## Abstract

We review the theoretical developments in the field of Rayleigh–Taylor instabilities and turbulent mixing, discuss what is known and what is not known about the phenomenon, and outline the features of similarity of the turbulent mixing process. Based on the physical intuition and on the results of rigorous theoretical studies, we put forward some new ideas on how to grasp the essentials of the mixing process and consider the influence of momentum transport on the invariants and on scaling and statistical properties of the unsteady turbulent mixing.

## 1. Rayleigh–Taylor phenomena

Turbulence is commonly considered to be the last unresolved problem of classical physics (Sreenivasan 1999). For years its complexity and universality have fascinated mathematicians and scientists, and have nourished the enthusiasm of philosophers (Reynolds 1883; Taylor 1935; Monin & Yaglom 1975; Frisch 1995). Similarity and isotropy are fundamental hypotheses that advanced our understanding of turbulent processes (Kolmogorov 1941*a*,*b*; Batchelor 1953; Barenblatt 1979). Still, the problem withstands the efforts applied, thus indicating a need for the development of new concepts to achieve better understanding and control of the irregular dynamics. Turbulent motions of real fluids are often characterized by non-equilibrium heat transport and sharp changes of density and pressure, and may be a subject to spatially varying and time-dependent acceleration (Taylor 1935; Monin & Yaglom 1975; Barenblatt 1979; Frisch 1995; Sreenivasan 1999). Turbulent mixing induced by the Rayleigh–Taylor instability (RTI) is a generic problem we encounter when trying to extend our knowledge of turbulent processes beyond the limit of idealized consideration (Abarzhi 2008). This article discusses some theoretical and empirical modelling approaches of RTI and turbulent mixing that have been developed over the recent decades and outlines some new ideas that may help us to grasp the essentials of the mixing process.

We observe the development of the RTI whenever two matters of different densities are accelerated against the density gradients (Rayleigh 1883; Davies & Taylor 1950). Extensive interfacial material mixing ensues with time (Read 1984; Schneider *et al.* 1998; Ramaprabhu & Andrews 2003; Meshkov 2006). The turbulent mixing plays a key role in a wide variety of natural phenomena, spanning astrophysical to micro-scales, as well as in technological applications (Abarzhi 2008). It influences the formation of the ‘hot spot’ in inertial confinement fusion (ICF) (Gamaly 1993; Bodner *et al.* 1998; Mima 2004; Remington *et al.* 2006; Besnard 2007), limits radial compression of imploding Z-pinches (Ryutov *et al.* 2000) and controls non-equilibrium heat transfer induced by ultra-fast high-power lasers in solids (Barnes *et al.* 1974; Lukyanchuk *et al.* 1993*a*,*b*; Perez & Lewis 2002). Rayleigh–Taylor (RT) turbulent mixing governs transports of mass, momentum and energy in mantle–lithosphere tectonics, ocean and atmosphere, interstellar molecular clouds, flashes of supernovae, and in flames and fires (Stone *et al.* 1996; Hillebrandt & Niemeyer 2000; Rosner 2003). Our ability to control the unsteady turbulent mixing is crucial for industrial applications in aerodynamics and aeronautics (e.g. Spalart & Wattmuff 1993; Gutman *et al.* 1995; Marmottant & Villermaux 2004), and in free-space optical telecommunications (e.g. Kaminow *et al.* 1996; Choi & Chan 2002). RTI is essential to occur under extreme conditions of high-energy density (Zel’dovich & Raizer 2002; Remington *et al.* 2006; Drake 2009), but one can still easily observe it in everyday life when looking at how salted and fresh water mix or how wine flows out from an overturned glass.

Mathematical aspects of the dynamics of RTI and turbulent mixing reveal intellectual richness of the theoretical problem (Abarzhi 2008). For instance, at atomistic and mesoscales, RT-induced material transport may depart from a standard scenario of Gibbs’ ensemble and quasi-static Boltzmann equation (Hoover 1991; Zhakhovskii *et al.* 2006; Kadau *et al.* 2007; Evans & Morriss 2008). On the other hand, at macroscopic scales singular character and similarity of the non-equilibrium dynamics are interplayed with the fundamental properties of three-dimensional Euler or Navier–Stokes equations and with the problem sensitivity to the boundary conditions at discontinuities and the initial conditions (Pumir & Siggia 1992; Baker *et al.* 1993; Tanveer 1993; Wu 2006). Furthermore, turbulent mixing is a multi-scale, anisotropic, non-local and statistically unsteady process, and its fundamental invariant properties depart dramatically from those in the classical Kolmogorov turbulence (Abarzhi *et al.* 2005). As in any turbulent process, contribution of all scales into the flow dynamics induces randomness of RT mixing (Frisch 1995). Yet, because of statistical unsteadiness and non-Gaussian character of fluctuations, capturing this randomness is a challenging task that necessitates further advancements of the methods of stochastic modelling (Abarzhi *et al.* 2007).

