2.1 Two Rayleigh–Bénard problems

We begin with the precise statement of the standard Rayleigh–Bénard problem. The most 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 top and bottom and insulating side walls (Bergé & Dubois Reference Bergé and Dubois1984; Grossmann & Lohse Reference Grossmann and Lohse2000; Ahlers et al. Reference Ahlers, Grossmann and Lohse2009; Chillà & Schumacher Reference Chillà and Schumacher2012):

(2.1) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x2202}_{t}\boldsymbol{u}+\boldsymbol{u}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u}=-\unicode[STIX]{x1D735}p+\unicode[STIX]{x1D708}\unicode[STIX]{x0394}\boldsymbol{u}+\unicode[STIX]{x1D6FC}gT\hat{\mathbf{z}}, & \displaystyle\end{eqnarray}$$

(2.2) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x2202}_{t}T+\boldsymbol{u}\boldsymbol{\cdot }\unicode[STIX]{x1D735}T=\unicode[STIX]{x1D705}\unicode[STIX]{x0394}T, & \displaystyle\end{eqnarray}$$

(2.3) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}=0, & \displaystyle\end{eqnarray}$$

with boundary conditions

(2.4) $$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{u}=0\quad \text{no slip at top, bottom and side walls}, & \displaystyle\end{eqnarray}$$

(2.5) $$\begin{eqnarray}\displaystyle & \displaystyle T|_{z=\pm H/2}=T_{top/bot}\quad \text{isothermal top/bottom walls}, & \displaystyle\end{eqnarray}$$

(2.6) $$\begin{eqnarray}\displaystyle & \displaystyle \hat{\boldsymbol{n}}\boldsymbol{\cdot }\unicode[STIX]{x1D735}T=0\quad \text{insulating/adiabatic side walls}, & \displaystyle\end{eqnarray}$$

where $T_{top}<T_{bot}$ are imposed space–time constant values which lead to convective instability. This standard Rayleigh–Bénard problem corresponds, mathematically, to a system with mixed Neumann and Dirichlet boundary conditions for the temperature, which is an active scalar. In the above equations, $\unicode[STIX]{x1D708}$ is the kinematic viscosity of the fluid, $\unicode[STIX]{x1D705}$ the thermal diffusivity, $g$ the acceleration due to gravity and the constant $\unicode[STIX]{x1D6FC}$ is the isobaric thermal expansion coefficient. Three dimensionless combinations of parameters characterize the system: the Rayleigh number, Prandtl number and aspect ratio, defined respectively by

(2.7a-c ) $$\begin{eqnarray}Ra=\frac{\unicode[STIX]{x1D6FC}gH^{3}\unicode[STIX]{x0394}T}{\unicode[STIX]{x1D705}\unicode[STIX]{x1D708}},\quad Pr=\frac{\unicode[STIX]{x1D708}}{\unicode[STIX]{x1D705}},\quad \unicode[STIX]{x1D6E4}=\frac{D}{H},\end{eqnarray}$$

where $\unicode[STIX]{x0394}T=T_{bot}-T_{top}$ and $D=\text{diam}(S)$ is the diameter of the cell cross-section. A major question of interest is the heat transport across the cell, quantified by the vertical heat flux averaged over volume $V$ and finite time interval $[0,t]$ :

(2.8) $$\begin{eqnarray}J\equiv \langle u_{z}T-\unicode[STIX]{x1D705}\unicode[STIX]{x2202}_{z}T\rangle _{V,t}.\end{eqnarray}$$

This is usually expressed as the dimensionless Nusselt number, which is the ratio between the total heat flux and the flux due to thermal conduction:

(2.9) $$\begin{eqnarray}Nu=\frac{J}{\unicode[STIX]{x1D705}\unicode[STIX]{x0394}T/H}.\end{eqnarray}$$

The object of many studies has been to determine the functional dependence of $Nu$ upon the dimensionless parameters $Ra,$ $Pr$ and $\unicode[STIX]{x1D6E4}$ of the problem.

An alternative problem is Rayleigh–Bénard convection with imposed heat flux at the boundary, which has been previously studied as a model for poorly conducting top and bottom walls (Otero et al. Reference Otero, Wittenberg, Worthing and Doering2002; Johnston & Doering Reference Johnston and Doering2009; Goluskin Reference Goluskin2015). In this case, the Dirichlet boundary condition (2.5) for the temperature is replaced by a Neumann condition:

(2.10) $$\begin{eqnarray}\left.-\unicode[STIX]{x1D705}\frac{\unicode[STIX]{x2202}T}{\unicode[STIX]{x2202}z}\right|_{z=\pm H/2}=J_{in}\quad \text{imposed flux at top/bottom walls},\end{eqnarray}$$

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 $t\rightarrow \infty$ with bounded temperature. The Nusselt number is still defined by (2.9) but the roles of $J$ and $\unicode[STIX]{x0394}T$ as response and control variables are reversed, with now $\unicode[STIX]{x0394}T=\langle T\rangle _{bot}-\langle T\rangle _{top}$ for $T$ space averaged over top and bottom walls and time averaged over $[0,t]$ . The natural dimensionless control variable in this problem is not the Rayleigh number but instead

(2.11) $$\begin{eqnarray}Ra_{\ast }=\frac{\unicode[STIX]{x1D6FC}gJH^{4}}{\unicode[STIX]{x1D705}^{2}\unicode[STIX]{x1D708}}=RaNu,\end{eqnarray}$$

as noted by Otero et al. (Reference Otero, Wittenberg, Worthing and Doering2002). Numerical studies of Johnston & Doering (Reference Johnston and Doering2009) and also Stevens, Lohse & Verzicco (Reference Stevens, Lohse and Verzicco2011) suggest that Rayleigh–Bénard convection with temperature b.c. and with flux b.c. (boundary conditions) exhibit essentially identical behaviour in the turbulent regime, including both $Nu$ – $Ra$ scaling and morphology of the flow, such as thermal plumes.

There is an important formal difference between temperature b.c. and flux b.c., however, in regard to thermal dissipative anomalies. In mathematical treatments, anomalies are often associated with dissipation rates of kinetic energy and thermal intensity non-vanishing in the formal limit $\unicode[STIX]{x1D708},$ $\unicode[STIX]{x1D705}\rightarrow 0,$ with all other parameters fixed. Such a formulation does not suffice with flux b.c. This can be seen already for the problem of pure conduction with vanishing velocity field. The exact steady-state solution has $\unicode[STIX]{x2202}T/\unicode[STIX]{x2202}z=-J/\unicode[STIX]{x1D705}=-\unicode[STIX]{x0394}T/H$ throughout the domain, and thus the mean thermal dissipation is $\langle \unicode[STIX]{x1D705}|\unicode[STIX]{x1D735}T|^{2}\rangle _{V}=J^{2}/\unicode[STIX]{x1D705}=\unicode[STIX]{x1D705}(\unicode[STIX]{x0394}T)^{2}/H$ (for more discussion of this example, see appendix B). For fixed $\unicode[STIX]{x0394}T$ as $\unicode[STIX]{x1D705}\rightarrow 0$ of course $\langle \unicode[STIX]{x1D705}|\unicode[STIX]{x1D735}T|^{2}\rangle _{V}\rightarrow 0,$ but for fixed $J$ instead $\langle \unicode[STIX]{x1D705}|\unicode[STIX]{x1D735}T|^{2}\rangle _{V}\rightarrow \infty$ ! This is a trivial divergence for pure conduction, which can be eliminated by instead holding the vertical temperature gradient $\unicode[STIX]{x2202}T/\unicode[STIX]{x2202}z=\unicode[STIX]{x1D6FD}$ fixed at $z=\pm H/2$ as $\unicode[STIX]{x1D705}\rightarrow 0.$ However, in the case of turbulent Rayleigh–Bénard convection with an imposed heat flux it is a priori unclear how to choose the $\unicode[STIX]{x1D705},\unicode[STIX]{x1D708}$ dependence of $J$ so that $\unicode[STIX]{x0394}T$ remains fixed in the limit $\unicode[STIX]{x1D705},\unicode[STIX]{x1D708}\rightarrow 0.$ Furthermore, even if $\unicode[STIX]{x0394}T$ is held fixed, the asymptotic behaviour of the thermal dissipation is unknown. This question has great importance, since it is known that the dissipation of kinetic energy and temperature fluctuations determine the scaling of $Nu$ at high $Ra$ for the fixed- $\unicode[STIX]{x0394}T$ problem (Siggia Reference Siggia1994; Ahlers et al. Reference Ahlers, Grossmann and Lohse2009). To understand the implications of our fluctuation–dissipation relations for Rayleigh–Bénard convection, we must therefore first discuss the role of dissipation, both of kinetic energy and temperature fluctuations, for $Nu$ – $Ra$ scaling in the fixed- $J$ problem. We thus turn to the mean balances for those quantities.

2.2 Mean balance equations

Global balances of conserved quantities impose simple but crucial constraints on turbulent Rayleigh–Bénard convection (Siggia Reference Siggia1994; Ahlers et al. Reference Ahlers, Grossmann and Lohse2009). Averaging the equation for conservation of kinetic energy over the cell volume and finite time interval $[0,t]$ gives

(2.12) $$\begin{eqnarray}\unicode[STIX]{x1D6FC}g\left(J-\frac{\unicode[STIX]{x1D705}}{H}\unicode[STIX]{x0394}T\right)=\unicode[STIX]{x1D708}\langle |\unicode[STIX]{x1D735}\boldsymbol{u}|^{2}\rangle _{V,t}+\left\langle \unicode[STIX]{x2202}_{t}\left(\frac{1}{2}u^{2}\right)\right\rangle _{V,t},\end{eqnarray}$$

where the left-hand side is input of kinetic energy by buoyancy force and the right-hand side is the sum of viscous dissipation and kinetic energy growth. Likewise, the mean temperature fluctuation balance is

(2.13) $$\begin{eqnarray}\frac{\unicode[STIX]{x0394}(TJ)}{H}=\unicode[STIX]{x1D705}\langle |\unicode[STIX]{x1D735}T|^{2}\rangle _{V,t}+\left\langle \unicode[STIX]{x2202}_{t}\left(\frac{1}{2}T^{2}\right)\right\rangle _{V,t},\end{eqnarray}$$

with $\unicode[STIX]{x0394}(TJ)=\langle TJ\rangle _{bot}-\langle TJ\rangle _{top},$ where the left-hand side is the input of temperature fluctuations from the boundary and the right-hand side is the sum of thermal dissipation and growth of mean temperature fluctuation. These equations hold for both temperature and flux b.c. However, for temperature b.c. $\unicode[STIX]{x0394}(TJ)=T_{bot}\langle J\rangle _{bot}-T_{top}\langle J\rangle _{top}$ whereas for flux b.c. $\unicode[STIX]{x0394}(TJ)=(\langle T\rangle _{bot}-\langle T\rangle _{top})J_{in}=\unicode[STIX]{x0394}TJ_{in}$ and $J_{in}$ imposed at the top and bottom is a priori distinct from $J$ defined as the volume average (2.8).

Further simplification occurs for the long-time $t\rightarrow \infty$ limit. In that case, the time derivative terms in (2.12)–(2.13) converge to zero, as long as volume-averaged kinetic energy and temperature fluctuation remain bounded uniformly in time at constant $\unicode[STIX]{x1D708},\unicode[STIX]{x1D705}>0.$ The balance equation for the mean of the temperature then gives the additional information that $\unicode[STIX]{x2202}J(z)/\unicode[STIX]{x2202}z=0$ where

(2.14) $$\begin{eqnarray}J(z)=\langle u_{z}T-\unicode[STIX]{x1D705}\unicode[STIX]{x2202}_{z}T\rangle _{A,\infty }=J\end{eqnarray}$$

is the vertical heat flux averaged over the cross-section $S$ of the cell at height $z$ and over an infinite interval of time. The constancy of the vertical heat flux with height implies that $J_{in}=J$ for flux b.c. and $\langle J\rangle _{top}=\langle J\rangle _{bot}=J$ for temperature b.c. The long-time global balance equations then become identical for temperature b.c. and flux b.c.:

(2.15) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6FC}g\left(J-\frac{\unicode[STIX]{x1D705}}{H}\unicode[STIX]{x0394}T\right)=\unicode[STIX]{x1D708}\langle |\unicode[STIX]{x1D735}\boldsymbol{u}|^{2}\rangle _{V,\infty }=\unicode[STIX]{x1D700}_{u}, & \displaystyle\end{eqnarray}$$

(2.16) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{J\unicode[STIX]{x0394}T}{H}=\unicode[STIX]{x1D705}\langle |\unicode[STIX]{x1D735}T|^{2}\rangle _{V,\infty }=\unicode[STIX]{x1D700}_{T}, & \displaystyle\end{eqnarray}$$

but the role of $J$ and $\unicode[STIX]{x0394}T$ as control and response variable is reversed for the two cases.

2.3 Anomalous dissipation and Kraichnan–Spiegel scaling

Now consider the limit $\unicode[STIX]{x1D708},\unicode[STIX]{x1D705}\rightarrow 0$ with $Pr$ fixed, as is common in mathematical treatments of the large $Ra$ limit. First note that $J-\unicode[STIX]{x1D705}(\unicode[STIX]{x0394}T/H)=J(1-1/Nu)\simeq J$ for $Nu\gg 1.$ Defining as usual the free-fall velocity

(2.17) $$\begin{eqnarray}U=(\unicode[STIX]{x1D6FC}g\unicode[STIX]{x0394}TH)^{1/2}\end{eqnarray}$$

and neglecting the small $1/Nu$ correction term in the energy balance,

(2.18) $$\begin{eqnarray}\frac{\unicode[STIX]{x1D700}_{u}}{U^{3}/H}=\frac{\unicode[STIX]{x1D700}_{T}}{(\unicode[STIX]{x0394}T)^{2}U/H}=\frac{J}{U\unicode[STIX]{x0394}T}=\frac{Nu}{\sqrt{RaPr}}.\end{eqnarray}$$

The Spiegel scaling law $Nu\sim C\cdot Ra^{1/2}Pr^{1/2}$ holds if and only if at fixed $Pr$

(2.19) $$\begin{eqnarray}\lim _{\unicode[STIX]{x1D708},\unicode[STIX]{x1D705}\rightarrow 0}\frac{\unicode[STIX]{x1D700}_{u}}{U^{3}/H}=\lim _{\unicode[STIX]{x1D708},\unicode[STIX]{x1D705}\rightarrow 0}\frac{\unicode[STIX]{x1D700}_{T}}{(\unicode[STIX]{x0394}T)^{2}U/H}=C>0.\end{eqnarray}$$

It is important to emphasize that this equivalence holds for both temperature b.c. and flux b.c. Our argument formalizes the dimensional reasoning of Spiegel (Reference Spiegel1971), whereas Kraichnan (Reference Kraichnan1962) based his conclusions on a conjectured turbulent shear layer at very high Rayleigh numbers and obtained a Nusselt number smaller by a factor of $\propto [\ln Ra]^{-3/2}.$ For standard temperature b.c. Rayleigh–Bénard convection, the relation (2.19) corresponds to both kinetic and thermal dissipation anomalies, if $\unicode[STIX]{x0394}T$ (and thus also $U$ ) is taken to be independent of $\unicode[STIX]{x1D708},\unicode[STIX]{x1D705}$ in the limit $\unicode[STIX]{x1D708},\unicode[STIX]{x1D705}\rightarrow 0$ . This is only true, strictly speaking, for the Spiegel (Reference Spiegel1971) dimensional predictions, whereas the Kraichnan (Reference Kraichnan1962) results correspond instead to dissipation both of thermal fluctuations and of kinetic energy vanishing very slowly (logarithmically) as $\unicode[STIX]{x1D708},\unicode[STIX]{x1D705}\rightarrow 0$ with all other parameters fixed.

