Rayleigh-Taylor Turbulence with Singular Nonuniform Initial Conditions

Rayleigh-Taylor turbulence with singular non-uniform initial conditions

L. Biferale,1 G. Boffetta,2 A.A. Mailybaev,3 and A. Scagliarini4 1Department of Physics and INFN, University of Rome “Tor Vergata”, Via della Ricerca Scientifica 1, 00133 Rome, Italy 2Department of Physics and INFN, University of Torino, via P. Giuria 1, 10125 Torino, Italy 3Instituto Nacional de Matemática Pura e Aplicada – IMPA, 22460-320 Rio de Janeiro, Brazil 4Istituto per le Applicazioni del Calcolo ’M. Picone’ – IAC-CNR, Via dei Taurini 19, 00185 Rome, Italy We perform Direct Numerical Simulations of three dimensional Rayleigh-Taylor turbulence with a non-uniform singular initial temperature background. In such conditions, the mixing layer evolves under the driving of a varying effective Atwood number; the long time growth is still self-similar, but not anymore proportional to t2 and depends on the singularity exponent c of the initial profile ∆T ∝ zc. We show that universality is recovered when looking at the efficiency, defined as the ratio of the variation rates of the kinetic energy over the heat flux. A closure model is proposed that is able to reproduce analytically the time evolution of the mean temperature profiles, in excellent agreement with the numerical results. Finally, we reinterpret our findings on the light of spontaneous stochasticity where the growth of the mixing layer is mapped in to the propagation of a wave of turbulent fluctuations on a rough background.

Introduction. Turbulent mixing is a mechanism of evolve in a non-homogeneous background. In particular utmost importance in many natural and industrial pro- we investigate analytically and by using direct numerical cesses, often induced by the Rayleigh-Taylor (RT) insta- simulations in three dimensions the generic case when the bility which takes place when a fluid is accelerated against initial unstable vertical temperature distribution is given a less dense one [1–5]. RT turbulence occurs in disciplines by the power law: as diverse as in astrophysics [7–9], atmospheric science [10], confined nuclear fusion [11, 12],plasma physics [13], z c T0(z) = (θ0/2) sgn(z) | | , (1) laser-matter interactions [14, 15] (see [4–6] for recent re- − L views). One important application of RT instability is the case of convective flow, in which density differences where L is a characteristic length scale and L z L. reflect temperature fluctuations of a single fluid and the The exponent of the singularity belongs to− the≤ interval≤ acceleration is provided by gravity. 1 < c < 1, where the upper limit corresponds to a In the simplest configuration of Boussinesq approxima- −smooth profile and the lower limit ensures that the po- tion for an incompressible flow, RT turbulence considers tential energy density, βgzT0(z), does not diverge near a planar interface which separates a layer of cooler (heav- the interface among the− two miscible fluids at z = 0. The ier, of density ρH ) fluid over a layer of hotter (lighter, of value c = 0 recovers the standard RT configuration. density ρ ) fluid under a constant body force such as L We develop a closure model based on the Prandtl Mix- gravity. The driving force is constant in time and pro- ing Length approach, which is able to reproduce with portional to g , where g is the acceleration due to the good accuracy the evolution of the mean temperature body force andA = (ρ ρ )/(ρ + ρ ) = βθ /2 is the H L H L 0 profile at all scales and for all values of the singularity Atwood number,A expressed− in term of the thermal expan- exponent c. Beside the importance of testing the robust- sion coefficient β and the temperature jump θ between 0 ness with respect to the initial configuration, the above the two layers. However, in some relevant circumstances setup allows us to investigate the idea that the Mixing one has to cope with time varying acceleration (as in in- Layer (ML) growth can be mapped to a traveling wave in ertial confinement fusion or in pulsating stars [16–18]) appropriate renormalized variables. This wave describes or with a varying Atwood number, that emerges when the self-similar evolution of the probability distribution the mixing proceeds over a non-uniform background as function (PDF) of turbulent fluctuations from small to

arXiv:1802.05021v2 [physics.flu-dyn] 13 Sep 2018 in thermally stratified atmosphere [19–21]. large scales in a rough background given by the initial In this work we address a question with both funda- singular profile [22, 23]. Such a description would then mental and applied importance: what happens when the naturally explain the universality of the ML evolution initial unstable profile is more general than the usual RT and its spontaneously stochastic behavior in the inertial step-function, as in the case of non-differentiable power- range [26]. We introduce a shell model for the RT evolu- law density profile. As a result, the mixing layer will tion to illustrate and quantify the ML statistical properties. Results for Navier–Stokes Equations. We con- Postprint version of the manuscript Phys. Rev. Fluids 3, sider the Boussinesq approximation for an incompress- 092601(R) (2018). ible velocity field u(r, t) coupled to the temperature field 2

