A Lagrangian fluctuation-dissipation relation for scalar turbulence, III. Turbulent Rayleigh-Bénard convection
aa r X i v : . [ phy s i c s . f l u - dyn ] J u l Under consideration for publication in J. Fluid Mech. A Lagrangian fluctuation-dissipation relationfor scalar turbulence, III.Turbulent Rayleigh-B´enard convection.
Gregory L. Eyink , and Theodore D. Drivas Department of Applied Mathematics & Statistics, The Johns Hopkins University, Baltimore,MD 21218, USA Department of Physics & Astronomy, The Johns Hopkins University, Baltimore, MD 21218,USA(Received ?; revised ?; accepted ?. - To be entered by editorial office)
A Lagrangian fluctuation-dissipation relation has been derived in a previous work to de-scribe the dissipation rate of advected scalars, both passive and active, in wall-boundedflows. We apply this relation here to develop a Lagrangian description of thermal dis-sipation in turbulent Rayleigh-B´enard convection in a right-cylindrical cell of arbitrarycross-section, with either imposed temperature difference or imposed heat-flux at the topand bottom walls. We obtain an exact relation between the steady-state thermal dissi-pation rate and the time for passive tracer particles released at the top or bottom wallto mix to their final uniform value near those walls. We show that an “ultimate regime”with the Nusselt-number scaling predicted by Spiegel (1971) or, with a log-correction, byKraichnan (1962) will occur at high Rayleigh numbers, unless this near-wall mixing timeis asymptotically much longer than the free-fall time, or almost the large-scale circula-tion time. We suggest a new criterion for an ultimate regime in terms of transition toturbulence of a thermal “mixing zone”, which is much wider than the standard thermalboundary layer. Kraichnan-Spiegel scaling may, however, not hold if the intensity andvolume of thermal plumes decrease sufficiently rapidly with increasing Rayleigh number.To help resolve this issue, we suggest a program to measure the near-wall mixing time,which we argue is accessible both by laboratory experiment and by numerical simulation.
1. Introduction
Turbulent thermal convection in Rayleigh-B´enard flows has been a focus of intenseinterest for decades, both because of its relevance to geophysics and astrophysics andalso because various theoretical, experimental, and numerical studies have suggested thepossibility of a universal scaling of the non-dimensionalized heat flux (Nusselt number)with external control parameters, such as the Rayleigh number, thermal Prandtl number,and cell aspect ratio. Most theoretical analyses of the problem have adopted an Eulerianfluid perspective, based upon balance equations for the mean temperature, temperaturefluctuations, and velocity fluctuations. A basic insight yielded by this approach is thatthe heat transport is directly related to the dissipation rates of temperature fluctua-tions and of kinetic energy (Siggia 1994; Ahlers et al. et al. (2009); Chill`a & Schumacher (2012) for dis-
G. L . Eyink and T. D. Drivas cussion of such flow structures in turbulent Rayleigh-B´enard convection. Attempts havebeen made to consider the effects of thermal plumes in phenomenological theory, such asGrossmann & Lohse (2004). Only a very few works have attempted, however, to applydetailed Lagrangian analysis to understand the thermal dissipation rate (e.g. Schumacher(2008)), and without any a priori theoretical foundation to guide the efforts.In two recent works of Drivas & Eyink (2017 a , b ) [hereafter, papers I and II] a La-grangian framework has been developed for studying turbulent scalar dissipation, validfor both passive and active scalars. The approach is based upon a representation of scalardiffusion effects by stochastic Lagrangian particle trajectories, and it yields an exact La-grangian fluctuation-dissipation relation (FDR), which equates the time-integrated scalardissipation rate to the variance of scalar inputs sampled by stochastic Lagrangian particletrajectories. The wide applicability of this FDR opens up a Lagrangian perspective onscalar dissipation for a great many situations. As a particular example in this paper, wepresent our FDR concretely and at length for thermal dissipation in turbulent Rayleigh-B´enard convection. We show in this situation that the space- and time-averaged thermaldissipation rate is exactly related to the time-correlations of the successive incidencesof stochastic Lagrangian trajectories on the top and bottom walls. In fact, the time-averaged scalar dissipation rate, apart from externally controlled parameters, is directlyproportional to a mixing time of near-wall particle distributions to a uniform distribution. The connection to thermal dissipation arises because the fluctuations in temperature in-put are due to the variable time which each particular stochastic Lagrangian trajectoryspends at the heated or cooled walls. When the particle distribution near these wallsrelaxes to uniform, then each trajectory carries the same temperature input and thefluctuations vanish. Although formulated in terms of stochastic particle trajectories, themixing time in our FDR can be measured by releasing a passive tracer, e.g. a dye, nearthe heated or cooled wall and observing its mixing to a uniform value near those walls.On the basis of this exact FDR, we discuss phenomenological Nusselt-Rayleigh scalinglaws for Rayleigh-B´enard. A scaling of Nusselt number
N u ∼ ( P r Ra ) / for a fluid ofPrandtl number P r at very high Rayleigh number Ra was first proposed by Kraichnan(1962), with a correction logarithmic in Ra , based on a theory assuming a turbulentlog-layer for the velocity, and on purely dimensional grounds by Spiegel (1971). Morerecently, such scaling has been predicted as an “ultimate regime” at sufficiently highRayleigh numbers in the “unifying theory” of Grossman-Lohse (regime IV l ), both asa pure power-law (Grossmann & Lohse 2000, 2001, 2002) with logarithmic corrections(Grossmann & Lohse 2011, 2012) different from those of Kraichnan. In honor of theoriginal proponents (and to avoid any theoretical presumptions), we shall speak here of“Spiegel scaling” for the hypothetical relation N u ∼ ( P r Ra ) / , of “Kraichnan-typescaling” for this relation modified by any logarithmic factor, and of Kraichnan-Spiegel(KS) scaling when we can ignore possible logarithms. † We shall employ the term “ulti-mate regime” in this work to mean the asymptotic scaling regime of
N u with Ra and P r as Ra → ∞ , without any presumption that the specific law is that of Kraichnan-Spiegelor some other law. A striking consequence of our FDR is that an “ultimate regime” ofconvection with KS -scaling will occur unless the near-wall mixing time is asymptoticallymuch larger than either the free-fall time or, what is nearly the same, the large-scale † As emphasized by Grossmann & Lohse (2011), the logarithmic corrections are not neces-sarily ignorable, even at quite large Rayleigh numbers, and may lead to “effective scaling laws” Nu ∼ Ra x with x < / Ra.
However, in the limit as Ra → ∞ , thepower-laws with and without log-corrections become nearly indistinguishable. It has also beensuggested that the turbulent boundary layers may become irrelevant at extremely high Ra andthat Spiegel scaling without logarithmic corrections may be attained (Lohse & Toschi 2003). luctuation-Dissipation Relation circulation time. Spiegel scaling can occur (in the strict sense, with no logarithmic cor-rections) if and only if there are dissipative anomalies for both temperature fluctuationsand kinetic energy in convective turbulence. If the mixing time greatly exceeds the free-fall time (as implied by Nusselt-number measurements at currently achievable Rayleighnumbers), then Lagrangian tracer particles spend this long mixing time traversing acentral region of the flow outside a near-wall “mixing zone”. The latter has a width ℓ T decreasing inversely to the square-root of the Nusselt number and which is thus muchgreater than the standard thermal boundary layer thickness δ T , that is inversely pro-portional to Nusselt number. This “mixing zone” might be identified with the similarconcept proposed by Castaing et al. (1989) and Procaccia et al. (1991). One possible ex-planation for the failure to observe KS-scaling at moderately large Rayleigh numbersis that turbulent transport in the central region is confined to eddies well outside the“mixing zone.” We suggest that an “ultimate regime” (with or without KS scaling) mayoccur when the Reynolds-number for eddies at distance ℓ T from the top/bottom wallsreaches the critical value for transition to turbulence. It is possible that KS-scaling isnot valid in this “ultimate regime”, if there is a mechanism which can lead to extremelylong mixing times at very high Rayleigh numbers. Some possible mechanisms suggestedby empirical observations are decreasing velocity of the global wind and decreasing vol-ume fraction of thermal plumes with increasing Rayleigh numbers. However, it is unclearif these effects can account quantitatively for the observed deviations from Kraichnanscaling even at Rayleigh numbers achieved heretofore. We argue that further empiricalstudies of the mixing time and of possible mechanisms for reduced near-wall mixing cancast significant new light on the Nusselt scaling.The detailed contents of this paper are as follows: In section 2 we review basic theory ofturbulent Rayleigh-B´enard convection, the two problems with imposed temperature dif-ference and imposed heat-flux ( § § § § § § § § §
2. Basic Theory of Rayleigh-B´enard Convection
We here summarize very briefly some of the basic theoretical relations that follow fromstandard Eulerian analyses of turbulent Rayleigh-B´enard convection. The primary resultsare the mean balance relations for the temperature, temperature variance, and kinetic en-ergy. A direct consequence of these balance relations is the connection between dissipative
G. L . Eyink and T. D. Drivas anomalies for kinetic energy and thermal fluctuations on the one hand, and the “ultimateregime” of convection with KS scaling on the other hand. We use the term “dissipativeanomaly” in the standard theoretical physics sense, to denote an energy dissipation ratewhich is non-vanishing in the infinite Reynolds-number limit, when non-dimensionalizedby large-scale velocity magnitude and correlation length (Falkovich et al. et al. (2002); Johnston & Doering (2009); Goluskin(2015) for previous studies of that case. In particular, the exact formulation of what con-stitutes a “dissipative anomaly” for constant-flux convection is somewhat more subtleand has not been discussed previously in the literature to our knowledge. We thereforeanalyze very carefully the problems of Rayleigh-B´enard convection both with imposedtemperature-difference and with imposed heat-flux.2.1.
