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Mixing and coherent structures in 2D viscous flows
Physica D 237 (2008) 1993–1997 www.elsevier.com/locate/physd
Mixing and coherent structures in 2D viscous flows H.W. Capel a , R.A. Pasmanter b,∗ a Inst. Theoretical Physics, University of Amsterdam, Valckenierstraat 65, 1018 XE Amsterdam, The Netherlands b KNMI, P.O. Box 201, 3730 AE, De Bilt, The Netherlands
Available online 29 April 2008
Abstract We introduce a dynamical description based on a probability density φ(σ, x, y, t) of the vorticity σ in two-dimensional viscous flows such that the average vorticity evolves according to the Navier–Stokes equations. A time-dependent mixing index is defined and the class of probability densities that R maximizes this index is studied. The time dependence of the Lagrange multipliers can be chosen in such a way that the masses m(σ, t) := dxdyφ(σ, x, y, t) associated with each vorticity value σ are conserved. When the masses m(σ, t) are conserved then (1) the mixing index satisfies an H-theorem and (2) the mixing index is the time-dependent analogue of the entropy employed in the statistical mechanical theory of inviscid 2D flows. In the context of our class of probability densities we also discuss the reconstruction of the probability density of the quasi-stationary coherent structures from the experimentally determined vorticity-stream function relations. c 2008 Elsevier B.V. All rights reserved.
PACS: 47.32.-y; 47.51.+a; 47.27.De; 47.10.ad; 47.10.ab Keywords: Viscous flows; Two-dimensional flows; Coherent structures; Stochastic dynamics
1. Introduction When studying the dynamics of two-dimensional fluid motion characterized by a vorticity field ω(x, y, t) it can be useful to turn to a probabilistic description with distributions φ(σ, x, y, t) for the microscopic vorticity σ such that the average value of σ over these distributions is equal to ω(x, y, t). The probability distribution represents an ensemble of systems, all the ensemble members satisfy the same constraints. The uncertainties associated with the probability distribution are due to, e.g., the finite experimental precision or to thermal fluctuations. In particular, this can be done in the description of the coherent structures, i.e. the quasistationary states (QSS), which are often reached in (numerical) experiments after fast mixing has taken place [5,8,12,19,22]. At high Reynolds’ numbers, the vorticity fields ω S (x, y) of these QSS’s satisfy ω–ψ relations to a good approximation, i.e., ω S (x, y) ' Ω (ψ(x, y)) where ψ(x, y) is the corresponding ∗ Corresponding author.
E-mail addresses: [email protected], [email protected] (R.A. Pasmanter). c 2008 Elsevier B.V. All rights reserved. 0167-2789/$ - see front matter doi:10.1016/j.physd.2008.04.016
stream-function. In other words, the QSS’s are approximate stationary solutions of the Euler equation. A statistical mechanical theory of inviscid two-dimensional steady flows was introduced in [10,11,15–18], an approach that can be traced back to earlier work of Lynden–Bell in 1967 [7]. Some outstanding aspects of this non-dissipative system are: (1) an infinite number of conserved quantities associated with each microscopic-vorticity value σ and (2) non-uniform equilibrium states (the coherent structures) which often correspond to negative-temperature states as already predicted by Onsager’s work on point vortices [13]. Theoretical predictions of the statistical mechanics approach to the coherent structures were compared with numerical simulations and with experimental measurements in quasi-two dimensional fluids, e.g., in [2,3, 8,9,19]. However, under standard laboratory conditions fluids are viscous and numerical simulations require the introduction of a non-vanishing (hyper)viscosity in order to avoid some numerical instabilities and other artifacts. In spite of this, in many cases it was found that the agreement between the theoretical predictions based on the statistical mechanics of Miller, Robert and Sommeria (MRS) [10,11,15–18] and (numerical) experiments was better than expected.
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In order to discuss these issues in a more dynamical setting we consider viscous flows and propose a family of model evolution equations for the vorticity distribution φ(σ, x, y, t) in Section 2. In Section 3 we discuss the class of time-dependent distributions that maximize a mixing index under certain constraints. In particular, it is shown that the time-dependent Lagrange multipliers appearing in these distributions can be chosen in such a way that the masses associated with each microscopic-vorticity value σ are conserved. When these masses are conserved, the mixing index satisfies an H-theorem. Moreover, the mixing index shows a minimal increase in time [Section 4]. The distribution φ S (σ, x, y) associated with a given QSS can be obtained, at least in principle, by addressing the reconstruction problem, i.e. how to extract its defining parameters from the QSS’s ω–ψ relation. This is discussed in Section 5. In doing so we provide a natural framework for a time-dependent statistical theory connecting an appropriate initial distribution to the QSS distribution associated with the experimental ω–ψ relation and evolving in agreement with the Navier–Stokes equation. The validity of the used assumptions should be tested, for example in numerical simulations. 2. Microscopic viscous models Let φ(σ, x, y, t)dσ be the probability of finding at time t a microscopic vorticity value in the range (σ, σ +dσ ) at a position (x, y). It should be non-negative and normalized Z dσ φ(σ, x, y, t) = 1. (1) The macroscopic vorticity field is Z ω(x, y, t) = hσ i := dσ σ φ(σ, x, y, t),
(2)
in which the pointed brackets denote averages over this distribution, In the inviscid case, the dynamics reduces to the advection of vorticity. Neglecting fluctuations in the velocity field, the time evolution of φ(σ, x, y, t) can be taken to be ∂φ(σ, x, y, t) + vE(x, y, t) · ∇φ(σ, x, y, t) = 0, ∂t
(3)
where the macroscopic, incompressible velocity field vE(x, y, t) z, is related to the macroscopic vorticity ω(x, y, t) by ∇×E v = ωe withe z a unit vector perpendicular to the (x, y)-plane. Extending this to the viscous case the models to be considered are of the form ∂φ(σ, x, y, t) + vE(x, y, t) · ∇φ(σ, x, y, t) = ν∆φ + νφ O,(4) ∂t with ν the fluid viscosity and O as yet undefined but constrained by (1) the conservation of the total probabilty R dσ φ(σ, x, y, t) = 1, and by (2) the macroscopic Navier–Stokes equation, i.e., ∂ω(x, y, t) + vE(x, y, t) · ∇ω(x, y, t) = ν∆ω(x, y, t) ∂t
(5)
should follow from the microscopic model. These two conditions are equivalent to,
O = 0, and σ O = 0. (6) It is convenient to introduce the “masses” m(σ, t) associated with each value σ of the microscopic vorticity, Z m(σ, t) := dxdyφ(σ, x, y, t). (7) In the inviscid case, ν = 0, Eq. (4) has a solution φ(σ, x, y, t) = δ(σ − ω(x, y, t)) therefore m(σ, t) is the area occupied by the vorticity field with value σ. As soon as we introduce a diffusion process, as it is implied by Eq. (4) with ν 6= 0, such an identification becomes impossible. In calling m(σ, t) a “mass” we stress the analogy between Eq. (4) and an advection–diffusion process of an infinite number of “chemical species”, one species for each value σ. Assuming that there is no leakage of φ(σ, x, y, t) through the boundary, the time derivative of a mass is Z ∂m(σ, t) = ν dxdy Oφ(σ, x, y, t). (8) ∂t The simplest viscous model satisfying the above requirements is the one with O ≡ 0. This model is instructive because, while it dissipates energy, it has an infinite number of conserved quantities, i.e., the masses m(σ, t). One of the consequences of the conservation laws for m(σ, R t) is that all the microscopicvorticity moments Mn (t) := dσ σ n m(σ, t) are constants of the motion, i.e., dMn /dt = 0. In the sequel we shall assume that the conservation of all the microscopic moments Mn implies in turn that the masses m(σ, t) are conserved. This is the case if certain technical conditions are satisfied, see e.g., [20]. By contraposition, all even moments of the macroscopic vorticity, Z Γ2n (t) := dxdyω2n (x, y, t), (9) are dissipated since, dΓ2n = −ν2n(2n − 1) dt
Z
dxdyω2(n−1) |∇ω|2 ≤ 0.