2 1.0 =2 100 γ .6

) =1 0.5 t γ Σ(

.67 =2 z 0 0 2 4 6 γ /L )

t (t t0(c))/t ( 1 − ∗ h 10−

25 .

=0 c =0 0.25 c c = −

FIG. 1. (Color online). Snapshots of the vertical section of 2 10− the temperature field T for three simulations of RT turbulence 0.5 1 5 10 with power law initial condition (1) with c = −0.25 (left), t/t c = 0 (center) and c = 0.25 (right) at three different times ∗ corresponding to the same mixing length h(t) ' 0.4Lz. High FIG. 2. Temporal evolution of the mixing layer h(t). From left (low) temperature is represented by yellow (blue). to right: c = −0.25 (green triangles), c = 0 (red squares) and c = 0.25 (blue circles). The three lines represent the power law predicted by the formula (5) with γ = 2/(1−c). Inset: Ef- T (r, t) by a buoyancy term: ficiency of kinetic energy production Σ = −(dE/dt)/(dP/dt) as a function of time for the three cases c = −0.25 (green tri- 2 ∂tu + u ∇u = ∇p + ν u βgT, (2) angles), c = 0 (red squares) and c = 0.25 (blue circles). The · − 2 ∇ − time axis is shifted by a time t0 which depends on c defining ∂tT + u ∇T = κ T, (3) · ∇ the onset of the self-similar growth. The bars indicate the where g = (0, 0, g) is the gravity acceleration, ν typical amplitude of fluctuations around the mean value in the plateau region. and κ are the kinematic− viscosity and thermal diffusivity respectively. The choice to rely on Navier-Stokes- Boussinesq equations for studies of RT at high Reynolds determined on dimensional grounds [31–33] from (1) and numbers is very widespread, and it is justified by the ob- (2) in the form servation that the turbulent Mach number has an upper 2 c bound [27], thus making RT turbulence an effectively in- u(t) /h(t) βgθ0(h(t)/L) , (4) compressible (or low compressible) phenomenon. On the ' other hand, it is known that, when detectable, compress- where u(t) is a large-scale velocity. Assuming that u ibility effects amount mainly to break the up-down sym- dh/dt, one ends with ' metry of the mixing layer growth, with ’spikes’ (down- 2 1+c ward falling temperature fluctuations) being on average t 1−c t 1−c faster than ’bubbles’ (upward rising); nevertheless, such h(t) L , u(t) U , (5) ' t∗ ' t∗ asymmetry is limited to the prefactor, while the scaling in time of the full mixing layer width, which is our main where U = L/t∗ and t∗ was defined above. Notice that interest here, is preserved [28, 29]. the first expression can be reinterpreted as a standard The initial condition for the velocity at position r = RT diffusion (x, y, z) is u(r, 0) = 0, while for the temperature field 2 T (r, 0) = T (z) we consider a generic power-law distri- h(t) = αc c(t)gt (6) 0 A bution given by (1). The only inviscid parameter that c 1/(1−c) 2 c/(1−c) relates spatial and temporal scales is ξ = βgθ0/L which where c(t) = (βθ0) gt /L is the time de- 1−c 2 A has physical dimensions of length /time . Thus, for pendent Atwood number and the pre-factor αc represents a given length L, the corresponding integral temporal the generalization of the standard RT α coefficient [34]. −1/2 (1−c)/2 p scale is given by t∗ = ξ L = L/(βgθ0). In order to test the above predictions, we performed di- The distribution (1) is unstable and the dimensional rect numerical simulations (DNS) of the system of equa- argument provides the inviscid growth exponent λ tions (2–3) in a periodic domain of size Lx Ly Lz 1/2 (1−c)/2 ' × × ξ k for the modes with wavenumber k, where with Ly = Lx and Lz = 4Lx by means of a fully parallel the dimensionless proportionality coefficient can be de- pseudo-spectral code at resolution 512 512 2048 for × × termined by solving the linear stability problem [30]. initial conditions (1) with different c and L = Lz. For all This dispersion relation predicts that the instability is runs we have βg = 1/2, θ0 = 1 and Pr = ν/κ = 1. RT in- driven by the smallest scales for all c < 1. stability is seeded by adding to the initial density field a −3 The nonlinear development of the RT instability pro- white noise of amplitude 10 θ0 and statistical quantities duces a mixing zone of width h(t). Its evolution can be are averaged over 10 independent runs. Figure 1 shows 3 examples of the vertical section of the temperature field 2 for three different initial conditions taken at three differ- 1.5 ent times corresponding to the same width of the mixing 1 )

