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as shown in Fig. 2. A closer inspection of the figure reveals that the model profile (7) is smoother than the actual profile at the edges of the mixing region (see inset of Fig. 2). The physical origin of this discrepancy is that turbulent mixing is not homogeneous within the mixing layer. Indeed turbulent velocity fluctuations decrease at the ends of the mixing region, and therefore a constant K overestimates the diffusivity in these regions. 

An improved model must therefore take into account a z -dependence of the diffusivity. Within the general framework of mixing length theory by Prandtl [1, 20], the eddy diffusivity can be written as K ( z, t ) ≃ H $^{2}$∂$_{z}$ V where H represents a length characteristic of mixing and V is the typical velocity fluctuation. Because velocity is driven by buoyancy at large scale, from equation (1) one can estimate that after a time t the typical velocity is V ∝ βgT t and taking H ∝ h ( t ) one obtains for the eddy diffusivity K ( z, t ) = a ( Ag ) $^{3}$t$^{5}$∂$_{z}$ c where a is again a dimensionless constant to be determined empirically. We remark that a similar approach, based on gradient dependent diffusivity, has been recently used for success- fully modeling mixing in stratified flows [21]. Inserting the above expression in (6) one obtains a nonlinear dif- fusive model for the mean temperature profile 

Observe that the non-linearity of (8) reflects the fact that temperature fluctuations are not passive in this problem as they drive velocity fluctuations in (1). 

Introducing the concentration derivative ϕ ( z, t ) = 3 / ( a ( Ag ) $^{3}$) ∂$_{z}$c ( z, t ) and a new time variable t ′ = t $^{6}$, (8) is rewritten in a more standard form 

with n = 1. Equation (9) represents a class of non- linear diffusion equations with concentration dependent diffusivity well studied in different fields such as thermal waves in plasma radiation [22] and diffusion problems in porous media where for our case n = 1 equation (9) is also known with the name of Boussinesq equation [23]. The value of n governs the behavior of the gradient when ϕ → 0 which is finite for the present case. The self-similar solution (for general n and dimensionality) is known [24] and gives for our case 

<img src='content_image/387631.jpg'>

where z$_{1}$ ( t ) = γAgt 2 with γ = (3 a/ 2) 1 / $^{3}$. 

Having the analytical expression (10) for the mean con- centration, the different definitions of the width of the mixing layer are all expressed in terms of z$_{1}$ and differ by a factor only (e.g. h$_{1}$ = 2 z$_{1}$ and h$_{M}$ = (3 / 4) z$_{1}$ for 

<img src='content_image/387628.jpg'>

<img src='content_image/387627.jpg'>

the tent function [5]). Figure 2 shows that the polyno- mial function (10) fits very well the mean concentration profile obtained from numerical simulations. Runs at dif- ferent resolutions (and viscosity, the only parameter in (1-2) when P r = 1) give analogous results. By fitting the numerical profiles at different times, one obtains the evolution of z$_{1}$ displayed in the inset of Fig. 2 which is consistent with the quadratic law z$_{1}$ = γAg ( t + t$_{0}$ ) 2 ( t$_{0}$ is the reference time as discussed above). The value ob- tained in this way for the coefficient is γ = 0 . 025 ± 0 . 002 which for the profile h$_{M}$ gives α = (3 / 4) γ ≃ 0 . 019 in agreement with previous numerical results [5, 10]. 

The nonlinear diffusion model can be extended from geometrical quantities to study the evolution of dynam- ical properties of turbulent convection. In particular, in the limit of small thermal diffusivity, from (5) and (8) one has an expression for the turbulent heat flux in terms of the mean temperature profile wT = a ( Ag ) $^{3}$t$^{5}$( ∂$_{z}$T ) $^{2}$/θ$_{0}$ . Figure 3 shows that the numerically measured profile of the heat flux is indeed quite close to the model predic- tion, a justification a posteriori of the proposed nonlin- ear closure scheme. Using the definition in (3) the loss of potential energy in kinetic energy (and dissipation) is written as − dP/dt = (4 / 5) γ $^{2}$( Ag ) $^{3}$t3 which shows that γ is a measure of the efficiency of conversion of available potential energy in the turbulent flow. 

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The relation between the heat flux and the profile ge- ometry can be reformulated in terms of dimensionless quantities. Indeed, integrating over the width of the mix- ing layer it gives a relation between the Nusselt number Nu = 〈 wT 〉 / ( κθ$_{0}$ ) (the ratio of convective to conductive heat transfer) and the Rayleigh number Ra = Agh $^{3}$/ ( νκ ) (the ratio of the buoyancy forces to diffusivities). Using 

