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5 Statistical Fluid Dynamics In the previous chapter, we considered fluid turbulence from a heuristic viewpoint, combining ideas based upon fundamental conservation laws with more qualitative ideas about irreversible behavior. However, there have been numerous attempts to study turbulence in a more quantitative and deductive fashion, beginning with the governing dynamical equations for the fluid. None of these attempts is, or could be, mathematically rigorous; all involve approximations that are difficult to justify and hard to test. In fact, these more deductive theories are perhaps best viewed as approximations in which the exact dynamical equations are replaced by stochastic-model equations having some (but not all) of the same physical properties as the exact equations, in much the same way as the quasigeostrophic equations contain only a part of the physics in the more exact primitive equations. In this chapter, we examine fluid turbulence from the standpoint of equilibrium and nonequilibrium statistical mechanics. This is a complicated and controversial subject, and our discussion will be introductory and elementary. In fact, we shall be much more concerned with the philosophy behind the statistical methods than with the detailed structure of the theory or with its successes and failures at describing real turbulence. This philosophy appears to be applicable to a much wider range of problems than so far considered. Readers who want a more thorough and complete description of statistical turbulence theory should consult more specialized sources.1

This chapter continues a theme, begun in Chapter 4, to be continued in Chapter 6, that much of our understanding of turbulence is based upon two general principles: the conservation principle, that quantities like energy and potential vorticity are conserved (apart from the effects of dissipation), and the irreversibility principle, that a turbulent system tends toward ever greater complexity. In Chapter 4, we met the irreversibility principle in the assumptions that a narrow spectral peak spreads out, and that nearby fluid particles tend to move apart. In this chapter, we encounter the irreversibility principle again, as a kind of macroscopic form of the Second Law. 1. The closure problem of turbulence First we re-state the fundamental closure problem of turbulence by means of an example. Consider a laboratory experiment in which water flows down a long, smooth, cylindrical pipe. An experimenter controls the average pressure gradient along the pipe. When this pressure gradient is sufficiently large, the flow is turbulent. As explained in Chapter 4, the details of this turbulent flow are erratic and unpredictable, but the average cross-sectional velocity is well described by a smooth, reproducible curve that can easily be fit with simple analytic functions. The fundamental question is this: Can this simple, smooth curve, obtained by averaging an enormous number of laboratory (or now perhaps numerical) data, instead be calculated directly from the governing Navier-Stokes equations, with no a posteriori averaging at all?

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The precise answer to this question is probably no. However, it might well be possible to get a very good approximation to the average velocity curve with much less work than that required to take and average all the measurements. Even more importantly, a successful approximation method would almost certainly illuminate the physics of turbulent flow. These are the hopes on which turbulence closure theory rests. An obvious way to begin is by forming the equations for the averages themselves. Readers of the previous chapter will readily accept that the governing equations of fluid dynamics can always be written in the abstract form dyi

dt= Aijkyj yk

j ,k∑ −ν iyi , (1.1)

where {yi} is a set of real numbers that defines the state of the whole fluid. For example, the vorticity equation

∂∇2ψ∂t

+ J ψ ,∇2ψ( ) = ν∇4ψ (1.2)

governs two-dimensional Navier-Stokes flow, where ψ(x,y,t) is the streamfunction for the flow, and

J A,B( ) ≡ ∂ A,B( )∂ x,y( ) (1.3)

is the horizontal Jacobian operator. If the flow is unbounded and 2π-periodic, then the Fourier transform of (1.2) is dψ k

dt= Ak,p,qψpψq

p, q∑ −νkψ k , (1.4)

where ψk(t) is the Fourier transform of ψ(x,t), ψ x,t( ) = ψ k t( )eik ⋅x

k∑ . (1.5)

Here, x=(x,y), k=(kx,ky), νk=νk2, k=|k|, and the interaction coefficients are given by Ak,p,q = 1

2 p × q( ) q2 − p2( )k−2δp+q =k , (1.6) where p×q≡pxqy-pyqx, and δ denotes the Kronecker delta. Eqn. (1.4) fits the form of (1.1) with i≡(α,k), and α=1(2) denoting the real (imaginary) part of ψk. In this case, the indices in (1.1) are triplets. As a second example, we consider two-dimensional flow bounded by a smooth, simply-connected curve in the (x,y) plane, and expand

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ψ x, y,t( ) = yi t( )ki−1φi x, y( )

i∑ (1.7)

in the orthonormal eigenfunctions φi(x,y) defined by ∇2φi x, y( ) = −ki

2φi x,y( ) , (1.8) the boundary condition φi=0, and the normalization, dx∫∫ φ i φ j = δij . (1.9) Now ki2 is the eigenvalue corresponding to eigenfunction φi(x). The ki-1-factor in the definition (1.7) of yi(t) is a matter of later convenience. Substituting (1.7) into (1.2), and using (1.8) and (1.9), we obtain an equation of the form (1.1) with coupling coefficients

Aijm =kj 2 − km2( )2kik jkm

dx φiJ φ j ,φm( )∫∫ . (1.10)

In both of these examples, the set {y1,y2,y3,......} (1.11) is countably infinite. If, on the other hand, the flow were spatially unbounded (and nonperiodic), we would replace (1.5) by ψ x,t( ) = dk∫∫ ψ k t( )eik ⋅x , (1.12) and obtain an equation like (1.4) in which integrals replace the sums over discrete indices. Then k varies continuously, and the set analogous to (1.11) is uncountably infinite. However, there is little practical difference between these countably and uncountably infinite sets. Even in the case of a bounded or periodic flow, the continuous-k representation (1.12) is a frequently useful idealization, because the higher wavenumbers differ by relatively small increments. This agrees with the physical notion that small-scale eddies are unaffected by distant boundaries. On the other hand, it will be very important to distinguish between systems with infinitely many degrees of freedom and systems in which the number N of dependent variables {y1,y2,y3,....,yN} (1.13) is finite. The latter frequently arise as numerical approximations to the differential equations governing fluid motion. For example, (1.4) with the sums truncated to exclude

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all the wavenumbers larger than a prescribed cutoff kc is a respectable numerical model of (1.2). Other examples include finite-difference approximations to (1.2), in which the yi(t) are the values of ψ (x,t) at the N gridpoints. Whether N is infinite or not, the dynamics (1.1) lead to a unclosed hierarchy of equations for the averages. The average of (1.1) yields N dynamical equations, d

dtyi = Aijk yj yk

j, k∑ − νi yi , (1.14)

for the N first moments <yi> in which the second moments <yjyk> also occur. Similarly, the equation,

ddt

yi yj = Aikm yjyk ym + Ajkm yiyk ym( )k ,m∑ − ν i +ν j( ) yi yj , (1.15)

for the second moments also contains the third moments <yjykym>, and so on. This hierarchy is mathematically unclosed, because the equations for the n-th moments always contain the (n+1)-th moments. Many theorists regard the difficulties associated with this unclosed hierarchy of equations as being equivalent to the general closure problem of turbulence. While this viewpoint may be too narrow, it provides a useful departure point from which to explore general statistical ideas that have been applied to turbulence. In any case, we shall see that the difficulties associated with the infinite hierarchy of equations are not mathematical difficulties in the technical sense, but inevitably involve additional physical hypotheses which at least partly amount to a refinement in the meaning of averaging. 2. The eddy-damped Markovian model Let the equations of fluid motion be dyi

dt= Aijkyj yk

j ,k∑ −ν iyi , (2.1)

with initial conditions yi 0( ) = yi0 . (2.2) With no loss in generality, we assume that the interaction coefficients are symmetric in their last two indices, Aijk = Aikj . (2.3) We also assume that Aijk vanishes whenever two of its indices are equal,

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Aijj = Ajij = Ajji = 0 . (2.4) The interaction coefficients (1.6) and (1.10) both have this property. In fact, (2.4) is a rather general property of fluid dynamical equations.2 Finally, we assume that Aijk + Ajki + Akij = 0 , (2.5) so that when ν=0, the dynamics (2.1) conserve the energy E ≡ yi

2

i∑ . (2.6)