layer h(t). We compute h(t) on the basis of the mean c ) R t ( 0.5 temperature proﬁle T (z, t) = T (x, y, z, t)dxdy as the c −3 region on which T (z, t) T (z, 0) > δθ0 with δ = 5 10 2)Λ 0 /

| − | × 0 [35]. In Fig. 2 we show that the evolution of h(t) is in θ

(( 0.5 good agreement with the power law predicted by scaling − T/ 1 (6) for the three different values of c. A small devia- − tion is observed for the largest c (which corresponds to 1.5 the faster growth) probably because of the short range of − 2 temporal scaling. This results confirms that the balance − 8 6 4 2 0 2 4 6 8 − − − − (4) gives the correct evolution of the mixing layer, even η over non-uniform backgrounds. c From (2-3) we derive the energy balance equation FIG. 3. Rescaled temperature profiles T/(θ0Λc(t) ) averaged over ten independent runs vs the vertical coordinate η (12), for dP dE c = 0 (squares), c = 0.25 (circles) and c = −0.25 (triangles) = βg wT = + εν (7) p 2/(1−c) − dt h i dt at three different times. Λc(t) = (1 − c)bc(t/t∗) is the time scaling factor of η in (12). The fitting parameters which defines the conversion of available potential energy −5 −6 −4 R are: b0 = 6 × 10 , b0.25 = 1.2 × 10 and b−0.25 = 7 × 10 . P (t) = βg zT (z, t)dz into turbulent kinetic energy c − 2 2 The solid lines represent the function −|η| fc(η), with fc(η) E(t) = (1/2) u(r, t) . εν = ν ( u) is the viscous given by equation (13). energy dissipationh andi representsh ∇ thei integral over the whole volume. Equationh•i (7) shows that not all the available potential energy is converted into turbulent ki- The effective diffusivity is expected to depend on time as (3+c)/(1−c) netic energy. It is therefore interesting to measure the uh, leading to K(t) = bcLU(t/t∗) with a free efficiency of the production of turbulent fluctuations, de- dimensionless parameter bc. In this case a self-similar fined as [36, 37] solution of (11) is obtained in the form (see the Supple- dE/dt mentary material) Σ = (8) c 2 − dP/dt − 1−c z z (t/t∗) T (z, t) = θ0 | | fc(η), η = p , (12) and to check how this is affected by the initial distribu- − L L (1 c)bc tion. The inset of Fig. 2 shows the time evolution of Σ, − which starts from a value close to 1 unit. When the tur- where the function bulent cascade develops we observe a peak in the energy Z |η| 2 sgn(η) −c −x2 dissipation which is reflected in the minimum of Σ. This fc(η) = 1−c x e dx (13) Γ 2 0 occurs at a time t0(c) which depends on the initial condition and which is used to shift the different cases. In the is such that fc 1 as η . For c = 0 (stan- → ± → ±∞ turbulent, self-similar regime, at t > t0, the efficiency of dard RT), the solution reduces to the error function conversion of potential energy into kinetic energy reaches f0(η) = erf(η) which is known to be a good fit for stan- an almost constant plateau Σ 0.5 which is independent, dard RT evolution [38]. In Fig. 3 we show that the ho- ' within the errors, on the initial density profile. mogeneous Prandtl approach works well also for c = 0 by At the level of local quantities, the evolution equation plotting the rescaled temperature profiles, for the6 three for the mean temperature profile reads different c’s considered, at three times as function of the rescaled coordinate η, as given by (12), superposed with ∂ T + ∂ wT = κ∂2 T. (9) t z zz the solution (13). Using a Prandtl Mixing Layer first-order closure with Results for Shell Models. Because of limitation in homogeneous eddy diffusivity K(t), the heat transfer is the resolution, DNS can access the turbulent dynamics of related to the local temperature gradient by [38]: the ML only in a limited range of scales. To get a more quantitative control of the multi-scale dynamical prop- wT = K(t) ∂zT cT /z . (10) − − erties, we use a shell model for the RT instability that was introduced in [26]. This system defines the dynam- In the above expression, the correction term cT /z ensures −n that wT vanishes outside the mixing zone, where T is ics at discrete vertical scales (“shells”) zn = 2 L with given by Eq. (1). Neglecting the diffusive term, equation n = 1, 2,..., where the associated variables ωn, Rn and (9) can be recast into Tn describe vorticity, horizontal and vertical temperature fluctuations, respectively. We modified the equations de- ∂tT = K(t) ∂z ∂zT cT /z . (11) scribed in [26] by using the complex nonlinearity of the − 4