(10)
Under appropriateR boundary conditions, e.g., periodic ones, the R energy E = 1/2 dxdy v 2 = 1/2 dxdyωψ, and its dissipation rate is dE/dt = −νΓ2 (t), where ψ(x, y, t) is the streamfunction associated with vE(x, y, t). 3. Time-dependent extremal distributions Equations like (4) have been studied extensively; see for example [14] and the references therein. Usually a timedependent velocity field vE(x, y, t) leads to chaotic trajectories, namely to the explosive growth of small-scale gradients. These small-scale gradients are then rapidly smoothed out by diffusion, the net result being a very large effective diffusion coefficient, large in comparison to the molecular coefficient ν. Based on these observations we will consider situations where during a period of time mixing takes place much faster
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than the changes in the masses m(σ, t). In order to quantify this weR introduce the degree of mixing of m(σ, t), s(σ, t) := −A−1 dxdyφ ln [Aφ/m(σ, t)] , and the corresponding total degree of mixing at time t, Z Z −1 dσ dxdyφ(σ, x, y, t) ln φ(σ, x, y, t) + S(t) = −A Z h i A−1 dσ m(σ, t) ln A−1 m(σ, t) . (11) The fast-mixing condition can thus be expressed as, |∂s(σ, t)/∂t| A−1 |∂m(σ, t)/∂t| .
(12)
This inequality is satisfied when the masses m(σ, t) are conserved, moreover in such a case the second term in (11) is constant in time so that the total degree of mixing S(t) is the time-dependent analogue of the entropy which is used in the MRS statistical mechanics theory and, as shown in the last paragraph of the present section, it satisfies an H-theorem. Accordingly, in the sequel we investigate the time-dependent distributions φ(σ, x, y, t) that maximize the total vorticity mixing S(t) under the following three constraints: (i) normalization, as in (1), (ii) given values of the masses m(σ, t), and (iii) a given distribution first moment hσ i = ω(x, y, t) which, by construction, evolves according to the Navier–Stokes equations. Introducing time-dependent Lagrange multipliers γ (x, y, t), e µ(σ, t) and χ (x, y, t) associated to the abovementioned constraints and denoting the maximizing distribution by φ M (σ, x, y, t), the vanishing of the first variation of S(t) with respect to φ leads to, φ M (σ, x, y, t) = Z −1 exp [µ(σ, t) + χ (x, y, t)σ ] , Z with Z (x, y, t) := dσ exp [σ χ (x, y, t) + µ(σ, t)] ,
(13)
and µ(σ, t) := −e µ(σ, t) + ln A−1 m(σ, t). The functions χ(x, y, t) and µ(σ, t) will be called the “potentials”. Two constraints given by (2) and (7) determine these potentials. For these distributions one has φ M (σ, x, y, t) =: e χ (x, y, t), t), therefore the (x, y)-dependence of ω as well φ(σ, n as that of the local moments m n := hσ i and the centered local n moments K n := (σ − ω) is only through χ (x, y, t), i.e., en (χ , t), Z (x, y, t) =: e K n (x, y, t) =: K Z (χ , t), ω(x, y, t) =: e e /∂χ = e Ω (χ, t), m n (x, y, t) =: m n (χ , t), and relations like ∂ Ω e K 2 (χ , t) hold. In the special case of a QSS at time TS the distribution obtained from the MRS approach is as in Eq. (13) with χ(x, y, TS ) = −βψ(x, y, TS ) where ψ(x, y, TS ) is the stream function at time TS and β is associated with an inverse temperature. This distribution is obtained by maximizing S(TS ) under the constraints (i) and (ii) and the constraint that the energy at time TS has some given value E(TS ). In the MRS inviscid approach the connection with the initial state is made by requiring that the masses at time TS and the energy at time TS equal their initial values, therefore the QSS can be predicted from the initial condition. In the context of our present work, for this to be approximately valid the following fast mixing
condition should be satisfied, dS(t)/dt E −1 (dE/dt) ,
(14)
on top of condition (12). Assume that at all times the probability density has the form given in (13). Omitting for convenience the subsript M, inserting (13) and (2) in the Navier–Stokes equation (5) and making use of simple algebraic equalities one shows that the time evolution of φ M is given by Eq. (4) with O(σ, x, y, t) given by, K3 2 |∇χ |2 − ω) − − ω) O(σ, x, y, t) = K 2 + (σ (σ K2 ∂µ ∂µ −1 ∂µ −1 (σ − ω) −ν . (15) − +ν (σ − ω) ∂t ∂t K2 ∂t
As one can check, O = 0 and σ O = 0 for all possible timedependences of ∂µ/∂t. From (15) it follows that the simplest viscous model with O(σ, x, y, t) ≡ 0, can be realized only under rather trivial conditions. Indeed, since O(σ, x, y, t) ≡ 0 has to hold for any value of σ, Eq. (15) implies that ∂µ/∂t must be quadratic in σ and that |∇χ |2 may be time-dependent but must be (x, y)independent. For general µ(σ, t) there is no conservation of the masses m(σ, t). However,R choosing a suitable time-dependence of µ(σ, t) such that dxdy φ O = 0, ensures the conservation of the masses m(σ, t), confer Eq. (8). This condition and Eq. (15) lead to a complicated integro-differential equation for the time-dependence However, using a Taylor P of µ(σ, t). k expansion µ(σ, t) = k µk (t)σ , we can derive an infinite set of linear differential equations for the dµk /dt. In fact, multiplying Eq. (15) by σ n φ(σ, x, y, t) and integrating it over σ one gets that: ∞ X
dµk ν σ n O = −ν |∇χ |2 h n2 + h nk dt k=2
(16)
with h nk = m k+n − m k m n − K 2−1 (m n+1 − ωm n ) (m k+1 − ωm k ) , where m n := hσ n i . The conservation R R
of the moments Mn = dxdy m n requires then that dxdy σ n O = 0 and hence (16) becomes, Z ∞ X dµk k=2
dt
dxdy h nk = ν
Z
dxdy h n2 |∇χ |2 .