A similar interpretation of Kraichnan–Spiegel scaling in terms of dissipative anomalies is also true for Rayleigh–Bénard convection with flux b.c., but it requires a bit more explanation. For the latter problem one can define velocity and temperature scales

(2.20a,b ) $$\begin{eqnarray}U_{\ast }=(\unicode[STIX]{x1D6FC}gJH)^{1/3},\quad \unicode[STIX]{x0394}T_{\ast }=J/U_{\ast }.\end{eqnarray}$$

Note that $Ra_{\ast }=Re_{\ast }^{3}Pr^{2}$ with $Re_{\ast }=U_{\ast }H/\unicode[STIX]{x1D708}.$ Neglecting again the $1/Nu$ correction in the energy balance,

(2.21a,b ) $$\begin{eqnarray}\frac{\unicode[STIX]{x1D700}_{u}}{U_{\ast }^{3}/H}=1,\quad \frac{\unicode[STIX]{x1D700}_{T}}{(\unicode[STIX]{x0394}T_{\ast })^{2}U_{\ast }/H}=\frac{\unicode[STIX]{x0394}T}{\unicode[STIX]{x0394}T}_{\ast }.\end{eqnarray}$$

If one chooses $J$ to be fixed as $\unicode[STIX]{x1D708},\unicode[STIX]{x1D705}\rightarrow 0,$ then the first equation has the seeming implication that there is necessarily a dissipative anomaly for kinetic energy in flux b.c. Rayleigh–Bénard 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 $\unicode[STIX]{x1D700}_{u}$ remains positive as $\unicode[STIX]{x1D708},\unicode[STIX]{x1D705}\rightarrow 0.$ Likewise, a ‘thermal dissipation anomaly’ is usually defined to occur when $\unicode[STIX]{x1D700}_{T}$ remains positive as $\unicode[STIX]{x1D708},\unicode[STIX]{x1D705}\rightarrow 0.$ The absence of ‘dissipative anomalies’ is then mathematically taken to mean that instead $\unicode[STIX]{x1D700}_{u},$ $\unicode[STIX]{x1D700}_{T}\rightarrow 0$ as $\unicode[STIX]{x1D708},\unicode[STIX]{x1D705}\rightarrow 0.$

However, the physically more natural interpretation of a ‘dissipative anomaly’ is that $\unicode[STIX]{x1D700}_{u}\propto U^{3}/H$ and $\unicode[STIX]{x1D700}_{T}\propto (\unicode[STIX]{x0394}T)^{2}U/H$ as $\unicode[STIX]{x1D708},\unicode[STIX]{x1D705}\rightarrow 0,$ with a non-zero and finite constant of proportionality. It is thus more reasonable to associate dissipative anomalies with non-vanishing of the dimensionless ratios

(2.22a,b ) $$\begin{eqnarray}\hat{\unicode[STIX]{x1D700}}_{u}=\frac{\unicode[STIX]{x1D700}_{u}}{U^{3}/H},\quad \hat{\unicode[STIX]{x1D700}}_{T}=\frac{\unicode[STIX]{x1D700}_{T}}{(\unicode[STIX]{x0394}T)^{2}U/H}\end{eqnarray}$$

as $\unicode[STIX]{x1D708},\unicode[STIX]{x1D705}\rightarrow 0$ rather than with non-vanishing of $\unicode[STIX]{x1D700}_{u},$ $\unicode[STIX]{x1D700}_{T}$ as $\unicode[STIX]{x1D708},\unicode[STIX]{x1D705}\rightarrow 0.$ If we adopt this physical definition, then by (2.18) the existence of ‘dissipative anomalies’ is exactly equivalent to 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 with vanishing of $\hat{\unicode[STIX]{x1D700}}_{u},$ $\hat{\unicode[STIX]{x1D700}}_{T}$ as $\unicode[STIX]{x1D708},\unicode[STIX]{x1D705}\rightarrow 0$ rather than with the vanishing of $\unicode[STIX]{x1D700}_{u},$ $\unicode[STIX]{x1D700}_{T}$ as $\unicode[STIX]{x1D708},\unicode[STIX]{x1D705}\rightarrow 0.$ If $\unicode[STIX]{x0394}T$ is held fixed as $\unicode[STIX]{x1D708},\unicode[STIX]{x1D705}\rightarrow 0,$ then the two formulations involving $\hat{\unicode[STIX]{x1D700}}_{u},$ $\hat{\unicode[STIX]{x1D700}}_{T}$ and $\unicode[STIX]{x1D700}_{u},$ $\unicode[STIX]{x1D700}_{T}$ are obviously equivalent to each other. As we now explain, however, these two formulations are not equivalent, if $J$ rather than $\unicode[STIX]{x0394}T$ is held fixed as $\unicode[STIX]{x1D708},\unicode[STIX]{x1D705}\rightarrow 0$ .

To see this, we use the definitions of $U,$ $U_{\ast }$ and $T_{\ast }$ to rewrite (2.18) as

(2.23) $$\begin{eqnarray}\frac{\unicode[STIX]{x1D700}_{u}}{U^{3}/H}=\frac{\unicode[STIX]{x1D700}_{T}}{(\unicode[STIX]{x0394}T)^{2}U/H}=\left(\frac{U_{\ast }}{U}\right)^{3}=\left(\frac{\unicode[STIX]{x0394}T_{\ast }}{\unicode[STIX]{x0394}T}\right)^{3/2}.\end{eqnarray}$$

First consider the case where Spiegel dimensional scaling holds. It follows from (2.23) that this scaling is equivalent to

(2.24a,b ) $$\begin{eqnarray}\lim _{\unicode[STIX]{x1D708},\unicode[STIX]{x1D705}\rightarrow \infty }\frac{U_{\ast }}{U}=C^{1/3},\quad \lim _{\unicode[STIX]{x1D708},\unicode[STIX]{x1D705}\rightarrow \infty }\frac{\unicode[STIX]{x0394}T_{\ast }}{\unicode[STIX]{x0394}T}=C^{2/3}.\end{eqnarray}$$

As a consequence, the velocities $U$ and $U_{\ast }$ differ only by a constant factor independent of Rayleigh number for $Ra\gg 1.$ The same is also true for temperature scales $\unicode[STIX]{x0394}T$ and $\unicode[STIX]{x0394}T_{\ast }$ . In this case, holding $J$ fixed as $\unicode[STIX]{x1D708},\unicode[STIX]{x1D705}\rightarrow 0$ is equivalent to holding $\unicode[STIX]{x0394}T$ fixed as $\unicode[STIX]{x1D708},\unicode[STIX]{x1D705}\rightarrow 0$ , and $\unicode[STIX]{x1D700}_{u},$ $\unicode[STIX]{x1D700}_{T}$ must converge to finite, positive values in the limit $\unicode[STIX]{x1D708},\unicode[STIX]{x1D705}\rightarrow 0$ . However, now consider the situation when Spiegel dimensional scaling does not hold. Rigorous upper bounds on Nusselt number of Doering & Constantin (Reference Doering and Constantin1996) for temperature b.c. and of Otero et al. (Reference Otero, Wittenberg, Worthing and Doering2002) for heat flux b.c. together with (2.18) imply then that $\hat{\unicode[STIX]{x1D700}}_{u},$ $\hat{\unicode[STIX]{x1D700}}_{T}\rightarrow 0$ . It follows from (2.23) that in that case $U_{\ast }/U\rightarrow 0$ and $\unicode[STIX]{x0394}T_{\ast }/\unicode[STIX]{x0394}T\rightarrow 0$ as $\unicode[STIX]{x1D708},\unicode[STIX]{x1D705}\rightarrow 0.$ According to the advocated interpretation above, this corresponds to a vanishing dissipative anomaly. However, it also follows that $\unicode[STIX]{x0394}T/\unicode[STIX]{x0394}T_{\ast }\rightarrow \infty$ , and comparing with (2.21) this means that when $J$ is held fixed as $\unicode[STIX]{x1D708},\unicode[STIX]{x1D705}\rightarrow 0,$ then $\unicode[STIX]{x1D700}_{T}\rightarrow \infty !$ It may appear odd to associate this behaviour with ‘absence of dissipative anomaly’, but it should be kept in mind that for fixed $J$ as $\unicode[STIX]{x1D708},\unicode[STIX]{x1D705}\rightarrow 0,$ then $\unicode[STIX]{x1D700}_{T}\rightarrow \infty$ even for the problem of pure heat conduction (see appendix B). Instead, a thermal dissipative anomaly in the physical sense is naturally associated with $\unicode[STIX]{x1D700}_{T}$ remaining finite for $\unicode[STIX]{x1D708},\unicode[STIX]{x1D705}\rightarrow 0$ when $J$ is held fixed.

It is easy to check by using the definitions of the various quantities that there is equivalence of the general scaling relations

(2.25) $$\begin{eqnarray}Nu\sim Ra^{x}Pr^{y}\quad \Longleftrightarrow \quad \frac{\unicode[STIX]{x0394}T}{\unicode[STIX]{x0394}T_{\ast }}\sim Ra_{\ast }^{z}Pr^{w},\end{eqnarray}$$

with

(2.26a,b ) $$\begin{eqnarray}x=\frac{1-3z}{2+3z},\quad y=\frac{1-3w}{2+3z}.\end{eqnarray}$$

These can be interpreted as a relation between $J$ and $\unicode[STIX]{x0394}T$ ,

(2.27) $$\begin{eqnarray}J\sim \frac{\unicode[STIX]{x1D705}^{1-2x}(\unicode[STIX]{x1D6FC}g)^{x}}{H^{1-3x}}Pr^{y-x}(\unicode[STIX]{x0394}T)^{1+x},\end{eqnarray}$$

which must be maintained with flux b.c. in order to hold $\unicode[STIX]{x0394}T$ fixed as $\unicode[STIX]{x1D708},\unicode[STIX]{x1D705}\rightarrow 0.$ It follows from the rigorous bounds of Doering & Constantin (Reference Doering and Constantin1996) and Otero et al. (Reference Otero, Wittenberg, Worthing and Doering2002) that $x\leqslant 1/2.$ If $x<1/2,$ below the Kraichnan–Spiegel value, then $J\rightarrow 0$ as $\unicode[STIX]{x1D708},\unicode[STIX]{x1D705}\rightarrow 0$ at fixed $\unicode[STIX]{x0394}T.$ In that case, $\unicode[STIX]{x0394}T/\unicode[STIX]{x0394}T_{\ast }\rightarrow \infty$ as $\unicode[STIX]{x1D708},\unicode[STIX]{x1D705}\rightarrow 0$ because the denominator $\unicode[STIX]{x0394}T_{\ast }\rightarrow 0.$

3.1 Rayleigh–Bénard convection with flux b.c.

The stochastic Lagrangian representation and FDR are simplest in form and easiest to analyse 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

(3.3) $$\begin{eqnarray}\displaystyle T(\boldsymbol{x},t) & = & \displaystyle \mathbb{E}[T_{0}(\tilde{\unicode[STIX]{x1D743}}_{t,0}(\boldsymbol{x}))+J(\tilde{\ell }_{t,0}^{top}(\boldsymbol{x})-\tilde{\ell }_{t,0}^{bot}(\boldsymbol{x}))]\nonumber\\ \displaystyle & = & \displaystyle \int \text{d}^{3}x_{0}T_{0}(\boldsymbol{x}_{0})p(\boldsymbol{x}_{0},0\mid \boldsymbol{x},t)\nonumber\\ \displaystyle & & \displaystyle -\,J\int _{0}^{t}\text{d}sp_{z}(H/2,s\mid \boldsymbol{x},t)+J\int _{0}^{t}\text{d}sp_{z}(-H/2,s\mid \boldsymbol{x},t),\end{eqnarray}$$

which follows directly from formulas (II:2.9) and (II:2.10) of paper II. We have introduced the backward-in-time transition probability

(3.4) $$\begin{eqnarray}p^{\unicode[STIX]{x1D708},\unicode[STIX]{x1D705}}(\boldsymbol{x}^{\prime },t^{\prime }\mid \boldsymbol{x},t)=\mathbb{E}[\unicode[STIX]{x1D6FF}^{d}(\boldsymbol{x}^{\prime }-\tilde{\unicode[STIX]{x1D743}}_{t,t^{\prime }}^{\unicode[STIX]{x1D708},\unicode[STIX]{x1D705}}(\boldsymbol{x}))]\quad t^{\prime }<t\end{eqnarray}$$

and the conditional probability density for the $z$ -component of the particle position:

(3.5) $$\begin{eqnarray}p_{z}(z^{\prime },s\mid \boldsymbol{x},t)=\iint _{S}\text{d}x^{\prime }\text{d}y^{\prime }p(x^{\prime },y^{\prime },z^{\prime },s\mid \boldsymbol{x},t)=\mathbb{E}[\unicode[STIX]{x1D6FF}(z^{\prime }-\tilde{\unicode[STIX]{x1D701}}_{t,s}(\boldsymbol{x}))].\end{eqnarray}$$

It is important to stress that the average $\mathbb{E}[\cdot ]$ appearing in these last two equations is over the Brownian motion $\widetilde{\boldsymbol{W}}_{s}$ only. In particular, the velocity $\boldsymbol{u}$ is a fixed (deterministic) field obtained as a solution of the Rayleigh–Bénard problem with flux b.c. (2.10). Nowhere in this paper shall we average over ensembles of $\boldsymbol{u}$ for random initial data $\boldsymbol{u}_{0},$ $T_{0}$ . It would be appropriate to add a subscript $\boldsymbol{u}$ to the transition probability densities defined above, in order to emphasize their dependence upon this explicit solution. Since this would lead to 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 $t\rightarrow \infty$ limit. Because of incompressibility of the velocity field and ergodicity of the stochastic Lagrangian flow for $\unicode[STIX]{x1D705}>0$ , the particle distributions must asymptotically become uniform over the flow domain:

(3.6a,b ) $$\begin{eqnarray}\lim _{t\rightarrow \infty }p(\boldsymbol{y},s\mid \boldsymbol{x},t)=\frac{1}{V},\quad \lim _{t\rightarrow \infty }p_{z}(z^{\prime },s\mid \boldsymbol{x},t)=\frac{1}{H},\end{eqnarray}$$

with $\boldsymbol{y},z^{\prime },s,$ and $\boldsymbol{x}$ fixed as $t\rightarrow \infty$ , where $V=HA$ is the volume of the cylindrical cell with cross-sectional area $A.$ Thus,

(3.7) $$\begin{eqnarray}\lim _{t\rightarrow \infty }\int \text{d}^{3}x_{0}T_{0}(\boldsymbol{x}_{0})p(\boldsymbol{x}_{0},0\mid \boldsymbol{x},t)=\frac{1}{V}\int \text{d}^{3}x_{0}T_{0}(\boldsymbol{x}_{0})=\langle T_{0}\rangle _{V}.\end{eqnarray}$$

For flux b.c. the volume average of $T(\boldsymbol{x},t)$ is conserved in time so that memory of the initial average must be preserved. We assume for simplicity that $\langle T_{0}\rangle _{V}=0,$ so that

(3.8) $$\begin{eqnarray}T(\boldsymbol{x},t)\simeq -J\int _{0}^{t}\text{d}s\left(p_{z}(H/2,s\mid \boldsymbol{x},t)-\frac{1}{H}\right)+J\int _{0}^{t}\text{d}s\left(p_{z}(-H/2,s\mid \boldsymbol{x},t)-\frac{1}{H}\right),\end{eqnarray}$$

where the term $Jt/H$ has been added and subtracted to make each of the integrands tend to zero for large time separations. From this expression one can see that the dependence upon the distant past for $s\ll t$ is negligible in comparison to the contribution from the recent past for $s\lesssim t.$ One can also understand why generally $T(\boldsymbol{x},t)>0$ for $z\gtrsim -H/2,$ slightly above the bottom wall, since then usually

(3.9a,b ) $$\begin{eqnarray}p_{z}(H/2,s\mid \boldsymbol{x},t)<\frac{1}{H},\quad p_{z}(-H/2,s\mid \boldsymbol{x},t)>\frac{1}{H},\quad s\lesssim t.\end{eqnarray}$$

Exactly the opposite inequalities generally hold for $z\lesssim H/2,$ slightly below the top wall. Of course, $T(\boldsymbol{x},t)$ evolves chaotically in time through the dependence of the transition probabilities upon $\boldsymbol{u},$ 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