Sabra model [39]. The resulting shell model retains scal- 10-2 e ing properties of the original Boussinesq equations (2– 1 L

10 = 3), along with some important inviscid invariants such 10-4 h as energy, helicity and entropy (see the Supplementary material). Having properties qualitatively similar to the 10-6 10-1 2=(1!c) full system, the shell model allows for numerical simula- ("t=t$) 10-8 tions in a very large range of scales, thus, serving as a h 10-8 10-6 10-4 10-2 natural playground for testing theoretical ideas in turbu- e 5 7 25 2 :5 : lence [40]. -3 0: : 0 0 10 ! 0 0 = = = = = At t = 0, the analogue of initial conditions (1) must be c c c c c chosen with vanishing vorticity and horizontal temperature variations ω (0) = R (0) = 0, while for the vertical n n -5 temperature variables we choose 10 10-3 10-2 10-1 100 101 z c T (0) = iθ n (14) "t n 0 L for all n. This initial condition leads to the same ex- FIG. 4. Evolution of the ML h in log-log scale for the shell 1/2 (1−c)/2 model with different c. The parameters L, βg and θ0 were plosive dispersion relation λn = ξ kn as the full 3 model (1–3) (see the Supplementary material). Phe- set to unity. The statistics was obtained from 10 evolutions, where a small random perturbation was added to the variables nomenological theory of the RT instability for the shell Rn at shells n ≥ 16. We use ∆t = t−t0 accounting for a small model is essentially identical to the one of the full initialization time t0. Inset: same curves presented as h/Le vs. 3D system [32], with turbulent fluctuations propagating 2/(1−c) (∆t/et∗) and compared to the universal approximation from small to large scales. It is convenient to charac- (15) shown with the dotted red line, where the larger deviation terize the size of the ML with the expression h(t) = corresponds to c = 0.7. P Tn(t)/Tn(0) 1 zn, which estimates the largest scale | − | zn at which the temperature profile Tn(t) deviates from its initial value Tn(0). This definition is in spirit of the commonly used integral formulas for the ML width [8]. By performing a large number of simulations with small dissipative coefficients and small random initial perturbations at small scales, we accurately verify the scaling law (5) for c = 0.25, 0, 0.25, 0.5, 0.7 in Fig. 4, where solid lines represent− the numerical results (averaged over realizations) and the green lines show the theoretical prediction. Here we use the small time shift t0 defining the typical time for the onset of self-similar growth. The results in Fig. 4 provide with high accuracy the dimensionless pre-factor αc for the power-law growth of the ML, see Eqs. (5-6). Numerical results show that the FIG. 5. PDFs (darker color for larger probability) for the ratios of temperature variables |T /T | as functions of time: dependence of αc on the singularity exponent c can be n n+1 −2/(1−c) n = 5 (upper) and n = 10 (lower) panels in the case c = 0.25. fitted well with the formula αc αLα . The ≈ t quantities αL 50 and αt 20 have a simple physical meaning: they≈ redefine the≈ dimensional length and tical line at time t , which means that unstable modes at time scales, Le = αLL and et = αtt∗, which reduce the ML e∗ width expression to the universal form all scales get excited simultaneously. It is argued [22, 23] that spontaneous stochastic tur- t 2/(1−c) bulent fluctuations develop