(17)
From this infinite set of equations, linear in dµ2 /dt, dµ3 /dt, . . ., the dµk /dt can, in principle, be solved. The solution describes a viscous model with an infinite number of conservation laws. Such a viscous model becomes physically more relevant by making it compatible with a quasi-stationary distribution eS (σ, χ ) corresponding to the Ω (ψ) relation at time TS , and φ with χ = −βψ(x, y), confer Section 5.
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Using Eq. (4) and the conservation of the masses m(σ, t) one obtains, # " Z ∇φ 2 ∂s(σ, t) ν − O ln Aφ . (18) = dxdyφ ∂t A φ When the distribution is of the form given in Eq. (13), using a Taylor expansion for µ(σ, t) one sees that after integration over σ the term containing O vanishes, therefore dS(t)/dt is nonnegative. As shown in Appendix A of Ref. [6] this H-theorem holds only for one specific measure of spatial mixing, namely for the one given in (11); see, e.g., Ref. [1,21]. The fast-mixing condition (14), divided by ν becomes then, Z Z D E 1 1 dxdy |∇ ln φ|2 dxdyω2 . (19) A E
E o . As we show below these experimental data can be used in order to determine the potential µ(σ, TS ) occuring in the distribution φ M (σ, x, y, TS ) as given by Eq. (13) with χ = −βψ(x, y, TS ). Once the distribution has been reconstructed from the experimental data we can then associate with it a time-dependent distribution function φ M (σ, x, y, t) as given by Eq. (13) which is a solution of the time-evolution Eq. (4) with suitable initial conditions and such that at time t = TS one has eS (σ, χ ) with χ = −βψ and with eM (σ, χ (x, y, TS ), TS ) = φ φ µ(σ, TS ) = µ(σ ). The time-dependence of µ(σ, t) is chosen as in Eq. (17) such that all masses m(σ, t) are constant in time. Here β can be defined such that M2 (TS ), the microscopicvorticity second moment of the QSS at time TS , is equal to Γ20 the enstrophy of the initial vorticity field ωo (x, y), i.e., −1 Z 0 S β = − Γ2 − Γ2 dxdy (dΩ /dψ) (21) A
4. Rate of mixing increase In order to see whether or not this fast-mixing condition (19)is satisfied by the time-dependent distributions φ M (σ, x, y, t), we derive a lower bound to h|∇ ln φ|2 i and we determine the extrema of the l.h.s. of Eq. (19). To this end notice that ∇ω = hσ ∇ ln φi can also be written as ∇ω = h(σ − ω) ∇ ln φi because h∇ ln φi = 0. Applying then the Cauchy–Schwartz inequality to |∇ω|2 = |h(σ − ω) ∇ ln φi|2 leads to |∇ω|2 ≤ h(σ − ω)2 ih|∇ ln φ|2 i ≡ K 2 h|∇ ln φ|2 i, i.e. to the desired lower bound, D E |∇ω|2 K 2 −1 ≤ |∇ ln φ|2 . (20) The lower bound on h|∇ ln φ|2 i that we have just found means that the fast-mixing condition ((19)) holds whenever Z Z |∇ω|2 1 1 dxdy dxdyω2 . A K2 E One can show that the family of probability distributions that reach the lower bound in (20) coincides with φ M (σ, x, y, t) as given by Eq. (13). Here, the input for the determination of the potential functions µ(σ, t) and χ (x, y, t) are the first and second σ -moments. The details concerning the derivation of this can be found in Appendix B of Ref. [6]. R Also the extrema of dxdyh|∇ ln φ|2 i can be investigated taking into account the constraints given by Eqs. (2) and (1). It turns out that in order to obtain sensible solutions, it is necessary to constrain also the distribution’s second moment hσ 2 i. We find R that all the extremizer distributions are local minima of dxdyh|∇ ln φ|2 i and coincide with φ M (σ, x, y, t), moreover they all reach the lower bound in (20). The details can be found in Appendix C of Ref. [6]. 5. Reconstructing µ(σ ) from experimental data Suppose that in an experiment one is given an initial vorticity field with its corresponding energy E o and that at a time TS one finds a quasi-stationary vorticity field ω S (x, y) = Ω (ψ(x, y)), with a monotonic Ω (ψ) and an energy E S ≤
Γ2S
R
dxdy ω2S (x,
= y) is the enstrophy of the QSS. All where the quantities on the r.h.s. of this formula are experimentally accessible. In order to determine the µ(σ ) potential from the experimental Ω (ψ) use can be made of ω = d ln e Z (χ )/dχ , at Z (χ ) given by (13) and β determined from χ = −βψ, with e Eq. (21). Below we illustrate this by considering some examples. A linear ω–ψ scatter plot at time TS , Ωl (ψ) = α1 ψ, with α1 > 0, corresponds to a Gaussian distribution centered on α1 ψ(x, y) and with a width α1 /|β|, i.e., µ(σ ) = βσ 2 /(2α1 ). In the present case the expression (21) for β reads, β = −α1 A/(Γ20 − Γ2S ) < 0. It is worthwhile noticing that a Gaussian distribution with only µ2 6= 0 cannot be preserved in the context of the models with conserved masses, i.e. those satisfying Eq. (17). In the case of nonlinear ω–ψ relations we first notice that, using vanishing boundary conditions at σ = ±∞, one has hdµ/dσ i = −χ (x, y). Introducing P into kthis equality the Taylor expansion µ(σ ) = k=2 µk σ , one gets P kµ m (χ ) = −χ . This is a nonlinear equation in χ k k−1 k=2 but since e Z (χ )m n = dn [ e Z (χ )]/dχ n it is equivalent to a linear equation in the partition function e Z (χ ), namely to X k=2
kµk
dk−1 e Z = −χ e Z. k−1 dχ
(22)
In general, Eq. (22) is of infinite order, however, it can be reduced to finite order when dµ/dσ is a rational function. For example if dµ/dσ = −2q 2 σ/[1 − q 2 σ 2 ] for kqσ k < 1 and 0 otherwise then e Z (χ ) satisfies a modified Bessel equation, for the details and more examples see Appendix D of Ref. [6]. It is often experimentally found that the ω–ψ plots satisfy Ω (−ψ) ' −Ω (ψ), or, µ(−σ ) = µ(σ ). Moreover, in many e (χ ) = f 1 χ + cases these plots are nearly linear so that, Ω 3 5 f 3 χ + f 5 χ , on an interval around ψ = 0 or χ = 0, with | f n+2 | χ 2 < | f n | for odd n and |µn+2 | < |µ2 µn | for even n. Inserting the corresponding powers expansions of e Z (χ ) and µ(σ ) into (22) allows us to express the {µn } in terms of the { f n } , i.e., to determine the probability density exp µ(σ )
