direct parametrizations of unclosed terms in general. The extension of the equations 1. Such PDF methods can be constructed by postulating a stochastic model for instantaneous velocities in analogy to the equations 1.

The significant advantage of this approach is that the closure problems related to the use of RANS equations can be overcome on the basis of a physically consistent model, see the detailed discussion in chapter 5. However, a problem that concerns the application of both RANS and PDF methods is related to the reference to ensemble-averaged fluid dynamic variables. According to the ergodic hypothesis, see the explanations given in Appendix 1A, such means have to be seen as relatively coarse quantities that represent mean values of fluid dynamic variables within volumes that are not always small compared to the flow domain considered.

This fact implies several problems. The first one concerns the modeling of the dynamics of fluctuations, this means the determination of a suitable model structure. The fact that fluctuations are taken with reference to ensemble averages implies the possibility that significant deviations from means occur. This may cause nonlinear fluctuation dynamics and non-trivial interactions between various fluctuations. The second problem is that such models cannot be universal. Usually, model parameters are related to standardized flow statistics, as specific components of the normalized Reynolds stress tensor.

Apparently, such parameters must depend on the flow considered, but they also may vary significantly in different regions of one flow. This poses the problem of finding optimal values for model parameters. Even if the modeling task is performed in an optimal way, the third problem is the impossibility to resolve several details of the physics of flows.

It is for instance impossible to resolve the structure of the turbulent mixing of scalars in mixing layers in a way that is comparable to the information obtained by DNS results. A way to overcome these problems related to the reference to ensemble averages is to apply a spatial filter operation with a filter width that is small compared to the length scale of large-scale fluid motions.

Accordingly, FDF methods are characterized by the combination of two essential advantages: large-scale fluid motions are treated without adopting modeling assumptions, and small-scale fluid motions are partly treated exactly and partly modeled on the basis of a physically consistent model. However, these advantages have their price. Such FDF calculations are found to require a computational effort that is less than that required for DNS, but this effort is still very expensive. An optimal solution to the problems related to the application of models for small- and large-scale turbulence is the construction of unified or bridging models, this means models that can be used to perform for instance DNS, LES or RANS simulations.

The development of such models would allow calibration of relatively coarse models by high-resolving simulations. The resulting optimally calibrated models could be used then to perform relatively inexpensive routine investigations. It is essential to emphasize that the construction of models that apply a variable filter width does not solve the problem of constructing bridging models.

With regard to this, the limit of a vanishing filter width does not pose any problem, but the limit of an extremely large filter width requires a careful consideration. Such questions related to the construction of unified turbulence models will be addressed in chapter 7 in conjunction with a discussion of further developments that may be expected in this field. Let us assume that we are interested in equations for the evolution of averaged quantities that characterize any flow considered.

From a practical point of view, the most convenient way to formulate a filter operation is the use of time-, space- or space-time averaging: such averages can be obtained at best by measurements. G r is a filter function with properties as described in chapter 6. The disadvantage of defining averages by means of the relation 1A.

Thus, equations derived in this way are faced again with a closure problem since there is no general methodology to determine G r. A simple way to avoid this problem is the use of ensemble averages as they are defined in the theory of probability. In contrast to the need to specify G r in equation 1A.

To define them, one considers an infinite number of equivalent systems which are called an ensemble. From a physical point of view, equivalent systems means that one considers systems with initial and boundary conditions that are the same according to our observation which is insufficient to distinguish all the details of these states, as for instance the concrete values of all the molecule velocities. Q n is the value of Q obtained for the nth flow, and N is the total number of flows. The subscript e is used here to distinguish between spatial filter operations defined by 1A.

This subscript will be neglected in the chapters 2 5 where only ensemble means are used to simplify the notation. Obviously, such a series of measurements is unavailable in general: one often has to use data derived from just one experiment. Usually, such data are given as spatially filtered values. This leads to the essential question about the consistency of ensemble means predicted by theory and quantities that can be obtained by measurements. This question is the concern of the ergodic theorem.

This question can be treated analytically for the case of homogeneous turbulence a homogeneous shear flow. The latter assumption can be considered as a reasonable approximation for small flow regions. As pointed out by Monin and Yaglom , 4. Xc may be seen as a "correlation volume" or "integral volume scale", this means a measure for the volume occupied by a typical eddy of the flow considered. In other words, ensemble means may be seen as box-averages where the boxes are large compared to the characteristic eddy volume Xc.

Stochastic variables. To develop equations for turbulent flows one needs knowledge about the characterization of stochastic variables at any time and properties of stochastic processes the structure of equations for the evolution of stochastic variables in time. The properties of stochastic variables will be pointed out in this chapter, and the most important features of stochastic processes will be discussed in chapter 3. The characterization of a single stochastic variable will be considered in sections 2.

