P = − βg 〈 zT 〉 as the energy balance indicates where ε$_{ν}$ = ν 〈 ( ∂$_{α}$v$_{β}$ ) $^{2}$〉 is the viscous energy dissipation and 〈〉 represents the integral over the physical domain. Assuming that in the turbulent state all quantities in (3) scale in the same way one can balance dv 2 $_{rms}$/dt ≃ βgθ$_{0}$v$_{rms}$ (because temperature fluctuations are bounded by the initial jump θ$_{0}$ ) and therefore one obtains the temporal scaling of velocity fluctuations v$_{rms}$ ≃ βgθ$_{0}$t ≃ Agt , i.e. a motion forced with constant acceleration g . The accelerated growth of the width of the mixing layer is one of the standard diagnostics in the stud- ies of RT turbulence [8, 14–16]. Several definitions for the width have been proposed, based on either local or global properties of the mean temperature profile T ( z, t ) ≡ 1 / ( L$_{x}$L$_{y}$ ) ∫ T ( x , t ) dxdy . The simplest mea- sure h$_{r}$ is based on the threshold value of z at which T ( z, t ) reaches a fraction r of the maximum value i.e. T ( ± h$_{r}$ ( t ) / 2 , t ) = ∓ rθ$_{0}$/ 2 [8]. This local definition of h can be rather noisy and therefore alternative definitions based on integral quantities have been proposed [5, 8, 17] number. The simplest approximation is to consider K independent on z . For our problem, being a diffusion co- efficient (i.e. a velocity time a scale) the eddy diffusivity is expected to depend on t as K ( t ) = b $^{2}$( Ag ) $^{2}$t3 with b a free dimensionless parameter. The self-similar solution to (6) with a step initial condition c ( z, 0) = θ ( z ) is where c = ( T$_{max}$ − T ) / ( T$_{max}$ − T$_{min}$ ) = 1 / 2 − T/θ$_{0}$ is the normalized dimensionless temperature (0 ≤ c ≤ 1) and M is a mixing function which has support on the mixing layer only, e.g. a logistic function M ( c ) = 4 c (1 − c ) [10] or a tent function M ( c ) = 2 c + (2 − 4 c ) θ ( c − 1 / 2) [5]. Di- mensionally, h is expected to grow with accelerated law h ( t ) = αAgt 2 with the dimensionless coefficient α which depends on the definition of h and apparently also on the form of the initial perturbation of the interface [16, 18]. Recent studies [5, 19] have shown that a more robust and consistent determination of α can be obtained if an initial time t$_{0}$ = 0 is taken into account (physically represent- ing the offset at which the t 2 law sets in) suggesting the possibility of a universal value, independent on the form of the initial perturbation. The evolution equation for the normalized tempera- ture profile c ( z, t ) is obtained by averaging (2) over the horizontal directions (assumed periodic) where w represent the vertical velocity. The thermal flux term wc makes (5) not closed. Following a common ap- proach in turbulence, we close this equation in terms of an eddy diffusivity K ( z, t ) so that (5) is rewritten as Molecular diffusivity κ , included additively in K ( z, t ), can be neglected for large scale properties at high P´ eclet The constant diffusivity solution (7) is a relatively good approximation of the actual profile obtained from the numerical simulations of the full set of equations (1-2), FIG. 3: (color online). Heat flux profile wT obtained at the same time of Fig. 2. Black line represents the prediction of the nonlinear diffusion model as discussed in the text. ∂$_{t}$c + ∂$_{z}$wc = κ∂ 2 $_{z}$c (5) ∂$_{t}$c = ∂$_{z}$K ( z, t ) ∂$_{z}$c (6) h$_{M}$ ≡ ∫ L z / 2 − L z / 2 M ( c ) dz (4) c ( z, t ) = 1 2 [ 1 + erf ( z bAgt 2 )] (7) FIG. 2: (color online). Normalized mean temperature profile c ( z ) computed by averaging over horizontal planes the tur- bulent temperature field of Fig. 1. Blue dotted line is the prediction of the linear diffusion model (7). Black contin- uous line is the fit with the nonlinear model (10). Lower inset: enlargement of the temperature profile at the edge of the mixing layer. Upper inset: evolution of z$_{1}$ obtained by fitting the temperature profile with (10) at different times and over four different realizations. The line represents the fit z$_{1}$ = γAg ( t + t$_{0}$ ) 2 which gives γ ≃ 0 . 025 and t$_{0}$ ≃ 3 . 3. − dP dt = βg 〈 wT 〉 = dE dt + ε$_{ν}$ (3) 2  0  0.2  0.4  0.6  0.8  1 -0.4 -0.2  0  0.2  0.4 c(z) z/L$_{z}$ 0.95 1 0.15 0.2 0 5 10 0 10 20 30 z 1 t  0  0.01  0.02  0.03  0.04  0.05 -0.4 -0.2  0  0.2  0.4 wT z/L$_{z}$