H.W. Capel, R.A. Pasmanter / Physica D 237 (2008) 1993–1997
e (χ ). For example, from the experimentally known scatter-plot Ω retaining terms up to f 5 and f 32 in the Taylor expansion of Eq. (22) one gets that µ2k = 0 for k > 3 and, e.g., that 1 3 15 µ2 = − f 1−1 − f 3 f 1−3 + f 5 f 1−4 − 12 f 32 f 1−5 . 2 2 2
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moreover, it is also the solution of Eq. (4) at time TS with conserved total moments Mn and starting from the initial condition φ(σ, x, y, to ) = δ(σ − ω0 (x, y)). When not all the yardstick relations are satisfied then the MRS approach can only give an approximate prediction of the experimental Ω (ψ) relation.
6. Discussion and conclusions
References
In this paper we exploited the fact that the viscous Navier–Stokes equations are compatible with the conservation of the microscopic vorticity masses. In Sections 3 and 4 we studied the family of maximally mixed states described by the distributions φ M (σ, x, y, t) given in Eq. (13). These distributions show also a minimal mixing increase among all distributions with the first and second microscopic moments,
same ω(x, y, t) and σ 2 . In Section 5 we addressed the problem of how to determine the QSS distribution φ S (σ, x, y) from an experimental ω–ψ relation observed at a time TS and β given by Eq. (21). Identifying this φ S (σ, x, y) with the distribution φ M (σ, x, y, TS ) of Eq. (13) and using a time-dependent µ(σ, t) satisfying Eq. (17) and such that µ(σ, TS ) = µ(σ ) we obtained a dynamical model that conserves the masses, i.e., with m(σ, t) = m(σ, to ), and connects the experimental ω–ψ relation found at time TS with an initial condition φ(σ, x, y, to ) of the form (13) with an appriopriate µ(σ, to ), confer Eq. (18). An extra bonus that follows from this methodology is that an H-theorem holds. There are some parallels with the inviscid, statistical mechanics approach of MRS in which the energy is conserved and so are the masses that, in the inviscid case, coincide with the areas occupied by specific vorticity values. In our dynamical models there is no a priori energy conservation and the masses are conserved by imposing Eq. (17). In order to assess the validity of the MRS approach in the case of highReynolds’ number flows, in Subsection III B of an earlier paper [4], we expressed the quantities Z Z δn := dσ dxdy σ n − ωnS (x, y) φ S (σ, x, y),
[1] C. Arndt, Information Measures, Springer, Berlin, 2001. [2] H. Brands, J. Stulemeyer, R.A. Pasmanter, T.J. Schep, A mean field prediction of the asymptotic state of decaying 2D turbulence, Phys. Fluids 9 (1998) 2815. [3] H. Brands, P.-H. Chavanis, R.A. Pasmanter, J. Sommeria, Maximum entropy versus minimum enstrophy vortices, Phys. Fluids 11 (1999) 3465. [4] H.W. Capel, R.A. Pasmanter, Evolution of the vorticity-area density during the formation of coherent structures in two-dimensional flows, Phys. Fluids 12 (2000) 2514. [5] G.J.F. van Heijst, J.B. Flor, Dipole formations and collisions in a stratified fluid, Nature 340 (1989) 212. [6] H.W. Capel, R.A. Pasmanter, Mixing and coherent structures in 2D viscous flows, (2007) http://xxx.arxiv.org/abs/physics/0702040. [7] D. Lynden–Bell, Statistical mechanics of violent relaxation in stellar systems, Mon. Not. R. Astrors. Soc. 136 (1967) 101. [8] D. Marteau, O. Cardoso, P. Tabeling, Equilibrium states of 2D turbulence: An experimental study, Phys. Rev. E 51 (1995) 5124–5127. [9] W.H. Matthaeus, W.T. Stribling, D. Martinez, S. Oughton, D. Montgomery, Selective decay and coherent vortices in two-dimensional incompressible turbulence, Phys. Rev. Lett. 66 (1991) 2731 and the references therein. [10] J. Miller, Statistical mechanics of Euler’s equation in two dimensions, Phys. Rev. Lett. 65 (1990) 2137. [11] J. Miller, P.B. Weichman, M.C. Cross, Statistical mechanics, Euler’s equation, and Jupiter’s red spot, Phys. Rev. A 45 (1992) 2328. [12] D. Montgomery, X. Shan, W. Matthaeus, Navier–Stokes relaxation to sinh-Poisson states at finite Reynolds numbers, Phys. Fluids A 5 (1993) 2207. [13] L. Onsager, Statistical hydrodynamics, Nuovo Cimento (Suppl. 6) (1949) 279–287. [14] J.M. Ottino, Mixing, chaotic advection and turbulence, Ann. Rev. Fluid Mech. 22 (1990) 207. [15] R. Robert, Etat d’´equilibre statistique pour l’´ecoulement bidimensionnel d’un fluide parfait, C. R. Acad. Sci. Paris 311 (S´erie I) (1990) 575. [16] R. Robert, Maximum entropy principle for two-dimensional Euler equations, J. Stat. Phys. 65 (1991) 531. [17] R. Robert, J. Sommeria, Relaxation towards a statistical equilibrium state in two-dimensional perfect fluid dynamics, Phys. Rev. Lett. 69 (1992) 2776. [18] R. Robert, J. Sommeria, Statistical equilibrium states for two dimensional flows, J. Fluid Mech. 229 (1991) 291. [19] E. Segre, S. Kida, Late states of incompressible 2D decaying vorticity fields, Fluid Dyn. Res. 23 (1998) 89. [20] J.A. Shohat, J.D. Tamarkin, The problem of moments, Amer. Math. Soc. Math. Surveys (no. 1) (1943). [21] C. Tsallis, Possible generalization of Boltzmann-Gibbs statistics, J. Stat. Phys. 52 (1988) 479. See also http://tsallis.cat.cbpf.br/biblio.htm. [22] J.C. McWilliams, The emergence of isolated coherent vortices in turbulent flow, J. Fluid Mech. 146 (1984) 21.
