Nonlinear diffusion model for Rayleigh-Taylor mixing G. Boffetta (1 , $^{2)}$, F. De Lillo$^{(1)}$, S. Musacchio(3) $^{(1)}$Dipartimento di Fisica Generale and INFN, Universit` a di Torino, via P.Giuria 1, 10125 Torino (Italy) (2) CNR-ISAC, Sezione di Torino, corso Fiume 4, 10133 Torino (Italy) (3) CNRS, Lab. J.A. Dieudonn´e UMR 6621, Parc Valrose, 06108 Nice (France) (Dated: October 24, 2018) The complex evolution of turbulent mixing in Rayleigh-Taylor convection is studied in terms of eddy diffusiviy models for the mean temperature profile. It is found that a non-linear model, derived within the general framework of Prandtl mixing theory, reproduces accurately the evolution of turbulent profiles obtained from numerical simulations. Our model allows to give very precise predictions for the turbulent heat flux and for the Nusselt number in the ultimate state regime of thermal convection. Turbulent thermal convection is one of the most im- portant manifestations of turbulence. It appears in many natural phenomena, from heat transport in stars to at- mosphere and oceanic mixing, and it also plays a funda- mental role in many technological applications [1]. This Letter is devoted to the study of turbulent convec- tion in the Rayleigh-Taylor (RT) setup, a paradigmatic configuration in which a heavy layer of fluid is placed on the top of a light layer. Gravitational instability at the in- terface of the two layers leads to a turbulent mixing zone which grows in time at the expenses of available potential energy [2]. Specific applications of RT convection range from cloud formation [3], to supernova explosion [4, 5] and solar corona heating [6]. Because of the absence of boundaries, the phenomenology of RT turbulence results simpler than other convective systems where the thermal forcing is provided by walls, such as the Rayleigh-Benard configuration. Recent theoretical work [7], confirmed by numerical simulations [5, 8–13], predicts for RT turbulence at small scales a turbulent cascade with Kolmogorov-Obukhov scaling (Bolgiano scaling in two dimensions). Here we concentrate on large scale features of RT turbulence. We propose a simple closure scheme based on the general framework of Prandtl mixing length theory and leading to a nonlinear diffusion model for temperature concen- tration. Our closure reproduces with high accuracy the spatial-temporal evolution of the mean temperature pro- file and allows to derive a prediction for the scaling law of Nu versus Ra which fits perfectly data obtained from direct numerical simulations. The equation of motion for the incompressible veloc- ity field v ( ∇ · v = 0) and temperature field T in the Boussinesq approximation is where β is the thermal expansion coefficient, ν the kinematic viscosity, κ the thermal diffusivity and g = (0 , 0 , − g ) is the gravitational acceleration. The initial condition (at t = 0) is a layer of cooler (heavier) fluid on the top of a hotter (lighter) layer at In the mixing layer turbulent kinetic energy E = (1 / 2) 〈 v $^{2}$〉 is produced at the expense of potential energy FIG. 1: (color online). Snapshot of a ( x, z ) section of the temperature field for Rayleigh-Taylor turbulence numerical simulation. White (black) represents hot, light (cold, heavy) fluid. Boussinesq equations (1-2) are integrated by a standard fully dealiased pseudo-spectral code at resolution N x × N y × N z with N y = N x and aspect ratio L x /L z = N x /N z = r (here N x = 1024 and r = 1). Other parameters are βg = 0 . 5, θ$_{0}$ = 1 ( Ag = 0 . 25), Pr = ν/κ = 1 and ν is chosen such that k$_{max}$η ≥ 1 . 2 in all runs at final time. Initial perturbation is seeded by adding a 10% of white noise to the initial temperature profile in a small layer around z = 0. arXiv:1001.2995v1 [nlin.CD] 18 Jan 2010 rest, i.e. v ( x , 0) = 0 and T ( x , 0) = − ( θ$_{0}$/ 2)sgn( z ) where θ$_{0}$ is the initial temperature jump which fixes the Atwood number A = (1 / 2) βθ$_{0}$ ( T = 0 is the reference mean tem- perature). This configuration is unstable and after the linear instability phase, the system develops a turbulent mixing zone which grows in time starting from the plane z = 0. An example of the turbulent temperature field ob- tained from high resolution direct numerical simulations of (1-2) is shown in Fig. 1. ∂$_{t}$ v + v · ∇ v = −∇ p + ν ∇ 2 v − β g T (1) ∂$_{t}$T + v T = κ ∇ $^{2}$T (2)