For example, the energy, dx ∇ψ ⋅∫∫ ∇ψ , (2.7) of two-dimensional flow takes the form (2.6) when the yi are defined by (1.7-9). The interaction coefficients (1.10) obey (2.5), and hence (1.2) conserves (2.7) when ν=0. The eddy-damped quasi-normal Markovian model (hereafter EDM) is an approximation to the moment hierarchy (1.14-15, etc.) arising from (2.1) that closes the hierarchy at the level of the second moments. We shall study EDM as a representative theory of nonequilibrium statistical mechanics.3 The EDM equations take a relatively simple form if the the turbulence is statistically homogeneous, that is, if the statistics do not depend on location in the flow. If the turbulence is homogeneous, then <ψ(x)> is a constant (conveniently assumed to be zero), and <ψ(x)ψ(x')> depends only on x-x'. For definiteness, we now regard (2.1) as the Fourier transformation of the equation (1.2) governing 2π-periodic, two-dimensional, homogeneous turbulence. The yi(t) are the coefficients of the functions sin nx +my( ) and cos nx + my( ) (2.8) in the expansion of ψ(x,y,t), and the subscript i denotes (m,n) and whether cosine or sine. Since the turbulence is homogeneous, it follows that <yi(t)>=0 and yi t( )yj t( ) = Yi t( )δij . (2.9) Hence the second-moment equations (1.15) take the form,

ddt

+ 2ν i⎛ ⎝ ⎜ ⎞

⎠ ⎟ Yi t( ) = 2 Aijk yiyj yk

j, k∑ . (2.10)

For the third moments, we obtain

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ddt

+ ν i + ν j +νk⎛ ⎝ ⎜ ⎞

⎠ ⎟ yiyjyk

= Aimn ymyn yjyk + Ajmn ymynyiyk + Akmn ymyn yiyj{ }m ,n∑

(2.11)

EDM closes (2.10-11) by treating the {yi} on the right-hand side of (2.11) as Gaussian random variables. If A, B, C, and D are Gaussian random variables with zero means, then4 ABC = 0 (2.12) and ABCD = AB CD + AC BD + AD BC . (2.13) Factoring the right-hand side of (2.11) in the same way as (2.13), making use of (2.9), and assuming (for simplicity) that the triple moments <yiyjyk> all vanish at the initial time t=0 (as if the initial conditions were completely random), we solve (2.11) and substitute the result into (2.10). The result is a closed equation, ddt

+ 2ν i⎛ ⎝ ⎜ ⎞

⎠ ⎟ Yi t( ) = ds

0

t

∫j,k∑ e− t− s( ) νi +ν j +νk( ){4Aijk

2Yj s( )Yk s( ) +8AijkAjikYk s( )Yi s( )}, (2.14)

for the covariances {Yi(t)}. Note that the assumptions leading to (2.14) have been applied somewhat inconsistently. If the {yi} were really Gaussian random variables, then according to (2.12), the right-hand side of (2.10) would vanish at all times, just as if the nonlinear terms in the original dynamical equations (2.1) had been entirely absent. Thus it is critical that the Gaussianity-assumption is applied only to the higher-order equations (2.11). Unfortunately, numerical integrations of (2.14) lead to large negative Yi when the Reynolds number is very large. Since Yi=<yi2> must be positive, this is clearly an unreasonable result. However, we can understand this calamity, and use the understanding to repair (2.14). To understand why (2.14) predicts negative Yi, suppose that the initial conditions {Yi(0)} correspond to a sharp peak in the wavenumber spectrum at t=0 (Figure 5.1). At small t>0, we expect the right-hand side of (2.14) to become large and negative for Yk inside the peak, and positive for Yk outside the peak. At later times, the curly bracket in the integrand of (2.14) should decrease in magnitude. However, since the time-integration in (2.14) runs all the way back to t=0, the large initial tendency for the spectral peak to decrease is never completely forgotten. The persistence of this large initial tendency is especially strong in the limit of very large Reynolds number (corresponding to small viscosity, ν i ,ν j ,νk → 0 ) because the values of the curly bracket in the distant past (s<<t) then weigh equally with those of the near past (s≈t) in the

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integrand of (2.14). In the limit of very large Reynolds number, the large initial tendencies are not forgotten, and consequently the spectral peak plunges right through zero. Clearly, this sling-shot behavior is caused by the unrealistically long memory in (2.14) as the {νi}→0. However, in real turbulent flows, past states are forgotten not because of the molecular viscosity, but because of the nonlinear-interaction terms in the equations of motion. The fluid forgets its past over roughly the same interval as future predictions become unreliable, and for much the same reason. At large Reynolds number, this memory time should be independent of the molecular viscosity, in contradiction with (2.14). These thoughts suggest that we can repair (2.14) by replacing the molecular viscosities νi, νj, νk in (2.11) with much larger values µi, µj, µk that parameterize the rapid memory loss of past states at high Reynolds number. These µi differ from the usual eddy viscosity coefficients in that they then enter (2.11) but not (2.10) and thus do not directly cause energy to decrease. The resulting covariance equation,

ddt

+ 2ν i⎛ ⎝ ⎜ ⎞

⎠ ⎟ Yi t( ) = ds

0

t

∫j,k∑ e− t− s( ) µi+ µ j +µ k( ){4Aijk

2Yj s( )Yk s( ) + 8AijkAjikYk s( )Yi s( )}, (2.15)

differs from (2.14) only in that µi replace the νi on the right-hand side. Once again, the µi parameterize the decrease in the triple correlations <yiyjyk> caused by the scrambling effect of the flow on itself. Hence the {µi} ought to increase with the Reynolds number. That is, the more turbulent the flow, the more rapidly triple correlations are destroyed. Therefore, at very large Reynolds number (large µi), most of the contribution to the integral in (2.15) should come from s near t. This in turn suggests that, at large Reynolds number, we can replace the curly-bracket term in (2.15) with its value at s=t. The result is

ddt

+ 2ν i⎛ ⎝ ⎜ ⎞

⎠ ⎟ Yi t( ) = θ ijk

j,k∑ {4Aijk

2Yj t( )Yk t( ) + 8Aijk AjikYk t( )Yi t( )} (2.16)

where θ ijk ≡ ds

0

t

∫ e− t− s( ) µi+µ j +µk( ) . (2.17) Eqn. (2.16) is the eddy-damped quasi-normal Markovian approximation (EDM). The words eddy-damped refer to the introduction of the µi. Markovian refers to the evaluation of the curly-bracket term in (2.15) at s=t. Normal refers to the use of (2.13). The approximation is termed quasi-normal because we have applied the Gaussian assumption inconsistently. Again, if the {yi} were indeed Gaussian with zero means, then <yiyjyk> would vanish in (2.10). It is easy to see that EDM does not allow the negative-Yi calamity. For, if Yi=0 for some particular i, with all other {Yj} positive, then, since the right-hand side of (2.16) is

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positive, Yi must be increasing. Using (2.5), one can show that when νi=0 EDM conserves the average energy, E = Yi

i∑ . (2.18)

We come back to this at length. We can regard µi-1 as the time interval over which yi remains correlated with itself, and θijk as the average persistence time for the product yiyjyk. According to (2.17), at short times θijk increases linearly with time, as the triple correlations build up from their assumed zero initial value. At long times, θ ijk ~

1µi + µ j + µk

t→ ∞( ) , (2.19)

assuming that the µi are constants or change slowly (on a time scale much longer than µi-1). Our derivation of EDM leaves the µi unspecified. However, by the reasoning of Chapter 4, the time scale Tk for the distortion of an eddy of size k -1 is given by Tk

−2 ~ ′ k 2E ′ k ( )d ′ k 0

k

∫ , (2.20) where E(k) is the energy spectrum. If we suppose that µk ∝Tk

−1 , (2.21) then the EDM equations corresponding to three-dimensional Navier-Stokes turbulence are consistent with Kolmogorov’s theory and predict an inertial-range spectrum E(k)~k -5/3.5 Readers familiar resonant wave-interaction theory will notice a strong resemblance between EDM and the equations for weakly interacting waves. Suppose we generalize (2.1) to dyi

dt+ iωi yi = Aijkyj yk

j ,k∑ −ν iyi , (2.22)

so that, in the limit of infinitesimal {yi}, the solutions are linear waves. Here, ωi is the (constant) wave frequency associated with mode yi, and i (where not a subscript) is the square root of -1. An equation of the form (2.22) arises from the generalization,

∂∇2ψ∂t

+ β∂ψ∂x

+ J ψ ,∇2ψ( ) = ν∇4ψ , (2.23)