On the side of the numerical simulations, modelling RT mixing is a severe test, as the numerical solutions appear sensitive to the initial and boundary conditions as well as to the influence of unresolved small-scale structures on the large-scale dynamics and to the anomalous character of energy transport (Gardner *et al.* 1988; Youngs 1991; He *et al.* 1999; Cook & Dimotakis 2001; Young *et al.* 2001; Kadau *et al.* 2007). Addressing these difficulties brings up the issue of predictive capability of the numerical models, their verification and validation as well as quantification of uncertainty of the solutions obtained (Calder *et al.* 2002; George *et al.* 2002). Synergy between theory, experiment and simulations thus appears crucial for obtaining an integrated description of the phenomenon and for grasping its universality features (Dimonte *et al.* 2004; Abarzhi 2008). On the side of the experiments, RT flows are hard to implement and systematically study in a well-controlled laboratory environment, as their sensitivity and transient character of the dynamics impose high requirements on accuracy, spatio-temporal resolution and dynamic range of the flow quantities as well as on data rate acquisition. One of the primary concerns for RT experiments is the influence of an observer on the results of observation and data interpretation (Dimonte 2000; Waddell *et al.* 2001; Jacobs & Krivets 2005; Abarzhi 2008). To provide reliable and repeatable datasets, experiments on the RTI and turbulent mixing in fluids and plasmas employ the most advanced diagnostic approaches (Dalziel *et al.* 1999; Aglitskiy *et al.* 2001; Kucherenko *et al.* 2003; Meshkov 2006; Remington *et al.* 2006; Orlov *et al.* 2010).

For many years now, extensive multi-faceted studies of RTIs have tried to capture the instability evolution, particularly, to understand what are the characteristic failure modes; what is the RTI growth rate under various physical conditions; how does the flow transition occur from initial to turbulent stages; and how can the mixing process be stabilized. Many of these questions have been answered in detail, especially at the early stages of the instability development (e.g. Sharp 1984; Kull 1991; Rupert 1992; Brouillette 2002; Meshkov 2006). Some others, such as RT dynamics in stratified and diffusive media or nonlinear evolution of the ablation front, are subjects of active research; many more await a fresh thoughtful look. With over a 100 RT-related papers published per year in peer-reviewed mathematical, computational, scientific and engineering journals, discussing this vast subject in its entire diversity and completeness would be a titanic task going beyond the intents of the present article. This work overviews the theoretical developments in the field of RTI, discusses what is known and what is not known about the phenomenon, and outlines the features of universality and similarity of the turbulent mixing process. Based on the physical intuition and on the results of rigorous theoretical studies, it puts forward some new ideas and a few provocative questions that, in the author’s hope, would attract the attention of an inquisitive mind and would further advance our understanding of turbulence.

## 2. Conservation principles governing dynamics of Rayleigh–Taylor instability and mixing

While looking at a wide variety of the RT flows, one can identify certain similarity features and somehow loosely separate three subsequent stages in the instability evolution. Initially, small perturbations at the fluid interface grow with time. For incompressible immiscible fluids subject to a sustained acceleration **g** (gravity), the growth is exponential (e.g. Rayleigh 1883; Kull 1991). In the nonlinear regime, a coherent structure of bubbles and spikes appears with light (heavy) fluid with density *ρ*_{l(h)} penetrating the heavy (light) fluid in bubbles (spikes) (e.g. Davies & Taylor 1950; Abarzhi 1998; Marinak *et al.* 1998; figure 1). The dynamics of the structure is governed by two, in general independent, length scales: the amplitude in the direction of gravity and the spatial period *λ* in the normal plane (Abarzhi *et al*. 2006). The horizontal scale *λ* is set by the mode of fastest growth or by the initial perturbation (Kull 1991), and it may increase, if the flow is two-dimensional and the initial perturbation is broadband and incoherent (Alon *et al.* 1994, 1995). The vertical scale rises asymptotically as power law with time. It is believed that in the mixing regime the growth is self-similar, , |**g**|=*g* (Read 1984; Youngs 1984; Schneider *et al.* 1998). The length scale can be regarded as the integral scale for energy dissipation in small-scale vortical structures, produced by shear and accumulated at the fluid interface (Abarzhi *et al.* 2005).

In miscible fluids, diffusion reduces effective buoyant force and advects small-scale structures from the interface into the bulk, thus slowing down the mixing process (Abarzhi *et al.* 2005). The growth rate of the amplitude depends also on the acceleration history, and RTI with impulsive acceleration is often considered to be the Richtmyer–Meshkov instability (RMI) developing when a shock wave refracts through the interface between two fluids with different values of the acoustic impedance (Richtmyer 1960; Meshkov 1969; Aleshin *et al.* 1990; Holmes *et al.* 1999). The evolution of the RTIs in low-energy density regimes (liquids or gases under ordinary conditions) differs from that in high-energy density regimes, for instance in supernova or laser-ablated plasmas, whereas shock-driven dynamics of compressible flows have their own specific features distinct from those in reactive materials (Remington *et al.* 2006; Drake 2009). In all these cases, however, RT flows are characterized by large-scale coherent structure, small-scale structures and energy transports to both large and small scales (Abarzhi 2008). The energy transfer to large scales is related, perhaps, to the influence of the initial conditions and that to small scales is induced by shear.