n = 2, 3, . . . in terms of spatial integrals of certain polynomials in Ω (ψ) and its derivatives {dr Ω /dψ r } . We then showed that the so-called yardstick relations (δn /∆Γn ) = 1, where ∆Γn := Γno − ΓnS are the total change in the n-th moments of the macroscopic vorticity over the time interval [to , TS ], are nontrivial checks of the validity of the statistical mechanics approach. Choosing β as in (21) the yardstick relation δ2 / Γ20 − Γ2S = 1 is automatically satisfied. When all the relations δn /(Γno − ΓnS ) = 1 hold then the quasi-stationary state predicted by the MRS approach is in agreement with the experimental ω–ψ relation,
Mixing and coherent structures in 2D viscous flows H.W. Capel a , R.A. Pasmanter b,∗ a Inst. Theoretical Physics, University of Amsterdam, Valckenierstraat 65, 1018 XE Amsterdam, The Netherlands b KNMI, P.O. Box 201, 3730 AE, De Bilt, The Netherlands
Available online 29 April 2008
Abstract We introduce a dynamical description based on a probability density φ(σ, x, y, t) of the vorticity σ in two-dimensional viscous flows such that the average vorticity evolves according to the Navier–Stokes equations. A time-dependent mixing index is defined and the class of probability densities that R maximizes this index is studied. The time dependence of the Lagrange multipliers can be chosen in such a way that the masses m(σ, t) := dxdyφ(σ, x, y, t) associated with each vorticity value σ are conserved. When the masses m(σ, t) are conserved then (1) the mixing index satisfies an H-theorem and (2) the mixing index is the time-dependent analogue of the entropy employed in the statistical mechanical theory of inviscid 2D flows. In the context of our class of probability densities we also discuss the reconstruction of the probability density of the quasi-stationary coherent structures from the experimentally determined vorticity-stream function relations. c 2008 Elsevier B.V. All rights reserved.
PACS: 47.32.-y; 47.51.+a; 47.27.De; 47.10.ad; 47.10.ab Keywords: Viscous flows; Two-dimensional flows; Coherent structures; Stochastic dynamics
1. Introduction When studying the dynamics of two-dimensional fluid motion characterized by a vorticity field ω(x, y, t) it can be useful to turn to a probabilistic description with distributions φ(σ, x, y, t) for the microscopic vorticity σ such that the average value of σ over these distributions is equal to ω(x, y, t). The probability distribution represents an ensemble of systems, all the ensemble members satisfy the same constraints. The uncertainties associated with the probability distribution are due to, e.g., the finite experimental precision or to thermal fluctuations. In particular, this can be done in the description of the coherent structures, i.e. the quasistationary states (QSS), which are often reached in (numerical) experiments after fast mixing has taken place [5,8,12,19,22]. At high Reynolds’ numbers, the vorticity fields ω S (x, y) of these QSS’s satisfy ω–ψ relations to a good approximation, i.e., ω S (x, y) ' Ω (ψ(x, y)) where ψ(x, y) is the corresponding ∗ Corresponding author.
E-mail addresses: [email protected], [email protected] (R.A. Pasmanter). c 2008 Elsevier B.V. All rights reserved. 0167-2789/$ - see front matter doi:10.1016/j.physd.2008.04.016
stream-function. In other words, the QSS’s are approximate stationary solutions of the Euler equation. A statistical mechanical theory of inviscid two-dimensional steady flows was introduced in [10,11,15–18], an approach that can be traced back to earlier work of Lynden–Bell in 1967 [7]. Some outstanding aspects of this non-dissipative system are: (1) an infinite number of conserved quantities associated with each microscopic-vorticity value σ and (2) non-uniform equilibrium states (the coherent structures) which often correspond to negative-temperature states as already predicted by Onsager’s work on point vortices [13]. Theoretical predictions of the statistical mechanics approach to the coherent structures were compared with numerical simulations and with experimental measurements in quasi-two dimensional fluids, e.g., in [2,3, 8,9,19]. However, under standard laboratory conditions fluids are viscous and numerical simulations require the introduction of a non-vanishing (hyper)viscosity in order to avoid some numerical instabilities and other artifacts. In spite of this, in many cases it was found that the agreement between the theoretical predictions based on the statistical mechanics of Miller, Robert and Sommeria (MRS) [10,11,15–18] and (numerical) experiments was better than expected.