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of (1.2) to include Rossby waves.6 By the same steps as before, we again obtain (2.16), but with

θ ijk ~µi + µ j + µ k

µi + µ j + µk( )2 + ω i +ω j +ω k( )2, t→ ∞ (2.24)

instead of (2.19). In the limit of strong turbulence (µ>>ω), (2.24) reduces to (2.19). However, in the limit of weak turbulence (µ<<ω), (2.24) reduces to θ ijk ~ π δ ω i +ω j +ω k( ) , (2.25) in agreement with resonant wave-interaction theory. In this limit, only triads (yi,yj,yk) whose three frequencies sum to zero can interact. However, (2.24) shows that even weak nonlinearity relaxes this criterion, allowing slightly off-resonant triads to transfer energy between modes. In fact, it seems very doubtful that one is ever justified in setting {µi}≡0 and eliminating the off-resonant interactions completely. 3. Stochastic model representation Thus far we have regarded EDM as an approximation to the moment hierarchy corresponding to the exact dynamics (2.1). Now we take a different viewpoint. We show that EDM is the exact result of averaging stochastic differential equations that resemble the exact dynamics. From this new viewpoint, the EDM approximation consists of replacing the exact dynamical equations with stochastic-model equations that are easier to analyze.7 Again we consider homogeneous turbulence governed by dyi

dt+ νi yi = Aijkyj yk

j ,k∑ , (3.1)

with initial conditions yi0. Again the interaction coefficients obey (2.3-5). But now we consider the stochastic model dyi

dt+ νi yi =W t( ) θ ijk

1/ 2Aijk yjGyk

G

j, k∑ , (3.2)

in which Gaussian random variables yiG replace the yi on the right-hand side of (3.1). These Gaussian random variables have the same (zero) means and covariances yi

G t( )yjG t( ) = yi t( )yj t( ) ≡ Yi t( )δij (3.3) as the {yj}. W(t) is a white-noise random variable, independent of the {yiG}, with zero mean and covariance

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W t( )W t'( ) = 2 δ t − t'( ) . (3.4) The presence of W(t) corresponds to the Markovianization step in the previous derivation of EDM. The θijk are positive nonrandom numbers with the dimensions of time, fully symmetric in their three subscripts. Unlike (3.1), (3.2) is a linear equation, with solution

yi t( ) = yi 0e− νi t + ds−∞

t

∫ e−ν i t− s( )W s( ) θijk1/ 2Aijk yj

G s( )ykG s( )j ,k∑ . (3.5)

From (3.5) and (3.2), and assuming (for simplicity) that yiG and yi0 are independent random variables, we obtain

dYidt

= 2 yidyidt

= −2νiYi + 2 ds−∞

t

∫ e−ν i t−s( ) W s( )W t( ) ×

×j ,k∑ θ ijk

1 / 2θ imn1 / 2Aijk Aimn yjG s( )ykG s( )ymG t( )ynG t( )

m,n∑

(3.6)

Then, using (2.3-5) and the property (2.13) of Gaussian random variables, we obtain the covariance equation,

ddt

+ 2ν i⎛ ⎝ ⎜ ⎞

⎠ ⎟ Yi t( ) = θ ijk

j,k∑ {4Aijk

2Yj t( )Yk t( )}, (3.7)

for the stochastic model (3.2). Unfortunately, (3.7) does not conserve the average energy (2.18) in the limit of vanishing molecular viscosity νi. In fact, when νi=0, (2.18) always increases, because the right-hand side of (3.7) is positive. The random driving terms on the right-hand side of (3.2) cause yi2 to increase, on average, for the same reason that a particle subject to a random forcing wanders steadily away, on average, from its initial location. To counteract the tendency of the { yj

G } to increase the model energy, we add an eddy-viscosity term to (3.2). That is, we replace (3.2) by the new stochastic model dyi

dt+ νi yi + ηi t( )yi =W t( ) θ ijk

1/ 2AijkyjGyk

G

j, k∑ , (3.8)

where ηi is a (time-dependent) eddy-viscosity coefficient. The stochastic model (3.8) leads to the covariance equation

ddt

+ 2ν i + 2ηi⎛ ⎝ ⎜ ⎞

⎠ ⎟ Yi t( ) = θijk

j ,k∑ {4Aijk

2Yj t( )Yk t( )} . (3.9)

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The eddy-coefficient ηi is to be prescribed in such a way that the average energy is conserved (apart from molecular viscosity). The choice ηi t( ) = −4 θ ijk Aijk AjkiYk t( )

jk∑ (3.10)

conserves energy (when νi=0) and makes (3.9) equivalent to (2.16). Thus EDM is the exact covariance equation for the stochastic model (3.8) with ηi(t) given by (3.10). The existence of a stochastic model (3.8) for which EDM is exact is important, because it guarantees that the statistics predicted by EDM are realizable, that is, correspond to kinematically possible states of the system {yi}. In particular, EDM can never predict negative Yi, because solutions of the EDM equations correspond to exact averages of (3.8) for which <yi2> can never be negative. The stochastic model also suggests an appealing alternative derivation of EDM: In the exact equations of motion, replace the nonlinear terms by terms containing Gaussian white-noise random variables, and add a compensating eddy viscosity. This stochastic-model derivation gives a clearer picture of the physical contents of EDM. In particular, it shows that EDM does include the effects of organized structures in the flow. However, in order to complete the new derivation of EDM that begins with the introduction of the stochastic model (3.8), we must offer a motivation for the prescription (3.10) of ηi(t). Since energy conservation is a desirable property of the resulting covariance equations, it is logical to require that (3.9) conserve energy in the mean. Unfortunately, this conservation property does not uniquely determine ηi(t). The reader will verify that both (3.10) and ηi t( ) = 2 θijk Ajki

2Yk t( )jk∑ (3.11)

conserve energy in the mean, but only the choice (3.10) agrees with the EDM of the previous section. Thus, from the standpoint of the stochastic-model derivation, at least two distinct covariance equations lay equal claim to our attention, or there is yet another criterion, besides the conservation property, that must be enforced. Fortunately, the latter proves to be the case, and the new criterion turns out to be the irreversibility principle in the form of the Second Law. 4. Entropy Let the dynamics be dyi

dt= Aijkyj yk

j ,k∑ −ν iyi , yi 0( ) = yi

0 , (4.1)

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as before, but now let the summations in (4.1) be truncated to a finite number N of modes. We regard (4.1) as the equations for a numerical model of the fluid. We also regard (4.1) as the equation for a trajectory in the N-dimensional phase space with coordinates y≡(y1, y2,...., yN). Every point in the phase space corresponds to a particular state of the whole system. Let P(y,t) be the density of system states in phase space. That is, let P(y,t)dy be the probability that the system is in phase-space volume dy centered on y at time t. Then, since the moving points that represent individual realizations of (4.1) can neither be created nor destroyed, the probability density P(y,t) obeys Liouville’s equation,

∂P∂t

+∂∂yi

˙ y iP( )i∑ = 0 , (4.2)

with initial condition, P y,0( ) = P0 y( ) , (4.3) where P0(y) is the probability density of the initial conditions yi0. Here ˙ y , the N-dimensional velocity in phase space, is given by (4.1). Eqn. (4.2) is analogous to the continuity equation,

∂ρ∂t

+∇ ⋅ vρ( ) = 0 , (4.4)

with P (the density of system-points in phase space) corresponding to ρ (the density of molecules), ˙ y corresponding to the fluid velocity v, and the summation over N phase-space dimensions corresponding to the summation over three physical-space dimensions. Now we divide the phase-space velocity ˙ y into two parts: an energy-conserving part, ˙ y i

c = Aijky jykj ,k∑ , (4.5)

and a non-conserving (viscous) part, ˙ y i

ν = −ν iyi . (4.6) Both (4.5) and (4.6) are steady velocity fields in phase space, but since Aijk vanishes whenever two of its indices are the same (cf. (2.4)), the conserving velocity (4.5) is also non-divergent,

∂˙ y ic

∂yii∑ = 0 . (4.7)

Thus if the viscosity vanishes, ˙ y is non-divergent, and (4.2) reduces to the analogue