(3.10) $$\begin{eqnarray}\displaystyle & & \displaystyle \langle T(\boldsymbol{x})\rangle _{\infty }=\lim _{t\rightarrow \infty }\frac{1}{t}\int _{0}^{t}\text{d}sT(\boldsymbol{x},s)\nonumber\\ \displaystyle & & \displaystyle \quad =-J\int _{-\infty }^{0}\text{d}s\left(p_{z}(H/2,s\mid \boldsymbol{x},0)-\frac{1}{H}\right)+J\int _{-\infty }^{0}\text{d}s\left(p_{z}(-H/2,s\mid \boldsymbol{x},0)-\frac{1}{H}\right),\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

whenever the latter integrals are convergent. If not, then taking the lower limit to $-\infty$ must be interpreted in the Cesàro mean sense, i.e. the limit of the time average of the integral with respect to its range of integration. These long-time averages will no longer depend upon the initial conditions $\boldsymbol{u}_{0}$ and $T_{0}$ at $s=-\infty$ if there is ergodicity of Eulerian dynamics for the Boussinesq fluid system at high $Ra.$

We can now present our Lagrangian fluctuation–dissipation relation for Rayleigh–Bénard convection with flux b.c., which expresses the volume- and time-averaged thermal dissipation rate as

(3.11) $$\begin{eqnarray}\unicode[STIX]{x1D705}\int _{0}^{t}\text{d}s\langle |\unicode[STIX]{x1D735}T(s)|^{2}\rangle _{V}=\frac{1}{2}\langle \text{Var}[T_{0}(\tilde{\unicode[STIX]{x1D743}}_{t,0})+J(\tilde{\ell }_{t,0}^{top}-\tilde{\ell }_{t,0}^{bot})]\rangle _{V},\end{eqnarray}$$

where $\text{Var}[\cdot ]$ denotes the variance over the Brownian motion $\widetilde{\boldsymbol{W}}_{s}.$ This result is a direct consequence of formulas (II:2.11), (II:2.13) of paper II. To make the result somewhat more concrete (but also more elaborate), we can decompose the variance on the right as

(3.12) $$\begin{eqnarray}\displaystyle & & \displaystyle \text{Var}[T_{0}(\tilde{\unicode[STIX]{x1D743}}_{t,0}(\boldsymbol{x}))+J(\tilde{\ell }_{t,0}^{top}(\boldsymbol{x})-\tilde{\ell }_{t,0}^{bot}(\boldsymbol{x}))]=\text{Var}[T_{0}(\tilde{\unicode[STIX]{x1D743}}_{t,0}(\boldsymbol{x}))]\nonumber\\ \displaystyle & & \displaystyle \quad +\,2\text{Cov}[T_{0}(\tilde{\unicode[STIX]{x1D743}}_{t,0}(\boldsymbol{x})),J(\tilde{\ell }_{t,0}^{top}(\boldsymbol{x})-\tilde{\ell }_{t,0}^{bot}(\boldsymbol{x}))]+\text{Var}[J(\tilde{\ell }_{t,0}^{top}(\boldsymbol{x})-\tilde{\ell }_{t,0}^{bot}(\boldsymbol{x}))],\qquad\end{eqnarray}$$

where $\text{Cov}[\cdot ,\cdot ]$ denotes the covariance of two random variables as functions of the Brownian motion $\widetilde{\boldsymbol{W}}_{s}.$ We can then provide explicit formulas for the three separate terms. As a special case of (II:2.18) of paper II we see that $\text{Var}[T_{0}(\tilde{\unicode[STIX]{x1D743}}_{t,0}(\boldsymbol{x}))]$ is given by the expression

(3.13) $$\begin{eqnarray}\displaystyle & & \displaystyle \text{Var}[T_{0}(\tilde{\unicode[STIX]{x1D743}}_{t,0}(\boldsymbol{x}))]=\int \text{d}^{d}x_{0}\int \text{d}^{d}x_{0}^{\prime }T_{0}(\boldsymbol{x}_{0})T_{0}(\boldsymbol{x}_{0}^{\prime })\nonumber\\ \displaystyle & & \displaystyle \quad \times \,[p_{2}^{\unicode[STIX]{x1D708},\unicode[STIX]{x1D705}}(\boldsymbol{x}_{0},0;\boldsymbol{x}_{0}^{\prime },0\mid \boldsymbol{x},t)-p^{\unicode[STIX]{x1D708},\unicode[STIX]{x1D705}}(\boldsymbol{x}_{0},0\mid \boldsymbol{x},t)p^{\unicode[STIX]{x1D708},\unicode[STIX]{x1D705}}(\boldsymbol{x}_{0}^{\prime },0\mid \boldsymbol{x},t)],\end{eqnarray}$$

where we have introduced the two-time (backward-in-time) transition probability density

(3.14) $$\begin{eqnarray}p_{2}^{\unicode[STIX]{x1D708},\unicode[STIX]{x1D705}}(\boldsymbol{y},s;\boldsymbol{y}^{\prime },s^{\prime }\mid \boldsymbol{x},t)=\mathbb{E}[\unicode[STIX]{x1D6FF}^{d}(\boldsymbol{y}-\tilde{\unicode[STIX]{x1D743}}_{t,s}^{\unicode[STIX]{x1D708},\unicode[STIX]{x1D705}}(\boldsymbol{x}))\unicode[STIX]{x1D6FF}^{d}(\boldsymbol{y}^{\prime }-\tilde{\unicode[STIX]{x1D743}}_{t,s^{\prime }}^{\unicode[STIX]{x1D708},\unicode[STIX]{x1D705}}(\boldsymbol{x}))],\quad s<t,\end{eqnarray}$$

which represents the joint probability for the particle to end up at $\boldsymbol{y}$ at time $s<t$ and at $\boldsymbol{y}^{\prime }$ at time $s^{\prime }<t,$ given that it started at $\boldsymbol{x}$ at $t$ . Likewise, the local time variance is given as a special case of (II:2.30) of paper II by

(3.15) $$\begin{eqnarray}\displaystyle & & \displaystyle \frac{1}{2}\text{Var}[J(\tilde{\ell }_{t,0}^{top}(\boldsymbol{x})-\tilde{\ell }_{t,0}^{bot}(\boldsymbol{x}))]\nonumber\\ \displaystyle & & \displaystyle \quad =\frac{J^{2}}{2}\int _{0}^{t}\text{d}s\int _{0}^{t}\text{d}s^{\prime }[p_{2z}(+H/2,s;+H/2,s^{\prime }\mid \boldsymbol{x},t)\nonumber\\ \displaystyle & & \displaystyle \qquad -\,p_{z}(+H/2,s\mid \boldsymbol{x},t)p_{z}(+H/2,s^{\prime }\mid \boldsymbol{x},t)\!]\nonumber\\ \displaystyle & & \displaystyle \qquad +\,\frac{J^{2}}{2}\int _{0}^{t}\text{d}s\int _{0}^{t}\text{d}s^{\prime }[\!p_{2z}(-H/2,s;-H/2,s^{\prime }\mid \boldsymbol{x},t)\nonumber\\ \displaystyle & & \displaystyle \qquad -\,p_{z}(-H/2,s\mid \boldsymbol{x},t)p_{z}(-H/2,s^{\prime }\mid \boldsymbol{x},t)\!]\nonumber\\ \displaystyle & & \displaystyle \qquad -\,\frac{J^{2}}{2}\int _{0}^{t}\text{d}s\int _{0}^{t}\text{d}s^{\prime }[\!p_{2z}(+H/2,s;-H/2,s^{\prime }\mid \boldsymbol{x},t)\nonumber\\ \displaystyle & & \displaystyle \qquad -\,p_{z}(+H/2,s\mid \boldsymbol{x},t)p_{z}(-H/2,s^{\prime }\mid \boldsymbol{x},t)\!]\nonumber\\ \displaystyle & & \displaystyle \qquad -\,\frac{J^{2}}{2}\int _{0}^{t}\text{d}s\int _{0}^{t}\text{d}s^{\prime }[\!p_{2z}(-H/2,s;+H/2,s^{\prime }\mid \boldsymbol{x},t)\nonumber\\ \displaystyle & & \displaystyle \qquad -\,p_{z}(-H/2,s\mid \boldsymbol{x},t)p_{z}(+H/2,s^{\prime }\mid \boldsymbol{x},t)\!],\end{eqnarray}$$

where

(3.16) $$\begin{eqnarray}p_{2z}(z^{\prime },s;z^{\prime \prime },s^{\prime }\mid \boldsymbol{x},t)=\iint _{S}\text{d}x^{\prime }\,\text{d}y^{\prime }\iint _{S}\text{d}x^{\prime \prime }\,\text{d}y^{\prime \prime }p_{2}(x^{\prime },y^{\prime },z^{\prime },s;x^{\prime \prime },y^{\prime \prime },z^{\prime \prime },s^{\prime }\mid \boldsymbol{x},t).\end{eqnarray}$$

Similarly to the preceding two terms, the cross-covariance is given by

(3.17) $$\begin{eqnarray}\displaystyle & & \displaystyle \hspace{-12.0pt}\text{Cov}[T_{0}(\tilde{\unicode[STIX]{x1D743}}_{t,0}(\boldsymbol{x})),J(\tilde{\ell }_{t,0}^{top}(\boldsymbol{x})-\tilde{\ell }_{t,0}^{bot}(\boldsymbol{x}))]\nonumber\\ \displaystyle & & \displaystyle \,=-J\int \text{d}^{3}x_{0}T_{0}(\boldsymbol{x}_{0})\int _{0}^{t}\text{d}s\iint _{S}\text{d}x^{\prime }\,\text{d}y^{\prime }\nonumber\\ \displaystyle & & \displaystyle \qquad \times \,[p_{2}(\boldsymbol{x}_{0},0;x^{\prime },y^{\prime },H/2,s\mid \boldsymbol{x},t)-p(\boldsymbol{x}_{0},0\mid \boldsymbol{x},t)p(x^{\prime },y^{\prime },H/2,s\mid \boldsymbol{x},t)]\nonumber\\ \displaystyle & & \displaystyle \qquad +\,J\int \text{d}^{3}x_{0}T_{0}(\boldsymbol{x}_{0})\int _{0}^{t}\text{d}s\iint _{S}\text{d}x^{\prime }\,\text{d}y^{\prime }\nonumber\\ \displaystyle & & \displaystyle \qquad \times \,[p_{2}(\boldsymbol{x}_{0},0;x^{\prime },y^{\prime },-H/2,s\mid \boldsymbol{x},t)-p(\boldsymbol{x}_{0},0\mid \boldsymbol{x},t)p(x^{\prime },y^{\prime },-H/2,s\mid \boldsymbol{x},t)].\qquad\end{eqnarray}$$

These formulas provide a purely Lagrangian representation of the thermal dissipation. It is notable that the entire expression vanishes if the particle positions are statistically independent at distinct times (or, in particular, if particle trajectories are deterministic).

The finite-time fluctuation–dissipation relation is a bit complicated because of the several terms. However, all of the $T_{0}$ -dependence disappears in the long-time limit where the following simpler relation holds:

(3.18) $$\begin{eqnarray}\displaystyle \langle \unicode[STIX]{x1D705}|\unicode[STIX]{x1D735}T|^{2}\rangle _{V,\infty } & = & \displaystyle \lim _{t\rightarrow \infty }\frac{1}{2t}\langle \text{Var}[J(\tilde{\ell }_{t,0}^{top}-\tilde{\ell }_{t,0}^{bot})]\rangle _{V}\nonumber\\ \displaystyle & = & \displaystyle \mathop{\sum }_{\unicode[STIX]{x1D706},\unicode[STIX]{x1D706}^{\prime }}\unicode[STIX]{x1D706}\unicode[STIX]{x1D706}^{\prime }\lim _{t\rightarrow \infty }\frac{J^{2}}{2t}\langle \text{Cov}[J(\tilde{\ell }_{t,0}^{\unicode[STIX]{x1D706}},\tilde{\ell }_{t,0}^{\unicode[STIX]{x1D706}^{\prime }})]\rangle _{V},\end{eqnarray}$$

with $\unicode[STIX]{x1D706},\unicode[STIX]{x1D706}^{\prime }=\pm$ , where we denote top/bottom walls by $+/-$ . Indeed, it is rigorously true that $\lim _{t\rightarrow \infty }(1/t)\text{Var}[T_{0}(\tilde{\unicode[STIX]{x1D743}}_{t,0}(\boldsymbol{x}))]=0$ for bounded initial data $T_{0},$ since the variance is then at most $2(\max |T_{0}|)^{2}.$ One can also argue that the contribution to the long-time average from the covariance (3.17) divided by $t$ gives a vanishing contribution, since particle positions at time zero and time $s$ will become independent for $s\gg 0$ and thus the $s$ -integrals in (3.17) are expected to converge for $t\rightarrow \infty$ . (Vanishing of the contribution from (3.17) follows also from the Cauchy–Schwarz inequality $|(1/2t)\text{Cov}[T_{0}(\tilde{\unicode[STIX]{x1D743}}_{t,0}(\boldsymbol{x})),J(\tilde{\ell }_{t,0}^{top}(\boldsymbol{x})-\tilde{\ell }_{t,0}^{bot}(\boldsymbol{x}))]|^{2}\leqslant (1/2t)\text{Var}[T_{0}(\tilde{\unicode[STIX]{x1D743}}_{t,0}(\boldsymbol{x}))]\times (1/2t)\text{Var}[J(\tilde{\ell }_{t,0}^{top}(\boldsymbol{x})-\tilde{\ell }_{t,0}^{bot}(\boldsymbol{x}))].$ ) For the same reason, the surviving contribution (3.18) is expected to be finite as $t\rightarrow \infty$ since in the double time integration over $s,s^{\prime }$ in (3.15) the integrand is non-negligible only for $s\simeq s^{\prime }.$

According to the general result (II:2.14) of paper II, the limit in (3.18) should furthermore be $\boldsymbol{x}$ -independent without averaging over space. This may in fact be shown by a direct argument, which yields the much simpler formula

(3.19) $$\begin{eqnarray}\lim _{t\rightarrow \infty }\frac{\unicode[STIX]{x1D706}\unicode[STIX]{x1D706}^{\prime }}{2t}\text{Cov}[\tilde{\ell }_{t,0}^{\unicode[STIX]{x1D706}}(\boldsymbol{x}),\tilde{\ell }_{t,0}^{\unicode[STIX]{x1D706}^{\prime }}(\boldsymbol{x})]=\frac{\unicode[STIX]{x1D706}\unicode[STIX]{x1D706}^{\prime }}{H}\int _{-\infty }^{0}\text{d}s\left[p_{z}(\unicode[STIX]{x1D706}H/2,s\mid \unicode[STIX]{x1D706}^{\prime }H/2,0)-\frac{1}{H}\right]_{\unicode[STIX]{x1D706},\unicode[STIX]{x1D706}^{\prime }},\end{eqnarray}$$

where $[\cdot ]_{\unicode[STIX]{x1D706},\unicode[STIX]{x1D706}^{\prime }}$ denotes symmetrization with respect to indices $\unicode[STIX]{x1D706},$ $\unicode[STIX]{x1D706}^{\prime }$ . Because derivation of the formula (3.19) is a bit technical, we present it in appendix A. We have defined

(3.20) $$\begin{eqnarray}p_{z}(z,s\mid z^{\prime },s^{\prime })=\frac{1}{A}\iint _{S}\text{d}x^{\prime }\,\text{d}y^{\prime }p_{z}(z,s\mid x^{\prime },y^{\prime },z^{\prime },s^{\prime }),\end{eqnarray}$$

which gives the transition probability for vertical heights of the particles, if at time $s^{\prime }$ a uniform distribution of particles is taken over the volume. Note that a uniform distribution over the volume implies a uniform distribution in area over the cross-section $S$ at each height $z^{\prime }.$ By incompressibility of the flow, the uniform distribution of particles over the volume is preserved in time and thus $\int \text{d}z^{\prime }p_{z}(z,s\mid z^{\prime },s^{\prime })=1$ for all $z,s,s^{\prime }$ . The formulas (3.18), (3.19) are our steady-state fluctuation–dissipation relation for Rayleigh–Bénard convection with flux b.c. The remarkable result is that the long-time average of the thermal dissipation is, in the Lagrangian sense, entirely due to statistical correlations of 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) in terms 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 argument directly from the Eulerian balance relation (2.16) for thermal fluctuations in the 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

(3.21) $$\begin{eqnarray}\bar{T}^{\unicode[STIX]{x1D706}^{\prime }}=-J\mathop{\sum }_{\unicode[STIX]{x1D706}=\pm }\unicode[STIX]{x1D706}\int _{-\infty }^{0}\text{d}s\left(p_{z}(\unicode[STIX]{x1D706}H/2,s\mid \unicode[STIX]{x1D706}^{\prime }H/2,0)-\frac{1}{H}\right)\end{eqnarray}$$

for $\unicode[STIX]{x1D706}^{\prime }=\pm .$ Thus, the steady-state temperature difference $\unicode[STIX]{x0394}T=\bar{T}^{-}-\bar{T}^{+}$ between the bottom and top plates is

(3.22) $$\begin{eqnarray}\unicode[STIX]{x0394}T=J\mathop{\sum }_{\unicode[STIX]{x1D706},\unicode[STIX]{x1D706}^{\prime }=\pm }\unicode[STIX]{x1D706}\unicode[STIX]{x1D706}^{\prime }\int _{-\infty }^{0}\text{d}s\left(p_{z}(\unicode[STIX]{x1D706}H/2,s\mid \unicode[STIX]{x1D706}^{\prime }H/2,0)-\frac{1}{H}\right).\end{eqnarray}$$

The result of substituting this Lagrangian expression for $\unicode[STIX]{x0394}T$ into the steady-state temperature balance $\unicode[STIX]{x1D700}_{T}=J\unicode[STIX]{x0394}T/H$ in (2.16) is completely equivalent to the combination of our equations (3.18) and (3.19). The additional information provided by our FDR is contained in the separate expression (3.19), which relates the four different $\unicode[STIX]{x0394}T$ -contributions in (3.22) for $\unicode[STIX]{x1D706},\unicode[STIX]{x1D706}^{\prime }=\pm$ to statistical correlations of boundary local time densities. We now discuss some of the significant implications of this fact.