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In order to discuss these issues in a more dynamical setting we consider viscous flows and propose a family of model evolution equations for the vorticity distribution φ(σ, x, y, t) in Section 2. In Section 3 we discuss the class of time-dependent distributions that maximize a mixing index under certain constraints. In particular, it is shown that the time-dependent Lagrange multipliers appearing in these distributions can be chosen in such a way that the masses associated with each microscopic-vorticity value σ are conserved. When these masses are conserved, the mixing index satisfies an H-theorem. Moreover, the mixing index shows a minimal increase in time [Section 4]. The distribution φ S (σ, x, y) associated with a given QSS can be obtained, at least in principle, by addressing the reconstruction problem, i.e. how to extract its defining parameters from the QSS’s ω–ψ relation. This is discussed in Section 5. In doing so we provide a natural framework for a time-dependent statistical theory connecting an appropriate initial distribution to the QSS distribution associated with the experimental ω–ψ relation and evolving in agreement with the Navier–Stokes equation. The validity of the used assumptions should be tested, for example in numerical simulations. 2. Microscopic viscous models Let φ(σ, x, y, t)dσ be the probability of finding at time t a microscopic vorticity value in the range (σ, σ +dσ ) at a position (x, y). It should be non-negative and normalized Z dσ φ(σ, x, y, t) = 1. (1) The macroscopic vorticity field is Z ω(x, y, t) = hσ i := dσ σ φ(σ, x, y, t),
(2)
in which the pointed brackets denote averages over this distribution, In the inviscid case, the dynamics reduces to the advection of vorticity. Neglecting fluctuations in the velocity field, the time evolution of φ(σ, x, y, t) can be taken to be ∂φ(σ, x, y, t) + vE(x, y, t) · ∇φ(σ, x, y, t) = 0, ∂t
(3)
where the macroscopic, incompressible velocity field vE(x, y, t) z, is related to the macroscopic vorticity ω(x, y, t) by ∇×E v = ωe withe z a unit vector perpendicular to the (x, y)-plane. Extending this to the viscous case the models to be considered are of the form ∂φ(σ, x, y, t) + vE(x, y, t) · ∇φ(σ, x, y, t) = ν∆φ + νφ O,(4) ∂t with ν the fluid viscosity and O as yet undefined but constrained by (1) the conservation of the total probabilty R dσ φ(σ, x, y, t) = 1, and by (2) the macroscopic Navier–Stokes equation, i.e., ∂ω(x, y, t) + vE(x, y, t) · ∇ω(x, y, t) = ν∆ω(x, y, t) ∂t
(5)
should follow from the microscopic model. These two conditions are equivalent to,
O = 0, and σ O = 0. (6) It is convenient to introduce the “masses” m(σ, t) associated with each value σ of the microscopic vorticity, Z m(σ, t) := dxdyφ(σ, x, y, t). (7) In the inviscid case, ν = 0, Eq. (4) has a solution φ(σ, x, y, t) = δ(σ − ω(x, y, t)) therefore m(σ, t) is the area occupied by the vorticity field with value σ. As soon as we introduce a diffusion process, as it is implied by Eq. (4) with ν 6= 0, such an identification becomes impossible. In calling m(σ, t) a “mass” we stress the analogy between Eq. (4) and an advection–diffusion process of an infinite number of “chemical species”, one species for each value σ. Assuming that there is no leakage of φ(σ, x, y, t) through the boundary, the time derivative of a mass is Z ∂m(σ, t) = ν dxdy Oφ(σ, x, y, t). (8) ∂t The simplest viscous model satisfying the above requirements is the one with O ≡ 0. This model is instructive because, while it dissipates energy, it has an infinite number of conserved quantities, i.e., the masses m(σ, t). One of the consequences of the conservation laws for m(σ, R t) is that all the microscopicvorticity moments Mn (t) := dσ σ n m(σ, t) are constants of the motion, i.e., dMn /dt = 0. In the sequel we shall assume that the conservation of all the microscopic moments Mn implies in turn that the masses m(σ, t) are conserved. This is the case if certain technical conditions are satisfied, see e.g., [20]. By contraposition, all even moments of the macroscopic vorticity, Z Γ2n (t) := dxdyω2n (x, y, t), (9) are dissipated since, dΓ2n = −ν2n(2n − 1) dt
Z
dxdyω2(n−1) |∇ω|2 ≤ 0.
(10)
Under appropriateR boundary conditions, e.g., periodic ones, the R energy E = 1/2 dxdy v 2 = 1/2 dxdyωψ, and its dissipation rate is dE/dt = −νΓ2 (t), where ψ(x, y, t) is the streamfunction associated with vE(x, y, t). 3. Time-dependent extremal distributions Equations like (4) have been studied extensively; see for example [14] and the references therein. Usually a timedependent velocity field vE(x, y, t) leads to chaotic trajectories, namely to the explosive growth of small-scale gradients. These small-scale gradients are then rapidly smoothed out by diffusion, the net result being a very large effective diffusion coefficient, large in comparison to the molecular coefficient ν. Based on these observations we will consider situations where during a period of time mixing takes place much faster
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than the changes in the masses m(σ, t). In order to quantify this weR introduce the degree of mixing of m(σ, t), s(σ, t) := −A−1 dxdyφ ln [Aφ/m(σ, t)] , and the corresponding total degree of mixing at time t, Z Z −1 dσ dxdyφ(σ, x, y, t) ln φ(σ, x, y, t) + S(t) = −A Z h i A−1 dσ m(σ, t) ln A−1 m(σ, t) . (11) The fast-mixing condition can thus be expressed as, |∂s(σ, t)/∂t| A−1 |∂m(σ, t)/∂t| .
