## Abstract

Turbulence is a supermixer. Turbulent mixing has immense consequences for physical phenomena spanning astrophysical to atomistic scales under both high- and low-energy-density conditions. It influences thermonuclear fusion in inertial and magnetic confinement systems; governs dynamics of supernovae, accretion disks and explosions; dominates stellar convection, planetary interiors and mantle-lithosphere tectonics; affects premixed and non-premixed combustion; controls standard turbulent flows (wall-bounded and free—subsonic, supersonic as well as hypersonic); as well as atmospheric and oceanic phenomena (which themselves have important effects on climate). In most of these circumstances, the mixing phenomena are driven by non-equilibrium dynamics. While each article in this collection dwells on a specific problem, the purpose here is to seek a few unified themes amongst diverse phenomena.

Compared with the vast variety of physical circumstances in which turbulent mixing occurs, our knowledge of it is still limited. The problem is relatively straightforward for a simple velocity field (e.g. a constant or periodic in two dimensions), but very few rigorous results exist if the fluid mixing is three-dimensional and fully turbulent. The case we do understand better, though incompletely, is the mixing of a passive scalar in isotropic and homogeneous turbulence at high Reynolds numbers, for which sound phenomenological understanding exists (e.g. Obukhov 1949; Yaglom 1949; Corrsin 1951; Batchelor 1959; Kraichnan 1968).

Most realistic mixing problems in nature and technology are more complex than passive scalar mixing and involve strong gradients of pressure and density, heat release, transfer of species and changes in chemical composition. They exhibit inherently non-equilibrium dynamics, especially under extreme conditions of high-energy density—as in fusion, interstellar molecular clouds or material transformation under impact (e.g. Zel’dovich & Raizer 2002; Remington *et al*. 2006). These phenomena are essentially anisotropic, involve non-local interactions among the many scales constituting the physical phenomena, as well as phase changes of matter, and are often driven by shocks or acceleration. In continuum approximation, their scaling laws, spectral shapes and invariant properties differ substantially from those of classical Kolmogorov (1941) turbulence. Singular aspects and similarity of their dynamics are interplayed with the fundamental properties of the Euler and Navier–Stokes equations (Ladyzhenskaya 1975; Scheffer 1976). In the kinetic limit, the non-equilibrium dynamics depart qualitatively from the standard scenario of the Gibbs ensemble and the Boltzmann weight (Hoover 1991; Evans & Morriss 2008). There is thus a strong need for advancing new theoretical methods well beyond the idealized (e.g. isotropic, homogeneous and statistically steady) domain.

The state-of-the-art numerical simulations have the potential for developing predictive modelling of the multi-scale non-equilibrium mixing dynamics. With computational power doubling every 18 months or so, larger simulations involving greater complexity of geometry and physical conditions are becoming possible. However, there are numerous issues relating to the sensitivity of the numerical solutions to the initial and boundary conditions, quantification of numerical errors, grid discretization and algorithms, data assimilation and processing, as well as adequate resolution of small-scale structures, especially in unfamiliar applications.

Experiments have, by and large, been the mainstay of our traditional knowledge of non-equilibrium systems, but they also have their limitations. The most important limitations concern the resolution and the inability to map out, with adequate accuracy, dynamic range and data-acquisition rate, the full spatio-temporal fields of the turbulent quantities dynamic range and data-acquisition rate, as well as challenges in reliable implementation and systematic study of the mixing dynamics in a well-controlled laboratory environment. Since novel physics is more likely to emerge from physical experiments than simulations, there is a definite need for more sophisticated experiments that yield accurate and reliable information on the non-equilibrium dynamics.