in the inverse cascade from h(t) = L . (15) e small to large scales. In the limit of large Reynolds num- et∗ bers, such behavior develops for rough (i.e., non-smooth) This relation is validated in Fig. 4 (inset). The ML velocity fields, in close analogy to the 1/3 Hölder conti- reaches the size Le at the time et independently of the nuity condition in the Onsager dissipation anomaly [24]. singularity exponent c; this can be seen in Fig. 4 as an Non-smooth temperature profile in the RT initial con- (approximately) common intersection point of the green ditions provides a natural rough background that can lines. In the limit c 1 (constant temperature gradient trigger similar effects in the RT turbulence. Existence with no singularity),→ the graph h(t) approaches the ver- of the inverse cascade of fluctuations must reflect in the 5 stochastic growth of the mixing layer independently of No. 302351/2015-9. G.B. acknowledges financial support the initial perturbation. Here, the stochastic component by the project CSTO162330 Extreme Events in Turbu- develops in the Eulerian evolution of velocity and tem- lent Convection. Numerical calculations have been made perature fields, unlike for the of turbulent Richardson possible through a CINECA-INFN agreement, providing dispersion where spontaneous stochasticity is predicted access to resources on MARCONI at CINECA. –and observed– for the separation of two Lagrangian trac- ers by a singular advecting velocity field [25]. It is hard to analyze such phenomenon with the DNS due to numerical limitations. However, it can be con- veniently studied in our shell model using the renormal- [1] D.H. Sharp, An overview of Rayleigh-Taylor instability, Physica D 12, 3 (1984). ized (logarithmic) space-time coordinates: n = log zn − 2 [2] H.J Kull, Theory of the Rayleigh-Taylor instability, Phys. and τ = log2 ∆t. To highlight the stochastic aspect, we Rep. 206, 197 (1991). choose to measure the probability distribution function [3] S.I. Abarzhi, Review of theoretical modelling approaches of the ratios among temperature fluctuations at adjacent of Rayleigh-Taylor instabilities and turbulent mixing, shells, Tn/Tn+1 , which are the equivalent of velocity Phil. Trans. Roy. Soc. London A 368, 1809 (2010). multipliers| used in| cascade description of fully developed [4] G. Boffetta and A.Mazzino, Incompressible Rayleigh- turbulence [41, 42]. Figure 5 presents the time-dependent Taylor turbulence, Annu. Rev. Fluid Mech. 49, 119 (2017). PDFs obtained numerically in the case c = 0.25 starting [5] Y. Zhou, Rayleigh-Taylor and Richtmyer-Meshkov insta- from many initial conditions different by a very small per- bility induced flow, turbulence, and mixing. I, Phys. Rep. turbation. These results support the idea that the ML 720-722, 1 (2017). growth can be mapped to a stochastic wave in appro- [6] Y. Zhou, Rayleigh-Taylor and Richtmyer-Meshkov insta- priate renormalized variables ( n, τ). The wave speed bility induced flow, turbulence, and mixing. II, Phys. is constant and given by the exponent− 2/(1 c) of ML Rep. 723-725, 1 (2017). width from Eq. (5). Such a wave represents a− front of the [7] M. Zingale, S.E. Woosley, C.A. Rendleman, M.S. Day and J.B. Bell, Three-dimensional numerical simulations turbulent fluctuations, which propagates into a determin- of Rayleigh-Taylor unstable flames in type Ia