In continuous matter approximation, the dynamics of RTI is governed by conservation principles that are nonlinear compressible Navier–Stokes or Euler equations with the initial and boundary conditions at the fluid interface and at the boundaries of the domain (Landau & Lifshitz 1987). In the bulk of the fluid, the equations of the conservation of mass and momentum are2.1where *ρ*, **v** and *p* are the density, velocity and pressure of the fluid, respectively; **S** denotes terms induced by viscous stress and other effects; and dot marks the partial derivative in time *t*. For compressible fluids, system (2.1) is augmented with the equation for energy transport and the equation of state. For miscible fluids, equations for concentration transport are also incorporated (Landau & Lifshitz 1987). For compressible and/or miscible fluids, the fluid ‘interface’ is considered as the region with strong density gradients, and its self-consistent description is a formidable theoretical task (Duff *et al.* 1962; Bodner 1974; Lukyanchuk *et al.* 1993*a*,*b*; Sanz 1994; Piriz 2001; Clarisse *et al.* 2008; Gauthier & Le Creurer 2010). Its resolution may require establishing new connections of continuous matter approximation to kinetic processes at atomistic scales and better understanding of the interplay between Eulerian and Lagrangian descriptions in the systems that are out of local thermodynamic equilibrium (Zhakhovskii *et al.* 2006; Kadau *et al.* 2007; Bell *et al.* 2007).

For incompressible immiscible fluids, the fluid interface is a discontinuity, so that *ρ*=*ρ*_{h}*H*(−*θ*)+*ρ*_{l}*H*(*θ*) and **v**=**v**_{h}*H*(−*θ*)+**v**_{l}*H*(*θ*), where *H* is the Heaviside step-function, *θ* is a scalar function on the coordinates and time with *θ*=0 at the interface, and **v**_{h(l)} is the velocity of the heavy (light) fluid located in the region *θ*<0 (*θ*>0) (Abarzhi *et al.* 2003). If there is no mass flow across the interface, the pressure and normal component of velocity are continuous at the interface2.2awhere the interface unit normal vector **n**=∇*θ*/|∇*θ*| (Landau & Lifshitz 1987). The conditions at the boundaries of the domain close the set of the governing equations. In particular, in spatially extended fluid systems, the flow has no mass sources,2.2band can be periodic in the plane (*x*,*y*) normal to the gravity axis *z* (figure 1).

To date, a rigorous theoretical description of the evolution of RTI and turbulent mixing for ‘all the times in the entire spatial domain’ has not been obtained. The ill-posedness of the problem, the nonlinearities and secondary instabilities that produce small-scale structures and result in the development of singularities in equations (2.1) and (2.2a,*b*) make the dynamics of RTI essentially non-local (Birkhoff & Carter 1957; Garabedian 1957; Baker *et al*. 1993; Tanveer 1993; Abarzhi 1998). On the one hand, the multi-scale mixing process maintains certain features of coherence and order, associated primarily with the dynamics of large scales and with the memory of the initial conditions (Marinak *et al*. 1998; He *et al*. 1999; Abarzhi *et al*. 2006). On the other hand, this is a non-deterministic process whose randomness results from contributions of small-scale structures and from interaction of all the scales (Frisch 1995; Abarzhi *et al*. 2007; Kadau *et al*. 2007).

In the limiting case of a single fluid with constant density, *ρ*=const., with disregarded influence of gravity, viscous stress and other terms, **g**=**S**=0, and in the asymptotic regime, when the initial and boundary conditions play no role and the dynamics is local and statistically steady, equation (2.1) describes isotropic and homogeneous turbulence (Frisch 1995). Kolmogorov was the first to discover in 1941 that the assumptions of locality, homogeneity and isotropy lead to space–time scale invariance of equation (2.1) with *l*→*lK*, *t*→*tK*^{1−n} and *v*→*vK*^{n}, where *v*=|**v**|, *n*=1/3 and *l* is the characteristic spatial scale. Understanding the invariant properties of equations (2.1) and (2.2a,*b*) governing non-local, inhomogeneous and anisotropic RT mixing would serve for extending the range of applicability of classical Kolmogorov’s consideration.

## 3. Theory of Rayleigh–Taylor dynamics: what is known and what is not known?

### (a) Linear and weakly nonlinear stages

Over the last 100 years, significant progress has been achieved in our understanding of the initial stages of RT evolution (e.g. Sharp 1984; Kull 1991; Abarzhi 2008; and references therein). Rayleigh (1883) was the first to analyse the stability of an incompressible inviscid fluid layer with the density gradient pointed against the pressure gradient. He found that the instability develops faster for smaller values of wavelength of the initial perturbation *λ*, as with the characteristic time scale, , where *A*=(*ρ*_{h}−*ρ*_{l})/(*ρ*_{h}+*ρ*_{l}) is the Atwood number. Later, the instability growth rate was derived for fluids with a sharp interface and with account for the effects of surface tension, viscosity and compressibility (e.g. Bellmann & Pennington 1954; Bernstein & Book 1983; Kull 1991; Mikaelian 1996). It has been shown, for instance, that viscosity induces in the dynamics a characteristic length scale *λ*_{m} associated with the mode of fastest growth, and *λ*_{m}∼(*v*^{2}/*Ag*)^{1/3}, where *v* is the characteristic kinematic viscosity. Among recent advances one should mention linear theories of compressible RTI (Gauthier & Le Creurer 2010), shock-driven RMI (Wouchuk 2001*a*,*b*) and ablative RTI in direct-drive ICF (Sanz 1994; Piriz 2001).