The extensions of these definitions to the case of many variables is the concern of section 2. It will be shown that stochastic variables are defined via their PDFs. An important problem is then the question of how it is possible to estimate the shape of a PDF.

This will be addressed in section 2. Examples for PDFs obtained in this way are then given in section 2. Unfortunately, such PDFs are not always easy to use. Thus, section 2. The states of turbulent systems are often extremely complex. They depend on many different factors which are unknown or not assessable for us. Usually, we are only able to recognize a small part of the complete state of turbulent systems, which is called the observable state. This inability to get knowledge about the complete state of turbulent systems may have serious consequences.

This is illustrated in Fig. What we have to do in this case is to consider the observable state [ as a stochastic variable, this means a variable that takes due to unknown factors values that we cannot predict accurately. Nevertheless, our experience shows that. The evolution of two complete system states ;1 and ;2 in time t is shown in a. At the time t, ;1 and ;2 have different [ values. The corresponding evolution of the variable [ is shown in b. Therefore, we may define stochastic variables by considering the range of all possible values and specifying the probability for the realization of certain values.

This sample space is used to determine the probability for the appearance of certain values of [. PDFs of one variable An example for the application of this procedure will be given in chapter 5, where vertical velocity PDFs are calculated in exactly this way for convective boundary layer turbulence. The ensemble average referred to by the brackets in 2. As pointed out above, it is usually more convenient to consider the PDF F[ for the appearance of [ values instead of p[ x, t. This PDF is given by dividing the probability p[ x, t by 'x,. The expression inside the sum is a discrete representation of the derivative of the theta function, which is given by the delta function see the explanations given in Appendix 2A.

In the limit 'x o 0, we obtain therefore the following formula for the calculation of F[,. Sometimes, the reference to [ leads to the question on which variables F[ depends. For given x and t, F[ represents the ensemble mean of a function of [ of the delta function. In other words, F[ is a deterministic and continuous function of x and t.

Further, it is worth noting that the PDF F[ and distribution function P[ are related according to their definitions in the following way,. The middle expression of the relation 2. This expression is consistent with 2. The last expression follows from 2. To prepare the following developments we consider some fundamental properties of the distribution function P[. These are given by. The relations 2. Relation 2.

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The fundamental properties of the PDF F[ are the following ones,. For x o f, the distribution function P[ monotonically tends to a constant, see 2. Therefore, its derivative F[ has to tend to zero, which implies 2. The validity of 2. It is worth noting that the properties 2. PDFs may have a variety of different forms. Very often, one is not so much interested in all this detailed information but first of all in a few global parameters that characterize essential characteristics of PDFs. Such parameters are given by the moments of a PDF. The characterization of PDFs by moments The dependence of [ on t is not indicated to simplify the following explanations.

The consistency of 2. The power n determines the order of a moment. It provides essential information: this is the value which we have to expect for [. However, most of the values of [ will have some deviations. To characterize such deviations, one applies the root of the variance V2 of [, which is defined by. Examples for the effect of V will be given in section 2. They are defined by. These parameters are standardized in terms of V such that they are dimensionless. The skewness is used to assess the asymmetry of PDFs.

For symmetric PDFs, the kurtosis indicates the deviation from a Gaussian PDF that describes usually the equilibrium distribution of stochastic variables. The function C[ is nothing but the Fourier transform of the PDF F[ if we assume that x runs from minus to plus infinity.

The expression 2. Unfortunately, there is no way to construct specific PDFs by truncating the series in 2. A way to construct at least some specific PDFs is to rewrite 2. The development of the exponential and comparison of coefficients of terms in growing order in iu shows that the first n cumulants can be expressed by the first n moments and vice versa. In contrast to the properties of moments, cumulants of high order may be seen to be less relevant than cumulants of low order.

This is caused by the fact that cumulants of higher than second order describe deviations from Gaussian PDFs see section 2. However, it is impossible to truncate 2. The theorem of Marcienkiewicz shows that one has two possibilities: i one considers zero K2 cannot be taken as equal to zero but has to be considered as a vanishing parameter, see the relation 2. In all the other cases the PDF cannot be positive everywhere. The implications of the relevant first case will be shown in the next subsection. Various Gaussian distribution functions and PDFs according to 2.

By neglecting all Kn with n t 3, the general expression 2. Examples are given in Fig. Another relevant feature of a Gaussian PDF is that F[ approaches to a delta function for a vanishing variance,. The definitions 2. PDFs of several variables The consideration of the joint statistics of several stochastic variables leads to a new question: how is it possible to assess the intensity of the coupling of different stochastic variables?