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∂P∂t

+ ˙ y i∂P∂yii

∑ = 0 (4.8)

of the equation Dρ

Dt= 0 (4.9)

for incompressible flow. The non-conserving velocity (4.6) is, on the other hand, convergent, so that if {νi}>0, then all the trajectories (4.1) eventually terminate at the origin, y=0. Now suppose that the non-conserving part of ˙ y is negligible, either because the viscosity actually vanishes, or because viscosity is so small that its effect can be temporarily neglected. Then, according to (4.8), every system-point keeps its initial value of P, and (since the phase flow is non-divergent), the volume of phase space with a particular value of P remains the same. From the analogy between (4.8) and the equation (4.9) for a conserved scalar, we anticipate that the field of P(y,t) becomes steadily more complex as t increases. Consider, for example, a two-dimensional phase-space in which P is initially uniform within a compact region, and zero outside (Figure 5.2). As t increases, the region of nonzero P typically spreads out by developing long filaments that gradually fill up the accessible parts of phase space. This evolving complexity of P means that (4.8) must be very hard to solve. However, it must also be true that exact solution of (4.8) is pointless, because (4.8), like (4.9) and indeed (4.1), typically exhibits unbounded sensitivity to its initial conditions. That is, the complicated structure of P(y,t) is also non-reproducible.8 However, P is typically sought for computing the average,

F = dy∫ F y( )P y,t( )∫∫∫ , (4.10)

of relatively smooth phase functions F(y). For example, <yi2> corresponds to F=yi2. Suppose that P actually evolves from t=0 to t=t1 as shown in Figure 5.2. While it may be practically impossible to calculate P(y,t1) accurately from (4.8) for use in (4.10), it is also obvious that, for any F(y) that depends smoothly on y, the average <F> at t1 can be calculated to good accuracy by replacing P(y,t1) with a probability density function that is uniform over the circular region in Figure 5.2. We therefore distinguish between P(y,t), the exact solution of (4.8), and ˆ P y,t( ) , a smoothed (sometimes also called coarse-grained) version of P. Nonequilibrium statistical mechanics seeks ˆ P y,t( ) but without first finding P(y,t). This is another way of posing the closure problem. Clearly, the stirring of P by the phase-space velocity ˙ y is what motivates the introduction of ˆ P . However, this same stirring property imposes the consistency requirement that ˆ P ought to be progressively more spread out at successively later times.9 This is a qualitative statement of the Second Law. To make the statement

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quantitative, we must introduce a measure of the spread of ˆ P , or, equivalently, of the uncertainty in y(t) represented by ˆ P y,t( ) . It turns out that the entropy

S ˆ P y,t( )[ ] = − dy ˆ P ln∫∫∫∫ ˆ P (4.11)

is the best measure of the uncertainty in y. Now we pause to explain why. Consider a random variable ξ that can assume any one of M discrete values {ξ1,ξ2,.....,ξM}. Let pi be the probability of ξi. We want a measure of the uncertainty in ξ. The measure

S p1, p2, ...., pM( ) = − pi ln pii=1

M

∑ (4.12)

has two desirable properties. First, it takes its maximum value when all the outcomes are equally probable, pi=1/M. And second, the measure (4.12) is additive in the following sense: If ξ is a composite random variable, for which each outcome consists of an outcome for η and an outcome for γ, where η and γ are independent random variables, then the uncertainty in ξ is the sum of the uncertainties in η and γ. The converse is also true: the only measure of uncertainty with these two properties is (4.12) (to within a constant factor).10 The definition (4.12) is useful by itself. For example, suppose we know only the first two moments of ξ,

ξ ≡ ξi pi = mi=1

M

∑ , ξ 2 ≡ ξi2 pi

i=1

M

∑ = Ξ . (4.13)

From m and Ξ we want to determine the best values of {pi} for use in estimating other statistics of ξ. According to information theory, the most objective estimate for {pi} is that which maximizes the entropy (4.12) subject to constraints that state everything actually known about {pi}. Maximizing (4.12) subject to the two constraints (4.13) and the normalization requirement

pi = 1i=1

M

∑ , (4.14)

we find that

pi = Cexp − αξi + βξi2( )

i=1

M

∑⎧ ⎨ ⎩

⎫ ⎬ ⎭

, (4.15)

where the three constants α, β and C are determined from m and Ξ by (4.13) and (4.14).

V-15

Now suppose that the random variable ξ can assume any value between 0 and 1. We are tempted to generalize (4.12) to S P ξ( )[ ] = − dξ P ξ( ) ln P ξ( )

0

1

∫ , (4.16) which has its maximum value when ξ is uniformly distributed on the interval [0,1]. However, we could apply the same logic to the random variable µ≡ξ2; then we would conclude that the greatest uncertainty corresponds to a uniform distribution of µ . These two conclusions obviously disagree: If ξ is uniformly distributed, then ξ2 cannot be. The proper generalization of (4.16) is S P ξ( )[ ] = − dξ m ξ( ) P ξ( ) ln P ξ( )

0

1

∫ , (4.17) where m(ξ), the measure of ξ-space, must be determined by some additional consideration. Similarly, the uncertainty in y(t) corresponding to the probability density ˆ P y,t( ) is

S ˆ P y,t( )[ ] = − dy m y,t( ) ˆ P ln∫∫∫∫ ˆ P , (4.18)

where m(y,t) is the undetermined measure of phase space. Clearly the entropy (4.11) corresponds to the choice m(y,t)≡constant in (4.18), but what justifies this choice? Consider the limiting case in which no smoothing of P to ˆ P occurs, either because the velocity ˙ y stirs P(y,t) ineffectively, or because the fine-grained probability density P is actually needed to compute the average of some very wiggly phase function. In this limiting case, P and ˆ P are the same, and (4.8) implies that (4.11) is conserved. Since no smoothing occurs, uncertainty does not increase, and the entropy stays the same. But (4.11) and (4.18) agree only if m(y,t) is a constant. Thus (4.7) fixes the measure of phase space: The measure is uniform in the phase-space coordinates for which the phase-space velocity is nondivergent. 5. The entropy principle Now we are ready to put the definition of entropy to work. First we show how to identify the imprecise description implicit in EDM with a particular ˆ P y,t( ) for which the entropy (4.11) can be evaluated. Then we verify that, according to the EDM, this entropy is a non-decreasing function of time when the viscosity vanishes. In EDM the dependent variables are the second moments, y1 t( )

2 , y2 t( )2 ,......, yN t( )

2{ } . (5.1)

V-16

Let the moments (5.1) have the values Yi at a fixed time t. Then, according to information theory, the most objective estimate for ˆ P y,t( ) is that which maximizes the entropy (4.11) subject to the N constraints yi

2 = Yi , (5.2) and the normalization requirement 1 = 1. (5.3) This easy variational problem leads to

ˆ P y,t( ) = exp λ − αi yi2

i∑⎧

⎨ ⎩

⎫ ⎬ ⎭

, (5.4)

where {αi} and λ are the Lagrange multipliers corresponding to (5.2) and (5.3). By (5.2) and (5.3),

λ = 12 ln α i

π⎛ ⎝ ⎜ ⎞

⎠ ⎟

i∑ and αi =

12Yi

. (5.5)

Then, by (4.11) and (5.4), S ˆ P y,t( )[ ] = 1

2 ln 2πYi( ) +1{ }i∑ . (5.6)

The entropy (5.6) measures the spread of the probability density (5.4). More precisely, (5.6) measures the uncertainty associated with a partial knowledge of the system state, namely, knowledge of the second moments alone. Now suppose that the viscosity vanishes. Then, by the reasoning of the previous section, the uncertainty should only increase, as the coarse-grained probability density spreads out. Thus dS

dt≥ 0 . (5.7)

By (5.6), the entropy law (5.7) is equivalent to d

dtlnYi

i∑ ≥ 0 . (5.8)

The inequality (5.8) is analogous to Boltzmann’s famous H-theorem. We shall show that EDM implies (5.8) in the same way that Boltzmann’s collision equation implies the H-theorem. In fact, EDM and the Boltzmann equation are themselves closely analogous.