Recent advances in theoretical description of non-equilibrium dynamics, laboratory experiments and diagnostics, involving low- and high-energy-density conditions, and large-scale numerical modelling and computations have served to bring turbulent mixing to a state of maturity, whose comprehension may have a significant impact on a broad range of disciplines in science, mathematics and technology. The success in solution of this highly fascinating problem cannot be achieved single-handedly in one laboratory. To encourage communications among experts in different fields, promote the exchange of ideas and suggestions on open problems and motivate rigorous discussions across theoretical, experimental, numerical and computational domains, a conference was held at the International Centre for Theoretical Physics in Trieste, Italy, in 2007 and then in 2009, including round tables and poster presentations. The subdisciplines represented at the conference included fluid dynamics, plasma physics, high-energy-density physics, astrophysics, combustion, material science, atmospheric and earth sciences, nonlinear and statistical physics, applied mathematics, probability and statistics, data processing and computations, nonlinear optics and cyber-physical systems. Some of the speakers of the 2007 meeting were invited to write up their presentations for inclusion in this special issue. The 12 articles to follow belong to this class. They were refereed in the standard way, and the authors had adequate time to respond to referees and rewrite their articles. Among them, these articles cover the mixing problem in several different manifestations of complexity. It is our expectation that the papers will serve as a proper introduction to the subject for experts and students alike.

The paradigms of turbulent mixing considered in this collection are the passive scalar mixing (Sreenivasan & Schumacher 2010) and mixing induced by hydrodynamic instabilities, including Rayleigh–Taylor (RT) and Richtmyer–Meshkov (RM) (Abarzhi 2010; Aglitskiy *et al*. 2010; Andrews & Dalziel 2010; Gauthier & Creurer 2010; Kadau *et al*. 2010; Nishihara *et al*. 2010). The passive scalar problem is standard, but the focus in this issue is the Lagrangian approach. The RT and RM mixing develops when fluids of different densities or acoustic impedances are accelerated against the density gradient or by passage of a shock wave (Rayleigh 1882; Davies & Taylor 1950; Taylor 1950; Richtmyer 1960; Meshkov 1969). A few other papers are related to methods. The paper by Kadau *et al*. (2010; see also Nishihara *et al*. 2010) focuses on atomistic calculations of fluid mixing, whereas that by Orlov *et al*. (2010) focuses on modern experimental methods using holographic technologies. Yet, others (Julien & Knobloch 2010; Khanin & Sobolevski 2010; Pétrélis & Fauve 2010; Pouquet & Minnini 2010) discuss various fundamental problems such as particle dynamics inside shocks, magnetic-field reversals, magneto-rotational instability and the role of helicity. These fundamental papers have broad implications for the turbulent-mixing problem.

Now is an exciting time to study mixing in a broader context. While individual groups and researchers will no doubt focus on specialized problems and make progress in their own setting, a common umbrella for problems of turbulent mixing and beyond under a variety of problems will be enormously helpful in advancing the theory, simulations and experiment. Significant progress can be achieved by bringing to bear the powerful merger of all these methods on more than one mixing problem.

In the following paragraphs, we provide a brief summary of the papers in the order of their appearance, followed by some general remarks on the outlook for the future.

The paper by Kadau *et al*. (2010) examines the power of molecular dynamics and Monte Carlo simulations for numerical modelling of fluid flows. These methods are more fundamental in comparison to continuum methods and can extend the modelling accuracy to a significantly wider range of scales and regimes than continuum-dynamics simulations. Recent large-scale atomistic simulations of the RT instability compare favourably with experiments. There are many similarities with continuum descriptions, but there is evidence of a significant difference between the continuum and atomistic simulations of non-equilibrium problems. Fluctuations, such as those that stem from molecular disorder and collisions, arise in a natural manner in atomistic methods, but are usually lacking in continuum descriptions; and small fluctuations, in particular those at the interface, can be amplified by many orders of magnitude.

In their paper, Sreenivasan & Schumacher (2010) dwell on the continuum basis of understanding turbulent mixing. Eulerian and Lagrangian points of view on the motion of fluids are equivalent, but cannot be related to each other in analytically tractable ways, except in a few special instances. Much of the existing work on mixing has been in the Eulerian framework, although the Lagrangian frame is more natural to the task. Accordingly, these authors focus on the Lagrangian view of passive scalar mixing, which has recently produced interesting results and interpretations. They review recent work on Lagrangian turbulence, especially those related to the so-called Kraichnan (1968) model for the advection of the passive scalar in synthetic turbulence. The synthetic velocity field in the model is temporally uncorrelated from one moment to the immediate next, but retains the generic character of spatial correlation statistics of the turbulent velocity field. While discussing possible implications for a better understanding of the passive scalar mixing in Navier–Stokes turbulence, the paper also lists some of the outstanding issues that need to be addressed beyond the Kraichnan model.