Two Rayleigh-B´enard Problems
We begin with the precise statement of the standard Rayleigh-B´enard problem. Themost well-studied situation is a Boussinesq fluid in a right cylindrical cell with height H and cross-section S of arbitrary but fixed shape, with temperature imposed at the topand bottom and insulating side-walls (Berg´e & Dubois 1984; Grossmann & Lohse 2000;Ahlers et al. ∂ t u + u · ∇ u = −∇ p + ν △ u + αgT ˆ z , (2.1) ∂ t T + u · ∇ T = κ △ T, (2.2) ∇ · u = 0 . (2.3)with boundary conditions: u = 0 no-slip at top, bottom, and side walls (2.4) T | z = ± H/ = T top/bot isothermal top/bottom walls (2.5)ˆ n · ∇ T = 0 insulating/adiabatic side walls (2.6)where T top < T bot are imposed space-time constant values which lead to convectiveinstability. This standard Rayleigh-B´enard problem corresponds, mathematically, to asystem with mixed Neumann and Dirichlet boundary conditions for the temperature,which is an active scalar. In the above equations, ν is the kinematic viscosity of the fluid, κ the thermal diffusivity, g the acceleration due to gravity and the constant α is theisobaric thermal expansion coefficient. Three dimensionless combinations of parameterscharacterize the system: the Rayleigh number, Prandtl number, and aspect ratio, definedrespectively by Ra = αgH ∆ Tκν , P r = νκ , Γ = DH (2.7)where ∆ T = T bot − T top and D = diam( S ) is the diameter of the cell cross-section. Amajor question of interest is the heat transport across the cell, quantified by the verticalheat flux averaged over volume V and finite time-interval [0 , t ]: J ≡ h u z T − κ∂ z T i V,t . (2.8) luctuation-Dissipation Relation N u = Jκ ∆ T /H . (2.9)The object of many studies has been to determine the functional dependence of
N u uponthe dimensionless parameters
Ra, P r, and Γ of the problem.An alternative problem is Rayleigh-B´enard convection with imposed heat flux at theboundary, which has been previously studied as a model for poorly conducting top andbottom walls (Otero et al. − κ ∂T∂z (cid:12)(cid:12)(cid:12)(cid:12) z = ± H/ = J in imposed flux at top/bottom walls (2.10)with J in a space-time constant value. As we see below, J in is the same as J given by(2.8) when a statistical steady-state exists at t → ∞ with bounded temperature. TheNusselt number is still defined by (2.9) but the roles of J and ∆ T as response and controlvariables are reversed, with now ∆ T = h T i bot − h T i top for T space-averaged over top andbottom walls and time-averaged over [0 , t ]. The natural dimensionless control variable inthis problem is not the Rayleigh number but instead Ra ∗ = αgJH κ ν = Ra N u, (2.11)as noted by Otero et al. (2002). Numerical studies of Johnston & Doering (2009) and alsoStevens et al. (2011) suggest that Rayleigh-B´enard convection with temperature-b.c. andwith flux-b.c. exhibit essentially identical behaviour in the turbulent regime, includingboth
N u - Ra scaling and morphology of the flow, such as thermal plumes.There is an important formal difference between temperature-b.c and flux-b.c., how-ever, with regard to thermal dissipative anomalies. In mathematical treatments, anoma-lies are often associated to dissipation rates of kinetic energy and thermal intensitynon-vanishing in the formal limit ν, κ → , with all other parameters fixed. Such aformulation does not suffice with flux b.c. This can be seen already for the problemof pure conduction with vanishing velocity field. The exact steady-state solution has ∂T /∂z = − J/κ = − ∆ T /H throughout the domain, and thus the mean thermal dis-sipation is h κ |∇ T | i V = J /κ = κ (∆ T ) /H (for more discussion of this example, seeAppendix B). For fixed ∆ T as κ → h κ |∇ T | i V → , but for fixed J instead h κ |∇ T | i V → ∞ ! This is a trivial divergence for pure conduction, which can be eliminatedby instead holding the vertical temperature-gradient ∂T /∂z = β fixed at z = ± H/ κ → . However, in the case of turbulent Rayleigh-B´enard convection with an imposedheat-flux it is a priori unclear how to choose the κ, ν dependence of J so that ∆ T remainsfixed in the limit κ, ν → . Furthermore, even if ∆ T is held fixed, the asymptotic behav-ior of the thermal dissipation is unknown. This question has great importance, since it isknown that the dissipation of kinetic energy and temperature fluctuations determine thescaling of N u at high Ra for the fixed-∆ T problem (Siggia 1994; Ahlers et al. N u - Ra scaling in the fixed- J problem. We thus turnto the mean balances for those quantities. G. L . Eyink and T. D. Drivas
Mean Balance Equations
Global balances of conserved quantities impose simple but crucial constraints on turbu-lent Rayleigh-B´enard convection (Siggia 1994; Ahlers et al. , t ] gives αg (cid:16) J − κH ∆ T (cid:17) = ν h|∇ u | i V,t + h ∂ t ( 12 u ) i V,t , (2.12)where the lefthand side is input of kinetic energy by buoyancy force and the righthandside is the sum of viscous dissipation and kinetic energy growth. Likewise, the meantemperature fluctuation balance is∆( T J ) H = κ h|∇ T | i V,t + h ∂ t ( 12 T ) i V,t (2.13)with ∆(
T J ) = h T J i bot − h T J i top , where the lefthand side is the input of temperaturefluctuations from the boundary and the righthand side is the sum of thermal dissipationand growth of mean temperature fluctuation. These equations hold for both temperature-and flux-b.c. However, for temperature-b.c. ∆( T J ) = T bot h J i bot − T top h J i top whereas forflux-b.c. ∆( T J ) = ( h T i bot − h T i top ) J in = ∆ T J in and J in imposed at the top and bottomis a priori distinct from J defined as the volume-average (2.8).Further simplification occurs for the long-time t → ∞ limit. In that case, the time-derivative terms in (2.12)-(2.13) converge to zero, as long as volume-average kinetic en-ergy and temperature fluctuation remain bounded uniformly in time at constant ν, κ > . The balance equation for the mean of the temperature then gives the additional infor-mation that ∂J ( z ) /∂z = 0 where J ( z ) = h u z T − κ∂ z T i A, ∞ = J (2.14)is the vertical heat flux averaged over the cross-section S of the cell at height z and overan infinite interval of time. The constancy of the vertical heat flux with height impliesthat J in = J for flux-b.c. and h J i top = h J i bot = J for temperature-b.c. The long-timeglobal balance equations then become identical for temperature-b.c. and flux-b.c.: αg (cid:16) J − κH ∆ T (cid:17) = ν h|∇ u | i V, ∞ = ε u (2.15) J ∆ TH = κ h|∇ T | i V, ∞ = ε T , (2.16)but the role of J and ∆ T as control and response variable is reversed for the two cases.2.3. Anomalous Dissipation and Kraichnan-Spiegel Scaling
Now consider the limit ν, κ → P r fixed, as common in mathematical treatmentsof the large Ra limit. First note that J − κ ∆ TH = J (1 − Nu ) ≃ J for N u ≫ . Definingas usual the free-fall velocity U = ( αg ∆ T H ) / (2.17)and neglecting the small Nu correction term in the energy balance, ε u U /H = ε T (∆ T ) U/H = JU ∆ T = N u √ Ra P r . (2.18)The Spiegel scaling law
N u ∼ C · Ra / P r / holds if and only if at fixed P r lim ν,κ → ε u U /H = lim ν,κ → ε T (∆ T ) U/H = C > . (2.19) luctuation-Dissipation Relation ∝ [ln Ra ] − / . For standard temperature-b.c. Rayleigh-B´enard convection, the relation (2.19) corre-sponds to both kinetic and thermal dissipation anomalies, if ∆ T (and thus also U ) istaken to be independent of ν, κ in the limit ν, κ →
0. This is only true, strictly speak-ing, for the Spiegel (1971) dimensional predictions, whereas the Kraichnan (1962) resultscorrespond instead to dissipation both of thermal fluctuations and of kinetic energy van-ishing very slowly (logarithmically) as ν, κ → U ∗ = ( αgJH ) / , ∆ T ∗ = J/U ∗ . (2.20)Note that Ra ∗ = Re ∗ P r with Re ∗ = U ∗ H/ν.
Neglecting again the 1 /N u correction inthe energy balance, ε u U ∗ /H = 1 , ε T (∆ T ∗ ) U ∗ /H = ∆ T ∆ T ∗ . (2.21)If one chooses J to be fixed as ν, κ → , then the first equation has the seeming implicationthat there is necessarily a dissipative anomaly for kinetic energy in flux-b.c. Rayleigh-B´enard convection, whenever a finite-energy long-time limit exists. Here we use the term“energy dissipation anomaly” in the most standard mathematical sense, namely, that ε u remains positive as ν, κ → . Likewise, a “thermal dissipation anomaly” is usually definedto occur when ε T remains positive as ν, κ → . The absence of “dissipative anomalies”is then mathematically taken to mean that instead ε u , ε T → ν, κ → . However, the physically more natural interpretation of a “dissipative anomaly” is that ε u ∝ U /H and ε T ∝ (∆ T ) U/H as ν, κ → , with a non-zero and finite constant ofproportionality. It is thus more reasonable to associate dissipative anomalies with non-vanishing of the dimensionless ratiosˆ ε u = ε u U /H , ˆ ε T = ε T (∆ T ) U/H (2.22)as ν, κ → ε u , ε T as ν, κ → . If we adopt this physicaldefinition, then by (2.18) the existence of “dissipative anomalies” is exactly equivalentto the validity of Spiegel dimensional scaling, for both temperature and heat-flux b.c.It is likewise physically more natural to associate absence of dissipative anomalies withvanishing of ˆ ε u , ˆ ε T as ν, κ → ε u , ε T as ν, κ → . If∆ T is held fixed as ν, κ → , then the two formulations involving ˆ ε u , ˆ ε T and ε u , ε T areobviously equivalent to each other. As we now explain, however, these two formulationsare not equivalent, if J rather than ∆ T is held fixed as ν, κ → U, U ∗ and T ∗ to rewrite (2.18) as ε u U /H = ε T (∆ T ) U/H = (cid:18) U ∗ U (cid:19) = (cid:18) ∆ T ∗ ∆ T (cid:19) / . (2.23)First consider the case where Spiegel dimensional scaling holds. It follows from (2.23) G. L . Eyink and T. D. Drivas that this scaling is equivalent tolim ν,κ →∞ U ∗ U = C / , lim ν,κ →∞ ∆ T ∗ ∆ T = C / , (2.24)As a consequence, the velocities U and U ∗ differ only by a constant factor independent ofRayleigh number for Ra ≫ . The same is also true for temperature scales ∆ T and ∆ T ∗ .In this case, holding J fixed as ν, κ → T fixed as ν, κ → ε u , ε T must converge to finite, positive values in the limit ν, κ →
0. However, nowconsider the situation when Spiegel dimensional scaling does not hold. Rigorous upperbounds on Nusselt number of Doering & Constantin (1996) for temperature b.c. and ofOtero et al. (2002) for heat-flux b.c. together with (2.18) imply then that ˆ ε u , ˆ ε T →
0. Itfollows from (2.23) that in that case U ∗ /U → T ∗ / ∆ T → ν, κ → . Accordingthe advocated interpretation above, this corresponds to a vanishing dissipative anomaly.However, it also follows that ∆
T / ∆ T ∗ → ∞ and comparing with (2.21) this means thatwhen J is held fixed as ν, κ → , then ε T → ∞ ! It may appear odd to associate thisbehavior with “absence of dissipative anomaly”, but it should be kept in mind that forfixed J as ν, κ → , then ε T → ∞ even for the problem of pure heat conduction (seeAppendix B). Instead, a thermal dissipative anomaly in the physical sense is naturallyassociated with ε T remaining finite for ν, κ → J is held fixed.It is easy to check by using the definitions of the various quantities that there isequivalence of the general scaling relations N u ∼ Ra x P r y ⇐⇒ ∆ T ∆ T ∗ ∼ Ra z ∗ P r w (2.25)with x = 1 − z z , y = 1 − w z . (2.26)These can be interpreted as a relation between J and ∆ TJ ∼ κ − x ( αg ) x H − x P r y − x (∆ T ) x (2.27)which must be maintained with flux-b.c. in order to hold ∆ T fixed as ν, κ → . It followsfrom the rigorous bounds of Doering & Constantin (1996) and Otero et al. (2002) that x / . If x < / , below the Kraichnan-Spiegel value, then J → ν, κ → T. In that case, ∆
T / ∆ T ∗ → ∞ as ν, κ → T ∗ → .
3. Lagrangian Fluctuation-Dissipation Relations
With this clear understanding of the relevance of dissipative anomalies for turbu-lent
N u - Ra scaling in Rayleigh-B´enard convection, we can now discuss the stochasticLagrangian representations for the temperature field and our fluctuation-dissipation re-lation (FDR) for the thermal dissipation. As discussed in paper II, the representationsinvolve the stochastic Lagrangian flow ˜ ξ ν,κt,s ( x ) reflected at the flow boundary and movingbackward in time, which satisfies for s < t a backward It¯o equation of the formˆd˜ ξ t,s ( x ) = u (˜ ξ t,s ( x ) , s ) d s + √ κ ˆd f W s − κ n (˜ ξ t,s ( x )) ˆd˜ ℓ t,s ( x ) (3.1)with ˜ ξ t,t ( x ) = x . Here f W s is a standard 3D Brownian motion (Wiener process), n ( x )is the inward-pointing unit normal vector at points x on the boundary, and ˜ ℓ t,s ( x ) isthe boundary local-time density . The latter quantity is discussed completely in paper II,but here we note that the local-time density appears in the stochastic representation luctuation-Dissipation Relation x of the boundary where there is a non-zero heat flux through thewall. In Rayleigh-B´enard convection, such points occur only at the top and bottom of theconvection cell, since sidewalls are assumed perfectly insulated. Using the notation ˜ ξ t,s =( ˜ ξ t,s , ˜ η t,s , ˜ ζ t,s ) and x = ( x, y, z ) for the Cartesian components, we can write separateexpressions for the local times densities at the top and bottom wall, as:˜ ℓ topt,s ( x ) = Z st dr δ (cid:18) ˜ ζ t,r ( x ) − H (cid:19) , ˜ ℓ bott,s ( x ) = Z st dr δ (cid:18) ˜ ζ t,r ( x ) + H (cid:19) . (3.2)These expressions are a special case of the general result of Paper II, eq.(2.6). For thepurposes of the present study, this completely specifies the stochastic Lagrangian flow˜ ξ ν,κt,s ( x ) . We have indicated here by superscripts the dependence of this flow on theparameters ν, κ, but, in order to avoid a too cluttered notation, we hereafter omit thesesuperscripts unless it is important to stress the dependence.3.1.
Rayleigh-B´enard Convection with Flux B. C.
The stochastic Lagrangian representation and FDR are simplest in form and easiest toanalyze for imposed heat-flux at the boundaries. We therefore discuss this case first.3.1.1.
Presentation of the Formulas
For this case the stochastic representation of the temperature field takes the form: T ( x , t ) = E h T (˜ ξ t, ( x )) + J (cid:16) ˜ ℓ topt, ( x ) − ˜ ℓ bott, ( x ) (cid:17)i = Z d x T ( x ) p ( x , | x , t ) − J Z t ds p z ( H/ , s | x , t ) + J Z t ds p z ( − H/ , s | x , t ) , (3.3)which follows directly from formulas (II;2.9) and (II;2.10) of Paper II. We have introducedthe backward-in-time transition probability p ν,κ ( x ′ , t ′ | x , t ) = E h δ d ( x ′ − ˜ ξ ν,κt,t ′ ( x )) i t ′ < t (3.4)and the conditional probability density for the z -component of the particle position: p z ( z ′ , s | x , t ) = Z Z S dx ′ dy ′ p ( x ′ , y ′ , z ′ , s | x , t ) = E h δ (cid:16) z ′ − ˜ ζ t,r ( x ) (cid:17)i . (3.5)It is important to stress that the average E [ · ] appearing in these last two equations isover the Brownian motion f W s only. In particular, the velocity u is a fixed (deterministic)field obtained as a solution of the Rayleigh-B´enard problem with flux b.c. (2.10). Nowherein this paper shall we average over ensembles of u for random initial data u , T . It wouldbe appropriate to add a subscript u to the transition probability densities defined above,in order to emphasize their dependence upon this explicit solution. Since this would leadto a more cumbersome notation, however, we shall refrain from doing so.It is illuminating to discuss the properties of this stochastic representation in the long-time t → ∞ limit. Because of incompressibility of the velocity field and ergodicity ofthe stochastic Lagrangian flow for κ >
0, the particle distributions must asymptoticallybecome uniform over the flow domain:lim t →∞ p ( y , s | x , t ) = 1 V , lim t →∞ p z ( z ′ , s | x , t ) = 1 H , (3.6)with y , z ′ , s, and x fixed as t → ∞ , where V = HA is the volume of the cylindrical cell0 G. L . Eyink and T. D. Drivas with cross-sectional area A. Thus,lim t →∞ Z d x T ( x ) p ( x , | x , t ) = 1 V Z d x T ( x ) = h T i V . (3.7)For flux-b.c. the volume average of T ( x , t ) is conserved in time so that memory of theinitial average must be preserved. We assume for simplicity that h T i V = 0 , so that T ( x , t ) ≃ − J Z t ds (cid:18) p z ( H/ , s | x , t ) − H (cid:19) + J Z t ds (cid:18) p z ( − H/ , s | x , t ) − H (cid:19) , (3.8)where the term Jt/H has been added and subtracted to make each of the integrands tendto zero for large time separations. From this expression one can see that the dependenceupon the distant past for s ≪ t is negligible in comparison to the contribution from therecent past for s . t. One can also understand why generally T ( x , t ) > z & − H/ , slightly above the bottom wall, since then usually p z ( H/ , s | x , t ) < H , p z ( − H/ , s | x , t ) > H , s . t. (3.9)Exactly the opposite inequalities generally hold for z . H/ , slightly below the top wall.Of course, T ( x , t ) evolves chaotically in time through the dependence of the transitionprobabilities upon u , and there can be rare fluctuations of temperature with the “wrong”sign near the top and bottom of the cell. Note that the long-time average is given by h T ( x ) i ∞ = lim t →∞ t Z t ds T ( x , s )= − J Z −∞ ds (cid:18) p z ( H/ , s | x , − H (cid:19) + J Z −∞ ds (cid:18) p z ( − H/ , s | x , − H (cid:19) , (3.10)whenever the latter integrals are convergent. If not, then taking the lower limit to −∞ must be interpreted in the Ces`aro mean sense, i.e. the limit of the time-average of theintegral with respect to its range of integration. These long-time averages will no longerdepend upon the initial conditions u and T at s = −∞ if there is ergodicity of Euleriandynamics for the Boussinesq fluid system at high Ra.
