4 which reproduces accurately the large scale phenomenol- ogy of Rayleigh-Taylor turbulence obtained from high- resolution numerical simulations. The model contains a single free parameter, a measure of the turbulence pro- duction efficiency, which is directly related to the rate of accelerated growth of the mixing layer. The proposed closure scheme represents an important step for a phe- nomenological description of RT turbulence as it connects the evolution of the Nusselt number to the growth of the mixing layer, a global geometrical quantity which can be easily obtained in experiments. the expression (10) and for the length h = h$_{1}$ = 2 z$_{1}$ , one obtains the temporal evolution laws for the two quanti- ties as Ra = 8 γ $^{3}$( Ag ) $^{4}$t$^{6}$/ ( νκ ), Nu = 2 γ $^{2}$( Ag ) $^{2}$t$^{3}$/ (5 κ ) and therefore the relation Equation (11) represents the well known Kraichnan’s prediction for the “ultimate state of thermal convection” [25, 26] which is a regime of turbulent convection ex- pected to hold when the contribution of thermal and kinetic boundary layers becomes negligible. Because of the absence of boundaries, RT turbulence is a natural candidate for the appearance of this regime which has indeed been observed recently in numerical simulations both in two and three dimensions [11, 12, 27]. Figure 4 shows that the prediction (11) with γ = 0 . 025 fits well the numerical data obtained from a set of simulations at different resolutions. The fact that Nu ≫ 1 is a posteri- ori confirmation of the negligible contribution of thermal diffusivity. It is interesting to observe that the above result for Nu satisfies a general bound which can be easily ob- tained starting from (5). Neglecting thermal diffusiv- ity and assuming a self-similar evolution of the pro- file c ( z, t ) = f ( z/z$_{1}$ ( t )) with the symmetry condition f ( − z ) = 1 − f ( z ), integrating (5) over the z domain [ L$_{z}$/ 2 , L$_{z}$/ 2] twice, one obtains Using the fact that for z > 0 c ( z, t ) > 1 / 2 and assuming that the flow is still unmixed, c ( z, t ) = 1 for z > z$_{1}$ , we get a bound If we now further assume the accelerated growth of the mixing layer, z$_{1}$ ( t ) = γAgt 2 we end with a bound on the growth of the Nusselt number which is indeed satisfied by our model. The physical in- terpretation of this bound is transparent: the growth of the heat flux follows the dimensional t 3 law with a coef- ficient which depends on the shape of the mean temper- ature profile. Maximum growth (14) is achieved when c ( z, t ) = 1 / 2 for − z$_{1}$ ≤ z ≤ z$_{1}$ which means a perfect mixing within the mixing layer. This would correspond to a coefficient ( γ/ 2) 1 / 2 in (11). In this Letter we have introduced a nonlinear diffu- sion model with a gradient dependent eddy diffusivity ∫ L z 2 − L z 2 dzwc = 2 ˙ z$_{1}$ z$_{1}$ [ 2 ∫ L z 2 0 dzzc ( z, t ) − L 2 z 4 ] (12) Nu = − 1 κ ∫ L z 2 − L z 2 dzwc ≤ 1 κ z$_{1}$ ˙ z$_{1}$ (13) Nu ≤ 2 κ γ $^{2}$( Ag ) $^{2}$t3 (14) Nu = 1 $_{5}$√$_{2}$ γ 1 / $^{2}$P r1 / $^{2}$Ra1 / 2 (11) [1] E. Siggia, Ann. Rev. Fluid Mech. 26 , 137 (1994). [2] D. H. Sharp, Physica D 12 , 3 (1984). [3] D. Schultz and et al., J. Atmos. Sci. 63 , 2409 (2006). [4] M. Zingale, S. Woosley, C. Rendleman, M. Day, and J. Bell, Astrophys. J. 632 , 1021 (2005). [5] W. Cabot and A. Cook, Nature Physics 2 , 562 (2006). [6] H. Isobe, T. Miyagoshi, K. Shibata, and T. Yokoyama, Nature 434 , 478 (2005). [7] M. Chertkov, Phys. Rev. Lett. 91 , 115001 (2003). [8] S. Dalziel, P. Linden, and D. Youngs, J. Fluid Mech. 399 , 1 (1999). [9] W. Cabot, Phys. Fluids 18 , 045101 (2006). [10] N. Vladimirova and M. Chertkov, Physics of Fluids 21 , 015102 (2009). [11] G. Boffetta, A. Mazzino, S. Musacchio, and L. Vozella, Phys. Rev. E 79 , 065301(R) (2009). [12] A. Celani, A. Mazzino, and L. Vozella, Phys. Rev. Lett. 96 , 134504 (2006). [13] T. Matsumoto, Phys. Rev. E 79 , 055301(R) (2009). [14] T. T. Clark, Phys. Fluids 15 , 2413 (2003). [15] Z. Huang, A. DeLuca, T. J. Atherton, M. Bird, C. Rosen- blatt, and P. Carl` es, Phys. Rev. Lett. 99 , 204502 (2007). [16] K. Kadau, C. Rosenblatt, J. Barber, T. Germann, Z. Huang, P. Carles, and B. Alder, Proc. Nat. Acad. Sci- ences 104 , 7741 (2007). [18] G. Dimonte and et al., Phys. Fluids 16 , 1668 (2004). [17] M. J. Andrews and D. B. Spalding, Phys. Fluids A 2 , 922 (1990). [19] J. Ristorcelli and T. Clark, J. Fluid Mech. 507 , 213 (2004). [20] L. Prandtl, Z. Angew. Math. Mech 5 , 136 (1925). [21] P. Odier, J. Chen, M. K. Rivera, and R. E. Ecke, Phys. Rev. Lett. 102 , 134504 (2009). [22] Y. Zel’Dovich and Y. Raizer, Ann. Rev. Fluid Mech. 1 , 385 (1969). [23] J. Bear, Dynamics of fluids in porous media (Dover Pub- lications, 1988). [24] R. E. Pattle, Quart. Journ. Mech. and Applied Math. 12 , 407 (1959). [25] R. Kraichnan, Phys. Fluids 5 , 1374 (1962). [26] S. Grossmann and D. Lohse, J. Fluid Mech. 407 , 27 (2000). [27] D. Lohse and F. Toschi, Phys. Rev. Lett. 90 , 034502 (2003).