The article by Khanin & Sobolevski (2010) considers the transport problem for a discontinuous velocity field as in shocks. This approach appears appropriate to non-equilibrium dissipative systems. The central issue in a study of nonlinear evolution of the Hamilton–Jacobi equation is to understand, both from the mathematical and physical points of view, the behaviour of the system after the formation of singularities. The paper goes beyond the regularization techniques available in one dimension. The authors show that the transport dynamics of particles can be defined uniquely, and that one can determine an effective velocity field. The solution is interpreted in terms of dynamics of Lagrangian particles ‘absorbed’ by shocks.

The paper by Pétrélis & Fauve (2010) presents a review of the different models on reversals of the magnetic field generated by a turbulent flow of an electrically conducting fluid (fluid dynamo). It has been known for many years that Earth’s magnetic field reverses its polarity aperiodically (while the same action occurs nearly periodically in the Sun). The reversals are somewhat surprising because the underlying flows are strongly turbulent, and advect and distort the magnetic field lines in complex ways. The authors describe a simple mechanism that explains several features of the paleomagnetic records of Earth’s magnetic field, in numerical simulations and in a recent dynamo experiment. This type of reversal also occurs in the large-scale velocity field in purely hydrodynamical flows (for turbulent convection see Sreenivasan *et al*. 2002), where such scales sometimes seem to be governed by low-dimensional dynamics. The authors believe that their model can also be used to understand such reversals developing in a turbulent background.

The paper authored by Julien & Knobloch (2010) studies the mechanism of the accretion process in astrophysics. The accretion can only occur in the presence of an efficient mechanism for the extraction of angular momentum. For the generation of substantial angular-momentum transport, a number of scenarios have been suggested, and a promising one is the magneto-rotational instability that may develop in a weak magnetic field and result in the appearance of thin sheets of matter redistributing the angular momentum. Magneto-rotational instabilities are similar to RT instabilities and have been the subject of numerous numerical simulations and, more recently, laboratory experiments. In the present review, the authors survey recent developments in the field, point out the multiple time-scale nature of the process and outline a new theoretical approach to understand the nonlinear saturation of the instabilities. The saturation of magneto-rotational instabilities is the key element for the accretion process.

The paper by Pouquet & Mininni (2010) is devoted to a study of how the invariant properties of physical systems drive a wide distribution of energy among modes, as observed in geophysical and astrophysical flows. In hydrodynamics, the role of helicity conservation (helicity being the correlation between velocity and its curl, measuring departures from mirror symmetry) remains unclear since it does not alter the energy spectrum. However, with solid-body rotation, significant differences emerge between helical and non-helical flows. This is the central theme of the paper, whose results point to the presence of a small parameter in the problem besides the Rossby number. This parameter can be associated with the non-dimensional ratio of the flux to small scales of the energy to that of helicity.

Andrews & Dalziel (2010) provide an outline of instability evolution and a comprehensive survey of how RT flows can be implemented in a fluid-dynamics laboratory. Owing to extreme sensitivity and the transient character of the dynamics and relatively fast transition to turbulence and mixing, the implementation and systematic study of RT flows in a laboratory environment is a formidable experimental task. The authors share their unique experience in one of the hardest problems in fluid-dynamics experiments. They focus their attention on incompressible fluids with similar densities (small Atwood number flows), seek to explain possible connections to experiments in fluids with highly contrasting densities (high Atwood number flows) and discuss issues of the design of an RT experiment.

Gauthier & Creuer’s (2010) paper is a review of compressibility effects in RT flows. The presence of compressibility modifies the phenomenology of the instability because the instability modes can be both convective and acoustic. The authors consider linear, nonlinear and turbulent regimes, and review results on stratified compressible flows for which instability criteria have been derived rigorously. They also review linear stability results for perfect fluids obtained from an analytical approach, as well as numerical results for viscous fluids, and include a survey of the effects of compressibility obtained by numerical simulations in both nonlinear and turbulent regimes.