V-17

In all of this we have assumed that {νi}=0. If the viscosity is nonzero, we replace (5.7) by

dSdtc

≥ 0 , (5.9)

where d/dtc is the rate of change caused by the non-viscous (i.e. energy conserving) terms in the EDM equation for dYi/dt. Even when viscosity is present, the energy-conserving part of the physics stirs the probability density function P in phase space, increasing the spread of ˆ P . On the other hand, the viscosity drives the system toward y=0,

dYidtν

= −2νiYi , (5.10)

causing the entropy (5.6) to decrease,

dSdtν

= −2 ν ii∑ < 0 . (5.11)

However, the general entropy law (5.9) holds whether or not the stirring actually dominates the viscosity. Now we verify that, when {νi}=0, EDM conserves the energy, d

dtYi = 0

i∑ , (5.12)

and obeys the entropy law in the form (5.8). Later we show that these energy and entropy properties virtually determine the form of EDM. We can prove (5.8) and (5.12) by direct manipulations on EDM, using the energy conservation property (2.5) of the exact dynamics. However, it is more illuminating to prove (5.8) and (5.12) by means of the following alternative derivation of EDM. The new derivation is rather formal; it is best presented as a series of steps. But the formal derivation offers the quickest way of seeing that EDM has the energy and entropy properties stated above. First, expand the exact (inviscid) dynamics in a Taylor series about t=0, to obtain

yi t( ) = yi 0( ) + t Aijk yj 0( )yk 0( )

j,k∑ + t2 Aijk Akmny j 0( )

m, n∑

j,k∑ ym 0( )yn 0( ) + (5.13)

Regard the initial conditions {yi(0)} as Gaussian random variables. Then squaring and averaging (5.13) yields the exact equation Yi t( ) = Yi 0( ) + 2t2 Aijk

2Yj 0( )Yk 0( )j ,k∑ + 4t2 Aijk AjikYi 0( )Yk 0( )

j ,k∑ +O t4( ) . (5.14)

V-18

Now, throw away the O(t4) terms in (5.14), differentiate the remaining terms with respect to time, and then replace the {Yj(0)} by {Yj(t)}. We obtain ˙ Y i t( ) = t 4Aijk

2Yj t( )Yk t( ) + 8Aijk AjikYi t( )Yk t( ){ }j,k∑ . (5.15)

Finally, replace the t-factor in (5.15) by the triad decorrelation time θijk. The result is (inviscid) EDM, dYi

dt= 4θ ijk Aijk (AijkYj + 2AjikYi)Yk

j ,k∑ . (5.16)

It is now easy to see how EDM inherits the energy conservation property of the exact dynamical equations. Since the exact (inviscid) dynamics have the same energy at all times, energy conservation holds term by term for the Taylor series expansion (5.14). In particular, (5.14) still conserves energy if all the terms except the t2-terms are dropped from the right-hand side. The replacement of {Yj(0)} by {Yj(t)} does not spoil energy conservation, because energy is conserved for arbitrary initial conditions {Yj(0)}. Hence (5.15) also conserves energy. Finally, energy conservation survives the replacement of t by θijk (provided that θijk is symmetric in its three indices) because the energy is conserved triad-wise if at all.11 By the same reasoning, EDM conserves any invariant of the exact dynamics whose average can be expressed as a linear combination of the Yi. For example, inviscid two-dimensional Euler flow governed by (1.2) (with ν=0) conserves (twice) the energy, E = dxdy∇ψ ⋅∇ψ∫∫ = yi

2

i∑ , (5.17)

and the enstrophy, Z = dxdy ∇2ψ( )2∫∫ = ki

2yi2

i∑ , (5.18)

where yi is defined by (1.7). Hence, the corresponding EDM equations conserve E = Yi

i∑ (5.19)

and Z = ki

2Yii∑ . (5.20)

By very similar reasoning, we show that EDM obeys the H-theorem (5.8).12 By definition,

V-19

S t( ) − S 0( ) = dy − ˆ P t( ) ln ˆ P t( ) + ˆ P 0( ) ln ˆ P 0( ){ }∫∫∫∫ . (5.21)

But, since the initial conditions are, by assumption, Gaussian, ˆ P 0( ) = P 0( ). Then, because

dy∫∫∫∫ P t( ) ln P t( ) = dy∫∫∫∫ P 0( ) ln P 0( ) (5.22)

by (4.8), (5.21) becomes

S t( ) − S 0( ) = dy − ˆ P t( ) ln ˆ P t( ) + P t( ) ln P t( ){ }∫∫∫∫ . (5.23)

But ˆ P y,t( ) is, by definition, the probability density that maximizes the entropy functional (4.11) for given {Yi(t)}. Hence (5.23) implies that S t( ) − S 0( ) ≥ 0 . (5.24) The exact equation (5.14) must be consistent with (5.24). But (5.15) is also exact in the limit t→0. Thus (5.15) must obey the H-theorem (5.8). Finally, (5.8) survives the replacement of t by θijk in (5.16) by the same reasoning as before. This proof of (5.8) shows that the H-theorem arises from the assumption of Gaussian initial conditions in the formal derivation of EDM. Of course we can also prove the H-theorem directly from (5.16). Since the H-theorem holds triadwise (if at all), we need only show that the entropy of a particular triad increases as a result of the interactions between the triad members alone. Let i, j, k denote the three members of a particular triad. Then, using only (5.16) and the energy-conservation property (2.5), we can show that dSdt

=ddtlnYi + lnYj + lnYk( ) = 8θ ijk

YiYjYkAijkYj Yi − Yk( ) + AjikYi Yj − Yk( )[ ]2 ≥ 0 . (5.25)

These results have meaning whether the viscosity vanishes or not. If the viscosity is nonzero, we can still say that the terms in EDM arising from the nonlinear-interaction terms in the original dynamics act by themselves to conserve energy and to cause entropy to increase. But suppose that the viscosity actually vanishes, so that the entropy lnYi

i∑ (5.26)

increases for an arbitrarily long time. How big can it get? Suppose that the energy (5.19) is the only conserved quadratic. Then the entropy cannot exceed the value obtained by maximizing (5.26) subject to the constraint (5.19). In this absolute equilibrium state,

V-20

Yi = E / N , (5.27) and the energy is equi-partitioned among the N modes. If however the exact dynamics (and hence EDM) have additional quadratic invariants, then the maximum achievable entropy will generally be lower. Suppose, for example, that the system conserves both the energy (5.19) and the enstrophy (5.20). Then the maximum entropy, obtained by maximizing (5.26) subject to (5.19) and (5.20), occurs when Yi =

1α + β ki

2 , (5.28)

where α and β are Lagrange multipliers determined by the values of <E> and <Z>. (Since the energy and enstrophy are conserved, these values are fixed by the initial conditions.) According to (5.28), a linear combination of the energy and enstrophy (namely αYi + β ki

2Yi) is equi-partitioned among the modes. Two things can prevent the system from attaining the absolute equilibrium state. First, if viscosity is present, then although the energy-conserving part of the physics still acts to increase the entropy, the viscosity drives the system toward the (minimum entropy) state of rest. Second, if the number of modes is infinite (N=∞), then absolute equilibrium cannot occur in a finite time, even if the viscosity actually vanishes. However, even in that case, the absolute-equilibrium states still represent informative target states towards which the nonlinear terms, acting alone, would drive the flow. We shall come back to this at length. Now we show that the entropy principle (5.8) is all that is needed to complete the stochastic-model derivation of EDM that began with (3.9). Assume, for simplicity, that the energy (5.19) is the only conserved quadratic. We want to show that if

ddt

+ 2ηi⎛ ⎝ ⎜ ⎞

⎠ ⎟ Yi = 4 θ ijkAijk

2YjYkj,k∑ , (5.29)

then (3.10) is the only choice of eddy-viscosity coefficient ηi that is consistent with the conservation principle (5.12) and the irreversibility principle (5.8). Since these properties must hold for (separately) arbitrary {θijk} and {Yj}, we assume that ηi takes the form ηi = ηijkθ ijkYk

j ,k∑ , (5.30)

where {ηijk} is a set of constants, to be determined. Again let i, j, k denote the three members of a particular triad. By requiring the energy of this triad to be conserved for arbitrary values of Yi, Yj, Yk, we obtain the three equations,

V-21

4Aijk2 = ηkij +η jik

4Ajik2 = ηkji +ηijk

4Akij2 = ηikj +η jki

(5.31)

Note that the last two equations are simply permutations of the first. Similarly, the absolute-equilibrium state (5.27) is a steady solution of (5.29) if and only if

4Aijk2 = ηijk +ηikj

4Ajki2 = η jik + η jki

4Akij2 = ηkij +ηkji

(5.32)