(12)
This inequality is satisfied when the masses m(σ, t) are conserved, moreover in such a case the second term in (11) is constant in time so that the total degree of mixing S(t) is the time-dependent analogue of the entropy which is used in the MRS statistical mechanics theory and, as shown in the last paragraph of the present section, it satisfies an H-theorem. Accordingly, in the sequel we investigate the time-dependent distributions φ(σ, x, y, t) that maximize the total vorticity mixing S(t) under the following three constraints: (i) normalization, as in (1), (ii) given values of the masses m(σ, t), and (iii) a given distribution first moment hσ i = ω(x, y, t) which, by construction, evolves according to the Navier–Stokes equations. Introducing time-dependent Lagrange multipliers γ (x, y, t), e µ(σ, t) and χ (x, y, t) associated to the abovementioned constraints and denoting the maximizing distribution by φ M (σ, x, y, t), the vanishing of the first variation of S(t) with respect to φ leads to, φ M (σ, x, y, t) = Z −1 exp [µ(σ, t) + χ (x, y, t)σ ] , Z with Z (x, y, t) := dσ exp [σ χ (x, y, t) + µ(σ, t)] ,
(13)
and µ(σ, t) := −e µ(σ, t) + ln A−1 m(σ, t). The functions χ(x, y, t) and µ(σ, t) will be called the “potentials”. Two constraints given by (2) and (7) determine these potentials. For these distributions one has φ M (σ, x, y, t) =: e χ (x, y, t), t), therefore the (x, y)-dependence of ω as well φ(σ, n as that of the local moments m n := hσ i and the centered local n moments K n := (σ − ω) is only through χ (x, y, t), i.e., en (χ , t), Z (x, y, t) =: e K n (x, y, t) =: K Z (χ , t), ω(x, y, t) =: e e /∂χ = e Ω (χ, t), m n (x, y, t) =: m n (χ , t), and relations like ∂ Ω e K 2 (χ , t) hold. In the special case of a QSS at time TS the distribution obtained from the MRS approach is as in Eq. (13) with χ(x, y, TS ) = −βψ(x, y, TS ) where ψ(x, y, TS ) is the stream function at time TS and β is associated with an inverse temperature. This distribution is obtained by maximizing S(TS ) under the constraints (i) and (ii) and the constraint that the energy at time TS has some given value E(TS ). In the MRS inviscid approach the connection with the initial state is made by requiring that the masses at time TS and the energy at time TS equal their initial values, therefore the QSS can be predicted from the initial condition. In the context of our present work, for this to be approximately valid the following fast mixing
condition should be satisfied, dS(t)/dt E −1 (dE/dt) ,
(14)
on top of condition (12). Assume that at all times the probability density has the form given in (13). Omitting for convenience the subsript M, inserting (13) and (2) in the Navier–Stokes equation (5) and making use of simple algebraic equalities one shows that the time evolution of φ M is given by Eq. (4) with O(σ, x, y, t) given by, K3 2 |∇χ |2 − ω) − − ω) O(σ, x, y, t) = K 2 + (σ (σ K2 ∂µ ∂µ −1 ∂µ −1 (σ − ω) −ν . (15) − +ν (σ − ω) ∂t ∂t K2 ∂t
As one can check, O = 0 and σ O = 0 for all possible timedependences of ∂µ/∂t. From (15) it follows that the simplest viscous model with O(σ, x, y, t) ≡ 0, can be realized only under rather trivial conditions. Indeed, since O(σ, x, y, t) ≡ 0 has to hold for any value of σ, Eq. (15) implies that ∂µ/∂t must be quadratic in σ and that |∇χ |2 may be time-dependent but must be (x, y)independent. For general µ(σ, t) there is no conservation of the masses m(σ, t). However,R choosing a suitable time-dependence of µ(σ, t) such that dxdy φ O = 0, ensures the conservation of the masses m(σ, t), confer Eq. (8). This condition and Eq. (15) lead to a complicated integro-differential equation for the time-dependence However, using a Taylor P of µ(σ, t). k expansion µ(σ, t) = k µk (t)σ , we can derive an infinite set of linear differential equations for the dµk /dt. In fact, multiplying Eq. (15) by σ n φ(σ, x, y, t) and integrating it over σ one gets that: ∞ X
dµk ν σ n O = −ν |∇χ |2 h n2 + h nk dt k=2
(16)
with h nk = m k+n − m k m n − K 2−1 (m n+1 − ωm n ) (m k+1 − ωm k ) , where m n := hσ n i . The conservation R R
of the moments Mn = dxdy m n requires then that dxdy σ n O = 0 and hence (16) becomes, Z ∞ X dµk k=2
dt
dxdy h nk = ν
Z
dxdy h n2 |∇χ |2 .
(17)
From this infinite set of equations, linear in dµ2 /dt, dµ3 /dt, . . ., the dµk /dt can, in principle, be solved. The solution describes a viscous model with an infinite number of conservation laws. Such a viscous model becomes physically more relevant by making it compatible with a quasi-stationary distribution eS (σ, χ ) corresponding to the Ω (ψ) relation at time TS , and φ with χ = −βψ(x, y), confer Section 5.
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Using Eq. (4) and the conservation of the masses m(σ, t) one obtains, # " Z ∇φ 2 ∂s(σ, t) ν − O ln Aφ . (18) = dxdyφ ∂t A φ When the distribution is of the form given in Eq. (13), using a Taylor expansion for µ(σ, t) one sees that after integration over σ the term containing O vanishes, therefore dS(t)/dt is nonnegative. As shown in Appendix A of Ref. [6] this H-theorem holds only for one specific measure of spatial mixing, namely for the one given in (11); see, e.g., Ref. [1,21]. The fast-mixing condition (14), divided by ν becomes then, Z Z D E 1 1 dxdy |∇ ln φ|2 dxdyω2 . (19) A E
E o . As we show below these experimental data can be used in order to determine the potential µ(σ, TS ) occuring in the distribution φ M (σ, x, y, TS ) as given by Eq. (13) with χ = −βψ(x, y, TS ). Once the distribution has been reconstructed from the experimental data we can then associate with it a time-dependent distribution function φ M (σ, x, y, t) as given by Eq. (13) which is a solution of the time-evolution Eq. (4) with suitable initial conditions and such that at time t = TS one has eS (σ, χ ) with χ = −βψ and with eM (σ, χ (x, y, TS ), TS ) = φ φ µ(σ, TS ) = µ(σ ). The time-dependence of µ(σ, t) is chosen as in Eq. (17) such that all masses m(σ, t) are constant in time. Here β can be defined such that M2 (TS ), the microscopicvorticity second moment of the QSS at time TS , is equal to Γ20 the enstrophy of the initial vorticity field ωo (x, y), i.e., −1 Z 0 S β = − Γ2 − Γ2 dxdy (dΩ /dψ) (21) A