supernovae, istic left state (delta function PDF) corresponding to the Astrophys. J. 632, 1021 (2005). initial power-law background (14), and leaves behind the [8] W.H. Cabot and A.W. Cook, Reynolds number effects stationary turbulent state on the right. This description on Rayleigh-Taylor instability with possible implications naturally explains the universality of the ML evolution for type Ia supernovae, Nat. Phys. 2, 562-568 (2006). and its spontaneously stochastic behavior in the inertial [9] J. Bell, M. Day, C. Rendleman, S. Woosley, M. Zin- range [26, 43]. gale, Direct numerical simulations of type Ia supernovae flames. II. The Rayleigh-Taylor instability, Astrophys. J. Conclusions. We have studied numerically and ana- 608, 883 (2004). lytically Rayleigh-Taylor turbulence with general power- [10] M.C. Kelley, G. Haerendel, H. Kappler, A. Valenzuela, law singular initial conditions, providing insight into sit- B.B. Balsley, D.A. Carter, W.L. Ecklund, C.W. Carl- uations when the mixing proceeds over a non-uniform son, B. Häusler and R. Torbert, Evidence for a Rayleigh- background, e.g. in thermally stratified atmosphere. We Taylor type instability and upwelling of depleted density have shown that independently of the singularity expo- regions during equatorial spread F, Geophys. Res. Lett. 3, 448-450 (1976). nent, the asymptotic self-similar growth of the ML is [11] J.D. Lindl, Inertial Confinement Fusion. Springer-Verlag, universal, if properly renormalized, i.e. by looking at New-York (1998). the mixing efficiency and at the mean rescaled Tempera- [12] S. Atzeni and J. Meyer-ter-Vehn, The Physics of Inertial ture profile. We show that a closure model based on the Fusion, Oxford University Press (2004). Prandtl mixing layer approach is able to reproduce ana- [13] H. Takabe, K. Mima, L. Montierth and R.L. Morse, Self- lytically the time evolution of the mean temperature pro- consistent growth rate of the Rayleigh-Taylor instability files. By using a shell model we have provided numerical in an ablatively accelerating plasma, Phys. Fluids 28, 3676 (1985). data supporting the above findings also at much larger [14] A. Sgattoni, S. Sinigardi, L. Fedeli, F. Pegoraro, and A. resolution both in time and scales. This model helped to Macchi, Laser-driven Rayleigh-Taylor instability: Plas- understand the behavior of prefactor in the ML growth monic effects and three-dimensional structures, Phys. process. Finally, we have shown that RT evolution can Rev. E 91, 013106 (2015). be reinterpreted in terms of the phenomenon known as [15] K. Shigemori, H. Azechi, M. Nakai, M. Honda, K. Me- spontaneous stochasticity where the growth of the mix- guro, N. Miyanaga, H. Takabe, and K. Mima, Measure- ing layer is mapped into the propagation of a wave of ments of Rayleigh-Taylor growth rate of planar targets irradiated directly by partially coherent light, Phys. Rev. turbulent fluctuations on a rough background. Lett. 78, 250 (1997). Acknowledgements. L.B. acknowledges financial sup- [16] G. Dimonte, P. Ramaprabhu and M. Andrews, Rayleigh- port from the European UnionâĂŹs Seventh Framework Taylor instability with complex acceleration history, Programme (FP7/2007-2013) under Grant Agreement Phys. Rev. E 76, 046313 (2007). No. 339032. A.A.M. was supported by the CNPq Grant [17] G. Dimonte and M. Schneider, Turbulent Rayleigh- 6