3.1.2 Mathematical consequences of the FDR

Let us denote the pointwise scalar variance as

(3.23) $$\begin{eqnarray}\langle \unicode[STIX]{x1D700}_{T}^{fluc}(\boldsymbol{x})\rangle _{t}\equiv \frac{1}{2t}\text{Var}[J(\tilde{\ell }_{t,0}^{top}(\boldsymbol{x})-\tilde{\ell }_{t,0}^{bot}(\boldsymbol{x}))],\end{eqnarray}$$

which is a spatially local measure in terms of statistical Lagrangian trajectories of the thermal dissipation averaged over the time interval $[0,t].$ It is related by our FDR to the space–time average thermal dissipation as

(3.24) $$\begin{eqnarray}\langle \unicode[STIX]{x1D700}_{T}^{fluc}\rangle _{\infty }\equiv \lim _{t\rightarrow \infty }\langle \unicode[STIX]{x1D700}_{T}^{fluc}(\boldsymbol{x})\rangle _{t}=\langle \unicode[STIX]{x1D705}|\unicode[STIX]{x1D735}T|^{2}\rangle _{V,\infty }=\unicode[STIX]{x1D700}_{T},\end{eqnarray}$$

so that knowledge of $\langle \unicode[STIX]{x1D700}_{T}^{fluc}(\boldsymbol{x})\rangle _{t}$ suffices to determine the global mean thermal dissipation. It may be expected for relatively short times that $\langle \unicode[STIX]{x1D700}_{T}^{fluc}(\boldsymbol{x})\rangle _{t}$ and $\langle \unicode[STIX]{x1D705}|\unicode[STIX]{x1D735}T(\boldsymbol{x},\boldsymbol{\cdot })|^{2}\rangle _{t}$ are well correlated in space, especially when the Boussinesq system with flux b.c. (2.10) is solved with initial condition $T_{0}=(\text{const.}),$ so that both $\text{Var}[T_{0}(\tilde{\unicode[STIX]{x1D743}}_{t,0}(\boldsymbol{x}))]$ and the covariance term in (3.17) vanish identically. For example, see appendix A.2 of paper II for pure heat conduction. Of course, by its definition $\langle \unicode[STIX]{x1D700}_{T}^{fluc}(\boldsymbol{x})\rangle _{t}\geqslant 0.$

In the long-time limit, each of the four terms in (3.19) for $\unicode[STIX]{x1D706},\unicode[STIX]{x1D706}^{\prime }=\pm$ should be positive separately. This can be seen from (3.19) since wall incidences of stochastic Lagrangian particles for near times $s\simeq s^{\prime }$ are correlated for same wall (homohedral) incidences but anti-correlated for opposite wall (heterohedral) incidences. Indeed, for $s=s^{\prime }$

(3.25) $$\begin{eqnarray}p_{z}(z,s\mid x,^{\prime },y^{\prime },z^{\prime },s)=\iint _{S}\text{d}x\,\text{d}y\unicode[STIX]{x1D6FF}^{3}(\boldsymbol{x}-\boldsymbol{x}^{\prime })=\unicode[STIX]{x1D6FF}(z-z^{\prime }),\end{eqnarray}$$

so that it must hold on the one hand that

(3.26) $$\begin{eqnarray}\lim _{s^{\prime }\rightarrow s}p_{z}(\pm H/2,s\mid \pm H/2,s^{\prime })=\infty ,\end{eqnarray}$$

and on the other hand that

(3.27) $$\begin{eqnarray}\lim _{s^{\prime }\rightarrow s}p_{z}(\mp H/2,s\mid \pm H/2,s^{\prime })=0.\end{eqnarray}$$

For $s\ll s^{\prime }$ the integrands of all four terms converge to zero as the incidences at widely separated times become independent and the (anti-)correlations decay. Because of the positive sign before the homohedral (variance) terms and the negative sign before the heterohedral (covariance) terms, all four contributions are presumably positive separately. Of the four terms in (3.19), positivity necessarily holds for the homohedral terms

(3.28) $$\begin{eqnarray}\displaystyle & \displaystyle \langle \unicode[STIX]{x1D700}_{T}^{fluc}\rangle _{\infty }^{++}=\lim _{t\rightarrow \infty }\frac{J^{2}}{2t}\text{Var}[\tilde{\ell }_{t,0}^{top}(\boldsymbol{x})]\geqslant 0, & \displaystyle\end{eqnarray}$$

(3.29) $$\begin{eqnarray}\displaystyle & \displaystyle \langle \unicode[STIX]{x1D700}_{T}^{fluc}\rangle _{\infty }^{--}=\lim _{t\rightarrow \infty }\frac{J^{2}}{2t}\text{Var}[\tilde{\ell }_{t,0}^{bot}(\boldsymbol{x})]\geqslant 0, & \displaystyle\end{eqnarray}$$

and is plausibly true by the previous argument for the heterohedral terms

(3.30) $$\begin{eqnarray}\langle \unicode[STIX]{x1D700}_{T}^{fluc}\rangle _{\infty }^{+-}=-\lim _{t\rightarrow \infty }\frac{J^{2}}{2t}\text{Cov}[\tilde{\ell }_{t,0}^{top}(\boldsymbol{x}),\tilde{\ell }_{t,0}^{bot}(\boldsymbol{x})]=\langle \unicode[STIX]{x1D700}_{T}^{fluc}\rangle _{\infty }^{-+}.\end{eqnarray}$$

Note that

(3.31) $$\begin{eqnarray}\langle \unicode[STIX]{x1D700}_{T}^{fluc}\rangle _{\infty }^{+-}=\langle \unicode[STIX]{x1D700}_{T}^{fluc}\rangle _{\infty }^{-+}\leqslant \frac{\langle \unicode[STIX]{x1D700}_{T}^{fluc}\rangle _{\infty }^{++}+\langle \unicode[STIX]{x1D700}_{T}^{fluc}\rangle _{\infty }^{--}}{2}\end{eqnarray}$$

as a direct consequence of the Young inequality $uv\leqslant (u^{2}+v^{2})/2$ and that

(3.32) $$\begin{eqnarray}\langle \unicode[STIX]{x1D700}_{T}^{fluc}\rangle _{\infty }^{--}=\langle \unicode[STIX]{x1D700}_{T}^{fluc}\rangle _{\infty }^{++}\end{eqnarray}$$

because of the exact symmetry $z\rightarrow -z,u_{z}\rightarrow -u_{z},T\rightarrow -T$ of the Boussinesq system with 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 fluctuation–dissipation contributions exactly for $\unicode[STIX]{x1D706},\unicode[STIX]{x1D706}^{\prime }=\pm$ as

(3.33a,b ) $$\begin{eqnarray}\langle \unicode[STIX]{x1D700}_{T}^{fluc}\rangle _{\infty }^{\unicode[STIX]{x1D706},\unicode[STIX]{x1D706}^{\prime }}=\frac{J^{2}}{H^{2}}\unicode[STIX]{x1D70F}_{mix}^{\unicode[STIX]{x1D706}\unicode[STIX]{x1D706}^{\prime }},\quad \unicode[STIX]{x1D70F}_{mix}^{\unicode[STIX]{x1D706}\unicode[STIX]{x1D706}^{\prime }}=\unicode[STIX]{x1D706}\unicode[STIX]{x1D706}^{\prime }\int _{-\infty }^{0}\text{d}s[H\cdot p_{z}(\unicode[STIX]{x1D706}H/2,s\mid \unicode[STIX]{x1D706}^{\prime }H/2,0)-1]_{\unicode[STIX]{x1D706},\unicode[STIX]{x1D706}^{\prime }}.\end{eqnarray}$$

Here $\unicode[STIX]{x1D70F}_{mix}^{\unicode[STIX]{x1D706}\unicode[STIX]{x1D706}^{\prime }}$ is an integral correlation time which measures the length of time required for the (anti-)correlations in (3.19) to decay for $|s-s^{\prime }|\rightarrow \infty ,$ that is, the time required for the particle distributions near the walls to relax to uniform $1/H$ distributions. Note that non-uniform particle distributions in these vicinities are required to feed temperature fluctuations into the flow. We use the suffix ‘mix’ because, as we see shortly, $\unicode[STIX]{x1D70F}_{mix}^{\unicode[STIX]{x1D706}\unicode[STIX]{x1D706}^{\prime }}$ also has the meaning of a near-wall mixing time of a passive tracer. The key implication of (3.33) is that long relaxation/mixing times correspond to large thermal dissipation. Summing over $\unicode[STIX]{x1D706},\unicode[STIX]{x1D706}^{\prime }=\pm ,$ we reach the important exact conclusion

(3.34) $$\begin{eqnarray}\unicode[STIX]{x1D700}_{T}=\frac{J^{2}}{H^{2}}\unicode[STIX]{x1D70F}_{mix},\end{eqnarray}$$

where $\unicode[STIX]{x1D70F}_{mix}=\unicode[STIX]{x1D70F}_{mix}^{hom}+\unicode[STIX]{x1D70F}_{mix}^{het}$ is the sum of all correlation/mixing times with

(3.35a,b ) $$\begin{eqnarray}\unicode[STIX]{x1D70F}_{mix}^{hom}=\unicode[STIX]{x1D70F}_{mix}^{++}+\unicode[STIX]{x1D70F}_{mix}^{--},\quad \unicode[STIX]{x1D70F}_{mix}^{het}=\unicode[STIX]{x1D70F}_{mix}^{+-}+\unicode[STIX]{x1D70F}_{mix}^{-+},\end{eqnarray}$$

and

(3.36) $$\begin{eqnarray}\unicode[STIX]{x1D70F}_{mix}^{het}\leqslant \unicode[STIX]{x1D70F}_{mix}^{hom}.\end{eqnarray}$$

The scaling of $\unicode[STIX]{x1D700}_{T}$ with physical parameters is therefore completely determined by the scaling of the total time $\unicode[STIX]{x1D70F}_{mix}.$

It is thus important to note that this time $\unicode[STIX]{x1D70F}_{mix}$ 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 thermal diffusivity of the fluid, or, equivalently, whose Schmidt number is equal to the fluid Prandtl number. If this tracer is released into the Rayleigh–Bénard cell at time $s$ with initial concentration $c_{s}(\boldsymbol{x},s)$ (mass per volume), then the concentration $c_{s}(\boldsymbol{x},t)$ at later times $t$ satisfies the passive advection–diffusion equation

(3.37) $$\begin{eqnarray}\unicode[STIX]{x2202}_{t}c_{s}+\boldsymbol{u}\boldsymbol{\cdot }\unicode[STIX]{x1D735}c_{s}=\unicode[STIX]{x1D705}\unicode[STIX]{x0394}c_{s}.\end{eqnarray}$$

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 analogous to (3.3) applies to any passive scalar, this means that the tracer concentration at time $t>s$ is given by the formula

(3.38) $$\begin{eqnarray}c_{s}(\boldsymbol{x},t)=\int \text{d}^{3}x_{s}c_{s}(\boldsymbol{x}_{s},s)p(\boldsymbol{x}_{s},s\mid \boldsymbol{x},t).\end{eqnarray}$$

Now assume that the initial concentration at time $s$ is in the form of an infinitesimally thin sheet released at the wall $z=\unicode[STIX]{x1D706}H/2,$ or

(3.39) $$\begin{eqnarray}c_{s}(\boldsymbol{x}_{s},s)=(1/A)\unicode[STIX]{x1D6FF}(z_{s}-\unicode[STIX]{x1D706}H/2).\end{eqnarray}$$

The normalization $1/A$ assumes that the total initial mass of the tracer is unity. In that case, it is easy to see from (3.38) for $t=0$ , (3.39), and the definition (3.20) of the vertical transition probability density that for any $s<0$

(3.40) $$\begin{eqnarray}p_{z}(\unicode[STIX]{x1D706}H/2,s\mid \unicode[STIX]{x1D706}^{\prime }H/2,0)=\iint _{S}\text{d}x\,\text{d}yc_{s}(\unicode[STIX]{x1D706}^{\prime }H/2,x,y,0).\end{eqnarray}$$

In other words, the transition probability that appears in the definition (3.33) of $\unicode[STIX]{x1D70F}_{mix}^{\unicode[STIX]{x1D706}\unicode[STIX]{x1D706}^{\prime }}$ has a direct physical interpretation as the integrated mass density measured at time $0$ on the wall $z=\unicode[STIX]{x1D706}^{\prime }H/2$ of a tracer that was released uniformly spread on the wall $z=\unicode[STIX]{x1D706}H/2$ at the earlier time $s<0.$ Thus, $\unicode[STIX]{x1D70F}_{mix}^{\unicode[STIX]{x1D706}\unicode[STIX]{x1D706}^{\prime }}$ is nothing other than the integral mixing time required for the integrated mass density of the tracer at the wall $z=\unicode[STIX]{x1D706}^{\prime }H/2$ measured at time 0 to its achieve its final uniform value $1/H$ as the release time $s\rightarrow -\infty .$ Although we derived our exact relation (3.34) between thermal dissipation rate $\unicode[STIX]{x1D700}_{T}$ and relaxation time $\unicode[STIX]{x1D70F}_{mix}$ 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 consider the case of pure thermal conduction, where $\boldsymbol{u}\equiv 0$ identically. It may be shown from formula (3.33) that for pure conduction

(3.41a,b ) $$\begin{eqnarray}\unicode[STIX]{x1D70F}_{mix}^{hom}=\frac{2}{3}\frac{H^{2}}{\unicode[STIX]{x1D705}},\quad \unicode[STIX]{x1D70F}_{mix}^{het}=\frac{1}{3}\frac{H^{2}}{\unicode[STIX]{x1D705}}.\end{eqnarray}$$

See appendix B. For pure conduction the near-wall mixing times scale as the time to diffuse across the cell height $H$ , reproducing the exact result for that problem that $\unicode[STIX]{x1D700}_{T}=J^{2}/\unicode[STIX]{x1D705}$ . Consistent with the general inequality (3.31), $\unicode[STIX]{x1D70F}_{mix}^{het}<\unicode[STIX]{x1D70F}_{mix}^{hom}$ . The fact that diffusive mixing is twice as fast at the opposite wall than at the wall where the tracer was released can be understood from the fact the tracer is already substantially mixed when it first diffuses across distance $H$ to the opposite wall, but the tracer must then diffuse back the distance $H$ to the original site of release in order to be mixed there. In thermal convection, equation (3.40) provides a means, in principle, to measure the near-wall mixing time $\unicode[STIX]{x1D70F}_{mix}$ in a laboratory experiment, by releasing a stream of such tracers at the top or bottom wall of the cell and then measuring their concentrations at both top and bottom walls at some much later time, designated as ‘time 0’. In fact, there are other methods for empirical determination of $\unicode[STIX]{x1D70F}_{mix}$ either by laboratory experiment or numerical simulation, which are probably more convenient. However, we shall delay our discussion of such measurement procedures (§ 4.4) until after we have discussed fully the physical implications of our exact relationship (§§ 4.1–4.3).