Exponential growth of the perturbation amplitude in the linear regime of RTI is superseded by its power-law time-dependency in the weakly nonlinear regime. This is so-called ‘saturation’ and it is believed to occur owing to nonlinear enhancement and interaction of higher order modes. The first empirical model of the saturation process was proposed by Haan (1991). Later, more rigorous description of the mode enhancement was suggested by Berning & Rubenchik (1998) and Ikegawa & Nishihara (2003). For weakly nonlinear RMI, significant progress has been achieved by Velikovich & Dimonte (1996) via the use of high-order perturbation theory. An important outcome of the linear and weakly nonlinear considerations is an indication of a non-local character of the instability evolution, and a necessity to account for ‘all-to-all’ interactions in order to derive to the leading order a reliable description of the flow dynamics (e.g. Velikovich & Dimonte 1996; Wouchuk 2001*a*,*b*).

Among the problems of early-time instability evolution still requiring rigorous theoretical considerations and important for experiments and simulations, one should mention linear RTI in compressible and stratified media with time-dependent acceleration and with thermal and/or mass diffusion, RTI and RMI under high-energy density conditions, including RTI and RMI of ablation front and with the effect of laser imprint in direct and indirect-drive ICF (Aglitskiy *et al*. 2001; Remington *et al*. 2006), and shear-driven RMI in solids (Piriz *et al*. 2005; Zybin *et al*. 2006). A crucial problem in the design of experiments and dataset interpretation is the coupling of horizontal *λ* and vertical scales in the weakly nonlinear regime of RTI and its dependency on the dispersion properties and the initial spectra, including interactions of modes that are in-phase, and modes whose phases are random (Kucherenko *et al.* 2003; Collins *et al.* 2004; Milovich *et al.* 2004).

To summarize, the theoretical studies of RTIs in linear and weakly nonlinear stages indicate: unstable RT flow has a characteristic length scale *λ*_{m} associated with the mode of fastest growth. Even at early times, the dynamics of RTI exhibits certain features of non-locality. The problem of coupling of horizontal *λ* and vertical scales and its dependency on the initial spectra and dispersion properties is open.

### (b) Nonlinear asymptotic dynamics

The nonlinear dynamics of RTI is important to study because it bridges the linear stage, where the perturbation growth rate is influenced by the initial conditions, , and turbulent mixing regime, whose evolution is believed to be self-similar, (Sharp 1984; Youngs 1984; Abarzhi 1996, 1998). Besides, most of the existing experiments and simulations on RTI do not transit completely to the turbulent regime (Dalziel *et al*. 1999; Kucherenko *et al*. 2003; Ramaprabhu & Andrews 2003; Meshkov 2006; Remington *et al.* 2006), whereas the steady motion of the nonlinear bubble of air rising through water with velocity has been documented by Taylor as early as in 1950 (Davies & Taylor 1950). A trustworthy theoretical description of the nonlinear RTI is required for capturing essentials of the turbulent mixing processes and for identification of robust diagnostic parameters for experiments and simulations (Dimonte *et al*. 2004; Abarzhi *et al*. 2006). For a detailed review of theoretical studies of the nonlinear RTI and RMI and the progress achieved in this field in recent decades, the reader is referred to Abarzhi (2008).

The title ‘Rayleigh–Taylor instability’ reflects the seminal contribution of G. I. Taylor to the field, as Davies & Taylor (1950) were the first to observe the nonlinear phenomenon for fluids with highly contrasting densities (*A*≈1) in cylindrical tubes, and suggested a simple model for their observations. The first theoretical studies of the nonlinear RTI were performed by Fermi & von Neuman (1951), who attempted to describe the evolution of the unstable free boundary by means of Lagrange dynamics. Later, Layzer (1955) suggested how to derive a nonlinear solution with velocity in a so-called single-mode approximation. Shortly after, Garabedian (1957) and Birkhoff & Carter (1957) applied conformal mapping techniques to account for the effect of singularity on the asymptotic dynamics and reported a non-uniqueness of the nonlinear steady solutions for RT bubbles.

In the 1990s, a number of important contributions advanced our knowledge of nonlinear evolution of RTI with *A*=1. One of these contributions was made by Inogamov (1992), who applied, for steady two-dimensional flow, the high-order spatial expansions and the Fourier series, attempting to bridge the results of Layzer & Garabedian at . Nearly at the same time Tanveer (1993) has shown that singularity in RT flow develops very slowly, logarithmically with time . For a study of the asymptotic free fall of the RT spike, the reader is referred to the work of Clavin & Williams (2005).

Sophisticated theoretical approaches developed in recent decades for nonlinear RT problems have substantially advanced the field of partial differential equations and have served to bring RTIs from an ‘application’ problem to a mature scientific problem. One of these approaches is group theory consideration, a rigorous theoretical framework that strictly obeys the conservation principles and is applicable for a broad class of RT-related nonlinear problems, including two-dimensional and three-dimensional RT and RM flows with various symmetries and for 0≪*A*≤1 (Abarzhi 2008).