We can now present our Lagrangian fluctuation-dissipation relation for Rayleigh-B´enard convection with flux-b.c., which expresses the volume- and time-averaged thermaldissipation rate as: κ Z t ds h|∇ T ( s ) | i V = 12 D Var h T (˜ ξ t, ) + J (cid:16) ˜ ℓ topt, − ˜ ℓ bott, (cid:17)i E V , (3.11)where Var[ · ] denotes the variance over the Brownian motion f W s . This result is a directconsequence of formulas (II;2.11), (II;2.13) of paper II. To make the result somewhatmore concrete (but also more elaborate), we can decompose the variance on the right asVar h T (˜ ξ t, ( x )) + J (cid:16) ˜ ℓ topt, ( x ) − ˜ ℓ bott, ( x ) (cid:17)i = Var h T (˜ ξ t, ( x )) i +2 Cov h T (˜ ξ t, ( x )) , J (cid:16) ˜ ℓ topt, ( x ) − ˜ ℓ bott, ( x ) (cid:17)i + Var h J (cid:16) ˜ ℓ topt, ( x ) − ˜ ℓ bott, ( x ) (cid:17)i . (3.12)where Cov[ · , · ] denotes the covariance of two random variables as functions of theBrownian motion f W s . We can then provide explicit formulas for the three separateterms. As a special case of eq.(II;2.18) of paper II we see that Var h T (˜ ξ t, ( x )) i is given luctuation-Dissipation Relation h T (˜ ξ t, ( x )) i = Z d d x Z d d x ′ T ( x ) T ( x ′ ) × h p ν,κ ( x , x ′ , | x , t ) − p ν,κ ( x , | x , t ) p ν,κ ( x ′ , | x , t ) i (3.13)where we have introduced the 2-time (backward-in-time) transition probability density p ν,κ ( y , s ; y ′ , s ′ | x , t ) = E h δ d ( y − ˜ ξ ν,κt,s ( x )) δ d ( y ′ − ˜ ξ ν,κt,s ′ ( x )) i , s < t (3.14)which represents the joint probability for the particle to end up at y at time s < t andat y ′ at time s ′ < t, given that it started at x at t . Likewise, the local-time variance isgiven as a special case of eq.(II;2.30) of paper II by:12 Var h J (cid:16) ˜ ℓ topt, ( x ) − ˜ ℓ bott, ( x ) (cid:17)i = J Z t ds Z t ds ′ h p z (+ H/ , s ; + H/ , s ′ | x , t ) − p z (+ H/ , s | x , t ) p z (+ H/ , s ′ | x , t ) i + J Z t ds Z t ds ′ h p z ( − H/ , s ; − H/ , s ′ | x , t ) − p z ( − H/ , s | x , t ) p z ( − H/ , s ′ | x , t ) i − J Z t ds Z t ds ′ h p z (+ H/ , s ; − H/ , s ′ | x , t ) − p z (+ H/ , s | x , t ) p z ( − H/ , s ′ | x , t ) i − J Z t ds Z t ds ′ h p z ( − H/ , s ; + H/ , s ′ | x , t ) − p z ( − H/ , s | x , t ) p z (+ H/ , s ′ | x , t ) i (3.15)where p z ( z ′ , s ; z ′′ , s ′ | x , t ) = Z Z S dx ′ dy ′ Z Z S dx ′′ dy ′′ p ( x ′ , y ′ , z ′ , s ; x ′′ , y ′′ , z ′′ , s ′ | x , t ) . (3.16)Similarly to the preceding two terms, the cross-covariance is given byCov h T (˜ ξ t, ( x )) , J (cid:16) ˜ ℓ topt, ( x ) − ˜ ℓ bott, ( x ) (cid:17)i = − J Z d x T ( x ) Z t ds Z Z S dx ′ dy ′ × h p ( x , x ′ , y ′ , H/ , s | x , t ) − p ( x , | x , t ) p ( x ′ , y ′ , H/ , s | x , t ) i + J Z d x T ( x ) Z t ds Z Z S dx ′ dy ′ × h p ( x , x ′ , y ′ , − H/ , s | x , t ) − p ( x , | x , t ) p ( x ′ , y ′ , − H/ , s | x , t ) i (3.17)These formulas provide a purely Lagrangian representation of the thermal dissipation.It is notable that the entire expression vanishes if the particle positions are statisticallyindependent at distinct times (or, in particular, if particle trajectories are deterministic).The finite-time fluctuation-dissipation relation is a bit complicated because of theseveral terms. However, all of the T -dependence disappears in the long-time limit wherethe following simpler relation holds: h κ |∇ T | i V, ∞ = lim t →∞ t D Var h J (cid:16) ˜ ℓ topt, − ˜ ℓ bott, (cid:17)i E V = X λ,λ ′ λλ ′ lim t →∞ J t D Cov h J (cid:16) ˜ ℓ λt, , ˜ ℓ λ ′ t, (cid:17)i E V (3.18)2 G. L . Eyink and T. D. Drivas with λ, λ ′ = ± , where we denote top/bottom walls by + / − . Indeed, it is rigorously truethat lim t →∞ t Var h T (˜ ξ t, ( x )) i = 0 for bounded initial data T , since the variance isthen at most 2(max | T | ) . One can also argue that the contribution to the long-timeaverage from the covariance (3.17) divided by t gives a vanishing contribution, sinceparticle positions at time 0 and time s will become independent for s ≫ s -integrals in (3.17) are expected to converge for t → ∞† . For the same reason,the surviving contribution (3.18) is expected to be finite as t → ∞ since in the doubletime-integration over s, s ′ in (3.15) the integrand is non-negligible only for s ≃ s ′ . According to the general result (II;2.14) of paper II, the limit in (3.18) should further-more be x -independent without averaging over space. This may in fact be shown by adirect argument, which yields the much simpler formulalim t →∞ λλ ′ t Cov h ˜ ℓ λt, ( x ) , ˜ ℓ λ ′ t, ( x ) i = λλ ′ H Z −∞ ds (cid:20) p z ( λH/ , s | λ ′ H/ , − H (cid:21) λ,λ ′ , (3.19)where (cid:2) · (cid:3) λ,λ ′ denotes symmetrization with respect to indices λ, λ ′ . Because derivationof the formula (3.19) is a bit technical, we present it in Appendix A. We have defined p z ( z, s | z ′ , s ′ ) = 1 A Z Z S dx ′ dy ′ p z ( z, s | x ′ , y ′ , z ′ , s ′ ) (3.20)which gives the transition probability for vertical heights of the particles, if at time s ′ a uniform distribution of particles is taken over the volume. Note that a uniformdistribution over the volume implies a uniform distribution in area over the cross-section S at each height z ′ . By incompressibility of the flow, the uniform distribution of particlesover the volume is preserved in time and thus R dz ′ p z ( z, s | z ′ , s ′ ) = 1 for all z, s, s ′ . Theformulas (3.18),(3.19) are our steady-state fluctuation-dissipation relation for Rayleigh-B´enard convection with flux-b.c. The remarkable result is that the long-time average ofthe thermal dissipation is, in the Lagrangian sense, entirely due to statistical correlationsof incidences of single fluid particles on the top and bottom walls at two distinct times.A bit further below we shall provide a more fluid-mechanical interpretation of (3.19) interms of mixing of a passive tracer, such as a dye, released near the top or bottom wall.The result of combining (3.18) and (3.19) can, in fact, be obtained by a simpler ar-gument directly from the Eulerian balance relation (2.16) for thermal fluctuations inthe long-time steady-state. To see this, use the Lagrangian formula (3.10) for the time-averaged temperature field and the definition (3.20) to write the area-averaged steady-state temperatures at the top/bottom walls as¯ T λ ′ = − J X λ = ± λ Z −∞ ds (cid:18) p z ( λH/ , s | λ ′ H/ , − H (cid:19) (3.21)for λ ′ = ± . Thus, the steady-state temperature difference ∆ T = ¯ T − − ¯ T + between thebottom and top plates is∆ T = J X λ,λ ′ = ± λλ ′ Z −∞ ds (cid:18) p z ( λH/ , s | λ ′ H/ , − H (cid:19) . (3.22)The result of substituting this Lagrangian expression for ∆ T into the steady-state tem-perature balance ε T = J ∆ T /H in (2.16) is completely equivalent to the combination of † Vanishing of the contribution from (3.17) follows also from the Cauchy-Schwartz inequality (cid:12)(cid:12)(cid:12) t Cov h T (˜ ξ t, ( x )) , J (cid:16) ˜ ℓ topt, ( x ) − ˜ ℓ bott, ( x ) (cid:17)i(cid:12)(cid:12)(cid:12) t Var h T (˜ ξ t, ( x )) i · t Var h J (cid:16) ˜ ℓ topt, ( x ) − ˜ ℓ bott, ( x ) (cid:17)i . luctuation-Dissipation Relation T -contributionsin (3.22) for λ, λ ′ = ± to statistical correlations of boundary local-time densities. We nowdiscuss some of the significant implications of this fact.3.1.2. Mathematical Consequences of the FDR
Let us denote the pointwise scalar variance as h ε flucT ( x ) i t ≡ t Var h J (cid:16) ˜ ℓ topt, ( x ) − ˜ ℓ bott, ( x ) (cid:17)i , (3.23)which is a spatially-local measure in terms of statistical Lagrangian trajectories of thethermal dissipation averaged over the time-interval [0 , t ] . It is related by our FDR to thespace-time average thermal dissipation as h ε flucT i ∞ ≡ lim t →∞ h ε flucT ( x ) i t = h κ |∇ T | i V, ∞ = ε T , (3.24)so that knowledge of h ε flucT ( x ) i t suffices to determine the global mean thermal dissipation.It may be expected for relatively short times that h ε flucT ( x ) i t and h κ |∇ T ( x , · ) | i t are wellcorrelated in space, especially when the Boussinesq system with flux b.c. (2.10) is solvedwith initial condition T = (const . ) , so that both Var[ T (˜ ξ t, ( x ))] and the covariance termin (3.17) vanish identically. For example, see II, Appendix A.2 for pure heat conduction.Of course, by its definition h ε flucT ( x ) i t > . In the long-time limit, each of the four terms in (3.19) for λ, λ ′ = ± should be positiveseparately. This can be seen from (3.19) since wall-incidences of stochastic Lagrangianparticles for near times s ≃ s ′ are correlated for same-wall ( homohedral ) incidences butanti-correlated for opposite-wall ( heterohedral ) incidences. Indeed, for s = s ′ p z ( z, s | x, ′ , y ′ , z ′ , s ) = Z Z S dx dy δ ( x − x ′ ) = δ ( z − z ′ ) , (3.25)so that it must hold on the one hand thatlim s ′ → s p z ( ± H/ , s | ± H/ , s ′ ) = ∞ , (3.26)and on the other hand that lim s ′ → s p z ( ∓ H/ , s | ± H/ , s ′ ) = 0 . (3.27)For s ≪ s ′ the integrands of all four terms converge to zero as the incidences at widelyseparated times become independent and the (anti-)correlations decay. Because of thepositive sign before the homohedral (variance) terms and the negative sign before theheterohedral (covariance) terms, all four contributions are presumably positive separately.Of the four terms in (3.19), positivity necessarily holds for the homohedral terms h ε flucT i ++ ∞ = lim t →∞ J t Var h ˜ ℓ topt, ( x ) i > , (3.28) h ε flucT i −−∞ = lim t →∞ J t Var h ˜ ℓ bott, ( x ) i > , (3.29)and is plausibly true by the previous argument for the heterohedral terms h ε flucT i + −∞ = − lim t →∞ J t Cov h ˜ ℓ topt, ( x ) , ˜ ℓ bott, ( x ) i = h ε flucT i − + ∞ . (3.30)4 G. L . Eyink and T. D. Drivas