3.2 Standard Rayleigh–Bénard convection

Having completed our discussion of the stochastic Lagrangian representation and the FDR for flux b.c., we now turn to the standard Rayleigh–Bénard problem with mixed boundary conditions (2.4)–(2.6).

A formula for the temperature field analogous to (3.3) can be derived as a special case of the general formula (II:3.29) of paper II, as

(3.42) $$\begin{eqnarray}\displaystyle T(\boldsymbol{x},t) & = & \displaystyle \mathbb{E}\left[T_{0}(\tilde{\unicode[STIX]{x1D743}}_{t,0}(\boldsymbol{x}))+\int _{0}^{t}J(\tilde{\unicode[STIX]{x1D743}}_{t,s}(\boldsymbol{x}),s)\,\hat{\text{d}}\tilde{\ell }_{t,s}^{top}(\boldsymbol{x})-\int _{0}^{t}J(\tilde{\unicode[STIX]{x1D743}}_{t,s}(\boldsymbol{x}),s)\,\hat{\text{d}}\tilde{\ell }_{t,s}^{bot}(\boldsymbol{x})\right]\nonumber\\ \displaystyle & = & \displaystyle \int \text{d}^{3}x_{0}T_{0}(\boldsymbol{x}_{0})p(\boldsymbol{x}_{0},0\mid \boldsymbol{x},t)\nonumber\\ \displaystyle & & \displaystyle -\,\mathop{\sum }_{\unicode[STIX]{x1D706}=\pm 1}\unicode[STIX]{x1D706}\int _{0}^{t}\text{d}s\iint _{S}\text{d}x^{\prime }\,\text{d}y^{\prime }J\left(x^{\prime },y^{\prime },\unicode[STIX]{x1D706}\frac{H}{2},s\right)p\left(x^{\prime },y^{\prime },\unicode[STIX]{x1D706}\frac{H}{2},s\biggm\vert\boldsymbol{x},t\right),\qquad\end{eqnarray}$$

where $J(\boldsymbol{x},t)=u_{z}(\boldsymbol{x},t)T(\boldsymbol{x},t)-\unicode[STIX]{x1D705}\unicode[STIX]{x2202}_{z}T(\boldsymbol{x},t)$ is the vertical heat flux, which becomes purely conductive at the top/bottom walls where $u_{z}\equiv 0$ . This representation has a hybrid Eulerian–Lagrangian character, since it involves both the particle probabilities $p(\boldsymbol{x}^{\prime },t^{\prime }\mid \boldsymbol{x},t)$ and the Eulerian field $J(\boldsymbol{x},t).$ A purely Lagrangian representation can be obtained 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 stochastic representation provides, however, only a strict lower bound on the thermal dissipation rate, 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) of paper II. In the steady-state (infinite-time), spatially local form analogous to (3.18) for the case of flux b.c., this FDR is

(3.43) $$\begin{eqnarray}\lim _{t\rightarrow \infty }\frac{1}{2t}\text{Var}\left[\int _{0}^{t}J(\tilde{\unicode[STIX]{x1D743}}_{t,s}(\boldsymbol{x}),s)\,\hat{\text{d}}\tilde{\ell }_{t,s}^{top}(\boldsymbol{x})-\int _{0}^{t}J(\tilde{\unicode[STIX]{x1D743}}_{t,s}(\boldsymbol{x}),s)\,\hat{\text{d}}\tilde{\ell }_{t,s}^{bot}(\boldsymbol{x})\right]=\langle \unicode[STIX]{x1D705}|\unicode[STIX]{x1D735}T|^{2}\rangle _{V,\infty }.\end{eqnarray}$$

As in the case of (3.18), the contributions of the initial data $T_{0}$ can be argued to become a spatial constant at long times and the limit becomes independent of $\boldsymbol{x}$ by the ergodicity of the stochastic Lagrangian flow. Arguments like those in § 3.1 and appendix A imply that

(3.44) $$\begin{eqnarray}\displaystyle & & \displaystyle \langle \unicode[STIX]{x1D705}|\unicode[STIX]{x1D735}T|^{2}\rangle _{V,\infty }=\lim _{t\rightarrow \infty }\frac{1}{Vt}\int _{-t}^{0}\text{d}s^{\prime }\int _{-\infty }^{0}\,\text{d}s\iint _{S}\text{d}x^{\prime \prime }\,\text{d}y^{\prime \prime }\iint _{S}\text{d}x^{\prime }\,\text{d}y^{\prime }\nonumber\\ \displaystyle & & \displaystyle \quad \times \,\mathop{\sum }_{\unicode[STIX]{x1D706},\unicode[STIX]{x1D706}^{\prime }=\pm 1}\unicode[STIX]{x1D706}\unicode[STIX]{x1D706}^{\prime }J\left(x^{\prime \prime },y^{\prime \prime },\unicode[STIX]{x1D706}^{\prime }\frac{H}{2},s^{\prime }\right)J\left(x^{\prime },y^{\prime },\unicode[STIX]{x1D706}\frac{H}{2},s\right)\nonumber\\ \displaystyle & & \displaystyle \quad \times \,\left[p\left(x^{\prime },y^{\prime },\unicode[STIX]{x1D706}\frac{H}{2},s\biggm\vert\hspace{0.0pt}x^{\prime \prime },y^{\prime \prime },\unicode[STIX]{x1D706}^{\prime }\frac{H}{2},0\right)-\frac{1}{V}\right]_{\unicode[STIX]{x1D706},\unicode[STIX]{x1D706}^{\prime }}\end{eqnarray}$$

analogous to (3.18), (3.19) for the flux b.c. case. The FDR (3.44), just like the representation (3.42) for the temperature, has a mixed Eulerian–Lagrangian character. It is consistent with the Eulerian balance relation $\unicode[STIX]{x1D700}_{T}=J\unicode[STIX]{x0394}T/H$ in (2.16) if one notes that for all values of $x^{\prime \prime },y^{\prime \prime }$

(3.45) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x0394}T & = & \displaystyle \mathop{\sum }_{\unicode[STIX]{x1D706},\unicode[STIX]{x1D706}^{\prime }=\pm 1}\unicode[STIX]{x1D706}\unicode[STIX]{x1D706}^{\prime }\int _{-\infty }^{0}\text{d}s\iint _{S}\text{d}x^{\prime }\,\text{d}y^{\prime }J\left(x^{\prime },y^{\prime },\unicode[STIX]{x1D706}\frac{H}{2},s\right)\nonumber\\ \displaystyle & & \displaystyle \qquad \times \,\left[p\left(x^{\prime },y^{\prime },\unicode[STIX]{x1D706}\frac{H}{2},s\biggm\vert\hspace{0.0pt}x^{\prime \prime },y^{\prime \prime },\unicode[STIX]{x1D706}^{\prime }\frac{H}{2},0\right)-\frac{1}{V}\right]\end{eqnarray}$$

by taking the (Cesàro-sense) limit $t\rightarrow \infty$ in the (3.42) for points $\boldsymbol{x}$ on the top or bottom wall where the temperature is held fixed. Here we used the asymptotic result

(3.46) $$\begin{eqnarray}\int _{0}^{t}\text{d}s\iint _{S}\text{d}x^{\prime }\,\text{d}y^{\prime }J\left(x^{\prime },y^{\prime },\unicode[STIX]{x1D706}\frac{H}{2},s\right)\sim Jt\end{eqnarray}$$

as $t\rightarrow \infty$ for $\unicode[STIX]{x1D706}=\pm$ with $J$ the time and area average of $J(x,y,z,t)$ (which is independent of $z$ ) in order to introduce the term $-1/V$ into the square bracket in (3.45). The result (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) into four different terms,

(3.47) $$\begin{eqnarray}\langle \unicode[STIX]{x1D705}|\unicode[STIX]{x1D735}T|^{2}\rangle _{V,\infty }=\mathop{\sum }_{\unicode[STIX]{x1D706},\unicode[STIX]{x1D706}^{\prime }=\pm }\langle \unicode[STIX]{x1D700}_{T}^{fluc}\rangle _{\infty }^{\unicode[STIX]{x1D706}\unicode[STIX]{x1D706}^{\prime }},\end{eqnarray}$$

where the four ‘fluctuational dissipations’ are the limiting covariances

(3.48) $$\begin{eqnarray}\langle \unicode[STIX]{x1D700}_{T}^{fluc}\rangle _{\infty }^{\unicode[STIX]{x1D706}\unicode[STIX]{x1D706}^{\prime }}=\lim _{t\rightarrow \infty }\frac{\unicode[STIX]{x1D706}\unicode[STIX]{x1D706}^{\prime }}{2t}\text{Cov}\left[\int _{0}^{t}J(\tilde{\unicode[STIX]{x1D743}}_{t,s}(\boldsymbol{x}),s)\,\hat{\text{d}}\tilde{\ell }_{t,s}^{\unicode[STIX]{x1D706}}(\boldsymbol{x}),\int _{0}^{t}J(\tilde{\unicode[STIX]{x1D743}}_{t,s}(\boldsymbol{x}),s)\,\hat{\text{d}}\tilde{\ell }_{t,s}^{\unicode[STIX]{x1D706}^{\prime }}(\boldsymbol{x})\right].\end{eqnarray}$$

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 again write these four terms as

(3.49) $$\begin{eqnarray}\langle \unicode[STIX]{x1D700}_{T}^{fluc}\rangle _{\infty }^{\unicode[STIX]{x1D706}\unicode[STIX]{x1D706}^{\prime }}=\frac{J^{2}}{H^{2}}\unicode[STIX]{x1D70F}_{mix}^{\unicode[STIX]{x1D706}\unicode[STIX]{x1D706}^{\prime }},\quad \unicode[STIX]{x1D706},\unicode[STIX]{x1D706}^{\prime }=\pm ,\end{eqnarray}$$

where the factors $\unicode[STIX]{x1D70F}_{mix}^{\unicode[STIX]{x1D706}\unicode[STIX]{x1D706}^{\prime }}$ 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, consider the tracer released at time $s$ as a point mass at one of the top or bottom walls:

(3.50) $$\begin{eqnarray}c_{s}(\boldsymbol{x}_{s},s)=\unicode[STIX]{x1D6FF}(x_{s}-x^{\prime })\unicode[STIX]{x1D6FF}(y_{s}-y^{\prime })\unicode[STIX]{x1D6FF}(z_{s}-\unicode[STIX]{x1D706}H/2).\end{eqnarray}$$

In that case, the tracer concentration field at time $t>s$ is

(3.51) $$\begin{eqnarray}c_{s}(\boldsymbol{x},t)=p(x^{\prime },y^{\prime },\unicode[STIX]{x1D706}H/2,s\mid \boldsymbol{x},t).\end{eqnarray}$$

The tracer concentration at time $t=0$ and evaluated at the top/bottom walls is thus equal to the transition probability density that appears in the integrand of the FDR (3.44). These concentrations will mix to values $1/V$ for $s\ll t$ and thus the times $\unicode[STIX]{x1D70F}_{mix}^{\unicode[STIX]{x1D706}\unicode[STIX]{x1D706}^{\prime }}$ introduced in (3.49) represent integral mixing times near the walls, averaged over both release 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 relates thermal dissipation rate and the near-wall mixing time:

(3.52) $$\begin{eqnarray}\unicode[STIX]{x1D700}_{T}=\frac{J^{2}}{H^{2}}\unicode[STIX]{x1D70F}_{mix},\end{eqnarray}$$

identical in form to that derived for flux b.c. From our previous discussion, the time $\unicode[STIX]{x1D70F}_{mix}$ for temperature b.c. is clearly much harder to measure empirically or to calculate theoretically, because the boundary values of $J(\boldsymbol{x},t)$ in space and time are unknown until the dynamical equations are solved. However, it is at least plausible that the turbulent Rayleigh–Bénard system with imposed temperature difference $\unicode[STIX]{x0394}T$ will behave very similarly to the system with imposed flux $J_{in},$ if the latter is adjusted so that $\unicode[STIX]{x0394}T$ remains unchanged as $\unicode[STIX]{x1D708},\unicode[STIX]{x1D705}\rightarrow 0.$ This must be the case for any result based solely upon the mean balance equations in § 2.2, since these are identical for the two boundary conditions. We expect that it is likely that $\unicode[STIX]{x1D70F}_{mix}$ scales with physical parameters such as $Ra,$ $Pr$ and $\unicode[STIX]{x1D6E4}$ in an identical manner for constant-flux Rayleigh–Bénard convection and for the standard problem with temperature b.c. at the top and bottom walls. Needless to say, this is open to question.

4.1 Mixing time and Nusselt–Rayleigh scaling

An immediate implication of the Lagrangian representations of the temperature difference, equation (3.22) for flux b.c. and (3.45) for temperature b.c., is the important exact result

(4.1) $$\begin{eqnarray}\unicode[STIX]{x1D70F}_{mix}=H\unicode[STIX]{x0394}T/J.\end{eqnarray}$$

This may be written as $\unicode[STIX]{x1D70F}_{mix}=H/U_{flux},$ by defining a thermal-flux velocity

(4.2) $$\begin{eqnarray}U_{flux}=J/\unicode[STIX]{x0394}T,\end{eqnarray}$$

which is the equivalent fluid velocity required to maintain a flux $J$ by convection of a temperature difference $\unicode[STIX]{x0394}T$ . The definition of Nusselt number $Nu=J/(\unicode[STIX]{x1D705}\unicode[STIX]{x0394}T/H)$ implies

(4.3a,b ) $$\begin{eqnarray}Nu=\frac{\unicode[STIX]{x1D70F}_{diff}}{\unicode[STIX]{x1D70F}_{mix}},\quad \unicode[STIX]{x1D70F}_{diff}=H^{2}/\unicode[STIX]{x1D705},\end{eqnarray}$$

where we now use $\unicode[STIX]{x1D70F}_{diff}$ 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 for pure conduction to the mixing time for convection. Thus, for large $Nu$ one has $\unicode[STIX]{x1D70F}_{mix}\ll \unicode[STIX]{x1D70F}_{diff}=H^{2}/\unicode[STIX]{x1D705}.$ Likewise, from the definitions of $U,U_{\ast },$ and $U_{flux}$ and (2.23) it follows immediately that

(4.4) $$\begin{eqnarray}U_{\ast }^{3}=U_{flux}U^{2}\Longrightarrow \frac{U_{flux}}{U}=\left(\frac{U_{\ast }}{U}\right)^{3}=\frac{\unicode[STIX]{x1D700}_{T}}{U(\unicode[STIX]{x0394}T)^{2}/H}.\end{eqnarray}$$

Hence, in the Spiegel (Reference Spiegel1971) scenario $U_{flux}\simeq U_{\ast }\simeq U$ (equality up to absolute constants) at high- $Ra$ and all of the times $\unicode[STIX]{x1D70F}_{mix},$ $H/U_{\ast }$ , and $H/U$ are of the same order. The theory of Kraichnan (Reference Kraichnan1962) assumed instead that the thermal boundary layer would become turbulent when $u_{\unicode[STIX]{x1D70F}}\unicode[STIX]{x1D6FF}_{T}/\unicode[STIX]{x1D705}=Pe_{T},$ where $Pe_{T}$ is a critical or transitional value (estimated to be approximately 3) of the Péclet number based upon the friction velocity $u_{\unicode[STIX]{x1D70F}}$ at the top/bottom walls and the thermal boundary layer thickness $\unicode[STIX]{x1D6FF}_{T}$ . Because of the standard relation $J=\unicode[STIX]{x1D705}\unicode[STIX]{x0394}T/2\unicode[STIX]{x1D6FF}_{T}$ , which essentially defines $\unicode[STIX]{x1D6FF}_{T}$ , Kraichnan’s assumption can be restated precisely in our framework as the conjecture that $U_{flux}=u_{\unicode[STIX]{x1D70F}}/2Pe_{T}$ . Appeal to the standard logarithmic law-of-the-wall led Kraichnan (Reference Kraichnan1962) to conclude that $u_{rms}\propto U/\ln (Ra)$ and $u_{\unicode[STIX]{x1D70F}}\propto U/\ln ^{3/2}(Ra)$ , so that these velocities differ only by a logarithmic factor. The theory of Grossmann & Lohse (Reference Grossmann and Lohse2012) likewise assumes turbulent boundary layers in an ‘ultimate regime’ at very high $Ra$ , but obtain the slightly different predictions that $u_{rms}\propto U$ (see (22) in Grossmann & Lohse Reference Grossmann and Lohse2011) and $u_{\unicode[STIX]{x1D70F}}\propto U/W(Re),$ where $W$ is Lambert’s function (equation (9) in Grossmann & Lohse Reference Grossmann and Lohse2011). As in the theory of Kraichnan (1962), the effective scaling exponents become indistinguishable from the dimensional predictions of Spiegel at extremely high Rayleigh numbers. For any other possible scaling behaviour than Kraichnan–Spiegel-type, $U_{flux}\ll U_{\ast }\ll U$ and $\unicode[STIX]{x1D70F}_{mix}\gg H/U_{\ast }\gg H/U,$ by factors growing faster than logarithms of $Ra$ .