The group theory consideration has enabled the study of local properties of nonlinear RTI, such as the asymptotic dynamics of the bubble front, as well as its global properties, such as the classification of interactions and pattern formation. It has shown that at late times, *t*/*τ*≫1, the dynamics of RT and RM flows exhibit certain universal features. In particular, both RT and RM bubbles conserve isotropy in the plane normal to the direction of acceleration. For highly symmetric three-dimensional flows, the asymptotic solutions coincide except for the difference in normalization factor, whereas three-dimensional–two-dimensional crossover is discontinuous (Abarzhi 1998, 2002). In both RTI and RMI, the nonlinear dynamics is essentially non-local and multi-scale (Abarzhi *et al.* 2003). The non-locality is exhibited in a multiplicity of regular asymptotic solutions that form a continuous family with a narrow stability interval (Abarzhi 1996). The multi-scale character implies that, for *t*/*τ*≫1, the period and amplitude of the front contribute independently to the dynamics (Abarzhi *et al.* 2006). Yet, despite these similarities, the shape of the RT bubble is curved as it moves steadily, and that of the RM bubble is flattened as it decelerates (Abarzhi *et al.* 2003; Herrmann *et al.* 2008). In the field of RTIs, group theory analysis has introduced the concept of coherent structures and has also shown that the requirements of stability and isotropy significantly limit the number of coherent patterns that may occur in the flow. Furthermore, there exist special symmetries (e.g. hexagonal symmetry in planar geometry; figure 1*b*) at which growth of horizontal scales may not be feasible and flow is regular (Abarzhi 2008).

Many qualitative and quantitative results found within group theory consideration have been confirmed by accurate observations, and many others have indicated a need in the further advancement of experimental diagnostics and design of experiments and simulations (e.g. Herrmann *et al.* 2008). Among important problems of late-time evolution of RTI it is worth emphasizing the effects of surface tension and viscosity on the dynamics; the nonlinear RTI in compressible, stratified and diffusive matters, especially under conditions of high-energy density; quantification of contributions of horizontal *λ* and vertical scales.

### (c) Empirical modelling of Rayleigh–Taylor turbulent mixing

An existence of self-similar regime in Rayleigh–Taylor flows with has been discussed by Fermi & von Neuman (1951) and by Belen’kii & Fradkin (1965). The first observations of the phenomenon were reported in the experiments of Read (1984) and in numerical simulations of Youngs (1984). These observations were focused on the diagnostics of vertical scale and on ascertainment of a similarity law, , where *α* is a constant. The scaling of the bubble velocity in the nonlinear regime (Davies & Taylor 1950) indicated that an increase in *λ* may lead to the flow acceleration. Thus, the growth of the horizontal scale *λ* with *λ*∼*gt*^{2} was suggested as a primary mechanism of the mixing development (Sharp 1984). To describe the turbulent mixing that ensued from the nonlinear regime and induced by the self-similar growth of the period *λ*, Glimm & Sharp (1990) applied renormalization group analysis and reported good agreement with the numerical simulations results. Shortly after, Alon and co-workers (1994, 1995) attempted to develop a universal empirical model of the turbulent mixing, which would be applicable for RTI and RMI for all the values of the Atwood number, 0<*A*≤1, in both two-dimensional and three-dimensional flows. The model was based on an empirical equation, which balanced with adjustable parameters the flow buoyancy, inertia and drag. The dynamics of RTI and RMI was interpreted as a single-scale process, solely governed by the spatial period *λ*. To derive the similarity solution, it was requested that .

Inspired by the simplicity of these models, several other empirical models were developed and indicated various dependencies of the mixing zone growth rate (Dimonte 2000). These developments have been well received in the community, as they implied that only a small fraction of the flow quantities should be diagnosed in experiments and simulations in order to capture the essentials of the mixing process. To identify the parameter *α*, so-called *α*-group collaboration has been established, and large resources have been used to measure the dynamics of the coarsest vertical and horizontal length scales (Dimonte *et al.* 2004). Despite significant efforts, the universality of scaling and *λ*/*gt*^{2} and the mechanism of the mixing process in RT flow still remain open issues. It is unclear whether the growth of horizontal scales is a sole mechanism of the mixing development; whether the mixing flow is fully disordered and the large-scale coherence is completely lost; and whether there is a parameter whose value controls the transition to mixing quantitatively (Abarzhi 2008 and references therein).

Having started from the end of the 1980s, other theoretical approaches have been developed and suggested to consider the RT and RM mixing as a slightly anisotropic turbulent flow (e.g. Gauthier & Bonnet 1990; Besnard *et al.* 1996; Steinkamp *et al.* 1999*a*,*b*; Ristorcelli & Clark 2004). These studies have been focused on low Atwood number flows, *A*≪1. In this case, small density differences can be imitated by the density fluctuations and are driven by the velocity field, similar to passive scalar mixing in classical turbulence. These important works have served to formulate the Rayleigh–Taylor problems within the general content of turbulent flows. Furthermore, these approaches helped in developing a number of sophisticated and accurate techniques for numerical modelling of the RTIs and RMIs and for comparison of the simulation results with the experiments (Steinkamp *et al.* 1999*a*,*b*; Ristorcelli & Clark 2004). More recently, Dimotakis (2000), Zhou *et al.* (2003) and Chertkov (2003) have attempted to apply and to modify the theory of Kolmogorov (1941*a*,*b*) for a description of the RTIs and to analyse the available experimental and numerical data in terms of turbulent power laws.