This question is very relevant in particular to theoretical analyses because the consideration of the limiting case of statistically independent stochastic variables is often a requirement for the applicability of approximations and the derivation of basic conclusions. This question of the intensity of the coupling of different stochastic variables is addressed in terms of correlation coefficients, as pointed out now. The ensemble-averaged product of these variables defines then their correlation coefficient. A simple analysis of h as a function of p reveals that h d 1. This is the unequality of Schwarz, which leads to many relevant implications.

This is for instance the case if two scalars are transported by the same turbulent eddy. It is very advantageous for some analyses to present joint PDFs of several variables such that one takes reference to the important special case that there is no correlation between different stochastic variables.

This can be achieved by introducing conditional PDFs. Regarding the joint PDF 2. By adopting 2. Conditional means can be defined in correspondence to 2. Hence, conditional means are calculated as means of the conditional PDF. The first rewriting of the left-hand side results from the properties of delta functions. Such expressions as 2. This relation explains conditional PDFs as means of corresponding conditional delta functions. After considering essential properties of PDFs, let us address now the question how specific PDFs can be constructed for a given case.

It was shown in section 2. Thus, a physical concept has to be involved. This can be done by taking reference to the predict- ability of the state of a stochastic variables considered, which is illustrated in Fig. This figure shows that the predictability of the state of a stochastic variable is determined by L.

For L o 0, the PDF becomes a delta function. The predictability is maximal in this case the uncertainty is minimal : we know that [ t will realize the value zero. For L o f, the predictability of the state of [ t is minimal the uncertainty is maximal : the probability of their realization is equal for all states. The concept of the predictability of states of stochastic variables can be used in the following way for the construction of PDFs. Very often we do not know anything about a PDF with the exception of a few low-order moments means and variances.

This knowledge is insufficient to determine the PDF, for which one would have to provide all its moments. What one can do in this case is to apply information about known moments combined with the constraint that the predict- ability related to the PDF to be constructed has to be minimal.

This approach makes only use of available information. Its details will be described now. The PDF of a uniformly L distributed stochastic variable [ t. L refers to the interval in which the PDF is nonzero. The first step to realize this concept is to define a unique measure S of the predictability of the state of stochastic variables.

To derive a constraint that enables the specification of the concrete structure of S one considers at best the special case of two independent variables [1 t and [2 t. In this case, S should satisfy the equation Shannon , Jaynes By inserting 2. To satisfy equation 2. Statistically most-likely PDFs This constant K is called the Boltzmann constant in the classical statistical mechanics, where it is introduced to give the entropy a suitable dimension.

Thus, we derive for S the expression. The appearance of the minus sign may be justified by inserting the Gaussian PDF 2. Obviously, V determines the uncertainty related to a Gaussian function in analogy to L in Fig. It is worth noting that the absolute value of S depends on the units of sample space variables.

This may be seen by rewriting 2. This results in the relation N being the dimension of the sample space. Nevertheless, entropy changes are unaffected by such transformations such that they can be used to study differences of the uncertainty related to various PDFs. Next, we construct a PDF according to the concept described above. This will be done for a PDF of only one variable. The extension to the case of several variables can easily be performed.

The goal is to construct a PDF which has s moments that agree with the given ones but maximizes the entropy S and therefore the uncertainty. The Pn are Lagrange multipliers which have to be chosen such that the conditions 2. The last term on the right-hand side of 2. The PDF 2. By introducing modified Lagrange multipliers On, relation 2.

By reformulating the On, one may see that 2. We assume that the first two moments are known,. Examples for statistically most-likely PDFs A simple way to adopt these relations for the calculation of Om and Omn is the use of partial integration. This results in. The remaining multiplier NF has to be calculated by adopting the constraint 2. This provides. The integration over z can be reduced then to the calculation of an one- dimensional integral.

This expression is nothing but the extension of the Gaussian PDF 2.

## COMPUTATIONAL AND SCIENTIFIC ENGINEERING

Essential properties of this PDF may be found again by partial integration. Hence, the triple correlations of Gaussian processes vanish, and the fourth-order moments are found to be parametrized by a symmetric combination of variances. Corresponding relations may be found for higher-order moments: all the odd moments are equal to zero and even moments are determined by symmetric combinations of variances.

As a second example, we assume that the first four PDF moments are known. The inclusion of third- and fourth-order moments permits the consideration of turbulence structure effects. The incorporation of third-order moments enables the simulation of bimodal turbulent motions transports in updrafts and downdrafts in convective boundary layer turbulence, see chapter 5. We consider again the case of one variable that is assumed to be unbounded.

This gives a mathematical reason for involving fourth-order moments. Such relations for Ok can be derived by partial integration in accordance with 2. Unfortunately, analytical expressions for Ok as functions of the first four moments of F[ cannot be obtained due to the appearance of the fifth- and sixth-order moments. However, the latter may be found by successive approximation: i appropriate initial values are chosen for them, ii O1, O2, O3, O4 are calculated by the equations 2.