Taylor instability experiments with variable acceleration, 104, 034505 (2010). Phys. Rev. E 54, 3740 (1996). [39] V.S. L’vov, E. Podivilov, A. Pomyalov, I. Procaccia and [18] D. Livescu, T. Wei and M.L. Petersen, Direct numeri- D. Vandembroucq, Improved shell model of turbulence, cal simulations of Rayleigh-Taylor instability, J. Phys.: Phys. Rev. E 58, 1811 (1998). Conf. Ser. 318, 082007 (2011). [40] L. Biferale, Shell models of energy cascade in turbulence, [19] A.S. Monin and A.M. Obukhov, Basic laws of turbulent Annual Rev. Fluid Mech. 35, 441 (2003). mixing in the atmospheric surface layer, Tr. Akad. Nauk [41] R. Benzi, L. Biferale and G. Parisi, On intermittency in a SSSR Geofiz. Insti. 24, 163–187 (1954). cascade model for turbulence, Physica D 65, 163 (1993). [20] B.A. Kader and A.M. Yaglom, Mean fields and fluctua- [42] Q. Chen, S. Chen, G.L. Eyink and K.R. Sreenivasan, Kol- tion moments in unstably stratified turbulent boundary mogorov’s third hypothesis and turbulent sign statistics, layers, J. Fluid Mech. 212, 637-662 (1990). Phys. Rev. Lett. 90, 254501 (2003). [21] L. Biferale, F. Mantovani, M. Sbragaglia, A. Scagliarini, [43] A.A. Mailybaev, Spontaneously stochastic solutions in F. Toschi and R. Tripiccione, Second-order closure in one-dimensional inviscid systems, Nonlinearity 29, 2238 stratified turbulence: Simulations and modeling of bulk (2016). and entrainment regions, Phys. Rev. E 84, 016305 (2011). [22] C.E. Leith and R.H. Kraichnan, Predictability of turbulent flows, J. Atm. Sci. 29, 1041 (1972). [23] G.L. Eyink, Turbulence noise, J. Stat. Phys. 83, 955 (1996). [24] G.L. Eyink and K.R. Sreenivasan, Onsager and the theory of hydrodynamic turbulence, Rev. Mod. Phys. 78, 87 (2006). [25] G. Falkovich, K. Gawędzki, and M. Vergassola, Particles and fields in fluid turbulence, Rev. Mod. Phys. 73, 913 (2001). [26] A.A. Mailybaev, Toward analytic theory of Rayleigh- Taylor instability: lessons from a toy model, Nonlinearity 30, 2466 (2017). [27] J.P. Mellado, S. Sarkar and Y. Zhou, Large-eddy sim- ulation of Rayleigh-Taylor turbulence with compressible miscible fluids, Phys. Fluids 17, 076101 (2005). [28] A. Scagliarini, L. Biferale, M. Sbragaglia, K. Sugiyama and F. Toschi, Lattice Boltzmann methods for thermal flows: Continuum limit and applications to compressible Rayleigh-Taylor systems, Phys. Fluids 22, 055101 (2010). [29] D. Livescu and J.R. Ristorcelli, Buoyancy-driven variable-density turbulence, J. Fluid Mech. 591, 43–71 (2007). [30] S. Chandrasekhar, Hydrodynamic and hydromagnetic stability. Courier Corporation, (2013). [31] A.W. Cook and P.E. Dimotakis, Transition stages of Rayleigh-Taylor instability between miscible fluids, J. Fluid Mech. 443, 69-99 (2001). [32] M. Chertkov, Phenomenology of Rayleigh-Taylor turbulence, Phys. Rev. Lett. 91, 115001 (2003). [33] J.R. Ristorcelli and T.T. Clark, Rayleigh-Taylor turbulence: self-similar analysis and direct numerical simulations, J. Fluid Mech. 507, 213-253 (2004). [34] G. Dimonte et al, A comparative study of the turbulent Rayleigh-Taylor instability using high-resolution three- dimensional numerical simulations: The Alpha-Group collaboration, Phys. Fluids 16, 1668-1693 (2004). [35] S.B. Dalziel, P.F. Linden and D.L. Youngs, Self-similarity and internal structure of turbulence induced by Rayleigh- Taylor instability, J. Fluid Mech. 399, 1 (1999). [36] G. Boffetta, A. Mazzino, S. Musacchio and L. Vozella, Statistics of mixing in three-dimensional Rayleigh-Taylor turbulence at low Atwood number and Prandtl number one, Phys. Fluids 22, 035109 (2010). [37] G. Boffetta, A. Mazzino and S. Musacchio, Rotating Rayleigh-Taylor turbulence, Phys. Rev. Fluids 1, 054405 (2016). [38] G. Boffetta, F. De Lillo and S. Musacchio, Nonlinear diffusion model for Rayleigh-Taylor mixing, Phys. Rev. Lett. 7