Note that h ε flucT i + −∞ = h ε flucT i − + ∞ h ε flucT i ++ ∞ + h ε flucT i −−∞ uv ( u + v ) / h ε flucT i −−∞ = h ε flucT i ++ ∞ (3.32)because of the exact symmetry z → − z, u z → − u z , T → − T of the Boussinesq systemwith flux-b.c (2.1)-(2.6), which transforms one solution into another solution.Since the natural scale of each of the probability density functions p z in (3.19) is 1 /H, one can write these fluctuational dissipation contributions exactly for λ, λ ′ = ± as h ε flucT i λ,λ ′ ∞ = J H τ λλ ′ mix , τ λλ ′ mix = λλ ′ Z −∞ ds (cid:2) H · p z ( λH/ , s | λ ′ H/ , − (cid:3) λ,λ ′ . (3.33)Here τ λλ ′ mix is an integral correlation time which measures the length of time required forthe (anti-)correlations in (3.19) to decay for | s − s ′ | → ∞ , that is, the time required forthe particle distributions near the walls to relax to uniform 1 /H distributions. Note thatnon-uniform particle distributions in these vicinities are required to feed temperaturefluctuations into the flow. We use the suffix “mix” because, as we see shortly, τ λλ ′ mix alsohas the meaning of a near-wall mixing time of a passive tracer. The key implicationof (3.33) is that long relaxation/mixing times correspond to large thermal dissipation.Summing over λ, λ ′ = ± , we reach the important exact conclusion ε T = J H τ mix (3.34)where τ mix = τ hommix + τ hetmix is the sum of all correlation/mixing times with τ hommix = τ ++ mix + τ −− mix , τ hetmix = τ + − mix + τ − + mix , (3.35)and τ hetmix τ hommix . (3.36)The scaling of ε T with physical parameters is therefore completely determined by thescaling of the total time τ mix . It is thus important to note that this time τ mix has a simple fluid-mechanical meaning,in addition to its probabilistic interpretation in terms of stochastic Lagrangian particles.Consider a passive tracer, e.g. a dye, whose molecular diffusivity is identical to the thermaldiffusivity of the fluid, or, equivalently, whose Schmidt number is equal to the fluidPrandtl number. If this tracer is released into the Rayleigh-B´enard cell at time s withinitial concentration c s ( x , s ) (mass per volume), then the concentration c s ( x , t ) at latertimes t satisfies the passive advection-diffusion equation ∂ t c s + u · ∇ c s = κ △ c s . (3.37)If one assumes that the cell walls are impermeable to the tracer, then equation (3.37)should be solved with no-flux b.c. Since a stochastic Lagrangian representation analogousto (3.3) applies to any passive scalar, this means that the tracer concentration at time t > s is given by the formula c s ( x , t ) = Z d x s c s ( x s , s ) p ( x s , s | x , t ) . (3.38)Now assume that the initial concentration at time s is in the form of an infinitesimally luctuation-Dissipation Relation z = λH/ , or c s ( x s , s ) = (1 /A ) δ ( z s − λH/ . (3.39)The normalization 1 /A assumes that the total initial mass of the tracer is unity. In thatcase, it is easy to see from (3.38) for t = 0, (3.39), and the definition (3.20) of the verticaltransition probability density that for any s < p z ( λH/ , s | λ ′ H/ ,
0) =
Z Z S dx dy c s ( λ ′ H/ , x, y, . (3.40)In other words, the transition probability that appears in the definition (3.33) of τ λλ ′ mix hasa direct physical interpretation as the integrated mass-density measured at time 0 on thewall z = λ ′ H/ z = λH/ s < . Thus, τ λλ ′ mix is nothing other than the integral mixing-time requiredfor the integrated mass density of the tracer at the wall z = λ ′ H/ /H as the release time s → −∞ . Although we derivedour exact relation (3.34) between thermal dissipation rate ε T and relaxation time τ mix using a stochastic formulation, both quantities have a direct fluid-mechanical meaning.To provide some further physical insight into these mixing times, it is useful to considerthe case of pure thermal conduction, where u ≡ τ hommix = 23 H κ , τ hetmix = 13 H κ . (3.41)See Appendix B. For pure conduction the near-wall mixing times scale as the time todiffuse across the cell height H , reproducing the exact result for that problem that ε T = J /κ . Consistent with the general inequality (3.31), τ hetmix < τ hommix . The fact thatdiffusive mixing is twice faster at the opposite wall than at the wall where the tracerwas released can be understood from the fact the tracer is already substantially mixedwhen it first diffuses across distance H to the opposite wall, but the tracer must thendiffuse back the distance H to the original site of release in order to be mixed there. Inthermal convection, eq. (3.40) provides a means, in principle, to measure the near-wallmixing time τ mix in a laboratory experiment, by releasing a stream of such tracers atthe top or bottom wall of the cell and then measuring their concentrations at both topand bottom walls at some much later time, designated as “time 0.” In fact, there areother methods for empirical determination of τ mix either by laboratory experiment ornumerical simulation, which are probably more convenient. However, we shall delay ourdiscussion of such measurement procedures (section 4.4) until after we have discussedfully the physical implications of our exact relationship ( sections 4.1-4.3).3.2. Standard Rayleigh-B´enard Convection
Having completed our discussion of the stochastic Lagrangian representation and theFDR for flux b.c., we now turn to the standard Rayleigh-B´enard problem with mixedboundary conditions (2.4)-(2.6).A formula for the temperature field analogous to (3.3) can be derived as a special caseof the general formula (II;3.29) of paper II, as: T ( x , t ) = E (cid:20) T (˜ ξ t, ( x )) + Z t J (˜ ξ t,s ( x ) , s ) ˆd˜ ℓ topt,s ( x ) − Z t J (˜ ξ t,s ( x ) , s ) ˆd˜ ℓ bott,s ( x ) (cid:21) = Z d x T ( x ) p ( x , | x , t )6 G. L . Eyink and T. D. Drivas − X λ = ± λ Z t ds Z Z S dx ′ dy ′ J ( x ′ , y ′ , λ H , s ) p ( x ′ , y ′ , λ H , s | x , t ) , (3.42)where J ( x , t ) = u z ( x , t ) T ( x , t ) − κ∂ z T ( x , t ) is the vertical heat flux, which becomespurely conductive at the top/bottom walls where u z ≡
0. This representation has ahybrid Eulerian-Lagrangian character, since it involves both the particle probabilities p ( x ′ , t ′ | x , t ) and the Eulerian field J ( x , t ) . A purely Lagrangian representation can beobtained from an alternative formula (II;3.5) of paper II, involving the first hitting-time on the heated (top/bottom) walls backward in time. This alternative stochasticrepresentation provides, however, only a strict lower bound on the thermal dissipationrate, not an equality FDR relation, and therefore is not as important here.A much more useful FDR for the thermal dissipation follows from formula (II;3.30) ofpaper II. In the steady-state (infinite-time), spatially-local form analogous to (3.18) forthe case of flux-b.c., this FDR is:lim t →∞ t Var (cid:20)Z t J (˜ ξ t,s ( x ) , s ) ˆd˜ ℓ topt,s ( x ) − Z t J (˜ ξ t,s ( x ) , s ) ˆd˜ ℓ bott,s ( x ) (cid:21) = h κ |∇ T | i V, ∞ . (3.43)As in the case of (3.18), the contributions of the initial data T can be argued to becomea spatial constant at long times and the limit becomes independent of x by the ergodicityof the stochastic Lagrangian flow. Arguments like those in section 3.1 and Appendix Aimply that h κ |∇ T | i V, ∞ = lim t →∞ V t Z − t ds ′ Z −∞ ds Z Z S dx ′′ dy ′′ Z Z S dx ′ dy ′ × X λ,λ ′ = ± λλ ′ J ( x ′′ , y ′′ , λ ′ H , s ′ ) J ( x ′ , y ′ , λ H , s ) × (cid:20) p ( x ′ , y ′ , λ H , s | x ′′ , y ′′ , λ ′ H , − V (cid:21) λ,λ ′ (3.44)analogous to (3.18),(3.19) for the flux-b.c. case. The FDR (3.44), just like the repre-sentation (3.42) for the temperature, has a mixed Eulerian-Lagrangian character. It isconsistent with the Eulerian balance relation ε T = J ∆ T /H in (2.16) if one notes thatfor all values of x ′′ , y ′′ ∆ T = X λ,λ ′ = ± λλ ′ Z −∞ ds Z Z S dx ′ dy ′ J ( x ′ , y ′ , λ H , s ) × (cid:20) p ( x ′ , y ′ , λ H , s | x ′′ , y ′′ , λ ′ H , − V (cid:21) (3.45)by taking the (Ces`aro-sense) limit t → ∞ in the eq.(3.42) for points x on the top orbottom wall where the temperature is held fixed. Here we used the asymptotic result Z t ds Z Z S dx ′ dy ′ J ( x ′ , y ′ , λ H , s ) ∼ Jt (3.46)as t → ∞ for λ = ± with J the time- and area-average of J ( x, y, z, t ) (which is indepen-dent of z ) in order to introduce the term − /V into the square bracket in (3.45). Theresult (3.46) is a consequence of (Eulerian) time-ergodicity of the system (2.4)-(2.6).Just as for the case of flux b.c., we may decompose the variance in our FDR (3.43) luctuation-Dissipation Relation h κ |∇ T | i V, ∞ = X λ,λ ′ = ± h ε flucT i λλ ′ ∞ (3.47)where the four “fluctuational dissipations” are the limiting covariances h ε flucT i λλ ′ ∞ = lim t →∞ λλ ′ t Cov (cid:20)Z t J (˜ ξ t,s ( x ) , s ) ˆd˜ ℓ λt,s ( x ) , Z t J (˜ ξ t,s ( x ) , s ) ˆd˜ ℓ λ ′ t,s ( x ) (cid:21) . (3.48)These are each expected to be positive separately and all of the results (3.28)-(3.32)follow with temperature b.c. using the same arguments as for flux b.c. We can againwrite these four terms as h ε flucT i λλ ′ ∞ = J H τ λλ ′ mix , λ, λ ′ = ± (3.49)where the factors τ λλ ′ mix each have the dimension of time. In fact, these represent J -weighted average near-wall mixing times of a passive tracer. To be more precise, considerthe tracer released at time s as a point-mass at one of the top or bottom walls: c s ( x s , s ) = δ ( x s − x ′ ) δ ( y s − y ′ ) δ ( z s − λH/ . (3.50)In that case, the tracer concentration field at time t > s is c s ( x , t ) = p ( x ′ , y ′ , λH/ , s | x , t ) . (3.51)The tracer concentration at time t = 0 and evaluated at the top/bottom walls is thusequal to the transition probability density that appears in the integrand of the FDR(3.44)). These concentrations will mix to values 1 /V for s ≪ t and thus the times τ λλ ′ mix introduced in (3.49) represent integral mixing times near the walls, averaged over bothrelease points and measurement points with respect to the heat-flux distributions.Summing the four terms gives again an exact relation for temperature b.c. which relatesthermal dissipation rate and the near-wall mixing time: ε T = J H τ mix , (3.52)identical in form to that derived for flux b.c. From our previous discussion, the time τ mix for temperature b.c. is clearly much harder to measure empirically or to calculatetheoretically, because the boundary values of J ( x , t ) in space and time are unknown untilthe dynamical equations are solved. However, it is at least plausible that the turbulentRayeigh-B´enard system with imposed temperature difference ∆ T will behave very sim-ilarly to the system with imposed flux J in , if the latter is adjusted so that ∆ T remainsunchanged as ν, κ → . This must be the case for any result based solely upon the meanbalance equations in section 2.2, since these are identical for the two b.c. We expect thatit is likely that τ mix scales with physical parameters such as Ra, P r and Γ in an identicalmanner for constant-flux Rayeigh-B´enard convection and for the standard problem withtemperature-b.c. at the top and bottom walls. Needless to say, this is open to question.