Some quantitative agreements have been reported between the observation data and the models that used adjustable parameters (e.g. Alon *et al.* 1994; Dimonte 2000; Dimotakis 2000; Zhou *et al.* 2003). Some qualitative features of RT mixing process still remain unclear. For instance, in high-energy density laboratory astrophysics experiments, aimed at the mixing enhancement for mimicking astrophysical phenomena (Drake 2009), despite very large values of the Reynolds number, the RT flows were keeping some degree of order (Robey *et al.* 2003; Zhou *et al.* 2003). To date, experiments and simulations did not provide a trustworthy guidance on whether the fundamental concepts of classical turbulence are applicable to an accelerating RT flow (e.g. Abarzhi *et al.* 2005 and references therein).

The recent success achieved in the theoretical description of RTI and its large-scale numerical modelling and laboratory experiments, and the similarity in behaviour of RT turbulent mixing in different physical regimes, both make this moment right for a theory to appear and to integrate the existing knowledge on RTIs, from initial to late stages. While revealing new qualitative characteristics of RT-driven phenomena, such a theory would account for the results of rigorous theoretical studies of the RTIs, and would serve to advance the methods of acquisition and interpretation of experimental and numerical data. Below, on the basis of physical intuition and accounting for the results of linear and nonlinear theories of RTI, we suggest to the reader some new ideas (Abarzhi *et al.* 2005, 2007) that may help to understand the mixing process and to advance our knowledge of turbulence beyond idealized, isotropic and homogeneous considerations.

## 4. Momentum-driven Rayleigh–Taylor mixing

A striking similarity of behaviour of RT mixing in the vastly different regimes indicates that this turbulent process has some features of universality and is eligible for the consideration of first principles. Indeed, as in all natural processes, turbulent transports obey the laws of conservation of mass, momentum and energy (equations (2.1) and (2.2a,*b*)). In canonical Kolmogorov turbulence, mass and momentum are conserved because of conditions of homogeneity, isotropy and locality, and the statistically steady turbulent flow is solely driven by the transport of kinetic energy (Kolmogorov 1941*a*,*b*). In contrast to the classical Kolmogorov’s scenario, RT turbulent mixing is governed by transports of mass, momentum, and potential and kinetic energies (equations (2.1) and (2.2a,*b*)), and, furthermore, is statistically unsteady. For momentum-driven RT mixing, the rate of momentum loss may be a better indicator of the dynamics than the rate of energy dissipation (Abarzhi *et al.* 2005). Below, we consider the influence of momentum transport on the mechanism of the mixing process, and on a number of scaling, invariant and statistical properties of the turbulent mixing flow.

### (a) Momentum transport in Rayleigh–Taylor turbulent mixing

In RT mixing flow, the dynamics of a parcel of fluid is governed by a balance per unit mass of the rate of momentum gain and the rate of momentum loss *μ* as4.1

Here *h* is the vertical length scale, *v* is the velocity, and and *μ* are the absolute values of vectors pointed in opposite directions and parallel to the direction of gravity. The system of equations (4.1) is a simplified, dimensional, ground-based model of the system of equations (2.1) and (2.2a,*b*) representing the conservation of mass and momentum.

The rate of momentum gain is the buoyant force, , where is the rate of energy gain induced by buoyancy and with *f*(*A*) being a function of the Atwood number. The rate of momentum loss is *μ*=*ε*/*v*, where *ε* is the rate of dissipation of kinetic energy. Without viscous scale and with *L* being the characteristic scale for energy dissipation, *ε*=*Cv*^{3}/*L* on the basis of dimensional grounds, where *C* is a constant. Hereafter, for the sake of simplicity, we re-scale *g* *f*(*A*)→*g*.

In system (4.1), the time scale *τ* is set by the initial conditions, and the time dependence of asymptotic solutions depends on whether the characteristic length scale of the flow is horizontal (*L*∼*λ*) or vertical (). If the relevant scale is horizontal, the solution is steady, and , as observed for nonlinear RTI (Davies & Taylor 1950). In this regime, the rates of momentum gain and loss are equal to one another, , and the rates of energy gain and dissipation are also in balance, . If the characteristic length scale is the vertical scale, *L*∼*h*, then asymptotically with time, *t*/*τ*≫1, the position and velocity change as *h*=*agt*^{2}/2 and *v*=*agt* with *a*=(1+2*C*)^{−1}. In this regime, the rates of energy gain and dissipation are time dependent with and *ε*=(1−*a*)*a* *g*^{2}*t*, whereas the rates of momentum gain and loss are time- and scale-invariant values, and *μ*=*Cv*^{2}/*h*=*g*(1−*a*) (Abarzhi *et al.* 2005).