The sets (5.31) and (5.32) comprise 6 equations in the six unknown constants {ηijk}. However, these six equations are not all independent; the sum of (5.31) equals the sum of (5.32). In fact, the general solution of (5.31-32) is ηijk = 2Aijk

2 + 2Ajki2 − 2Akij

2 + Cε ijk , (5.33) where εijk is the permutation symbol, and C is an arbitrary constant. However, unless C=0, the solutions of (5.29) oscillate about the maximum entropy state (5.27), violating the entropy principle (5.8). The choice C=0 corresponds to EDM, (5.16), after use of the conservation relations (2.5). We see that the conservation principle and the irreversibility principle virtually determine the form of the EDM approximation. 6. Equilibrium statistical mechanics We began with the definition of entropy, a measure of the uncertainty about the state of the physical system under consideration. Then, on account of the instability property of the exact dynamics (that is, on account of the stirring of probability density in phase space), we argued that the entropy ought to increase with time. Next we showed that EDM obeys this law of entropy increase and, moreover, that the entropy law is one of two essential ingredients (the other being conservation of energy) that largely determine the form of EDM. Finally we discussed the absolute-equilibrium states of maximum entropy. Acknowledging that these absolute-equilibrium states are unrealizable in the cases of most interest to us, we argued that they nevertheless reveal the direction toward which nonlinear interactions (that is, the stirring in phase space) drive the system. In the next two sections, we examine these absolute-equilibrium states more closely. Absolute equilibrium is the subject of equilibrium statistical mechanics, which predicts the final state toward which isolated systems evolve when the number of degrees of freedom is finite. Equilibrium statistical mechanics does not predict the path of evolution to absolute equilibrium, nor can it predict the kind of equilibrium that results from a balance between the external forcing of some modes in the system and the damping of others. These are the province of nonequilibrium statistical mechanics,

V-22

exemplified by EDM. The nonequilibrium theory includes the equilibrium theory as a limiting case. Unfortunately, the nonequilibrium theory is much more complicated than the equilibrium theory and is in fact highly developed (i.e. non-controversial) only for systems near absolute equilibrium, which (as we shall see) certainly excludes fluid turbulence. However, insofar as the more easily calculable absolute-equilibrium states offer some insight on nonequilibrium behavior, equilibrium statistical mechanics has a considerable value of its own. In this section, we consider equilibrium statistical mechanics as a separate theory, emphasizing its logical simplicity and avoiding references to the nonequilibrium theory.13 As an example, we calculate the equilibrium state of a classical ideal gas with a fixed number of molecules, thereby repaying a debt incurred in Chapter 1. In the following section, we consider the much more problematic equilibrium state corresponding to a macroscopic fluid with an infinite number of modes. Consider the general dynamical system described by ˙ y i = fi y( ), i = 1, N , (6.1) where N is finite. Once again, we regard (6.1) as the equation for a trajectory in the N-dimensional phase space with coordinates y ≡ y1, y2, ..........., yN( ) . (6.2) If the phase-space velocity (6.1) is nondivergent,

∂fi∂yii

∑ = 0 , (6.3)

then the probability density P(y,t) of system states obeys

∂P∂t

+ fi y( ) ∂P∂yii

∑ = 0 . (6.4)

According to the fundamental principle of equilibrium statistical mechanics, in systems governed by (6.1) and satisfying (6.3), the smoothed (or coarse-grained) version of P(y,t) eventually becomes uniform over all the accessible parts of phase space. That is, the dynamics stirs P until it is practically equivalent to a uniform distribution in phase coordinates for which the phase-space velocity ˙ y is nondivergent. Again we note that the assumption of a uniform, static probability density is consistent with (6.4) only if the phase-space motion is nondivergent; otherwise an initially uniform probability density would tend to acquire local extrema at the points of convergence and divergence. On the other hand, if the probability density is uniform in one set of phase-space coordinates satisfying (6.3), then it is also uniform in any other set of coordinates for which the motion is nondivergent, because the two coordinate sets then have a constant Jacobian of transformation.

V-23

We must still explain the meaning of accessible. According to equilibrium statistical mechanics, all the final states with the same value of the energy (and other conserved quantities if present) as the initial state are considered accessible. Thus, in a system for which the energy E(y) is the only conserved quantity, the equilibrium probability density is P y( ) = Cδ E y( ) − E0( ) , (6.5) where δ( ) is Dirac’s delta-function, E0 is the initial energy, and C is a normalization constant determined by

∫∫∫ dy P =∫ 1. (6.6)

The probability density (6.5) is called the microcanonical ensemble. (From now on, we omit the distinction between P and ˆ P , with the understanding that we now always mean the smoothed probability density.) If we regard the delta-function as a sequence of functions in the usual way, then (6.5) states that the probability density is (roughly speaking) uniform on an (N-1)-dimensional hypersurface of varying thickness. Now, P is typically wanted for computing the averages of functions involving only a small number of the phase coordinates. However, to compute <yi2> (for example), we only need the partial density,

Pi yi( ) ≡ dy1...dyi−1 dyi+1∫∫∫∫ ...dyN P , (6.7)

obtained by integrating over all the modes except yi. If P is given by (6.5), and if E(y) is a sum of functions each depending on only a few coordinates, e.g. E y( ) = E1 y1( ) + E2 y2( ) ++ EN yN( ) , (6.8) then it can be shown that (6.7) takes the approximate form Pi yi( ) = Ci exp −α Ei yi( )[ ] , (6.9) where α and Ci are constants.14 But if (6.9) holds for each partial density, and if we want the total probability density P(y) only for evaluating the averages of sum functions like (6.8), then we can use P y( ) = Cexp −α E y( )[ ] (6.10) as an approximation to (6.5). C is the normalization constant, and α is determined by the average energy,

V-24

∫∫∫ dy E y( )P y( ) =∫ E0 . (6.11)

The probability density (6.10), called the macrocanonical ensemble, is often also justified by assuming that the whole system is in contact with a very large thermal bath, but the assumption of a thermal bath is not very compatible with the applications we have in mind. Nowhere in the theory based upon uniform probability density in phase space does the concept of entropy necessarily arise. However, we can define the entropy of the equilibrium probability density to be

S = − ∫∫∫ dy P lnP∫ . (6.12)

Then the macrocanonical density (6.10) can also be obtained by maximizing the entropy (6.12) subject to the constraints (6.6) and (6.11). Similarly, the microcanonical density (6.5) corresponds to the maximum of (6.12) subject to (6.6), (6.11) and

∫∫∫ dy E y( ) − E0( )2 P y( ) =∫ Ξ , (6.13)

in the limit Ξ→0. This limit corresponds to the limiting sequence of functions represented by the delta function in (6.5). In Section 18 of Chapter 1, we introduced Boltzmann’s definition of the entropy, S = lnΩ , (6.14) where Ω is (proportional to) the number of microstates with the given energy. Now we show that these two definitions of entropy, (6.12) and (6.14), are essentially the same. First suppose that probability density takes the constant value P0 over finite volume of phase space and vanishes everywhere else. Then, bearing the normalization requirement (6.6) in mind, we see that P0-1 is the volume of phase space in which P=P0. This volume is proportional to the number of microstates Ω. Thus, according to (6.14), S = ln 1

P0. (6.15)

If P is non-uniform in phase space, then (6.15) generalizes to

S = ln 1

P= − ∫∫∫ dy P lnP∫ , (6.16)

which agrees with (6.12). Thus the two definitions of entropy are essentially the same. To illustrate the machinery of equilibrium statistical mechanics, we calculate the absolute equilibrium state of an monatomic ideal gas inside a rigid container. The phase-space coordinates are the locations and momenta of each gas molecule,

V-25

y = q1,,q3M , p1,, p3M{ } , (6.17) where M is the number of molecules. The subscripts on q and p identify both the molecule and the direction in physical space. In this case the dimension of the phase space, N=6M, is truly finite; no artificial truncation is required. The phase-space velocity ˙ y is nondivergent because the molecules obey Hamilton’s equations,

dqidt

=∂H∂pi, dpi

dt= −

∂H∂qi

. (6.18)

Note that (6.18) implies (6.3) but that the converse is generally untrue. However, equilibrium statistical mechanics only demands (6.3); canonical coordinates are not generally required. The energy of the monatomic ideal gas is a sum function,