4. Physical Implications of the Fluctuation-Dissipation Relation
We now discuss some of the physical implications of our Lagrangian fluctuation-dissipation relation for turbulent Rayleigh-B´enard convection.8
G. L . Eyink and T. D. Drivas
Mixing Time and Nusselt-Rayleigh Scaling
An immediate implication of the Lagrangian representations of the temperature differ-ence, (3.22) for flux b.c. and (3.45) for temperature b.c., is the important exact result τ mix = H ∆ T /J. (4.1)This may be written as τ mix = H/U flux , by defining a thermal flux velocity U flux = J/ ∆ T, (4.2)which is the equivalent fluid velocity required to maintain a flux J by convection of atemperature difference ∆ T . The definition of Nusselt number N u = J/ ( κ ∆ T /H ) implies
N u = τ diff τ mix , τ diff = H /κ, (4.3)where we now use τ diff to denote the near-wall mixing time for pure conduction (or,pure diffusion). The Nusselt number is seen to be just a ratio of the mixing time forpure conduction to the mixing time for convection. Thus, for large N u one has τ mix ≪ τ diff = H /κ. Likewise, from the definitions of
U, U ∗ , and U flux and eq.(2.23) it followsimmediately that U ∗ = U flux U = ⇒ U flux U = (cid:18) U ∗ U (cid:19) = ε T U (∆ T ) /H . (4.4)Hence, in the Spiegel (1971) scenario U flux ≃ U ∗ ≃ U (equality up to absolute constants)at high- Ra and all of the times τ mix , H/U ∗ , and H/U are of the same order. The theoryof Kraichnan (1962) assumed instead that the thermal boundary-layer would becometurbulent when u τ δ T /κ = P e T , where P e T is a critical or transitional value (estimatedto be about 3) of the P´eclet number based upon the friction velocity u τ at the top/bottomwalls and the thermal boundary-layer thickness δ T . Because of the standard relation J = κ ∆ T / δ T , which essentially defines δ T , Kraichnan’s assumption can be restated preciselyin our framework as the conjecture that U flux = u τ / P e T . Appeal to the standardlogarithmic law-of-the-wall led Kraichnan (1962) to conclude that u rms ∝ U/ ln( Ra ) and u τ ∝ U/ ln / ( Ra ) , so that these velocities differ only by a logarithmic factor. The theoryof Grossmann & Lohse (2012) likewise assumes turbulent boundary layers in an “ultimateregime” at very high Ra , but obtain the slightly different predictions that u rms ∝ U (see eq.(22) in Grossmann & Lohse (2011)) and u τ ∝ U/W ( Re ) , where W is Lambert’sfunction (eq.(9) in Grossmann & Lohse (2011)). As in the theory of Kraichnan (1962),the effective scaling exponents become indistinguishable from the dimensional predictionsof Spiegel at extremely high Rayleigh numbers. For any other possible scaling behaviorthan Kraichnan-Spiegel-type, U flux ≪ U ∗ ≪ U and τ mix ≫ H/U ∗ ≫ H/U, by factorsgrowing faster than logarithms of Ra .The most illuminating form of the relation τ mix = H ∆ T /J in (4.1) is obtained bycombining it with the result from the non-dimensionalized balance equations (2.18) that
J/U ∆ T = N u/ √ RaP r, that is, the ratio of the true Nusselt number and the Spiegelprediction. From this it is easy to see that τ mix = HU √ RaP rN u . (4.5)
Thus, τ mix differs from the free-fall time τ free = H/U by precisely the same factor that theSpiegel dimensional prediction differs from the true Nusselt number.
This is our key con-clusion for Rayleigh-B´enard turbulence. Recall that the free-fall velocity U is observedempirically to be of the same order as the velocity U lsc of the large-scale circulation luctuation-Dissipation Relation et al. τ free is roughly of the same order as the large-scale circulation time τ lsc = L/U lsc .For example, it follows directly from the definition of the free-fall velocity that the U -based Reynolds number is U H/ν = Ra / P r − / , whereas experiments show that U lsc H/ν scales very similarly in Ra , with an exponent only slightly smaller than 1 / et al. (2009), section IV) and nu-merical simulations give similar results (e.g. Scheel & Schumacher (2014), section 2). Asmentioned above, U lsc ≃ U is also predicted by Grossmann & Lohse (2012) at very high Ra.
Our key conclusion is thus nearly equivalent to the statement that existence of dis-sipative anomalies and validity of Spiegel scaling requires τ mix of order the large-scaleturnover time τ lsc , whereas for any other scaling τ mix ≫ τ free ≃ τ lsc (and, in fact, even τ mix ≫ H/U ∗ ) and a passive tracer released at one wall remains unmixed near both wallsfor many large-scale circulation times. It is a priori quite surprising that wall-incidencescould remain correlated over so many large-scale circulation times. This requires somediscussion of the underlying Lagrangian mechanisms.4.2. Stochastic Lagrangian Dynamics and the “Mixing Zone”
As we have argued in this paper and I,II, the extension of Lagrangian methods to realis-tic, non-ideal fluid flows naturally requires stochastic particle trajectories. These methodstake their simplest form, furthermore, when the stochastic trajectories are evolved back-ward in time (cf. our eq.(3.1)). The backward evolution may appear artificial at first sight,but it arises from the physical fact that a scalar such as temperature undergoing bothadvection and diffusion is an average of its past values rather than its future values. Herewe shall apply the stochastic Lagrangian framework to gain new insight into the thermaldissipation physics of turbulent Rayleight-B´enard convection. While forward stochas-tic particle evolution (or even deterministic evolution of a diffusive, passively-advectedtracer, as in eqs.(3.37)-(3.40)) can be substituted for the backward stochastic evolution,this involves considerable more complexity of the description and loss of insight. Thetime-asymmetry between forward and backward time-evolutions is a consequence of thefundamental irreversibility of the dissipation process. As we shall see below, one of theessential features of the Lagrangian description is this asymmetry in time.Thermal plumes play a well-known role in the Lagrangian dynamics of the tempera-ture field (for recent reviews, see Ahlers et al. (2009); Chill`a & Schumacher (2012) andreferences therein). In particular, plumes should transport thermal fluctuations from theboundary into the interior and create large values of h ε flucT ( x ) i t at points x within thebulk of the flow well way from the walls. If a plume reaches the interior point x withinthe time t, then going backward in time the particle is transported to very near the wallof origin of the plume, at which point the stochastic noise ∝ √ κ can then allow theparticle to hit the wall. Note that, since the fluid velocity u vanishes smoothly at thewall for ν, κ > , advection on its own can never produce a wall incidence in finite time.The non-vanishing of the boundary local time densities ˜ ℓ top/bott, ( x ) for points x in thebulk of the flow is thus presumably due largely to the mediation of the plumes when Ra ≫
1. In particular, one expects enhanced values of h ε flucT ( x ) i t at times t for which x is within a plume. Observe that enhanced values of the thermal dissipation ε T ( x , t ) itselfare observed within plumes (e.g. see Emran & Schumacher (2012)).However, to understand possible mechanisms that could give τ mix ≫ τ free it is moreimportant to understand the Lagrangian dynamics that could lead stochastic particles toescape from the top/bottom wall backward in time or, equivalently, to hit the top/bottomwalls forward in time. Note that τ hommix ≫ τ free means that a particle currently at thetop/bottom wall had a high probability to be at the same wall over a very long period0 G. L . Eyink and T. D. Drivas of earlier times (many free-fall times). Similarly, τ hetmix ≫ τ free means that a particlecurrently at the top/bottom wall had a very low probability to be at the opposite wallover a very long period of earlier times (many free-fall times). Thus, turbulent convectionmust not be very efficient at bringing particles close to the wall if τ mix ≫ τ free , else theparticles would readily escape backward in time. The obvious fluid motions which carrystochastic Lagrangian particles from interior points to points close to the wall consist ofthe large-scale circulation and “old” plumes which have completed a transit between topand bottom. These motions will carry particles between points near the top and bottomin a turnover time τ lsc , which is only slightly larger than τ free . While much previous workhas focused on the role of thermal plumes, the backward stochastic evolution reveals adirect connection of the thermal dissipation rate with these weaker fluid motions.The fact that τ mix ≫ τ free means that these motions must not typically bring theparticles close enough to actually hit the wall, and instead the interior particles willcirculate many, many free-fall times before finally hitting the top or bottom wall. Inorder to be guaranteed to hit the wall in time τ mix , the particles must be advected to adistance ℓ T = √ κτ mix from the top/bottom wall, since thermal diffusion then suffices tocarry the particle the remaining distance in time τ mix . We shall thus refer to ℓ T as the“thermal diffusion length,” over the mixing time τ mix . Using Eq.(4.3) gives ℓ T = √ κτ mix = H/ √ N u. (4.6)Consider any distance δ asymptotically much smaller than ℓ T for Ra ≫ , in particular,the traditional outer thermal boundary layer thickness δ T = H/ N u.
Our results implythat bulk fluid particles cannot reach such a distance δ from the top/bottom walls inany time less than τ mix , with τ mix ≫ τ free if KS-scaling fails. For example, the time forparticles at the top/bottom wall to diffuse the distance δ T across the standard thermalboundary layer is δ T /κ = τ mix / N u, which is much shorter than the time τ mix to transiteffectively the bulk.The thermal-diffusion length ℓ T appears to be a new length-scale, not previously iden-tified in Rayleigh-B´enard convection. The region within distance ℓ T of the top/bottomwalls but further away from those walls than δ T might possibly be identified with the“plume mixing zone” proposed by Castaing et al. (1989); see also Procaccia et al. (1991).The inner boundary of this “mixing zone” was considered to be δ T , while the outerboundary distance (denoted ℓ m or d m ) was predicted to scale as ℓ m ∼ H/Ra / . SinceCastaing et al. (1989) and Procaccia et al. (1991) also predicted
N u ∼ Ra / , this gives ℓ m ∼ H/ √ N u , consistent with our result (4.6) for the scaling of ℓ T . If we adopt the ter-minology of Castaing et al. (1989), then it is in the “central region”, at distances furtherthan ℓ T from the top/bottom walls, where particles must be “trapped” for many free-falltimes, if KS-scaling fails. Note, however, that Castaing et al. (1989) proposed the “mixingzone” to be a region with a small fraction of the volume occupied by thin “plumes” witha thickness δ T , whose large temperature differences relative to the surrounding fluid gavethem a strong vertical motion. In this “mixing zone” of thickness ℓ m (or d m ), the heatflux was considered to be carried mainly by this bouyant motion of “plumes”, whereas inthe thermal boundary layer the heat transport was purely conductive. By contrast, thethermal diffusion length ℓ T in our analysis is identified precisely by the condition thatdiffusion alone suffices to transport termperature fluctuations across the “mixing zone”over the (very long) time-scale τ mix . Thus, the agreement in scaling of ℓ m = d m and ℓ T with N u may be purely coincidental † . † Let us examine this issue in more detail.The plausible equations u rms ∼ ( αgHT rms ) / [(3.2)] and J ∼ u rms T rms [(3.3)] of Castaing et al. (1989) together with their proposed relation luctuation-Dissipation Relation δ T , their basic conjecture (see Grossmann & Lohse (2004),eq.(36)) is that the plume shedding frequency scales as f shed ∼ J/δ T (∆ T ) . (4.7)Combining this theorized result with our exact relation (4.1), it follows that τ mix ∼ ( N u ) τ shed , (4.8)where τ shed = 1 /f shed is the mean time-interval between shedding events. Since τ diff =( N u ) τ shed (e.g. see again Grossmann & Lohse (2004), eq.(36)), eq.(4.8) can be equiv-alently stated as τ mix ∼ √ τ shed τ diff . In either form, eq.(4.8) implies that τ mix ≫ τ shed for N u ≫ . To make connection with either the free-fall time or the large-scale circula-tion time within the approach of Grossmann & Lohse (2004), one must relate τ shed withthe times τ free = H/U or τ lsc = H/U lsc . This requires a relation between the Nusseltnumber and the Reynolds and P´eclet numbers. By appealing to Prandtl-Blasius laminarboundary layer theory, Grossmann & Lohse (2004) argued that
N u ∼ √ f Re P r, where f ( P r ) is a factor which interpolates between f = 1 for P r ≪ f ∼ P r − / for P r ≫ . Combining this relation with τ diff = ( N u ) τ shed , Grossmann & Lohse (2004)derived their eq.(37): τ shed ∼ H/f U lsc = τ lsc /f, (4.9)so that the shedding-time differs from large-scale circulation time only by the factor f. The final impication of the Grossmann & Lohse (2004) theory for the mixing-time is that τ mix ∼ N u τ lsc /f, (4.10)with τ mix larger than τ lsc by the factor N u/f.
Note that this conclusion is expected to beunchanged if the kinetic boundary becomes turbulent while the thermal boundary stayslaminar. Indeed, the scaling law
N u ∼ √ f Re P r is also characteristic of the “backgrounddominated” regime in Grossmann & Lohse (2011). In the “plume-dominated” regimethen instead
N u ∼ ( Re P r ) / (see eq.(11) of Grossmann & Lohse (2011)) which iseasily checked to imply that τ shed ∼ N u τ lsc and thus τ mix ∼ ( N u ) τ lsc . In that case,the mixing time is an even larger multiple of the large-scale circulation time.The prediction (4.10) of the Grossmann & Lohse (2004, 2011) hypotheses and our ex-act relation (4.5) is empirically testable, e.g. see the end of section 3.1.2, and section 4.4below. A direct investigation of (4.10) would illuminate the Lagrangian bases of the “uni-fying theory” of Grossmann & Lohse, which have not yet been subject to stringent test. δ T /ℓ m ∼ T rms / ∆ T ∼ Re γ [(3.18)] imply for a general scaling law Nu ∼ Ra β that γ = (2 β − / γ = − β/ ℓ m ∼ ℓ T is only satisfied for thechoice β = 2 / ℓ m ∼ H/ √ Nu in general. A possible way out of thisconclusion was indicated by Castaing et al. (1989), p.20: “To introduce a different scaling, onewould need to assume that the thermals fill only a vanishing fraction of the space, at large Ra .”Accordingly, one can modify their (3.3) to J ∼ u rms T rms f where the volume fraction f ∼ Ra − δ . If one leaves (3.2) and (3.18) of Castaing et al. (1989) unchanged, their (3.4) is replaced by ε = (1 + γ ) / , γ = (2 β + 2 δ − / , where ε is the Reynolds exponent in Re rms ∼ Ra ε . If oneimposes our condition γ = − β/ , one obtains δ = (2 − β ) / . In that case, δ < β > / β . = 0 . − .