Comparing the rates of energy and momentum, one can suggest two distinct mechanisms of how a transition may occur from the nonlinear to turbulent regime of RTI. In the former case, the energy dissipation rate *ε* is comparable in the nonlinear and turbulent regimes, *ε*∼*g*^{3/2}*λ*^{1/2}∼*g*^{2}*t*, so that the horizontal scale *λ* may grow with time as *λ*∼*gt*^{2}. This scenario agrees with the results of the merger models (Glimm & Sharp 1990; Alon *et al.* 1994; Dimonte *et al.* 2004). The growth of horizontal scale is possible; however, it is not a necessary condition for the mixing to occur. Indeed, according to observations, the values of *a* are rather small, *a*∼0.1–0.2 (Dimonte *et al.* 2004). This implies that, in RT mixing flow, almost all energy induced by the buoyant force dissipates, , and the rates of momentum gain and loss cause only a slight misbalance between one another, , with . The turbulent mixing may thus develop if the vertical *h* scale dominates the flow and if it is regarded as the scale for energy dissipation that occurs in small-scale structures at the fluid interface (Abarzhi *et al*. 2005).

### (b) Invariant quantities, scaling properties and symmetries of Rayleigh–Taylor turbulent mixing

In canonical isotropic and homogeneous turbulence, the concepts of non-dissipative energy cascade and existence of inertial interval are compatible with the time and scale invariance of the rate of energy dissipation. Indeed, the energy injected at large scales, *ε*∼*v*^{2}(*v*/*L*), is transferred without loss through the inertial range and is dissipated at small scales, *ε*∼(*vL*)(*v*/*L*)^{2} (Kolmogorov 1941*a*,*b*; Batchelor 1953). In RT mixing flow, the energy dissipation rate is time dependent and statistically unsteady, *ε*∼*g*^{2}*t*, and these fundamental concepts may be inapplicable. On the other hand, in RT turbulent mixing, the rate of momentum loss *μ*∼*v*^{2}/*L* is a time- and scale-invariant value, whereas in isotropic Kolmogorov turbulence this is not a diagnostic parameter. These distinctions indicate that invariants and symmetry properties of RT turbulent mixing and isotropic Kolmogorov turbulence differ substantially. Indeed, Kolmogorov turbulence is Galilean invariant, whereas RT turbulent mixing is non-inertial.

In canonical turbulence, time and scale invariance of the energy dissipation rate defines how the flow quantities scale. If at large scale *L* the characteristic velocity is *v* and at small scale *l* the characteristic velocity is *v*_{l}, then the invariance of energy dissipation rate yields the velocity scaling as *v*_{l}/*v*∼(*l*/*L*)^{1/3}. Similarly, the global Reynolds number, *Re*=*vL*/*ν*, and the local Reynolds number, *Re*_{l}=*v*_{l}*l*/*ν*, scale as *Re*_{l}∼*Re*(*l*/*L*)^{4/3}. The viscous length scale, at which viscosity governs the dynamics, corresponds to *Re*_{l}∼1 and is determined by the value of energy dissipation rate as *l*∼(*ν*^{3}/*ε*)^{1/4} (Kolmogorov 1941*a*,*b*; Batchelor 1953; Frisch 1995).

In RT turbulent mixing, the energy dissipation rate is time dependent, *ε*∼*g*^{2}*t*, and the scaling properties of the flow quantities are defined by the time- and scale-invariant rate of momentum loss as . In particular, fluid velocity scales as *v*_{l}/*v*∼(*l*/*L*)^{1/2}. This indicates that turbulent mixing flow is more ordered than Kolmogorov turbulence. In RT turbulent mixing the global Reynolds number is time dependent, *Re*=*vL*/*ν*∼*g*^{2}*t*^{3}/*ν*, and the local Reynolds number, *Re*_{l}=*v*_{l}*l*/*ν*, scales as *Re*_{l}∼*Re*(*l*/*L*)^{3/2}. The viscous length scale, at which viscosity dominates the dynamics, corresponds to *Re*_{l}∼1 and is determined by the rate of momentum loss as *l*∼(*ν*^{2}/*μ*)^{1/3}. Remarkably, that due to , the viscous length scale, is comparable to the mode of fastest growth *l*∼(*ν*^{2}/*μ*)^{1/3}∼(*ν*^{2}/*g*)^{1/3}, a characteristic length of the RT flow (Youngs 1984; Kull 1991; Abarzhi 2008). This indicates a self-consistency of our momentum-based consideration. Similarly, one can analyse the dissipative scales and surface tension effects, as well as the spectral properties of RT turbulent flow. It is worth emphasizing that the flow helicity **v**⋅[∇×**v**] is the invariant quantity in both RT mixing flow and canonical Kolmogorov turbulence. Studying this quantity and its influence on the turbulent dynamics may serve for obtaining an integrated description of the highly anisotropic (RTI) and isotropic (Kolmogorov) turbulent flows, and for further extension of the range of applicability of the classical approaches (Frisch 1995).