E =12m

pi2

i=1

3M

∑ . (6.19)

The macrocanonical distribution (6.10) implies that the mean-square momenta are

pi2 =

pi 2 exp −α pi 2 / 2m{ }dpi−∞

∞

∫exp −α pi 2 / 2m{ }dpi−∞

∞

∫=mα

. (6.20)

Hence the pressure is

p = ρp12

m2 =ρm1α= nNA

1α

, (6.21)

where n is the number of moles per unit volume and NA is Avogodro’s number. Equation (6.21) agrees with the equation of state, p = n R*T , (6.22) for an ideal gas (where R*=kNA is the universal gas constant and k is Boltzmann’s constant; see Chapter 1) provided that α =

1kT

. (6.23)

Thus the factor α appearing in the exponent of (6.10) has the physical interpretation of inverse temperature. For non-monatomic ideal gases, we also arrive at (6.22); the

V-26

additional degrees of freedom contribute to the energy (and heat capacity) but only the translational degrees of freedom contribute to the pressure. From the macrocanonical density (6.10) and the definition (6.12) of entropy, we can calculate the fundamental relation between entropy, (average) energy, and the volume of the gas, obtaining the result given in Section 18 of Chapter 1. As emphasized in Chapter 1, this fundamental relation tells us everything there is to know about the thermodynamic equilibrium state of the gas. Of course, the ideal gas is very special because its energy is a sum function, so that all the probability integrals can be factored into simple integrals. In all the really interesting problems studied with equilibrium statistical mechanics, the energy contains interaction terms requiring sophisticated approximation methods. Systems undergoing phase changes offer some of the biggest challenges. In the following section, we re-examine the absolute-equilibrium states corresponding to the equations for a macroscopic fluid. In the cases we consider, the energy and other conserved quantities are also sum functions, and the calculations are therefore as easy as in the case of an ideal gas. The fundamental difficulty turns out to be one of interpretation — of properly understanding the behavior of the equilibrium states as the number of modes becomes infinite. 7. The meaning of absolute equilibrium Once again we consider the Navier-Stokes equations in the abstract form (4.1). If the fluid is two-dimensional, the {yi} could be the coefficients in a Fourier- or eigenmode-expansion of the streamfunction ψ(x,y,t); see (1.5) and (1.7). The subscript denotes the wavevector or eigenmode. If the fluid is three-dimensional, then the {yi} could be the Fourier amplitudes of each directional component of velocity. In that case the subscript denotes the wavevector, the direction, and the real or imaginary part. Let the exact dynamics (4.1) be truncated to a finite number of modes. Again, one could simply discard all the modes corresponding to wavevector magnitudes larger than some cutoff kmax. Alternatively, the truncated version of (4.1) might represent a finite-difference approximation to the fluid equations on a uniform grid, with the grid-spacing corresponding to kmax-1. If viscosity is present, and if kmax is sufficiently large (for the given viscosity), then the truncation in wavenumber has a negligible effect on the dynamics, because viscosity wipes out the high-wavenumber components of the flow. But suppose that the viscosity vanishes. Then, as can always be checked, the phase-space flow is nondivergent in the coordinates {yi}, and the methods of equilibrium statistical mechanics apply to the truncated system. Provided that the conserved quantities take the form of sum functions, we can work out the absolute-equilibrium states. As already remarked, these states correspond to the final states towards which numerical models (having a finite number of degrees of freedom) would evolve in the absence of forcing and viscosity. The absolute-equilibrium states indicate the general direction in which nonlinear interactions drive numerical models, even when forcing and viscosity are present. Whether or not this information has practical value remains to be settled. Clearly, the assumptions of vanishing viscosity and a truncation in modes are incompatible insofar as

V-27

the exact, untruncated dynamics is concerned. The question, then, is whether we learn anything at all from the absolute-equilibrium states of numerical models. In the case of inviscid, truncated three-dimensional turbulence, the only significant invariant is the total energy,15 and equilibrium statistical mechanics predicts equi-partition of the energy among the modes. In three dimensions, the number of modes with wavevector magnitude within dk of k is proportional to k2dk. Thus, equilibrium statistical mechanics predicts an energy spectrum of the form E k( ) = Ck2, 0 < k < kmax , (7.1) where C is a constant determined by the total energy. Now, suppose we increase kmax by increments, but allow the truncated system to reach equilibrium after each change in kmax. Then the spectrum assumes the sequence of shapes shown in Figure 5.3. As kmax increases, the energy moves into higher and higher wavenumbers, but E(k) approaches no definite limit; there is no way to give equal amounts of energy to an infinite number of modes, and then end up with a finite total energy! This lack of convergence is closely related to the ultraviolet catastrophe that occurs when equilibrium statistical mechanics is applied to the equations of classical electrodynamics. What then do we learn from (7.1)? Only that three-dimensional turbulence wants to transfer its energy into very high wavenumbers. The Kolmogorov theory presented in Chapter 4 predicts a k -5/3 energy spectrum in which the energy moves from low wavenumbers to high wavenumbers. Now we see that any spectrum with a slope less than that of (7.1) could be expected to transfer its energy to high wavenumbers. Certainly this conclusion is rather tame. However, we shall see that the corresponding absolute equilibrium states of two-dimensional and (especially) quasigeostrophic turbulence are much more interesting and instructive. As a general rule, absolute-equilibrium theory becomes more illuminating as the corresponding dynamics (and dynamical invariants) become richer and more complex. The two-dimensional absolute equilibrium states are interesting because they can be made to incorporate at least one of the dynamical invariants corresponding to the conservation of vorticity on fluid particles. Consider the truncated, inviscid dynamical equations corresponding to inviscid two-dimensional flow governed by

∂∂t

∇2ψ + J ψ ,∇2ψ( ) = 0, (7.2)

in a finite domain. Let the {yi} be the eigenmode amplitudes defined by (1.7-9). Let kmin be the lowest eigenvalue (i.e. wavenumber) in the system. Then kmin-1 is the size of the domain. Since Aijk vanishes whenever two of its subscripts are equal, (6.3) holds, and the flow in phase space is nondivergent. The dynamical invariants include (twice) the energy

E = dx ∇ψ ⋅∫∫ ∇ψ = yi2

i=1

N

∑ , (7.3)

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and the enstrophy

Z = dx ∇2ψ( )2∫∫ = ki2yi

2

i=1

N

∑ . (7.4)

The macrocanonical ensemble based only on (7.3) and (7.4) is

P y( ) = Cexp − α + β ki2( )yi 2

i=1

N

∑⎧ ⎨ ⎩

⎫ ⎬ ⎭

, (7.5)

where the constants C, α, and β are determined by

yi2

i=1

N

∑ = E, ki2 yi

2

i=1

N

∑ = Z, and 1 =1 , (7.6)

and E, Z are the prescribed energy and enstrophy of the system. By a general theorem of statistical mechanics,16 these constants are uniquely determined for all realizable values of E and Z, that is, for all E > 0 and kmin2 E < Z < kmax2 E. (7.7) We can regard α and β as inverse temperatures (or chemical potentials) corresponding to the energy and enstrophy, respectively. According to (7.5), <yi>=0, and <yi2> is given by (5.28). In two dimensions, the number of modes with wavevector magnitude within dk of k is proportional to k dk. Thus the absolute-equilibrium spectrum for bounded, inviscid, two-dimensional turbulence is E k( ) = k

α + βk 2, kmin < k < kmax , (7.8)

where α and β are determined from E and Z by

E k( )kmin

kmax

∫ dk =12βln kmax2 +α β

kmin2 + α β

⎛

⎝ ⎜

⎞

⎠ ⎟ = E (7.9)

and

k 2E k( )kmin

kmax

∫ dk =12β

kmax2 − kmin

2( ) − αβln kmax2 +α β

kmin 2 + α β⎛

⎝ ⎜

⎞

⎠ ⎟

⎡

⎣ ⎢

⎤

⎦ ⎥ = Z . (7.10)

In writing (7.8-10), we assume that the mode-spacing is sufficiently dense that all the sums can be replaced by integrals. This assumption, which is least accurate for the

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modes near kmin, is an unnecessary convenience; our main conclusions also apply to the case of perfectly discrete modes. As the energy E and enstrophy Z vary over the range (7.7) of realizable values, the corresponding inverse temperatures α (E,Z) and β (E,Z) take on all the values for which α + β k2 > 0 (kmin