43 at current highest Rayleigh numbers.One is led to an unreasonable conclusion that f should increase as a positive power of Ra ! Thus,either one of the assumptions (3.2), (3.3), or (3.18) of Castaing et al. (1989) is wrong (we findthe last the most dubious) or else the identification ℓ T ∼ ℓ m is generally invalid. G. L . Eyink and T. D. Drivas
In fact, it is far from clear to us how to reconcile the quantitative prediction (4.10) withthe more physical reasoning invoked by Grossmann & Lohse (2004) to justify the con-clusions (4.7), (4.9) cited above. For example, Grossmann & Lohse (2004) have equated τ shed with the “traveling time τ travel ∼ L/U f [or
H/U lsc f in our notations] of (hot)plumes from the bottom to the top”. Grossmann & Lohse (2004) also state that, withintheir boundary-layer analysis, the “temperature is assumed to be passive”. With thesetheoretical assumptions, it is not obvious why the mixing time τ mix of a passive scalarreleased uniformly at one wall should scale any differently than the above “travelingtime” τ travel ∼ τ lsc within the Grossmann & Lohse (2004) theory. The problem of ac-counting physically for the discrepancy in magnitudes of τ mix on the one hand and τ free or τ lsc on the other hand exists not only within the Grossmann-Lohse phenomenologybut whenever theory, experiment, or simulation indicates a Nusselt number N u muchsmaller than the Spiegel dimensional prediction √ Ra P r.
We shall discuss below somephysical mechanisms that might allow the ratio τ mix /τ free to become large.At extremely high Rayleigh numbers, the Grossmann-Lohse theory posits an “ulti-mate regime” in which both kinetic and thermal bounday layers transition to turbu-lence with logarithmic profiles. In that case, the thermal boundary layer is predicted byGrossmann & Lohse (2011) to permeate the cell and attain thickness δ T ∼ H . The rela-tion (4.7) which assumed a laminar thermal boundary layer then becomes inapplicable,and presumably must be replaced by f shed ∼ J/H (∆ T ) . (4.11)Together with our basic result (4.1), this leads to the identification τ mix ≃ τ shed . (4.12)Grossmann & Lohse (2011) predict in this regime that N u ∼ √
Ra P r L ( Re ) [see theireq.(23)], where L ( Re ) is a slowly-vanishing logarithmic factor. Together with our equation(4.5) this implies that τ mix ∼ τ free / L ( Re )so that the mixing time is predicted to be longer than the free-fall time only by the factor1 / L ( Re ) . Recall that τ free ≃ τ lsc in the Grossmann & Lohse (2011) theory, because ofthe cancellation of log-corrections in their eq.(22). This is an example of a Kraichnan-type theory in which τ mix is only logarithmically larger than τ free and τ lsc . If such anear-equality of time-scales indeed holds at extremely high Ra , then the Lagrangianmechanisms which make τ mix /τ free large at lower Ra must be somehow less effective.4.3. Criterion for an “Ultimate Regime”
Our considerations do not allow us to make a definite theoretical prediction for
N u as afunction of Ra and P r, so that we must consider briefly the observations. There seem tohave been relatively few empirical studies of turbulent Rayleigh-B´enard convection withwell-controlled flux-b.c. The 2D numerical study of Johnston & Doering (2009) exhibiteda scaling law
N u ∼ Ra x with non-KS exponent x . = 0 .
285 up to Ra = 10 . This studyalso found identical scaling with temperature-b.c. in the turbulent regime and suggestedthat the two types of boundary conditions would exhibit generally equivalent physicalbehavior in the turbulent regime also in 3D. A subsequent 3D simulation of Stevens et al. (2011) with temperature b.c at the top wall and flux-b.c. at the bottom wall providessome corroboration of this hypothesis. With this assumption, we may also compare withthe much larger body of work for temperature-b.c. In contrast to the empirical results forturbulent dissipation of passive scalars summarized by Donzis et al. (2005), experimental luctuation-Dissipation Relation show that thermal dissipationin turbulent Rayleigh-B´enard convection scales as ε T (∆ T ) U/H ∼ Ra z (4.13)with a negative exponent z . = − .
20 (Emran & Schumacher 2008, 2012; He et al.
N u ∼ Ra x with exponent x = 0 . z . = 0 . < / , which representsa 40% difference from the KS scaling exponent. Experiments at even higher Rayleighnumbers see somewhat larger exponents x . = 0 . .
43 but apparently still at least 10-20% smaller than the KS exponent x = 0 . et al. Ra ∼ (Niemela et al. et al. et al. Re ℓ ≡ u ( ℓ T ) ℓ T /ν as most appropriate to signalthe possible transition. Here u ( ℓ T ) is a typical (e.g. rms) velocity at distance ℓ T fromthe top/bottom walls. When the Reynolds number Re ℓ becomes critical, then turbu-lent mixing reaches to eddies at distance ℓ T from the top/bottom walls, and one mayplausibly expect a strong effect on the mixing rate and perhaps even that τ mix ∼ τ free , implying KS-scaling. This is similar to the arguments of Niemela & Sreenivasan (2003),Niemela & Sreenivasan (2006), and also Grossmann & Lohse (2011), He et al. (2012),but replacing the kinetic boundary layer thickness δ v = aH/ √ Re with the thermal diffu-sion length ℓ T = bH/ √ N u . Here we have introduced a dimensionless prefactor b of orderunity, whose precise value could be important at moderately large Rayleigh numbers. Itmust hold that b .
1, since the time to diffuse across the distance ℓ T should be somewhatsmaller than τ mix . A reasonable choice would be perhaps to set b = 1 by convention, sothat ℓ T / κ = τ mix . Based on the data published in Roche et al. (2010) and He et al. (2012), one can estimate Re ℓ = u ( ℓ T ) ℓ T /ν at the apparent onset of a new Nusselt-scalingin their experiments to be of the order of several hundreds, in the range where transitionto turbulence is expected.In order to compare our proposed criterion for an “ultimate regime” based upon thethermal diffusion length ℓ T with criteria based on the kinetic boundary-layer thickness δ v , one must know the relative magnitudes of these two lengths. If KS scaling holds, thenboth δ v and ℓ T scale with Rayleigh number as Ra − / (up to possible log-corrections),so that at fixed Prandtl number one may essentially identify δ v ∼ ℓ T . However, alternatetheories predict
N u ∼ Ra x with x < /
2, and the rigorous upper bound of Otero et al. (2002) for flux-b.c. requires x /
2. In that case, ℓ T /δ v ∼ ( b/a ) Ra (1 − x ) / (where weassume the approximate scaling Re ∼ Ra / ), and ℓ T > δ v at sufficiently high Rayleighnumbers. At lower Rayleigh numbers, one may have instead ℓ T < δ v if b < a. For example,assuming the predicted scaling exponent x = 1 / b/a = 1 /
10 at
P r = 1 wouldlead to ℓ T < δ v up to Ra = 10 . It is thus unclear whether our suggested criterion foran “ultimate regime” is more stringent or weaker than those based on standard shearReynolds numbers (Niemela & Sreenivasan (2003, 2006), Grossmann & Lohse (2011),He et al. (2012)). It also is not clear that an “ultimate regime” based on our criterionmust have KS-scaling. It is quite plausible that “trapping” of fluid particles will disappear4
G. L . Eyink and T. D. Drivas once bulk turbulence reaches down to the “mixing zone” at distance ℓ T from the wall, butthe effect may not be to make τ mix ∼ τ free . Instead, a critical value of Re ℓ may signalthe transition to an ultimate N u - Ra scaling which is asymptotically valid for Ra → ∞ ,but still with τ mix ≫ τ free .A possible explanation of deviations from KS-scaling even at arbitrarily large valuesof Ra is provided by the numerical results of Emran & Schumacher (2012), who foundthat the thermal plumes occupy a smaller volume of the flow (and are associated withdecreased thermal dissipation) for increasing Ra.
Thus, convective transport across thecell height is less efficient and larger τ mix values are required to achieve near-wall verticalmixing. However, the decrease of the volume fraction of plumes f pl observed in Fig. 6of Emran & Schumacher (2012) is rather weak, declining only from 75% to 65% in 3.5decades, roughly f pl ∼ Ra − . . If we assume that the “effective transport velocity” is U eff = U f pl , then from (4.5) and the empirical results cited above τ mix ∼ HU eff Ra . , (4.14)and still τ mix ≫ H/U eff or, equivalently, U flux ≪ U eff for Ra ≫ . Another possibleexplanation of the deviations from KS-scaling is weakness of the large-scale circulationor convective wind, so that U flux ≃ U lsc while U lsc ≪ U. Observations indicate, however,that this is a very slight effect at accessible Rayleigh numbers, too small to account forobserved deviations from KS-scaling. For example, the two empirical expressions for theplume shedding frequency in Niemela & Sreenivasan (2002) [see p.206 there, denoted ω p ]combine to give U lsc /U ≃ . P r − / Ra − . which implies a near equality U lsc ≃ U wellbeyond Ra = 10 . Likewise, the measurements of Re lsc in the experiment of Qiu & Tong(2002) reviewed by Ahlers et al. (2009) imply that U lsc /U ∼ Ra − . , and the numericalresults of Scheel & Schumacher (2014) on the Reynolds number Re rms imply an rms flowvelocity u rms /U ∼ Ra − . . The result of these consistent observations is that τ mix ∼ HU lsc Ra . − . , (4.15)and still τ mix ≫ τ lsc or U flux ≪ U lsc . Yet another potential explanation for a small ratio U flux /U ≪ U flux ≃ u τ of Kraichnan (1962). However,the result u τ /U ∼ A/ ln / ( Ra ) predicted by Kraichnan (1962) or the similar prediction u τ /U ∼ ¯ κ/W ( Re ) of Grossmann & Lohse (2011) show only a very weak (logarithmic)decrease with Ra and are not expected to become relevant until Ra & . None ofthese effects, alone or in combination, seem clearly sufficient to explain the large value τ mix ∼ ( H/U ) Ra . − . that is inferred from existing Nusselt-number measurements † .As a concrete illustration of the size of τ mix , consider the experimental results of He etal. (2012), which have been interpreted as evidence for transition to an ultimate regimewith KS-scaling. For the largest value Ra = 1 . × reported and with P r = 0 . N u = 5631 and (4.5) give τ mix /τ free = 5397. There is thus notonly an issue to account for the observed deviations from KS-scaling, but even to explainhow current empirical results can be consistent with our exact relation (4.5)! The veryslow mixing rate inferred for all current experiments and simulations by the observeddeviations from Kraichnan-Spiegel scaling remains to be explained.One fact which may be relevant is that our long-time FDR (3.19) as derived in Ap-pendix A is only valid for times t ≫ τ mix . If τ mix ≫ τ free , as evidence suggests, then † However, with different thresholds in defining “plumes”, Emran & Schumacher (2012) findan f pl which is smaller and declines faster. For example, with the parameter δ = 1 of that papertheir Fig. 7 shows a decline of f pl from 50% to 20% over 3.5 decades or f pl ∼ Ra − . . luctuation-Dissipation Relation ‡ . Notethat the possibility raised here is distinct from the breakdown of time-ergodicity ofthe turbulent steady-state, which was suggested on general grounds by Frisch (1986)and later supported by experimental observations of a turbulent Taylor-Couette flow(Huisman et al. t ≫ τ mix ≫ τ free required for time-averagesto converge to their unique steady-state value may be much longer than the time-seriesavailable in typical experimental or numerical studies. Although infinite-time ergodicitywould hold, the consequences would be similar to ergodicity-breaking, with averages overavailable time-series exhibiting multiple values depending upon precise initial conditionsand experimental details, such as the relative sizes of t/τ mix and Ra.