### (c) Stochastic modelling of statistically unsteady turbulent mixing process

As in any turbulent process, RT-mixing dynamics has some random character resulting from the contribution of small-scale structures and interactions of all the scales (e.g. Frisch 1995; Abarzhi *et al*. 2007; Kadau *et al*. 2007). Capturing this randomness is a complex task. In Kolmogorov turbulence, random character of flow dissipation is induced by velocity fluctuations with the energy dissipation rate being a statistic invariant (Frisch 1995). In RT-mixing flow, both the velocity and the length scale fluctuate and the energy dissipation rate grows with time. We account for the random character of dissipation in RT flow on the basis of the idea that, even in a statistically unsteady process (whose fluctuating quantities are time dependent and non-Gaussian), there exist time- and scale-invariant values fluctuating about their means, particularly the rate of momentum loss *μ* (Abarzhi *et al.* 2007).

To study the effect of fluctuations on the mixing dynamics, equation (4.1) is represented in differential form4.2awith differentials of momentum gain and loss d*M*=*C*(*v*^{2}/*h*) d*t* and with *C* being a stochastic process (Abarzhi *et al.* 2007). This process is, in general, time dependent, *C*=*C*(*t*), and is characterized by a time scale *τ*_{C}, showing how fast the distribution *C*(*t*) approaches a stationary probability density function, *p*(*C*). The function *p*(*C*) is non-symmetric, *C*>0, with the mean value 〈*C*〉, with the mode *C*_{max} corresponding to the highest value of *p*(*C*), and with the standard deviation *σ* describing the fluctuations’ intensity. For stochastic processes with log-normal distribution, , the mean , the mode , and4.2bwith d*W* being a standard Weiner process.

The stochastic modelling results indicate that fluctuations do not change the asymptotic time dependence of the dynamics, so that *h*∼*gt*^{2}/2 as , yet they influence the coefficient, *a*=*h*/*gt*^{2} (Abarzhi *et al*. 2007). Depending on the shape of the distribution *p*(*C*) and on the fluctuations’ intensity *σ*, the mean value of *a* may vary in several folds, and, furthermore, it saturates slowly with time for (*t*/*τ*)≫1. This result qualitatively explains the several-fold scatter in the values of in the experiments and simulations of Dimonte *et al*. (2004). It indicates that the growth-rate parameter *α* is sensitive to the dissipation statistics and it is a significant parameter, not because it is ‘deterministic’ or ‘universal’, but because its value is rather small, *α*≪1/2 (Abarzhi *et al*. 2007). Found in many experiments and simulations, the small *α* implies that in RT flows almost all energy induced by the buoyant force dissipates, and a slight misbalance between the rates of momentum loss *μ* and gain is sufficient for the mixing development. We emphasize that the rate of momentum loss *μ*(*t*)=*Cv*^{2}/*h* is relatively insensitive to the effect of fluctuations, and monitoring the momentum transport thus has crucial importance for grasping the essentials of the mixing process.

The foregoing results can be expanded to the case of time-dependent and spatially varying acceleration. It can be shown that the ratio remains the same as long as acceleration is a similarity function on space and time.

### (d) Comparison with experiments

Time and scale invariance of *μ* suggests that, for describing the accelerated turbulent mixing, the rate of momentum may be a better indicator than the rate of energy. Energy is conjugated with time, momentum is conjugated with space and the transport of momentum can be captured from the spatial distributions of the velocity and density fields. On the other hand, RT mixing flows are characterized by power-law dependencies and time-dependent quantities. Their reliable quantification in the experiments and simulations requires substantial dynamic range and high spatio-temporal resolution and high data acquisition rate. These outstanding tasks have not been resolved so far and may be addressed in the future.

### (e) Summary

In this section we have considered the influence of momentum transport on a number of scaling, invariant and statistical properties of RT mixing flow. It is shown that the rate of momentum loss may be a better indicator of the unsteady turbulent dynamics than the rate of energy dissipation. Our consideration does not presume a single-scale character of the mixing dynamics and distinguishes between the evolution of horizontal and vertical scales. The results obtained indicate two possible mechanisms for the mixing development. The first is the traditional ‘merge’ associated with the growth of horizontal scales. The second is associated with the production of small-scale structures and with the growth of the vertical scale, which plays the role of the scale for energy dissipation. Based on invariance of the rate of momentum loss, we have shown that the fundamental invariant and scaling properties of RT turbulent mixing depart substantially from the classical Kolmogorov scenario. Particularly, turbulent mixing flow appears more ordered than isotropic turbulence, and its viscous scale is comparable to the mode of fastest growth. Invariance of helicity may serve for obtaining an integrated description of highly anisotropic (RTI) and isotropic (Kolmogorov) turbulent flows and for further extension of the range of applicability of the canonical approaches. The stochastic modelling results indicate that the growth-rate parameter of the mixing zone is sensitive to statistical properties of dissipation. The momentum-based consideration of RT mixing resonates with the properties rigorously derived from the conservation principles by the linear and nonlinear theories (Abarzhi 2008), including the existence of characteristic length-scale and multi-scale nonlinear dynamics, and indicate a principal opportunity of regularization of the mixing process.

## 5. Conclusion

We have overviewed the theoretical and empirical modelling approaches of RTI and the turbulent mixing that have been developed over recent decades, and outlined some new ideas that may help in grasping the essentials of the mixing process.

## Acknowledgements

The author thanks Drs S. I. Anisimov, L. P. Kadanoff, R. Rosner and K. R. Sreenivasan for discussions.

## Footnotes

One contribution of 13 to a Theme Issue ‘Turbulent mixing and beyond’.

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