The most illuminating form of the relation $\unicode[STIX]{x1D70F}_{mix}=H\unicode[STIX]{x0394}T/J$ in (4.1) is obtained by combining it with the result from the non-dimensionalized balance equations (2.18) that $J/U\unicode[STIX]{x0394}T=Nu/\sqrt{RaPr},$ that is, the ratio of the true Nusselt number and the Spiegel prediction. From this it is easy to see that

(4.5a,b ) $$\begin{eqnarray}\unicode[STIX]{x1D70F}_{mix}=\unicode[STIX]{x1D70F}_{free}\frac{\sqrt{RaPr}}{Nu},\quad \unicode[STIX]{x1D70F}_{free}:=\frac{H}{U}.\end{eqnarray}$$

Thus, $\unicode[STIX]{x1D70F}_{mix}$ differs from the free-fall time $\unicode[STIX]{x1D70F}_{free}$ by precisely the same factor that the Spiegel dimensional prediction differs from the true Nusselt number. This is our key conclusion for Rayleigh–Bénard turbulence. Recall that the free-fall velocity $U$ is observed empirically to be of the same order as the velocity $U_{lsc}$ of the large-scale circulation or global wind (Niemela & Sreenivasan Reference Niemela and Sreenivasan2002; Ahlers et al. Reference Ahlers, Grossmann and Lohse2009), so that the free-fall time $\unicode[STIX]{x1D70F}_{free}$ is approximately of the same order as the large-scale circulation time $\unicode[STIX]{x1D70F}_{lsc}=L/U_{lsc}$ . For example, it follows directly from the definition of the free-fall velocity that the $U$ -based Reynolds number is $UH/\unicode[STIX]{x1D708}=Ra^{1/2}Pr^{-1/2},$ whereas experiments show that $U_{lsc}H/\unicode[STIX]{x1D708}$ scales very similarly in $Ra$ , with an exponent only slightly smaller than $1/2$ (Niemela & Sreenivasan (Reference Niemela and Sreenivasan2003, appendix D); Ahlers et al. (Reference Ahlers, Grossmann and Lohse2009), § IV) and numerical simulations give similar results (e.g. Scheel & Schumacher (Reference Scheel and Schumacher2014), § 2). As mentioned above, $U_{lsc}\simeq U$ is also predicted by Grossmann & Lohse (Reference Grossmann and Lohse2012) at very high $Ra.$ Our key conclusion is thus nearly equivalent to the statement that existence of dissipative anomalies and validity of Spiegel scaling requires $\unicode[STIX]{x1D70F}_{mix}$ of order the large-scale turnover time $\unicode[STIX]{x1D70F}_{lsc},$ whereas for any other scaling $\unicode[STIX]{x1D70F}_{mix}\gg \unicode[STIX]{x1D70F}_{free}\simeq \unicode[STIX]{x1D70F}_{lsc}$ (and, in fact, even $\unicode[STIX]{x1D70F}_{mix}\gg H/U_{\ast }$ ) and a passive tracer released at one wall remains unmixed near both walls for many large-scale circulation times. It is a priori quite surprising that wall incidences could remain correlated over so many large-scale circulation times. This requires some discussion of the underlying Lagrangian mechanisms.

4.2 Stochastic Lagrangian dynamics and the ‘mixing zone’

As we have argued here and in papers I and II, the extension of Lagrangian methods to realistic, non-ideal fluid flows naturally requires stochastic particle trajectories. These methods take their simplest form, furthermore, when the stochastic trajectories are evolved backward in time (cf. our (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 both advection and diffusion is an average of its past values rather than its future values. Here we shall apply the stochastic Lagrangian framework to gain new insight into the thermal dissipation physics of turbulent Rayleigh–Bénard convection. While forward stochastic particle evolution (or even deterministic evolution of a diffusive, passively advected tracer, as in (3.37)–(3.40)) can be substituted for the backward stochastic evolution, this involves considerable more complexity of the description and loss of insight. The time asymmetry between forward and backward time evolutions is a consequence of the fundamental irreversibility of the dissipation process. As we shall see below, one of the essential features of the Lagrangian description is this asymmetry in time.

Thermal plumes play a well-known role in the Lagrangian dynamics of the temperature field (for recent reviews, see Ahlers et al. (Reference Ahlers, Grossmann and Lohse2009), Chillà & Schumacher (Reference Chillà and Schumacher2012) and references therein). In particular, plumes should transport thermal fluctuations from the boundary into the interior and create large values of $\langle \unicode[STIX]{x1D700}_{T}^{fluc}(\boldsymbol{x})\rangle _{t}$ at points $\boldsymbol{x}$ within the bulk of the flow well way from the walls. If a plume reaches the interior point $\boldsymbol{x}$ within the time $t,$ then going backward in time the particle is transported to very near the wall of origin of the plume, at which point the stochastic noise $\propto \sqrt{\unicode[STIX]{x1D705}}$ can then allow the particle to hit the wall. Note that, since the fluid velocity $\boldsymbol{u}$ vanishes smoothly at the wall for $\unicode[STIX]{x1D708},\unicode[STIX]{x1D705}>0,$ advection on its own can never produce a wall incidence in finite time. The non-vanishing of the boundary local time densities $\tilde{\ell }_{t,0}^{top/bot}(\boldsymbol{x})$ for points $\boldsymbol{x}$ in the bulk of the flow is thus presumably due largely to the mediation of the plumes when $Ra\gg 1$ . In particular, one expects enhanced values of $\langle \unicode[STIX]{x1D700}_{T}^{fluc}(\boldsymbol{x})\rangle _{t}$ at times $t$ for which $\boldsymbol{x}$ is within a plume. Observe that enhanced values of the thermal dissipation $\unicode[STIX]{x1D700}_{T}(\boldsymbol{x},t)$ itself are observed within plumes (e.g. see Emran & Schumacher Reference Emran and Schumacher2012).

However, to understand possible mechanisms that could give $\unicode[STIX]{x1D70F}_{mix}\gg \unicode[STIX]{x1D70F}_{free}$ it is more important to understand the Lagrangian dynamics that could lead stochastic particles to escape from the top/bottom wall backward in time or, equivalently, to hit the top/bottom walls forward in time. Note that $\unicode[STIX]{x1D70F}_{mix}^{hom}\gg \unicode[STIX]{x1D70F}_{free}$ means that a particle currently at the top/bottom wall had a high probability to be at the same wall over a very long period of earlier times (many free-fall times). Similarly, $\unicode[STIX]{x1D70F}_{mix}^{het}\gg \unicode[STIX]{x1D70F}_{free}$ means that a particle currently at the top/bottom wall had a very low probability to be at the opposite wall over a very long period of earlier times (many free-fall times). Thus, turbulent convection must not be very efficient at bringing particles close to the wall if $\unicode[STIX]{x1D70F}_{mix}\gg \unicode[STIX]{x1D70F}_{free}$ , else the particles would readily escape backward in time. The obvious fluid motions which carry stochastic Lagrangian particles from interior points to points close to the wall consist of the large-scale circulation and ‘old’ plumes which have completed a transit between top and bottom. These motions will carry particles between points near the top and bottom in a turnover time $\unicode[STIX]{x1D70F}_{lsc},$ which is only slightly larger than $\unicode[STIX]{x1D70F}_{free}.$ While much previous work has focused on the role of thermal plumes, the backward stochastic evolution reveals a direct connection of the thermal dissipation rate with these weaker fluid motions.

The fact that $\unicode[STIX]{x1D70F}_{mix}\gg \unicode[STIX]{x1D70F}_{free}$ means that these motions must not typically bring the particles close enough to actually hit the wall, and instead the interior particles will circulate many, many free-fall times before finally hitting the top or bottom wall. In order to be guaranteed to hit the wall in time $\unicode[STIX]{x1D70F}_{mix},$ the particles must be advected to a distance $\ell _{T}=\sqrt{\unicode[STIX]{x1D705}\unicode[STIX]{x1D70F}_{mix}}$ from the top/bottom wall, since thermal diffusion then suffices to carry the particle the remaining distance in time $\unicode[STIX]{x1D70F}_{mix}$ . We shall thus refer to $\ell _{T}$ as the ‘thermal diffusion length’, over the mixing time $\unicode[STIX]{x1D70F}_{mix}.$ Using (4.3) gives

(4.6) $$\begin{eqnarray}\ell _{T}=\sqrt{\unicode[STIX]{x1D705}\unicode[STIX]{x1D70F}_{mix}}=H/\sqrt{Nu}.\end{eqnarray}$$

Consider any distance $\unicode[STIX]{x1D6FF}$ asymptotically much smaller than $\ell _{T}$ for $Ra\gg 1,$ in particular, the traditional outer thermal boundary layer thickness $\unicode[STIX]{x1D6FF}_{T}=H/2Nu.$ Our results imply that bulk fluid particles cannot reach such a distance $\unicode[STIX]{x1D6FF}$ from the top/bottom walls in any time less than $\unicode[STIX]{x1D70F}_{mix},$ with $\unicode[STIX]{x1D70F}_{mix}\gg \unicode[STIX]{x1D70F}_{free}$ if KS scaling fails. For example, the time for particles at the top/bottom wall to diffuse the distance $\unicode[STIX]{x1D6FF}_{T}$ across the standard thermal boundary layer is $\unicode[STIX]{x1D6FF}_{T}^{2}/\unicode[STIX]{x1D705}=\unicode[STIX]{x1D70F}_{mix}/4Nu,$ which is much shorter than the time $\unicode[STIX]{x1D70F}_{mix}$ to transit effectively the bulk.

The thermal diffusion length $\ell _{T}$ appears to be a new length scale, not previously identified in Rayleigh–Bénard convection. The region within distance $\ell _{T}$ of the top/bottom walls but further away from those walls than $\unicode[STIX]{x1D6FF}_{T}$ might possibly be identified with the ‘plume mixing zone’ proposed by Castaing et al. (Reference Castaing, Gunaratne, Kadanoff, Libchaber and Heslot1989); see also Procaccia et al. (Reference Procaccia, Ching, Constantin, Kadanoff, Libchaber and Wu1991). The inner boundary of this ‘mixing zone’ was considered to be $\unicode[STIX]{x1D6FF}_{T},$ while the outer boundary distance (denoted $\ell _{m}$ or $d_{m}$ ) was predicted to scale as $\ell _{m}\sim H/Ra^{1/7}.$ Since Castaing et al. (Reference Castaing, Gunaratne, Kadanoff, Libchaber and Heslot1989) and Procaccia et al. (Reference Procaccia, Ching, Constantin, Kadanoff, Libchaber and Wu1991) also predicted $Nu\sim Ra^{2/7},$ this gives $\ell _{m}\sim H/\sqrt{Nu}$ , consistent with our result (4.6) for the scaling of $\ell _{T}$ . If we adopt the terminology of Castaing et al. (Reference Castaing, Gunaratne, Kadanoff, Libchaber and Heslot1989), then it is in the ‘central region’, at distances further than $\ell _{T}$ from the top/bottom walls, where particles must be ‘trapped’ for many free-fall times, if KS scaling fails. Note, however, that Castaing et al. (Reference Castaing, Gunaratne, Kadanoff, Libchaber and Heslot1989) proposed the ‘mixing zone’ to be a region with a small fraction of the volume occupied by thin ‘plumes’ with a thickness $\unicode[STIX]{x1D6FF}_{T},$ whose large temperature differences relative to the surrounding fluid gave them a strong vertical motion. In this ‘mixing zone’ of thickness $\ell _{m}$ (or $d_{m}$ ), the heat flux was considered to be carried mainly by this buoyant motion of ‘plumes’, whereas in the thermal boundary layer the heat transport was purely conductive. By contrast, the thermal diffusion length $\ell _{T}$ in our analysis is identified precisely by the condition that diffusion alone suffices to transport temperature fluctuations across the ‘mixing zone’ over the (very long) time scale $\unicode[STIX]{x1D70F}_{mix}.$ Thus, the agreement in scaling of $\ell _{m}=d_{m}$ and $\ell _{T}$ with $Nu$ may be purely coincidental. (Let us examine this issue in more detail. The plausible equations $u_{rms}\sim (\unicode[STIX]{x1D6FC}gHT_{rms})^{1/2}$ (3.2) and $J\sim u_{rms}T_{rms}$ (3.3) of Castaing et al. (Reference Castaing, Gunaratne, Kadanoff, Libchaber and Heslot1989) together with their proposed relation $\unicode[STIX]{x1D6FF}_{T}/\ell _{m}\sim T_{rms}/\unicode[STIX]{x0394}T\sim Re^{\unicode[STIX]{x1D6FE}}$ (3.18) imply for a general scaling law $Nu\sim Ra^{\unicode[STIX]{x1D6FD}}$ that $\unicode[STIX]{x1D6FE}=(2\unicode[STIX]{x1D6FD}-1)/3$ (3.4). The condition $\unicode[STIX]{x1D6FE}=-\unicode[STIX]{x1D6FD}/2$ that follows from identifying $\ell _{m}\sim \ell _{T}$ is only satisfied for the choice $\unicode[STIX]{x1D6FD}=2/7$ and thus one does not obtain $\ell _{m}\sim H/\sqrt{Nu}$ in general. A possible way out of this conclusion was indicated by Castaing et al. (Reference Castaing, Gunaratne, Kadanoff, Libchaber and Heslot1989), p. 20: ‘To introduce a different scaling, one would 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 $J\sim u_{rms}T_{rms}f$ where the volume fraction $f\sim Ra^{-\unicode[STIX]{x1D6FF}}.$ If one leaves (3.2) and (3.18) of Castaing et al. (Reference Castaing, Gunaratne, Kadanoff, Libchaber and Heslot1989) unchanged, their (3.4) is replaced by $\unicode[STIX]{x1D700}=(1+\unicode[STIX]{x1D6FE})/2,$ $\unicode[STIX]{x1D6FE}=(2\unicode[STIX]{x1D6FD}+2\unicode[STIX]{x1D6FF}-1)/3,$ where $\unicode[STIX]{x1D700}$ is the Reynolds exponent in $Re_{rms}\sim Ra^{\unicode[STIX]{x1D700}}.$ If one imposes our condition $\unicode[STIX]{x1D6FE}=-\unicode[STIX]{x1D6FD}/2,$ one obtains $\unicode[STIX]{x1D6FF}=(2-7\unicode[STIX]{x1D6FD})/4.$ In that case, $\unicode[STIX]{x1D6FF}<0$ for any $\unicode[STIX]{x1D6FD}>2/7$ and, in particular, for the observed values $\unicode[STIX]{x1D6FD}\doteq 0.3-0.43$ 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. (Reference Castaing, Gunaratne, Kadanoff, Libchaber and Heslot1989) is wrong (we find the last the most dubious) or else the identification $\ell _{T}\sim \ell _{m}$ is generally invalid.)