By choosing the calculated fifth- and sixth-order moments as new initial values, this procedure has to be repeated until the third- and fourth-order moments provided by 2. A unique solution of that variation problem always exists. The possibility of such variations is restricted. The unequality of Schwarz 2.

In addition to this, one has the constraint Ku d 3 regarding the Ku-variations in Fig. This follows as a consequence of the fact that O4 has to be non-negative to assure that F[ vanishes at high x. These figures reveal that Ku and Sk variations enable the simulation of a variety of processes that may be seen as a superposition of two modes. Applications of 2. This is relevant to simulations of scalar fluctuations that are usually bounded for instance mass fractions of substances are bounded by zero and unity due to their definition.

As in section 2. Their calculation is now more complicated than for the case of unbounded variables. Instead of the relations 2. An example for such a PDF is given in Fig. Thus, one often applies other PDFs, which enable the simulation of similar PDF features but such that simple algebraic relations between PDF parameters and moments exist. With regard to unbounded variables, a simple way to construct such PDFs is to apply a superposition of two Gaussian modes, see for instance Luhar et al.

Ways to simulate such non-Gaussian PDFs for bounded variables will be pointed out next. Examples for other PDFs A PDF, which is often applied to simulate scalar distributions for instance the distribution of the frequency of turbulence, see Pope , is the gamma PDF. The parameters D and E of the beta function are related to the first two moments of F[ by. Several characteristic shapes of the beta PDF are shown in Fig. The advantage of using the beta function is that its parameters are related to PDF moments in a much simpler way.

This enables more efficient flow calculations. The application of such beta functions for reacting flow simulations is described for instance by Wall et al. Appendix 2A: Theta and delta functions The theta and delta function will be introduced first with regard to the case of one variable: the extension of these expressions to the many-variable case is very simple, see the explanations given in section 2A.

There are many different ways to define a theta function T y of one variable. Examples for TN y are given in Fig. This figure reveals that TN y tends for large N to. Examples for GN y are also given in Fig. This figure shows that G vanishes for all y z 0, and it diverges for y o 0.

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For that reason, the delta function is referred to as a generalized function or a distribution. These are functions which do not have to exist for all arguments, but integrals over such functions multiplied with other functions have to exist. An inspection of the properties of delta functions reveals that they have all the properties of PDFs. The normalization condition 2. Obviously, the other two constraints 2. The delta function is characterized by the following essential integral properties g y and h y are any test function and a is any number ,.

Relation 2A. Obviously, the right-hand side of this relation reduces to g a , as may be seen by means of 2A. Due to the properties of the delta function at infinity, the integral over this derivative must vanish. In relation 2A.

## Turbulence, and Fluid Mechanics in General

These derivatives are assumed to exist. To prove 2A. The second rewriting applies in the first-order of approximation the Taylor series of g at the corresponding root yi the zeroth-order terms vanish. The neglect of yi compared to the infinite L and the consideration of the cases that gi may be positive or negative results in the fourth rewriting. The right-hand side of 2A. It is worth noting that the properties 2A6.

The relations 2A. Stochastic processes. Next, we turn to the modeling of the evolution of stochastic variables. Out of equilibrium phase transitions. Prediction of the bistability of turbulent flows for the 2D Navier Stokes Equations. Classification of phase transitions. The 2D Euler equations are an example of systems with long range interactions. Systems with long range interactions are not additive, which can lead to inequivalence between the microcanonical and canonical ensembles. The microcanonical ensemble may show richer behavior than the canonical one, including negative heat capacities the temperature decrease when the energy is increased and other non-common behaviors like negative temperature jumps when the energy is increased, at a microcanonical critical point.

We have proposed a generalization of Landau classification for systems with long range interactions that describes all the possible phase transitions associated with situations of ensemble inequivalence. The phenomenology for such phase transitions is richer than the classical one. We have then predicted new ensemble inequivalence situations that have never been observed yet and others than have been observed only after our work. Simpler variational problems for the RSM statistical equilibria.

The Robert-Sommeria-Miller equilibrium statistical mechanics predicts the final organization of two dimensional flows. This powerful theory is difficult to handle practically, due to the complexity associated with an infinite number of constraints.

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We have established the relations between all these variational problems, justifying the use of simpler formulations. This provides a drastic mathematical simplifications for the study of equilibria, and increases our physical insight by justifying new physical analogies. Review paper on the statistical mechanics of geophysical flows: F. We predict theoretically and observe numerically out of equilibrium phase transitions: the turbulent flow switches randomly from a state with a large scale dipole to a state with a unidirectional flow. Similar theoretical considerations lead to the prediction of out of equilibrium phase transitions in a large class of other geometries, and also for geostrophic, high rotating flow experiments or 2D magnetic flows.