Measuring the Near-Wall Mixing Time
All of these issues would be greatly illuminated by direct experimental and numericalmeasurements of τ mix at currently achievable Rayleigh numbers using passive tracers.We have already discussed how this may be done, in principle, both for flux b.c. [seeeqs.(3.37)-(3.40)] and for temperature b.c. [see eqs.(3.50)-(3.51)]. However, the methodsdiscussed previously are unwieldy, because a stream of tracers must be released contin-uously at one wall (top/bottom) with the tracer released at each time s distinguishablefrom those released at other times s ′ . This requirement will greatly complicate the designof any possible experiment.An alternative method to measure the mixing-time follows from the observation thatthe transition probability densities which appear in the definitions of τ mix satisfy theKolmogorov equation( ∂ t ′ + u ( x ′ , t ′ ) · ∇ ′ x ) p ( x ′ , t ′ | x , t ) = − κ △ ′ x p ( x ′ , t ′ | x , t ) , t ′ < t. (4.16)in the variables x ′ , t ′ (e.g. see Risken (2012), section 4.7). Thus, solving the backwardadvection-diffusion equation for the concentration of a passive tracer( ∂ t ′ + u ( x ′ , t ′ ) · ∇ ′ x ) c ( x ′ , t ′ ) = − κ △ ′ x c ( x ′ , t ′ ) (4.17)with a delta-sheet initial condition for the scalar at either the top or bottom wall c ( x ,
0) = (1 /A ) δ ( z − λ ′ H/ , (4.18) ‡ This might be the case if there is no finite-time singularity for the ideal Boussinesq equations,but if singularities do appear in the opposite limit t → ∞ first and then ν, κ → ν, κ → t → ∞ without a finite-time dissipative anomaly for the kinetic energy. This can beseen from eq.(2.12) for mean energy balance over a finite time t, which shows that, if there is noanomaly at finite time as ν, κ →
0, then αgJ = h ∂ t ( u ) i V and the space-average kinetic energymust grow linearly in time for fixed J . Of course, there could still be a dissipative anomaly inthe statistical steady-state obtained in the limit first t → ∞ and then ν, κ →
0, because thetwo limits need not commute. Indeed, we know from the steady-state relations (2.21) that whenheat flux J is fixed as ν, κ → , then ε u must remain positive. This is not conclusive, however,because it is possible that ∆ T → ∞ and U → ∞ in this limit, in which case the dimensionlesskinetic energy dissipation ˆ ε u defined in (2.22) tends to zero. In this case one expects also that U rms → ∞ , so that the steady-state kinetic energy becomes infinite as ν, κ → G. L . Eyink and T. D. Drivas one gets p z ( λH/ , s | λ ′ H/ ,
0) =
Z Z S dx ′ dy ′ c ( x ′ , y ′ , λH/ , s ) . Thus, the time τ λλ ′ mix defined by (3.33) for flux b.c. is the integral-time for mixing to auniform value 1 /H of the integrated mass-density observed at the wall z = λH/ z = λ ′ H/ . A similar interpretationis possible for temperature b.c, but now the tracer satisfying the backward equation(4.17) is released as a point-mass on the wall z = λ ′ H/ τ λλ ′ mix defined in (3.49) is J -weighted. The great advantage of this alternative formulation,is that a single type of tracer may be released at at single instant of time 0 , and not astream of distinguishable tracers continuously in time. The only price to be paid is thatthe experiment must be run backward in time!Since it is obviously impossible to run a laboratory experiment backward in time, thisalternative formulation appears to have dubious merit. However, the forward-in-timeversion of this experiment appears far more feasible. In such an experiment the tracerwould be released as a thin, uniform sheet at one wall, say, z = + H/ , at time 0 and thenits concentration at both the walls z = ± H/ τ ++ mix and ˜ τ + − mix for forward mixing of the tracer could be estimated. This experiment hasan attractive feature that only observations at the top/bottom walls are required, andno observations are needed from internal probes that might alter the fluid motions. Ofcourse, it is not clear that the forward mixing times ˜ τ ++ mix , ˜ τ + − mix measured in this mannerare the same as the backward mixing times τ ++ mix , τ + − mix that are rigorously connected byour FDR’s to the thermal dissipation rate. In particular, forward-in-time mixing awayfrom the wall will presumably be dominated by new thermal plumes, while backward-in-time mixing will be dominated by “old plumes” and the large-scale-circulation. Thus, itwill be no surprise if these times are quantitatively different. However, it seems reasonableto conjecture that the forward and backward mixing times will scale in the same mannerwith Ra, P r, and Γ , even if the prefactors for the scaling laws are distinct.Numerical simulations can directly study the backward-in-time mixing as describedby eqs.(4.17)-(4.18) above. One approach would be to use standard PDE methods suchas finite-difference or finite-element schemes to solve the backward advection-diffusionequation (4.17). This may be difficult, however, because of the very singular initial con-dition (4.18) required for the tracer concentration. An easier approach is probably touse a numerical Monte Carlo method with stochastic Lagrangian particles satisfying thebackward It¯o equation (3.1) with reflection at the boundary. As discussed in Paper II,numerical methods exist to implement these equations. By releasing a large number ofsuch particles uniformly distributed across the top or bottom wall, one could then esti-mate the position probability densities backward-in-time required in our FDR. In eitherapproach, one must have available the solution of the Rayleigh-B´enard equations over along interval of past time. Because of the parabolic character of the Bousinesq equations,it is not possible to solve them backward in time. One option, of the type exploited by usin Papers I and II, is to utilize a computer database storing an entire Rayleigh-B´enardsolution in space-time. Storing a full space-time history of a simulation is, of course, verycostly, especially over the very long times (many turnover times) required. Alternatively,one might store the solution frames only at a discrete set of times t k or “checkpoints”,separated by some relatively large time-interval ∆ t, and then use these frames to recon-struct the solution as needed over each interval [ t k , t k +1 ] by forward integration of the luctuation-Dissipation Relation t k as initial data † . Of course, conventional nu-merical simulations of Rayleigh-B´enard convection can also measure the forward mixingtimes in the same manner as a laboratory experiment.
5. Summary and Discussion
In this paper, we have worked out in detail the consequences of our Lagrangianfluctuation-dissipation relation for turbulent Rayleigh-B´enard convection. We have shown,in particular, that the thermal dissipation rate is controlled by a near-wall mixing timeof stochastic Lagrangian particles (or, equivalently, of a passive tracer concentration)due to the stirring by convective motions. Nevertheless, we have shown that this mixingtime at large Rayleigh numbers must be orders of magnitude longer than the large-eddy circulation time, if Kraichnan-Spiegel scaling is invalid. One possibility is that themixing time will become proportional to the turnover time in an “ultimate regime” ofvery high Rayleigh number, at which point Spiegel dimensional scaling would necessarilybecome valid. This transition could plausibly occur when bulk turbulence reaches a dis-tance ℓ T = H/ √ N u from the top/bottom walls and tracer concentration can be rapidlyadvected into and out of this “mixing zone.” However, even in the range of Rayleigh num-bers that is currently accessible to empirical study, there is an outstanding issue of howto account for the very long mixing times required by available Nusselt measurements.We have considered possible explanations, such as decreasing volume-fraction of plumesor decreased convective wind-speed relative to the free-fall velocity, as the Rayleigh num-ber is increased. However, none of these effects is obviously large enough to explain theobserved departures of
N u from Kraichnan-Spiegel predictions. It is even possible thatour theoretical relations do not apply to existing observations from experiments andsimulations, because these studies do not average over the long interval of times (manymixing times) that are necessary in order for our exact steady-state relations to apply.In this situation, we believe that a programme to measure the mixing time empiri-cally would be very valuable. We have outlined methods to do so, both by laboratoryexperiment and numerical simulation. A direct measurement would determine whetherour exact steady-state relations apply and, thus, whether experiments and simulationsare averaging over long enough time-intervals to observe a steady-state regime. Suchempirical studies would also be able to illuminate the Lagrangian mechanisms of mix-ing, for example, by applying flow-visualization techniques to learn how the tracer getstransported by thermal plumes and trapped by the large-scale circulation in the bulk.Further theoretical studies may also be useful. The “mixing rate” that we consider in thiswork is similar to that studied in the previous papers of Chertkov & Lebedev (2003) andLebedev & Turitsyn (2004), which considered the rate of homogenization of a passivescalar in the near-wall regions of a turbulent flow. Those studies averaged over ensemblesof velocities with a very short time-correlation and also assumed that homogenizationoccurs first away from the walls, which may not be true for a tracer released at the walland remaining near the wall for long times. However, it would be interesting if some ofthese earlier theoretical analyses could be modified to give information about the relevantscalar mixing time that appears in our exact relation.We would like to emphasize the universality of our conclusions. Our fluctuation-dissipation relation (3.52) requires only flow incompressibility and is otherwise indepen-dent of the velocity field. The relation holds in the same form with boundary conditionson velocity distinct from stick. b.c., with equations for velocity other than incompressible † We thank David Goluskin for suggesting this possibility G. L . Eyink and T. D. Drivas
Navier-Stokes, and even with synthetic velocity fields satisfying no dynamical equationat all. The basic formula (4.1) for τ mix in terms of J and ∆ T is derived using only (3.52)and the steady-state Eulerian balance relation (2.16) for temperature fluctuations. Itthus likewise holds for any advecting, divergence-free velocity. Results like (4.5) relating τ mix to τ free also hold in general, with an appropriate analogue of “free-fall velocity”.We give here brief details of several concrete examples:
1. Rayleigh-B´enard convection with free-slip velocity . The set-up is the same as standardRayleigh-B´enard described by the equations (2.1)-(2.6) but with the stick b.c. (2.4) on thevelocity replaced by the free-slip (no-stress) conditions ˆ n · u = 0 , ∂ u T /∂n = 0 where u T = u − (ˆ n · u )ˆ n . This problem has been studied analytically in 2D by Whitehead & Doering(2011), who derived an upper bound on Nusselt number
N u . Ra / , uniform in P r < ∞ . With this estimate, our equation (4.5) gives the rigorous bound τ mix /τ free > . Ra / P r / . (5.1)For this problem, therefore, the scalar mixing time at the walls becomes unboundedlylarge compared with the free-fall time as Ra → ∞ . There must be some mechanism (e.g.decreasing volume and intensity of thermal plumes, weakening large-scale circulation,etc.) to explain the inefficiency of scalar mixing near the walls.
2. Convection in a porous medium.
This problem is the same as standard Rayleigh-B´enard described by the equations (2.1)-(2.6) but with the incompressible Navier-Stokesequation (2.1) replaced with Darcy’s law: u = Kν ( −∇ p + αgT ˆ z ) , where K is the permeability coefficient with dimension of (length) . For example, seeOtero et al. (2004). In this case the Rayleigh number is defined as Ra = αgKH (∆ T ) /νκ and the free-fall velocity as U = αg (∆ T ) K/ν.
The dimensionless Eulerian balance equa-tions for kinetic energy and temperature fluctuations that replace (2.18) are ε u P r U /H = ε T (∆ T ) U/H = N uRa , (5.2)and, with τ free = H/U, our Lagrangian relation (4.1) implies that τ mix = τ free RaN u , (5.3)replacing (4.5). The dimensional prediction
N u ∼ Ra by Howard (1966) holds if andonly if there are dissipative anomalies for kinetic energy and temperature fluctuations.Otero et al. (2004) demonstrated the upper bound N u . × Ra, which with equa-tion (5.3) implies that τ mix > (33 . τ free . The numerical results of Hewitt et al. (2012)for Ra × are consistent with the prediction Ra ∼ N u, which, if true, implies that τ mix ≃ τ free and that near-wall scalar mixing is efficient in porous medium convection.
3. Optimal scalar transport with enstrophy constraint.
The papers of Hassanzadeh et al. (2014) and Tobasco & Doering (2016) have studied synthetic advecting velocity fieldsthat are constructed to maximize
N u with an enstrophy constraint of the form h|∇ u | i V = ( P e ) /τ diff . (5.4)where P e , the “Peclet number”, is any specified constant. There is some arbitrarinesshere in the definition of the “free velocity” U , but a natural choice is made by imposing P r U H = ν h|∇ u | i V , (5.5) luctuation-Dissipation Relation U = ( κ/H ) P e / . For this choice of U and τ free = H/U, our Lagrangianrelation (4.1) gives τ mix = τ free ( P e ) / N u . (5.6)A rigorous upper bound
N u c ( P e ) / has been proved for all divergence-free advectingvelocity fields satisfying (5.4) as a constraint (Souza 2016). Furthermore, Tobasco & Doering(2016) have constructed an “optimal transport field” u ∗ that gives N u & P e / / (log P e ) / , differing from the upper bound by a logarithmic factor only. By (5.6) τ mix for this opti-mal field is larger than τ free by at most a logarithm of P e and near-wall mixing occurswith almost maximal efficiency as
P e → ∞ .In each of these examples † just as in standard Rayleigh-B´enard convection, our La-grangian fluctuation-dissipation relation (3.52) is valid and relates the near-wall scalarmixing time τ mix and an appropriate “free time” τ free . When dissipative anomalies existas ν, κ → , then dimensional scaling of N u holds and τ mix ≃ τ free . If instead
N u scales non-dimensionally, then necessarily τ mix ≫ τ free as ν, κ → , signalling very slownear-wall scalar mixing. As the above examples make clear, our relation is consistentwith both scenarios and by itself cannot decide between them. While we can offer nofinal resolution of the puzzles raised here, we believe that our approach will have greatvalue for further theoretical and empirical investigations of turbulent convection, becauseit provides an exact connection between the thermal dissipation rate and the underly-ing Lagrangian fluid mechanisms. In general, for all wall-bounded turbulent flows, ourfluctuation-dissipation relation can be exploited to provide an exact link between scalardissipation and Lagrangian fluid mechanisms. † Another example of a quite different type is homogeneous Rayleigh-B´enard convection .This problem is described by the Boussinesq equations (2.1)-(2.3) in a cubic domain but withperiodic boundary conditions replacing (2.4)-(2.6) and with the temperature equation (2.2)replaced with ∂ t T + u · ∇ T = κ △ T + w ∆ TH , where w is the vertical fluid velocity and − ∆ T /H isan imposed vertical temperature gradient. For example, see Lohse & Toschi (2003). Assumingthat a long-time steady-state exists (which is a delicate assumption in this problem, e.g. seestudies of Calzavarini et al. (2006); Schmidt et al. (2012)), then the Eulerian mean balanceequations (2.15)-(2.16) hold just as in standard Rayleigh-B´enard convection. Then, existenceof kinetic energy and thermal dissipative anomalies is equivalent to Spiegel dimensional scalingof Nu with Ra, P r.