Another important point of comparison for our Lagrangian relation (4.5) is theoretical work of Grossmann & Lohse (Reference Grossmann and Lohse2004) to explicate the role of thermal plumes in their ‘unifying theory’ of turbulent Rayleigh–Bénard convection. Assuming laminar scalar boundary layers of thickness $\unicode[STIX]{x1D6FF}_{T},$ their basic conjecture (see Grossmann & Lohse (Reference Grossmann and Lohse2004), equation (36)) is that the plume shedding frequency scales as

(4.7) $$\begin{eqnarray}f_{shed}\sim J/\unicode[STIX]{x1D6FF}_{T}(\unicode[STIX]{x0394}T).\end{eqnarray}$$

Combining this theorized result with our exact relation (4.1), it follows that

(4.8) $$\begin{eqnarray}\unicode[STIX]{x1D70F}_{mix}\sim (Nu)\unicode[STIX]{x1D70F}_{shed},\end{eqnarray}$$

where $\unicode[STIX]{x1D70F}_{shed}=1/f_{shed}$ is the mean time interval between shedding events. Since $\unicode[STIX]{x1D70F}_{diff}=(Nu)^{2}\unicode[STIX]{x1D70F}_{shed}$ (e.g. see again Grossmann & Lohse (Reference Grossmann and Lohse2004), equation (36)), equation (4.8) can be equivalently stated as $\unicode[STIX]{x1D70F}_{mix}\sim \sqrt{\unicode[STIX]{x1D70F}_{shed}\unicode[STIX]{x1D70F}_{diff}}$ . In either form, equation (4.8) implies that $\unicode[STIX]{x1D70F}_{mix}\gg \unicode[STIX]{x1D70F}_{shed}$ for $Nu\gg 1.$ To make connection with either the free-fall time or the large-scale circulation time within the approach of Grossmann & Lohse (Reference Grossmann and Lohse2004), one must relate $\unicode[STIX]{x1D70F}_{shed}$ with the times $\unicode[STIX]{x1D70F}_{free}=H/U$ or $\unicode[STIX]{x1D70F}_{lsc}=H/U_{lsc}.$ This requires a relation between the Nusselt number and the Reynolds and Péclet numbers. By appealing to Prandtl–Blasius laminar boundary layer theory, Grossmann & Lohse (Reference Grossmann and Lohse2004) argued that $Nu\sim \sqrt{fRePr},$ where $f(Pr)$ is a factor which interpolates between $f=1$ for $Pr\ll 1$ and $f\sim Pr^{-1/3}$ for $Pr\gg 1.$ Combining this relation with $\unicode[STIX]{x1D70F}_{diff}=(Nu)^{2}\unicode[STIX]{x1D70F}_{shed},$ Grossmann & Lohse (Reference Grossmann and Lohse2004) derived their (37):

(4.9) $$\begin{eqnarray}\unicode[STIX]{x1D70F}_{shed}\sim H/fU_{lsc}=\unicode[STIX]{x1D70F}_{lsc}/f,\end{eqnarray}$$

so that the shedding time differs from large-scale circulation time only by the factor $f.$ The final implication of the Grossmann & Lohse (Reference Grossmann and Lohse2004) theory for the mixing time is that

(4.10) $$\begin{eqnarray}\unicode[STIX]{x1D70F}_{mix}\sim Nu\unicode[STIX]{x1D70F}_{lsc}/f,\end{eqnarray}$$

with $\unicode[STIX]{x1D70F}_{mix}$ larger than $\unicode[STIX]{x1D70F}_{lsc}$ by the factor $Nu/f.$ Note that this conclusion is expected to be unchanged if the kinetic boundary becomes turbulent while the thermal boundary stays laminar. Indeed, the scaling law $Nu\sim \sqrt{fRePr}$ is also characteristic of the ‘background-dominated’ regime in Grossmann & Lohse (Reference Grossmann and Lohse2011). In the ‘plume-dominated’ regime then instead $Nu\sim (RePr)^{1/3}$ (see (11) of Grossmann & Lohse (Reference Grossmann and Lohse2011)) which is easily checked to imply that $\unicode[STIX]{x1D70F}_{shed}\sim Nu\unicode[STIX]{x1D70F}_{lsc}$ and thus $\unicode[STIX]{x1D70F}_{mix}\sim (Nu)^{2}\unicode[STIX]{x1D70F}_{lsc}$ . 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 (Reference Grossmann and Lohse2004, Reference Grossmann and Lohse2011) hypotheses and our exact relation (4.5) is empirically testable, e.g. see the end of §§ 3.1.2 and 4.4 below. A direct investigation of (4.10) would illuminate the Lagrangian bases of the ‘unifying theory’ of Grossmann and Lohse, which have not yet been subject to stringent test. In fact, it is far from clear to us how to reconcile the quantitative prediction (4.10) with the more physical reasoning invoked by Grossmann & Lohse (Reference Grossmann and Lohse2004) to justify the conclusions (4.7), (4.9) cited above. For example, Grossmann & Lohse (Reference Grossmann and Lohse2004) have equated $\unicode[STIX]{x1D70F}_{shed}$ with the ‘traveling time $\unicode[STIX]{x1D70F}_{travel}\sim L/Uf$ (or $H/U_{lsc}f$ in our notations) of (hot) plumes from the bottom to the top’. Grossmann & Lohse (Reference Grossmann and Lohse2004) also state that, within their boundary-layer analysis, the ‘temperature is assumed to be passive’. With these theoretical assumptions, it is not obvious why the mixing time $\unicode[STIX]{x1D70F}_{mix}$ of a passive scalar released uniformly at one wall should scale any differently than the above ‘travelling time’ $\unicode[STIX]{x1D70F}_{travel}\sim \unicode[STIX]{x1D70F}_{lsc}$ within the Grossmann & Lohse (Reference Grossmann and Lohse2004) theory. The problem of accounting physically for the discrepancy in magnitudes of $\unicode[STIX]{x1D70F}_{mix}$ on the one hand and $\unicode[STIX]{x1D70F}_{free}$ or $\unicode[STIX]{x1D70F}_{lsc}$ on the other hand exists not only within the Grossmann–Lohse phenomenology but whenever theory, experiment, or simulation indicates a Nusselt number $Nu$ much smaller than the Spiegel dimensional prediction $\sqrt{RaPr}.$ We shall discuss below some physical mechanisms that might allow the ratio $\unicode[STIX]{x1D70F}_{mix}/\unicode[STIX]{x1D70F}_{free}$ to become large.

At extremely high Rayleigh numbers, the Grossmann–Lohse theory posits an ‘ultimate regime’ in which both kinetic and thermal boundary layers transition to turbulence with logarithmic profiles. In that case, the thermal boundary layer is predicted by Grossmann & Lohse (Reference Grossmann and Lohse2011) to permeate the cell and attain thickness $\unicode[STIX]{x1D6FF}_{T}\sim H$ . The relation (4.7) which assumed a laminar thermal boundary layer then becomes inapplicable, and presumably must be replaced by

(4.11) $$\begin{eqnarray}f_{shed}\sim J/H(\unicode[STIX]{x0394}T).\end{eqnarray}$$

Together with our basic result (4.1), this leads to the identification

(4.12) $$\begin{eqnarray}\unicode[STIX]{x1D70F}_{mix}\simeq \unicode[STIX]{x1D70F}_{shed}.\end{eqnarray}$$

Grossmann & Lohse (Reference Grossmann and Lohse2011) predict in this regime that $Nu\sim \sqrt{RaPr}{\mathcal{L}}(Re)$ (see their (23)), where ${\mathcal{L}}(Re)$ is a slowly vanishing logarithmic factor. Together with our equation (4.5) this implies that

(4.13) $$\begin{eqnarray}\unicode[STIX]{x1D70F}_{mix}\sim \unicode[STIX]{x1D70F}_{free}/{\mathcal{L}}(Re)\end{eqnarray}$$

so that the mixing time is predicted to be longer than the free-fall time only by the factor $1/{\mathcal{L}}(Re).$ Recall that $\unicode[STIX]{x1D70F}_{free}\simeq \unicode[STIX]{x1D70F}_{lsc}$ in the Grossmann & Lohse (Reference Grossmann and Lohse2011) theory, because of the cancellation of log corrections in their (22). This is an example of a Kraichnan-type theory in which $\unicode[STIX]{x1D70F}_{mix}$ is only logarithmically larger than $\unicode[STIX]{x1D70F}_{free}$ and $\unicode[STIX]{x1D70F}_{lsc}$ . If such a near equality of time scales indeed holds at extremely high $Ra$ , then the Lagrangian mechanisms which make $\unicode[STIX]{x1D70F}_{mix}/\unicode[STIX]{x1D70F}_{free}$ 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 $Nu$ as a function of $Ra$ and $Pr,$ so that we must consider briefly the observations. There seem to have been relatively few empirical studies of turbulent Rayleigh–Bénard convection with well-controlled flux b.c. The 2-D numerical study of Johnston & Doering (Reference Johnston and Doering2009) exhibited a scaling law $Nu\sim Ra^{x}$ with non-KS exponent $x\doteq 0.285$ up to $Ra=10^{10}.$ This study also found identical scaling with temperature b.c. in the turbulent regime and suggested that the two types of boundary conditions would exhibit generally equivalent physical behaviour in the turbulent regime also in three dimensions. A subsequent 3-D simulation of Stevens et al. (Reference Stevens, Lohse and Verzicco2011) with temperature b.c. at the top wall and flux b.c. at the bottom wall provides some corroboration of this hypothesis. With this assumption, we may also compare with the much larger body of work for temperature b.c. In contrast to the empirical results for turbulent dissipation of passive scalars summarized by Donzis, Sreenivasan & Yeung (Reference Donzis, Sreenivasan and Yeung2005), experimental and numerical studies up to Rayleigh numbers of order $10^{10}$ show that thermal dissipation in turbulent Rayleigh–Bénard convection scales as

(4.14) $$\begin{eqnarray}\frac{\unicode[STIX]{x1D700}_{T}}{(\unicode[STIX]{x0394}T)^{2}U/H}\sim Ra^{z},\end{eqnarray}$$

with a negative exponent $z\doteq -0.20$ (Emran & Schumacher Reference Emran and Schumacher2008, Reference Emran and Schumacher2012; He, Tong & Xia Reference He, Tong and Xia2007; He & Tong Reference He and Tong2009). Thus, empirical evidence in this range of Rayleigh numbers does not support existence of a thermal dissipative anomaly. Consistently, these experiments and simulations observe $Nu\sim Ra^{x}$ with exponent $x=0.5+z\doteq 0.3<1/2,$ which represents a 40 % difference from the KS scaling exponent. Experiments at even higher Rayleigh numbers see somewhat larger exponents $x\doteq 0.32$ – $0.43$ but apparently still at least 10–20 % smaller than the KS exponent $x=0.5$ (Ahlers et al. Reference Ahlers, Grossmann and Lohse2009; Chillà & Schumacher Reference Chillà and Schumacher2012). Such deviations apparently persist up to $Ra\sim 10^{17}$ (Niemela et al. Reference Niemela, Skrbek, Sreenivasan and Donnelly2000). On the other hand, two experimental groups that observed the largest exponents (Roche et al. Reference Roche, Gauthier, Kaiser and Salort2010; He et al. Reference He, Funfschilling, Nobach, Bodenschatz and Ahlers2012) have interpreted their results, not implausibly, as transitions to an ‘ultimate regime’ with logarithmic corrections of the sort predicted by Kraichnan (Reference Kraichnan1962), Grossmann & Lohse (Reference Grossmann and Lohse2011) and others.

If an ‘ultimate regime’ does exist at very high Rayleigh numbers, then our Lagrangian analysis suggests the Reynolds number $Re_{\ell }\equiv u(\ell _{T})\ell _{T}/\unicode[STIX]{x1D708}$ as most appropriate to signal the possible transition. Here $u(\ell _{T})$ is a typical (e.g. r.m.s.) velocity at distance $\ell _{T}$ from the top/bottom walls. When the Reynolds number $Re_{\ell }$ becomes critical, then turbulent mixing reaches to eddies at distance $\ell _{T}$ from the top/bottom walls, and one may plausibly expect a strong effect on the mixing rate and perhaps even that $\unicode[STIX]{x1D70F}_{mix}\sim \unicode[STIX]{x1D70F}_{free},$ implying KS scaling. This is similar to the arguments of Niemela & Sreenivasan (Reference Niemela and Sreenivasan2003), Niemela & Sreenivasan (Reference Niemela and Sreenivasan2006) and also Grossmann & Lohse (Reference Grossmann and Lohse2011), He et al. (Reference He, Funfschilling, Nobach, Bodenschatz and Ahlers2012), but replacing the kinetic boundary layer thickness $\unicode[STIX]{x1D6FF}_{v}=aH/\sqrt{Re}$ with the thermal diffusion length $\ell _{T}=bH/\sqrt{Nu}$ . Here we have introduced a dimensionless prefactor $b$ of order unity, whose precise value could be important at moderately large Rayleigh numbers. It must hold that $b\lesssim 1$ , since the time to diffuse across the distance $\ell _{T}$ should be somewhat smaller than $\unicode[STIX]{x1D70F}_{mix}.$ A reasonable choice would be perhaps to set $b=1$ by convention, so that $\ell _{T}^{2}/2\unicode[STIX]{x1D705}=\unicode[STIX]{x1D70F}_{mix}/2.$ Based on the data published in Roche et al. (Reference Roche, Gauthier, Kaiser and Salort2010) and He et al. (Reference He, Funfschilling, Nobach, Bodenschatz and Ahlers2012), one can estimate $Re_{\ell }=u(\ell _{T})\ell _{T}/\unicode[STIX]{x1D708}$ at the apparent onset of a new Nusselt scaling in their experiments to be of the order of several hundreds, in the range where transition to turbulence is expected.

In order to compare our proposed criterion for an ‘ultimate regime’ based upon the thermal diffusion length $\ell _{T}$ with criteria based on the kinetic boundary layer thickness $\unicode[STIX]{x1D6FF}_{v},$ one must know the relative magnitudes of these two lengths. If KS scaling holds, then both $\unicode[STIX]{x1D6FF}_{v}$ and $\ell _{T}$ scale with Rayleigh number as $Ra^{-1/4}$ (up to possible log corrections), so that at fixed Prandtl number one may essentially identify $\unicode[STIX]{x1D6FF}_{v}\sim \ell _{T}.$ However, alternate theories predict $Nu\sim Ra^{x}$ with $x<1/2$ , and the rigorous upper bound of Otero et al. (Reference Otero, Wittenberg, Worthing and Doering2002) for flux b.c. requires $x\leqslant 1/2$ . In that case, $\ell _{T}/\unicode[STIX]{x1D6FF}_{v}\sim (b/a)Ra^{(1-2x)/4}$ (where we assume the approximate scaling $Re\sim Ra^{1/2}$ ), and $\ell _{T}>\unicode[STIX]{x1D6FF}_{v}$ at sufficiently high Rayleigh numbers. At lower Rayleigh numbers, one may have instead $\ell _{T}<\unicode[STIX]{x1D6FF}_{v}$ if $b<a.$ For example, assuming the predicted scaling exponent $x=1/3$ of Malkus (Reference Malkus1954), Priestley (Reference Priestley1959) and Kraichnan (Reference Kraichnan1962) at intermediate Rayleigh numbers, a ratio $b/a=1/10$ at $Pr=1$ would lead to $\ell _{T}<\unicode[STIX]{x1D6FF}_{v}$ up to $Ra=10^{12}.$ It is thus unclear whether our suggested criterion for an ‘ultimate regime’ is more stringent or weaker than those based on standard shear Reynolds numbers (Niemela & Sreenivasan (Reference Niemela and Sreenivasan2003, Reference Niemela and Sreenivasan2006), Grossmann & Lohse (Reference Grossmann and Lohse2011), He et al. (Reference He, Funfschilling, Nobach, Bodenschatz and Ahlers2012)). It is also not clear that an ‘ultimate regime’ based on our criterion must have KS scaling. It is quite plausible that ‘trapping’ of fluid particles will disappear once bulk turbulence reaches down to the ‘mixing zone’ at distance $\ell _{T}$ from the wall, but the effect may not be to make $\unicode[STIX]{x1D70F}_{mix}\sim \unicode[STIX]{x1D70F}_{free}.$ Instead, a critical value of $Re_{\ell }$ may signal the transition to an ultimate $Nu$ – $Ra$ scaling which is asymptotically valid for $Ra\rightarrow \infty$ , but still with $\unicode[STIX]{x1D70F}_{mix}\gg \unicode[STIX]{x1D70F}_{free}$ .