Orlov *et al*. (2010) consider the importance of high-precision experiments and the available technologies for achieving such accuracies, with special attention paid to holographic technologies. They invoke, as the inspiration for precision measurements, those of Lord Rayleigh whose measurements were instrumental in the discovery of argon. The article emphasizes the new opportunities provided by modern technology in a variety of research areas of fluid dynamics. These opportunities should enable a better understanding of complex mixing flows under realistic conditions, thus allowing for better comparisons to be made with results from new theoretical work and large-scale numerical simulations. The authors argue that these technologies can be applied in a variety of problems to the study fundamental properties of flow–particle interactions, rotating flows, non-canonical boundary layers and RT mixing.

The next two papers are on the RM instability. The paper by Aglitskiy *et al*. (2010) is closely connected to direct-drive inertial confinement fusion (ICF) studies, where the possibility of ignition and high-energy gain are influenced largely by the RM instability that inherently occurs during the implosion process. The state of the art radiographic diagnostic techniques developed by the authors in the recent decade have enabled direct observations of the RM-type effects in the ICF-relevant conditions and have stimulated the advancement of the theory of the RM instability in high-energy-density phenomena. The authors review the progress in experimental and theoretical studies, including issues connected with finite-thickness targets, re-shock and re-rarefaction of the RM-unstable material interfaces, ablative RM instability, feedout and perturbation development associated with impulsive loading.

In the next paper, Nishihara *et al*. (2010) present a theoretical framework for the studying linear and nonlinear RM instability. Because the contact surface is rippled, the transmitted and reflected wave fronts are also corrugated, and some circulation is generated at the boundary. This circulation is progressively modified by the acoustic field radiated by the wavefronts. The authors consider the growth of the instability to be driven by both the initial circulation at the material interface and its subsequent variation in time owing to the sonic field of the deformed shocks. In addition to visiting an exact analytical model to determine the asymptotic linear growth rate, the authors consider the ablative RM instability and its stabilization mechanisms. They explore, via molecular-dynamics simulations, the RM instability in solids and liquids for planar and cylindrical geometries, and show the generation of vortical structures.

The concluding article in the issue is by Abarzhi (2010). It briefly overviews recent theoretical developments in the field, discusses the known and unknown aspects of the RT phenomena and outlines features of similarity of the turbulent-mixing process in vastly different regimes. Based on the physical intuition and the results of rigorous theoretical studies, this work puts forward some new ideas on how to grasp the essentials of the mixing process on the basis of momentum transport, and considers the influence of the transport on the invariant, scaling and statistical properties of the turbulent-mixing flow. Presenting the RT instability as an intellectually rich problem, this review also discusses connections of the RT instability to kinetic processes, Eulerian and Lagrangian formalisms, symmetries of canonical turbulence, magneto-hydrodynamics, evolution of shock-driven flows and the physics of high-energy-density phenomena.

We hope that this special issue will expose the phenomena of Turbulent mixing and beyond to a wide scientific community, and that an inquisitive mind will be captured and fascinated by the opportunity to grasp the universal features of the multi-scale, non-equilibrium dynamics and to consider new concepts, whose lucidity and simplicity can cut through the complexity of the problem. We further hope that this issue will serve to integrate our knowledge on the subject and enrich its development.

## Acknowledgements

We are grateful to the following organizations for supporting the conferences in 2007 and 2009 on Turbulent mixing and beyond: National Science Foundation, USA; International Centre for Theoretical Physics, Italy; Air Force Office of Scientific Research, USA; European Office of Aerospace Research and Development of the AFOSR, UK; Commissariat à l’Energie Atomique, France; Department of Energy, USA; Lawrence Livermore National Laboratory, USA; Argonne National Laboratory, USA; Los Alamos National Laboratory, USA; Institute for Laser Engineering, Japan; University of Chicago, USA; ASC Alliance Center for Astrophysical Thermonuclear Flashes, USA; and Photron (Europe) Ltd., UK.

## Footnotes

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

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