Our relation (3.52) involving the local times at the cell wall does not,of course, apply. However, another version of our Lagrangian fluctuation-dissipation relationapplies in a periodic domain: see (I;2.15) of paper I. When this appropriate relation is appliedto homogeneous Rayleigh-B´enard convection, it yields the result for thermal dissipation ratethat ε T = κ T (∆ T /H ) with (vertical) “turbulent diffusivity” κ T given by a Taylor-like formula: κ T = R −∞ dt (cid:10) h ˜ w L (0) ˜ w L ( t ) i ⊤ E,V (cid:11) ∞ . Here ˜ w L ( x , t ) = w (˜ ξ ,t ( x ) , t ) is the (vertical) Lagrangianvelocity sampled backward in time along stochastic trajectories ˜ ξ ,t ( x ) satisfying the analogueof eq.(3.1), h·i ⊤ E,V is the truncated 2-time correlation averaged over Brownian motions andcell volume, and h·i ∞ is an infinite-time average. Thus, the condition of anomalous thermaldissipation, ε T ≃ (∆ T ) U/H, will hold if and only if κ T ≃ UH.
In general, one can write κ T = U τ corr , where τ corr is an integral correlation-time of the stochastic Lagrangian velocity˜ w L ( t ) . Then a thermal dissipative anomaly exists if and only if τ corr ≃ τ free as ν, κ → , andotherwise τ corr ≪ τ free . Numerical results of Lohse & Toschi (2003) and Schmidt et al. (2012)seem to support Spiegel dimensional scaling in this homogeneous situation. G. L . Eyink and T. D. Drivas
Acknowledgements
We would like to thank Charlie Doering and David Goluskin for useful discussions, andthe comments of several anonymous reviewers that have helped to improve the paper.We would also like to thank the Institute for Pure and Applied Mathematics (IPAM) atUCLA, where this paper was partially completed during the fall 2014 long program on“Mathematics of Turbulence”. G.E. is partially supported by a grant from NSF CBET-1507469 and T.D. was partially supported by the Duncan Fund and a Fink Award fromthe Department of Applied Mathematics & Statistics at the Johns Hopkins University.
Appendix A. Steady-State FDR for Turbulent Convection
We here derive (3.19) and (3.44) in the main text. Various ergodicity assumptions arerequired in the argument, both for the stochastic Lagrangian flow in physical space andfor the Boussinesq fluid system in phase-space. These assumptions are carefully statedwhere required, but not proved a priori . We give full details only for the relation (3.19)with heat-flux b.c., because the derivation of (3.44) with temperature b.c. is very similar.The derivation is very similar to that given in paper I, Appendix B.1 for the steady-stateFDR in a periodic domain, with no wall but with an interior scalar source. The localtime density at the wall in the present case plays a role similar to the scalar source there.The starting point of our argument islim t →∞ t Cov h ˜ ℓ λt, ( x ) , ˜ ℓ λ ′ t, ( x ) i = lim t →∞ t Z t ds Z t ds ′ × h p z ( λH/ , s ; λ ′ H/ , s ′ | x , t ) − p z ( λH/ , s | x , t ) p z ( λ ′ H/ , s ′ | x , t ) i , (A 1)which is a restatement of (3.15). First, we use symmetry of the integrand in ( s, λ ) , ( s ′ , λ ′ )to restrict the time-integration range to s < s ′ < t and then use the Markov property ofthe backward stochastic flow to write p z ( λH/ , s ; z ′ , s ′ | x , t ) = Z Z S dx ′ dy ′ p z ( λH/ , s | x ′ , y ′ , z ′ , s ′ ) p ( x ′ , y ′ , z ′ , s ′ | x , t ) , (A 2)givinglim t →∞ t Cov h ˜ ℓ λt, ( x ) , ˜ ℓ λ ′ t, ( x ) i = lim t →∞ t Z t ds Z ts ds ′ Z Z S dx ′ dy ′ × h(cid:16) p z ( λH/ , s | x ′ , y ′ , λ ′ H/ , s ′ ) − p z ( λH/ , s | x , t ) (cid:17) p ( x ′ , y ′ , λ ′ H/ , s ′ | x , t ) i λ,λ ′ , (A 3)where (cid:2) A λλ ′ (cid:3) λλ ′ = (cid:0) A λλ ′ + A λ ′ λ (cid:1) denotes the symmetrization with respect to λ, λ ′ .Next, we divide the triangular region R = { ( s, s ′ ) : 0 < s < s ′ < t } into three subregions: R I = { ( s, s ′ ) : 0 < s < s ′ < t − nτ } R II = { ( s, s ′ ) : 0 < s < t − nτ, t − nτ < s ′ < t } R III = R \ ( R I ∪ R II ) (A 4)where τ is the scalar mixing time (essentially the same as τ mix ) and n is a positive integer.Region R III gives a contribution which is O ( n τ /t ) and can be neglected in the limit luctuation-Dissipation Relation t → ∞ . In Region R II one has both t − s > nτ and s ′ − s > nτ, so that as n → ∞ onemay use the ergodicity of the Lagrangian flow in physical space to obtain p z ( λH/ , s | x ′ , y ′ , λ ′ H/ , s ′ ) → H , p z ( λH/ , s | x , t ) → H , (A 5)for λ, λ ′ = ± , which give canceling contributions in (A 3). Thus, this region makesan arbitrarily small contribution for sufficiently large n. Finally, in region R I one has t − s > t − s ′ > nτ, so that again by ergodicity of the Lagrangian flow as n → ∞ p z ( λH/ , s | x , t ) → H , p ( x ′ , y ′ , λ ′ H/ , s ′ | x , t ) → AH . (A 6)Taking the limit t → ∞ and using the independence of the limit value upon n one obtainslim t →∞ t Cov h ˜ ℓ λt, ( x ) , ˜ ℓ λ ′ t, ( x ) i = 1 H lim t →∞ t Z t ds Z ts ds ′ h p z ( λH/ , s | λ ′ H/ , s ′ ) − H i λ,λ ′ (A 7)Finally, to simplify still further, change the order of the integrals over s, s ′ and thenset s → s − s ′ to obtainlim t →∞ t Cov h ˜ ℓ λt, ( x ) , ˜ ℓ λ ′ t, ( x ) i = 1 H lim t →∞ t Z t ds ′ Z s ′ ds h p z ( λH/ , s | λ ′ H/ , s ′ ) − H i λ,λ ′ = 1 H lim t →∞ t Z t ds ′ Z − s ′ ds h p z ( λH/ , s ′ + s | λ ′ H/ , s ′ ) − H i λ,λ ′ = 1 H lim s ′ →∞ Z − s ′ ds h p z ( λH/ , s ′ + s | λ ′ H/ , s ′ ) − H i λ,λ ′ = 1 H lim s ′ →∞ Z − s ′ ds h p z ( λH/ , s ′′ + s | λ ′ H/ , s ′′ ) − H i λ,λ ′ (A 8)for any s ′′ , since the integral should be the same for any pair of times s ′′ + s, s ′′ replacing s ′ + s, s ′ in the transition probability. The underlying assumption here is that this integralover an infinitely long time interval (in the past) should be independent of the state atthe present time. This is an assumption about the ergodicity of the Eulerian dynamicsdefined by the Boussinesq equation, i.e. the existence of a unique statistical steady-stateobtained by an infinite time-average. Under this hypothesis, one can take in particular s ′′ = 0 to obtain thatlim t →∞ t Cov h ˜ ℓ λt, ( x ) , ˜ ℓ λ ′ t, ( x ) i = 1 H Z −∞ ds h p z ( λH/ , s | λ ′ H/ , − H i λ,λ ′ , (A 9)as was claimed in (3.19). The lower limit s = −∞ in the time-integral must be taken inthe Ces´aro mean sense, as shown by the second line of (A 8). Appendix B. Steady-State FDR for Pure Conduction
Pure thermal conduction is interesting as a “control experiment” to assess the effectsof fluid advection. Heat conduction in a right cylindrical cell with arbitrary cross-sectionand no-flux conditions on the sidewalls reduces to a 1D problem. Therefore, without lossof generality, we consider the Neumann problem for the heat equation on the interval oflength H. In order to simplify the analysis (so that cosine series suffice), we consider not2
G. L . Eyink and T. D. Drivas the symmetric interval [ − H/ , H/
2] but instead [0 , H ] . The problem considered is thusprecisely the same as that treated in Appendix A of Paper II: ∂ t T = κ∂ x T for x ∈ [0 , H ] κ∂ x T = − J at x = 0 , H (B 1) T = 0 at t = 0 . In Paper II we considered the limit of κ → t fixed. We here consider theopposite limit of t → ∞ with κ fixed, appropriate to steady-state conduction.The temperature field will be obtained from eq. (3.3) in the main text. The transitionprobability density for a Brownian motion in the interval [0 , H ] that is reflected at theendpoints is given by the cosine series in eq.(A3) of paper II [hereafter eq.(II:A3)]. Usingthis series, the large- t asymptotics of the mean local times appearing in (3.3) are − E h ˜ ℓ σHt, ( x ) i = tH + 2 Hκπ ∞ X n =1 ( − σH cos (cid:0) nπH x (cid:1) n (cid:16) − e − κ ( nπH ) t (cid:17) ∼ tH + 2 Hκπ ∞ X n =1 ( − σH cos (cid:0) nπH x (cid:1) n , t ≫ H /κ (B 2)The above cosine series can be computed by standard methods as: − E h ˜ ℓ t, ( x ) i = tH + H κ (cid:20) (cid:16) xH (cid:17) − (cid:16) xH (cid:17) + 2 (cid:21) , − E h ˜ ℓ Ht, ( x ) i = tH + H κ (cid:20) (cid:16) xH (cid:17) − (cid:21) . With these, one obtains using the representation (3.3) of the temperature field: T ( x, t ) = − Jκ (cid:18) x − H (cid:19) as t → ∞ . (B 3)This is indeed the exact steady-state solution of (B 1) (unique up to an arbitrary con-stant). From this one gets the thermal dissipation field h ε T ( x ) i ∞ ≡ h κ | ∂ x T | i ∞ = J κ , (B 4)pointwise in x and averaged over a time-interval [0 , t ] with t ≫ H /κ .We next evaluate the mean scalar fluctuation variance in the limit t → ∞ . For thispurpose, we could use the general infinite-time limit formula (3.19) that is derived in Ap-pendix A. Here we shall instead proceed directly from eq.(A 3) in the previous subsection.After minor manipulation, this formula becomes here: J t Var h J (cid:16) ˜ ℓ Lt, ( x ) − ˜ ℓ Rt, ( x ) (cid:17)i = J t Z t ds Z ts ds ′ n (cid:18) H (cid:19) h δp (0 , s | , s ′ ) − δp (0 , s | H, s ′ ) i + h δp ( H, s | H, s ′ ) − δp ( H, s | x, t ) i δp ( H, s ′ | x, t ) + h δp (0 , s | , s ′ ) − δp (0 , s | x, t ) i δp (0 , s ′ | x, t ) − h δp ( H, s | , s ′ ) − δp ( H, s | x, t ) i δp (0 , s ′ | x, t ) − h δp (0 , s | H, s ′ ) − δp (0 , s | x, t ) i δp ( H, s ′ | x, t ) o where δp ( a, s | x, t ) = p ( a, s | x, t ) − /H was introduced to eliminate the common contri-butions arising from the 1 /H term in the transition densities. All terms in the aboveformula which are x -dependent vanish in the infinite-time limit. To see this explicitly, fortimes t ≫ H /κ we obtain by substituting the cosine series (II:A3) that Z t ds Z ts ds ′ δp ( σH, s | σ ′ H, s ′ ) δp ( σ ′ H, s ′ | x, t ) luctuation-Dissipation Relation −→ (cid:18) π (cid:19) H (2 κ ) ∞ X n,m =1 ( − ( σ + σ ′ ) m + σ ′ n cos (cid:0) nπH x (cid:1) n m (B 5) Z t ds Z ts ds ′ δp ( σH, s | x, s ′ ) δp ( σ ′ H, s ′ | x, t ) −→ (cid:18) π (cid:19) H (2 κ ) ∞ X n,m =1 ( − σm + σ ′ n cos (cid:0) nπH x (cid:1) cos (cid:0) mπH x (cid:1) n m . (B 6)For all combinations of σ, σ ′ = 0 ,
1, these sums are absolutely convergent and can becalculated by the same means as the average local times. However, the important pointis that they are independent of time and therefore their contribution to the total variancevanishes like 1 /t . The only non-vanishing contribution to h ε flucT ( x ) i ∞ arises from the first x -independent term, which is identical to the general infinite-time result (3.19). Thereare both homohedral and heterohedral contributions, which may be calculated again bysubstituting the cosine series (II:A3) and performing the time-integrals to obtain for t ≫ H /κ ( − σ H Z t ds Z ts ds ′ δp (0 , s | σH, s ′ ) ∼ tκ (cid:18) π (cid:19) ∞ X n =1 ( − σ ( n +1) n = t κ × ( σ = 01 σ = 1In terms of the homohedral and heterohedral correlation times as defined by (3.33) , ourresult states: τ hommix = 23 H κ , τ hetmix = 13 H κ (B 7)in agreement with the general inequality (3.36) discussed in Section 4 of the main text.Finally, the scalar variance is calculated as h ε flucT ( x ) i ∞ = J H (cid:0) τ hommix + τ hetmix (cid:1) = J κ , (B 8)As dictated by the general result (3.19), the infinite-time fluctuational dissipation (B 8)for all points x ∈ [0 , H ] equals the space-average thermal dissipation (B 4) calculatedfrom the exact steady-state solution.The thermal dissipation rate is here positive, and in fact diverges, as κ → . Asdiscussed in Section 2.3, this is not an anomaly in the true sense, but is simply anartifact of keeping J fixed as κ →
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