4. Rate of mixing increase In order to see whether or not this fast-mixing condition (19)is satisfied by the time-dependent distributions φ M (σ, x, y, t), we derive a lower bound to h|∇ ln φ|2 i and we determine the extrema of the l.h.s. of Eq. (19). To this end notice that ∇ω = hσ ∇ ln φi can also be written as ∇ω = h(σ − ω) ∇ ln φi because h∇ ln φi = 0. Applying then the Cauchy–Schwartz inequality to |∇ω|2 = |h(σ − ω) ∇ ln φi|2 leads to |∇ω|2 ≤ h(σ − ω)2 ih|∇ ln φ|2 i ≡ K 2 h|∇ ln φ|2 i, i.e. to the desired lower bound, D E |∇ω|2 K 2 −1 ≤ |∇ ln φ|2 . (20) The lower bound on h|∇ ln φ|2 i that we have just found means that the fast-mixing condition ((19)) holds whenever Z Z |∇ω|2 1 1 dxdy dxdyω2 . A K2 E One can show that the family of probability distributions that reach the lower bound in (20) coincides with φ M (σ, x, y, t) as given by Eq. (13). Here, the input for the determination of the potential functions µ(σ, t) and χ (x, y, t) are the first and second σ -moments. The details concerning the derivation of this can be found in Appendix B of Ref. [6]. R Also the extrema of dxdyh|∇ ln φ|2 i can be investigated taking into account the constraints given by Eqs. (2) and (1). It turns out that in order to obtain sensible solutions, it is necessary to constrain also the distribution’s second moment hσ 2 i. We find R that all the extremizer distributions are local minima of dxdyh|∇ ln φ|2 i and coincide with φ M (σ, x, y, t), moreover they all reach the lower bound in (20). The details can be found in Appendix C of Ref. [6]. 5. Reconstructing µ(σ ) from experimental data Suppose that in an experiment one is given an initial vorticity field with its corresponding energy E o and that at a time TS one finds a quasi-stationary vorticity field ω S (x, y) = Ω (ψ(x, y)), with a monotonic Ω (ψ) and an energy E S ≤
Γ2S
R
dxdy ω2S (x,
= y) is the enstrophy of the QSS. All where the quantities on the r.h.s. of this formula are experimentally accessible. In order to determine the µ(σ ) potential from the experimental Ω (ψ) use can be made of ω = d ln e Z (χ )/dχ , at Z (χ ) given by (13) and β determined from χ = −βψ, with e Eq. (21). Below we illustrate this by considering some examples. A linear ω–ψ scatter plot at time TS , Ωl (ψ) = α1 ψ, with α1 > 0, corresponds to a Gaussian distribution centered on α1 ψ(x, y) and with a width α1 /|β|, i.e., µ(σ ) = βσ 2 /(2α1 ). In the present case the expression (21) for β reads, β = −α1 A/(Γ20 − Γ2S ) < 0. It is worthwhile noticing that a Gaussian distribution with only µ2 6= 0 cannot be preserved in the context of the models with conserved masses, i.e. those satisfying Eq. (17). In the case of nonlinear ω–ψ relations we first notice that, using vanishing boundary conditions at σ = ±∞, one has hdµ/dσ i = −χ (x, y). Introducing P into kthis equality the Taylor expansion µ(σ ) = k=2 µk σ , one gets P kµ m (χ ) = −χ . This is a nonlinear equation in χ k k−1 k=2 but since e Z (χ )m n = dn [ e Z (χ )]/dχ n it is equivalent to a linear equation in the partition function e Z (χ ), namely to X k=2
kµk
dk−1 e Z = −χ e Z. k−1 dχ
(22)
In general, Eq. (22) is of infinite order, however, it can be reduced to finite order when dµ/dσ is a rational function. For example if dµ/dσ = −2q 2 σ/[1 − q 2 σ 2 ] for kqσ k < 1 and 0 otherwise then e Z (χ ) satisfies a modified Bessel equation, for the details and more examples see Appendix D of Ref. [6]. It is often experimentally found that the ω–ψ plots satisfy Ω (−ψ) ' −Ω (ψ), or, µ(−σ ) = µ(σ ). Moreover, in many e (χ ) = f 1 χ + cases these plots are nearly linear so that, Ω 3 5 f 3 χ + f 5 χ , on an interval around ψ = 0 or χ = 0, with | f n+2 | χ 2 < | f n | for odd n and |µn+2 | < |µ2 µn | for even n. Inserting the corresponding powers expansions of e Z (χ ) and µ(σ ) into (22) allows us to express the {µn } in terms of the { f n } , i.e., to determine the probability density exp µ(σ )
H.W. Capel, R.A. Pasmanter / Physica D 237 (2008) 1993–1997
e (χ ). For example, from the experimentally known scatter-plot Ω retaining terms up to f 5 and f 32 in the Taylor expansion of Eq. (22) one gets that µ2k = 0 for k > 3 and, e.g., that 1 3 15 µ2 = − f 1−1 − f 3 f 1−3 + f 5 f 1−4 − 12 f 32 f 1−5 . 2 2 2
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moreover, it is also the solution of Eq. (4) at time TS with conserved total moments Mn and starting from the initial condition φ(σ, x, y, to ) = δ(σ − ω0 (x, y)). When not all the yardstick relations are satisfied then the MRS approach can only give an approximate prediction of the experimental Ω (ψ) relation.
6. Discussion and conclusions
References
In this paper we exploited the fact that the viscous Navier–Stokes equations are compatible with the conservation of the microscopic vorticity masses. In Sections 3 and 4 we studied the family of maximally mixed states described by the distributions φ M (σ, x, y, t) given in Eq. (13). These distributions show also a minimal mixing increase among all distributions with the first and second microscopic moments,
same ω(x, y, t) and σ 2 . In Section 5 we addressed the problem of how to determine the QSS distribution φ S (σ, x, y) from an experimental ω–ψ relation observed at a time TS and β given by Eq. (21). Identifying this φ S (σ, x, y) with the distribution φ M (σ, x, y, TS ) of Eq. (13) and using a time-dependent µ(σ, t) satisfying Eq. (17) and such that µ(σ, TS ) = µ(σ ) we obtained a dynamical model that conserves the masses, i.e., with m(σ, t) = m(σ, to ), and connects the experimental ω–ψ relation found at time TS with an initial condition φ(σ, x, y, to ) of the form (13) with an appriopriate µ(σ, to ), confer Eq. (18). An extra bonus that follows from this methodology is that an H-theorem holds. There are some parallels with the inviscid, statistical mechanics approach of MRS in which the energy is conserved and so are the masses that, in the inviscid case, coincide with the areas occupied by specific vorticity values. In our dynamical models there is no a priori energy conservation and the masses are conserved by imposing Eq. (17). In order to assess the validity of the MRS approach in the case of highReynolds’ number flows, in Subsection III B of an earlier paper [4], we expressed the quantities Z Z δn := dσ dxdy σ n − ωnS (x, y) φ S (σ, x, y),
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n = 2, 3, . . . in terms of spatial integrals of certain polynomials in Ω (ψ) and its derivatives {dr Ω /dψ r } . We then showed that the so-called yardstick relations (δn /∆Γn ) = 1, where ∆Γn := Γno − ΓnS are the total change in the n-th moments of the macroscopic vorticity over the time interval [to , TS ], are nontrivial checks of the validity of the statistical mechanics approach. Choosing β as in (21) the yardstick relation δ2 / Γ20 − Γ2S = 1 is automatically satisfied. When all the relations δn /(Γno − ΓnS ) = 1 hold then the quasi-stationary state predicted by the MRS approach is in agreement with the experimental ω–ψ relation,