2 < k2 < kmax2 ) . (7.11)

The range (7.11) includes, at its extremes, two negative temperature regimes. The first occurs when α < 0, β > 0, α ≈ −β kmin

2 , (7.12) and corresponds to an energy spectrum in which all the energy is concentrated at the lowest wavenumber kmin. The second negative-temperature regime occurs when α > 0, β < 0, α ≈ −β kmax

2 , (7.13) and corresponds to a concentration of energy at the highest wavenumbers in the truncated system. Now let E and Z be given constants, and consider again the limit kmax→∞, with kmin held fixed. From (7.9) and (7.10), it follows that17

β ~ kmax2

2 Z − kmin2E( ) kmax → ∞( ) (7.14)

and

αβ~ −kmin

2 + kmax2 exp −

kmax2EZ − kmin

2E( )⎧ ⎨ ⎩

⎫ ⎬ ⎭

kmax → ∞( ) . (7.15)

Thus the limit of progressively finer spatial resolution always yields the state of lowest negative α. Let k0 be any fixed wavenumber between kmin and kmax. Then as kmax→∞ (with E, Z, kmin, and k0 held fixed), (7.14) and (7.15) imply that E k( )dk

kmin

k0

∫ → E (7.16)

and k 2E k( )dk

kmin

k0

∫ → kmin2E . (7.17)

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Since (7.16) holds for all k0>kmin, it follows that, as kmax→∞, all the energy moves into the lowest-wavenumber mode. If the energy at kmin is E, then the enstrophy at kmin must be kmin2E. Then, because k0 can also be arbitrarily large, (7.17) implies that all the remaining enstrophy moves toward infinite wavenumber as kmax→∞. Thus, as the spatial resolution increases, a single domain-filling vortex absorbs all of the energy in the system, and all the enstrophy not required by this vortex appears in eddies at the smallest resolved lengthscales. Of course, this equilibrium picture gives no information about the path by which the system reaches absolute equilibrium. In Chapter 4, we noted that the energy transfer to low wavenumbers in two-dimensional turbulence sometimes occurs through the formation and gradual coalescence of relatively isolated axisymmetric vortices. Equilibrium statistical mechanics cannot explain such disequilibrium behavior.18 Absolute-equilibrium theory based only on (7.6) is open to the further criticism that the dynamics (7.2) actually conserves an infinite number of quantities — the vorticity of each moving fluid particle. Consequently, (7.2) conserves every quantity of the form dx F ∇2ψ( )∫∫ , (7.18) where F( ) is an arbitrary function. Particular attention attaches to the powers, dx ∇2ψ( )n∫∫ , (7.19) where n is any integer. The enstrophy corresponds to n=2. Absolute-equilibrium theory can easily accommodate the average-vorticity (n=1) invariant, but the higher (n>2) powers lead to integrals that cannot be performed. The neglect of these higher powers is sometimes excused by the observation that their conservation typically does not survive the truncation in modes. However, this excuse is not so much a defense of (7.5) as it is a criticism of the truncated dynamical equations. But suppose that the exponent in (7.5) did include an arbitrary number of the higher powers (7.19). The resulting probability density would be a very complicated, fine-grained function whose lower moments might anyway be insensitive to the prescribed values of these higher powers. Somewhat surprisingly, there is a truncated dynamics that conserves every quantity of the form (7.18), and, moreover, exactly satisfies (7.2). This is the system composed of N point vortices at locations {(xi(t),yi(t))}. Let Γi be the (constant) circulation around the i-th vortex. In an unbounded fluid, the governing dynamical equations take the canonical form

Γidxidt

= −∂H∂yi, Γi

dyidt

=∂H∂xi

, (7.20)

where

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H =12π

Γi Γ ji> j∑ ln xi − x j( ) . (7.21)

is the Hamiltonian. Hence the motion (7.20) is nondivergent in the phase space spanned by {xi, yi}. The truncated dynamics (7.20-21) conserves the energy H and the vorticity on every fluid particle. This vorticity is singular on the fluid particles at xi(t), and vanishes on every other particle. Thus the conservation of all the vorticity invariants is built into the numbering of the vortices. The macrocanonical ensemble, P x1,…,xN , y1,…,yN( ) = Ce−α H , (7.22) thus recognizes the energy and all the vorticity invariants. In a classic paper, Onsager (1949) considered this absolute-equilibrium state, and predicted the clumping of like-signed vortices in solutions of (7.20). Unfortunately, the energy (7.21) couples every vortex to every other vortex, making averages with (7.22) very difficult to compute. The literature on the equilibrium statistical mechanics of point vortices is large, rewarding, and incomplete.19 However, the predictions of (7.22) seem very similar to those of (7.5), supporting the idea that the higher powers of vorticity are of secondary importance. In the next chapter, we apply many of the ideas in this chapter to quasigeostrophic turbulence. Although we shall not use nonequilibrium theories like EDM, we shall make extensive use of their two most important ingredients: the conservation principle for energy and (potential) vorticity, and the irreversibility principle. The irreversibility principle can take many useful forms, but we shall find the entropy principle and, particularly, the theory of absolute equilibrium to be especially informative. Notes for Chapter 5. 1. Leslie (1973) and McComb (1990) offer a much more extensive treatment of this subject. See also Lesieur (1990) and the extensive reviews by Orszag (1977) and Rose and Sulem (1978). For a very general overview of statistical turbulence theory, see Monin and Yaglom (1971,1975). 2. Even in the cases where (2.4) do not seem to hold, it is often possible to satisfy (2.4) by a change in the definition of the yi. For example, in the case of a finite-difference approximation to the shallow-water equations, (2.4) hold if the yi are the gridded values of h, hu, and hv. The property (2.4) seems to be a consequence of the underlying Hamiltonian structure of inviscid fluid mechanics. 3. Section 2 closely follows Orszag’s (1970, 1977) derivation of EDM. One can also regard EDM as an abridgment of Kraichnan’s (1959) direct-interaction approximation; see Leslie (1973). 4. For a discusion of these and other properties of Gaussian random variables, see for example Frisch (1995, Chapter 4). 5. See Lesieur (1990, p. 176 and 276). For a comparison between EDM and direct numerical simulations, see Herring et al. (1974), Herring and McWilliams (1985), and Herring (1990). 6. See Holloway and Hendershott (1977).

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7. Section 3 is based upon the papers by Leith (1971) and Kraichnan (1971a). See also Frisch et al. (1974). 8. Many of these ideas in were expressed by Orszag (1970,1977), who stressed that exact solutions of the full moment hierarchy has the instability property of the original equations and is time-reversible. 9. It would, after all, be illogical to assume that one could lose (by replacing P with ˆ P ) the information contained in the detailed structure of P(y,t) and then recover it at a later time. 10. See, for example, Khinchin (1957, pp 9-13). This book is an excellent mathematical introduction to information theory, a subject invented by Shannon and Weaver (1949) and made a basis for statistical mechanics by Jaynes (1957). The textbooks by Tribus (1961) and Katz (1967) develop statistical mechanics from the information-theory viewpoint. This viewpoint seems ideally suited to closure theory, which is based upon an incomplete statistical description of the flow. 11. To see this, consider the initial condition in which only the three yi corresponding to a particular triad are nonzero. 12. See Montgomery (1976) and Carnevale et al. (1981). Carnevale (1982) demonstrated that direct numerical simulations of two-dimensional turbulence obeyed the H-theorem. 13. For a brief mathematical introduction to statistical mechanics, see Thompson (1972). 14. The various proofs of (6.9) differ in complexity, depending on the strength of the exact hypotheses; see Khinchin (1949), Grad (1952) or Thompson (1972). 15. Kraichnan (1973) argued that the helicity is much less important than the energy. 16. See, for example, Katz (1967, pp. 47-50) 17. See Carnevale and Frederiksen (1987). 18. However, it has been claimed that equilibrium statistical mechanics can explain the structure of the isolated vortices that arise in two-dimensional turbulence. See Miller (1990) and Robert and Sommeria (1991). 19. See Kraichanan and Montgomery (1980, sec. 3.3) and references therein. Kraichnan (1975) discusses the relationship between the absolute-equilibrium states of the truncated continuum and the system of discrete vortices.