SUPPLEMENTAL MATERIAL

∗ ∗ 2 R˙ n = ω Rn+1 ωn−1Rn−1 + ωnT κRn/z , (25) Derivation of the mean proﬁle solution, T (z, t) n − n − n

∗ ∗ 2 Let us write (11) as T˙n = ω Tn+1 ωn−1Tn−1 ωnR κTn/z . (26) n − − n − n c T This system defines the dynamics at discrete vertical ∂tT = K(t) ∂z z ∂z c . (16) −n | | z scales (“shells”) zn = 2 L with n = 1, 2,..., where the | | associated variables ωn, Rn and Tn describe vorticity, Substituting T from (12) and dropping the common fac- horizontal and vertical temperature fluctuations, respec- c tor θ0( z /L) yields tively. − | | Equations (24–26) are analogous to those proposed K(t) c ∂tfc = ∂z ( z ∂zfc) , (17) in [26], except for the fact that here we used the more z c | | | | popular Sabra model nonlinearity [39, 40] for the vorticity Eq. (24), where ω = u /z and u are the ve- where fc = fc(η), with η given by the second expression n n n n in (12) as locity shell variables for the Sabra model. Notice that usually in shell model literature the equations are writ- − 2 1 z t 1−c ten using kn = 1/zn to denotes scales in Fourier space. η(z, t) = p . (18) Equation (24) without the buoyancy term has energy (1 c)bc L t∗ P 2 P n − E = un , and the helicity H = ( 1) unωn as | | − | | We can write Eq. (17) in the form inviscid invariants in agreement with 3D Navier-Stokes equations. Equations (25) and (26) possess the inviscid 2 P 2 2 dfc c dfc d fc invariant S = Rn + Tn , which can be interpreted ∂tη = K(t) + ∂zη ∂zη. (19) | | | | dη z dη dη2 as the entropy. One can show that the initial condition (14) with van- Using (18) and the definition of ishing ωn(0) = Rn(0) = 0 lead to the exponentially growing modes [26]. Let us consider small perturbations, L2 t (3+c)/(1−c) ∆ωn and ∆Rn, and neglect the dissipative terms. Then, K(t) = bc (20) t∗ t∗ Eqs. (24) and (25) linearized near the initial state read in Eq. (19) leads, after a long but elementary derivation, c iβg ˙ zn to: ∆ω ˙ n = ∆Rn, ∆Rn = iθ0 ∆ωn. (27) zn − L 2 d fc c dfc Solution of these equations provide one unstable mode 2 + + 2η = 0. (21) dη η dη for each “wavenumber” kn = 1/zn with the corresponding positive Lyapunov exponent Denoting gc = dfc/dη we can recast the above expression in to: 1/2 (1−c)/2 c λn = ξ kn , ξ = βgθ0/L , (28) dg c c + + 2η g = 0. (22) in the direct analogy with the RT instability of the full dη η c 3D system. The general solution of Eq. (22) has the form In summary, the shell model (24–26) mimics spatial variations of the vorticity and temperature fields at a −c −η2 gc(η) = C η e (23) wide range of scales zn in a way that closely repro- | | duces important properties of the full RT instability. with an arbitrary pre-factor C. Finally, the solution Such description can be adapted for both two and three R for fc(η) = gc(η)dη takes the form (13), where C = spatial dimensions, by tuning the model coefficients to 1−c 2/Γ 2 is determined from the condition fc 1 as conserve the respective invariants [26]. It should be η . → ± stressed that the resulting models feature most phe- → ±∞ nomenological properties of the RT turbulence described by Chertkov [32]. Shell Model for RT evolution For numerical analysis, we consider dimensionless for- mulation with the parameters L, βg and θ0 set to unity We introduce the RT shell model equations in the form and very small dissipative parameters ν = κ = 10−10.

∗ ∗ We simulated numerically the model with 30 shells. As ω˙ n = ωn+2ωn+1/4 + ωn+1ωn−1/2 − (24) c 1, the growth of small-scale linear modes is depleted, 2 → +2ωn−1ωn−2 + iβgRn/zn νωn/z , aﬀecting the length of the power-law interval; see Fig. 4. − n 8

0.05 For example, one has λn kn for c = 0.9. In this case ers the model with a larger number of shells and much the power-law interval is∝ not observed unless one consid- smaller dissipative parameters.