A possible explanation of deviations from KS scaling even at arbitrarily large values of $Ra$ is provided by the numerical results of Emran & Schumacher (Reference Emran and Schumacher2012), who found that the thermal plumes occupy a smaller volume of the flow (and are associated with decreased thermal dissipation) for increasing $Ra.$ Thus, convective transport across the cell height is less efficient and larger $\unicode[STIX]{x1D70F}_{mix}$ values are required to achieve near-wall vertical mixing. However, the decrease of the volume fraction of plumes $f_{pl}$ observed in figure 6 of Emran & Schumacher (Reference Emran and Schumacher2012) is rather weak, declining only from $75$ to $65\,\%$ in 3.5 decades, roughly $f_{pl}\sim Ra^{-0.02}.$ If we assume that the ‘effective transport velocity’ is $U_{eff}=Uf_{pl},$ then from (4.5) and the empirical results cited above

(4.15) $$\begin{eqnarray}\unicode[STIX]{x1D70F}_{mix}\sim \frac{H}{U_{eff}}Ra^{0.18},\end{eqnarray}$$

and still $\unicode[STIX]{x1D70F}_{mix}\gg H/U_{eff}$ or, equivalently, $U_{flux}\ll U_{eff}$ for $Ra\gg 1.$ Another possible explanation of the deviations from KS scaling is weakness of the large-scale circulation or convective wind, so that $U_{flux}\simeq U_{lsc}$ while $U_{lsc}\ll U.$ Observations indicate, however, that this is a very slight effect at accessible Rayleigh numbers, too small to account for observed deviations from KS scaling. For example, the two empirical expressions for the plume shedding frequency in Niemela & Sreenivasan (Reference Niemela and Sreenivasan2002) (see p. 206 there, denoted $\unicode[STIX]{x1D714}_{p}$ ) combine to give $U_{lsc}/U\simeq 2.5Pr^{-1/2}Ra^{-0.03}$ which implies a near equality $U_{lsc}\simeq U$ well beyond $Ra=10^{13}.$ Likewise, the measurements of $Re_{lsc}$ in the experiment of Qiu & Tong (Reference Qiu and Tong2002) reviewed by Ahlers et al. (Reference Ahlers, Grossmann and Lohse2009) imply that $U_{lsc}/U\sim Ra^{-0.04},$ and the numerical results of Scheel & Schumacher (Reference Scheel and Schumacher2014) on the Reynolds number $Re_{rms}$ imply a root-mean-square (r.m.s.) flow velocity $u_{rms}/U\sim Ra^{-0.01}.$ The result of these consistent observations is that

(4.16) $$\begin{eqnarray}\unicode[STIX]{x1D70F}_{mix}\sim \frac{H}{U_{lsc}}Ra^{0.16-0.19},\end{eqnarray}$$

and still $\unicode[STIX]{x1D70F}_{mix}\gg \unicode[STIX]{x1D70F}_{lsc}$ or $U_{flux}\ll U_{lsc}.$ Yet another potential explanation for a small ratio $U_{flux}/U\ll 1$ is the theoretical suggestion $U_{flux}\simeq u_{\unicode[STIX]{x1D70F}}$ of Kraichnan (Reference Kraichnan1962). However, the result $u_{\unicode[STIX]{x1D70F}}/U\sim A/\ln ^{3/2}(Ra)$ predicted by Kraichnan (Reference Kraichnan1962) or the similar prediction $u_{\unicode[STIX]{x1D70F}}/U\sim \bar{\unicode[STIX]{x1D705}}/W(Re)$ of Grossmann & Lohse (Reference Grossmann and Lohse2011) show only a very weak (logarithmic) decrease with $Ra$ and are not expected to become relevant until $Ra\gtrsim 10^{13}.$ None of these effects, alone or in combination, seem clearly sufficient to explain the large value $\unicode[STIX]{x1D70F}_{mix}\sim (H/U)Ra^{0.1-0.2}$ that is inferred from existing Nusselt number measurements. (However, with different thresholds in defining ‘plumes’, Emran & Schumacher (Reference Emran and Schumacher2012) find an $f_{pl}$ which is smaller and declines faster. For example, with the parameter $\unicode[STIX]{x1D6FF}=1$ of that paper, their figure 7 shows a decline of $f_{pl}$ from 50 to 20 % over 3.5 decades or $f_{pl}\sim Ra^{-0.11}$ .) As a concrete illustration of the size of $\unicode[STIX]{x1D70F}_{mix}$ , consider the experimental results of He et al. (2012), which have been interpreted as evidence for transition to an ultimate regime with KS scaling. For the largest value $Ra=1.075\times 10^{15}$ reported and with $Pr=0.859$ , their measured value $Nu=5631$ and (4.5) give $\unicode[STIX]{x1D70F}_{mix}/\unicode[STIX]{x1D70F}_{free}=5397$ . There is thus not only an issue to account for the observed deviations from KS scaling, but even to explain how current empirical results can be consistent with our exact relation (4.5)! The very slow mixing rate inferred for all current experiments and simulations by the observed deviations 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 appendix A is only valid for times $t\gg \unicode[STIX]{x1D70F}_{mix}.$ If $\unicode[STIX]{x1D70F}_{mix}\gg \unicode[STIX]{x1D70F}_{free},$ as evidence suggests, then no experiments or simulations are averaging over the very long time intervals required to see the infinite-time limit. Thus, a more exotic speculation is that all current experiments and simulations may be observing very long lived but transient regimes with no dissipative anomalies and different scaling from the true infinite-time limit. (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 $t\rightarrow \infty$ first and then $\unicode[STIX]{x1D708},\unicode[STIX]{x1D705}\rightarrow 0$ (similar to what is expected for 2-D turbulence). No finite energy steady state can exist in the limit first $\unicode[STIX]{x1D708},\unicode[STIX]{x1D705}\rightarrow 0$ and then $t\rightarrow \infty$ without a finite-time dissipative anomaly for the kinetic energy. This can be seen from (2.12) for mean energy balance over a finite time $t,$ which shows that, if there is no anomaly at finite time as $\unicode[STIX]{x1D708},\unicode[STIX]{x1D705}\rightarrow 0$ , then $\unicode[STIX]{x1D6FC}gJ=\langle \unicode[STIX]{x2202}_{t}(u^{2})/2\rangle _{V}$ and the space-averaged kinetic energy must grow linearly in time for fixed $J$ . Of course, there could still be a dissipative anomaly in the statistical steady state obtained in the limit first $t\rightarrow \infty$ and then $\unicode[STIX]{x1D708},\unicode[STIX]{x1D705}\rightarrow 0$ , because the two limits need not commute. Indeed, we know from the steady-state relations (2.21) that when heat flux $J$ is fixed as $\unicode[STIX]{x1D708},\unicode[STIX]{x1D705}\rightarrow 0,$ then $\unicode[STIX]{x1D700}_{u}$ must remain positive. This is not conclusive, however, because it is possible that $\unicode[STIX]{x0394}T\rightarrow \infty$ and $U\rightarrow \infty$ in this limit, in which case the dimensionless kinetic energy dissipation $\hat{\unicode[STIX]{x1D700}}_{u}$ defined in (2.22) tends to zero. In this case one expects also that $U_{rms}\rightarrow \infty ,$ so that the steady-state kinetic energy becomes infinite as $\unicode[STIX]{x1D708},\unicode[STIX]{x1D705}\rightarrow 0$ .) Note that the possibility raised here is distinct from the breakdown of time ergodicity of the turbulent steady state, which was suggested on general grounds by Frisch (Reference Frisch, Diner, Fargue and Lochak1986) and later supported by experimental observations of a turbulent Taylor–Couette flow (Huisman et al. Reference Huisman, van der Veen, Sun and Lohse2014). Multiple steady states have likewise been hypothesized in highly turbulent Rayleigh–Bénard convection (Grossmann & Lohse Reference Grossmann and Lohse2011). However, even if the Boussinesq dynamical system describing turbulent Rayleigh–Bénard convection is time ergodic, it is possible that the intervals $t\gg \unicode[STIX]{x1D70F}_{mix}\gg \unicode[STIX]{x1D70F}_{free}$ required for time averages to converge to their unique steady-state value may be much longer than the time series available in typical experimental or numerical studies. Although infinite-time ergodicity would hold, the consequences would be similar to ergodicity breaking, with averages over available time series exhibiting multiple values depending upon precise initial conditions and experimental details, such as the relative sizes of $t/\unicode[STIX]{x1D70F}_{mix}$ and $Ra.$

4.4 Measuring the near-wall mixing time

All of these issues would be greatly illuminated by direct experimental and numerical measurements of $\unicode[STIX]{x1D70F}_{mix}$ at currently achievable Rayleigh numbers using passive tracers. We have already discussed how this may be done, in principle, both for flux b.c. (see (3.37)–(3.40)) and for temperature b.c. (see (3.50)–(3.51)). However, the methods discussed previously are unwieldy, because a stream of tracers must be released continuously at one wall (top/bottom) with the tracer released at each time $s$ distinguishable from those released at other times $s^{\prime }$ . This requirement will greatly complicate the design of any possible experiment.

An alternative method to measure the mixing time follows from the observation that the transition probability densities which appear in the definitions of $\unicode[STIX]{x1D70F}_{mix}$ satisfy the Kolmogorov equation

(4.17) $$\begin{eqnarray}(\unicode[STIX]{x2202}_{t^{\prime }}+\boldsymbol{u}(\boldsymbol{x}^{\prime },t^{\prime })\boldsymbol{\cdot }\unicode[STIX]{x1D735}_{x}^{\prime })p(\boldsymbol{x}^{\prime },t^{\prime }\mid \boldsymbol{x},t)=-\unicode[STIX]{x1D705}\unicode[STIX]{x1D6E5}_{x}^{\prime }p(\boldsymbol{x}^{\prime },t^{\prime }\mid \boldsymbol{x},t),\quad t^{\prime }<t,\end{eqnarray}$$

in the variables $\boldsymbol{x}^{\prime },t^{\prime }$ (e.g. see Risken (Reference Risken2012), § 4.7). Thus, solving the backward advection–diffusion equation for the concentration of a passive tracer

(4.18) $$\begin{eqnarray}(\unicode[STIX]{x2202}_{t^{\prime }}+\boldsymbol{u}(\boldsymbol{x}^{\prime },t^{\prime })\boldsymbol{\cdot }\unicode[STIX]{x1D735}_{x}^{\prime })c(\boldsymbol{x}^{\prime },t^{\prime })=-\unicode[STIX]{x1D705}\unicode[STIX]{x1D6E5}_{x}^{\prime }c(\boldsymbol{x}^{\prime },t^{\prime }),\end{eqnarray}$$

with a delta-sheet initial condition for the scalar at either the top or bottom wall

(4.19) $$\begin{eqnarray}c(\boldsymbol{x},0)=(1/A)\unicode[STIX]{x1D6FF}(z-\unicode[STIX]{x1D706}^{\prime }H/2),\end{eqnarray}$$

one gets

(4.20) $$\begin{eqnarray}p_{z}(\unicode[STIX]{x1D706}H/2,s\mid \unicode[STIX]{x1D706}^{\prime }H/2,0)=\iint _{S}\text{d}x^{\prime }\,\text{d}y^{\prime }c(x^{\prime },y^{\prime },\unicode[STIX]{x1D706}H/2,s).\end{eqnarray}$$

Thus, the time $\unicode[STIX]{x1D70F}_{mix}^{\unicode[STIX]{x1D706}\unicode[STIX]{x1D706}^{\prime }}$ defined by (3.33) for flux b.c. is the integral time for mixing to a uniform value $1/H$ of the integrated mass density observed at the wall $z=\unicode[STIX]{x1D706}H/2$ when the tracer is released as a thin, uniform sheet at wall $z=\unicode[STIX]{x1D706}^{\prime }H/2.$ A similar interpretation is possible for temperature b.c., but now the tracer satisfying the backward equation (4.18) is released as a point mass on the wall $z=\unicode[STIX]{x1D706}^{\prime }H/2$ at time 0, and the mixing time $\unicode[STIX]{x1D70F}_{mix}^{\unicode[STIX]{x1D706}\unicode[STIX]{x1D706}^{\prime }}$ 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 single instant of time $0,$ and not a stream of distinguishable tracers continuously in time. The only price to be paid is that the experiment must be run backward in time!

Since it is obviously impossible to run a laboratory experiment backward in time, this alternative formulation appears to have dubious merit. However, the forward-in-time version of this experiment appears far more feasible. In such an experiment the tracer would be released as a thin, uniform sheet at one wall, say, $z=+H/2,$ at time $0$ and then its concentration at both the walls $z=\pm H/2$ would be monitored at later times at many observation points distributed uniformly across those walls. In this way, the integral times $\tilde{\unicode[STIX]{x1D70F}}_{mix}^{++}$ and $\tilde{\unicode[STIX]{x1D70F}}_{mix}^{+-}$ for forward mixing of the tracer could be estimated. This experiment has an attractive feature that only observations at the top/bottom walls are required, and no observations are needed from internal probes that might alter the fluid motions. Of course, it is not clear that the forward mixing times $\tilde{\unicode[STIX]{x1D70F}}_{mix}^{++},$ $\tilde{\unicode[STIX]{x1D70F}}_{mix}^{+-}$ measured in this manner are the same as the backward mixing times $\unicode[STIX]{x1D70F}_{mix}^{++},$ $\unicode[STIX]{x1D70F}_{mix}^{+-}$ that are rigorously connected by our FDRs to the thermal dissipation rate. In particular, forward-in-time mixing away from 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, it will be no surprise if these times are quantitatively different. However, it seems reasonable to conjecture that the forward and backward mixing times will scale in the same manner with $Ra,$ $Pr$ and $\unicode[STIX]{x1D6E4},$ even if the prefactors for the scaling laws are distinct. (Analogously, numerical studies find that mean-squared particle dispersion in homogeneous turbulence is time asymmetric, but the dispersion grows as $t^{3}$ both forward and backward in time and only the prefactors in front of the power laws differ. This holds both at short times (Jucha et al. Reference Jucha, Xu, Pumir and Bodenschatz2014) and also at longer times displaying Richardson dispersion (Sawford, Yeung & Borgas Reference Sawford, Yeung and Borgas2005; Berg et al. Reference Berg, Lüthi, Mann and Ott2006; Eyink Reference Eyink2011; Buaria, Yeung & Sawford Reference Buaria, Yeung and Sawford2016).)

Numerical simulations can directly study the backward-in-time mixing as described by (4.18)–(4.19) above. One approach would be to use standard partial differential equation (PDE) methods such as finite-difference or finite-element schemes to solve the backward advection–diffusion equation (4.18). This may be difficult, however, because of the very singular initial condition (4.19) required for the tracer concentration. An easier approach is probably to use a numerical Monte Carlo method with stochastic Lagrangian particles satisfying the backward Itō 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 of such particles uniformly distributed across the top or bottom wall, one could then estimate the position probability densities backward in time required in our FDR. In either approach, one must have available the solution of the Rayleigh–Bénard equations over a long interval of past time. Because of the parabolic character of the Boussinesq equations, it is not possible to solve them backward in time. One option, of the type exploited by us in papers I and II, is to utilize a computer database storing an entire Rayleigh–Bénard solution in space–time. Storing a full space–time history of a simulation is, of course, very costly, 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 $\unicode[STIX]{x0394}t,$ and then use these frames to reconstruct the solution as needed over each interval $[t_{k},t_{k+1}]$ by forward integration of the fluid equations with the frame at time $t_{k}$ as initial data (we thank David Goluskin for suggesting this possibility). Of course, conventional numerical simulations of Rayleigh–Bénard convection can also measure the forward mixing times in the same manner as a laboratory experiment.