In diverse areas of science and technology, including inertial confinement fusion (ICF), astrophysics, geophysics, and engineering processes, turbulent mixing induced by hydrodynamic instabilities is of scientific interest as well as practical significance. Because of the fundamental roles they often play in ICF and other applications, three classes of hydrodynamic instability-induced turbulent flows—those arising from the Rayleigh-Taylor, Richtmyer-Meshkov, and Kelvin-Helmholtz instabilities—have attracted much attention. ICF implosions, supernova explosions, and other applications illustrate that these phases of instability growth do not occur in isolation, but instead are connected so that growth in one phase feeds through to initiate growth in a later phase. Essentially, a description of these flows must encompass both the temporal and spatial evolution of the flows from their inception. Hydrodynamic instability will usually start from potentially infinitesimal spatial perturbations, will eventually transition to a turbulent flow, and then will reach a final state of a true multiscale problem. Indeed, this change in the spatial scales can be vast, with hydrodynamic instability evolving from just a few microns to thousands of kilometers in geophysical or astrophysical problems. These instabilities will evolve through different stages before transitioning to turbulence, experiencing linear, weakly, and highly nonlinear states. The challenges confronted by researchers are enormous. The inherent difficulties include characterizing the initial conditions of such flows and accurately predicting the transitional flows. Of course, fully developed turbulence, a focus of many studies because of its major impact on the mixing process, is a notoriously difficult problem in its own right. In this pedagogical review, we will survey challenges and progress, and also discuss outstanding issues and future directions.

## I. INTRODUCTION

The related topics of fluid dynamical instabilities, the transition of laminar flow fields to turbulence, and the evolution of fully turbulent flow, as well as the effects these flows have on mixing rates, have presented significant challenges to researchers for over 100 years. Although substantial progress has been made in these fields, each topic remains an active topic of research.

In this paper, we will limit our discussion to turbulence (Fig. 1) and mixing of fluids or collision-dominated (dense) plasmas induced by three classes of instabilities—the Rayleigh-Taylor Instability (RTI),^{1,2} the Richtmyer-Meshkov Instability (RMI),^{3,4} and the Kelvin-Helmholtz Instability (KHI).^{5,6}

This work complements and further elaborates the recent comprehensive reviews on hydrodynamic instability induced flows.^{7,8} In these flow scenarios, an initial, perhaps infinitesimal, perturbation at an interface induces an instability. For the cases of RTI and RMI, the fluid may be initially quiescent, whereas the initial state of the KHI is a laminar flow. The instability may lead to a transitionally chaotic state exhibiting the emergence of both small-scale and increasingly larger scales of fluid motions. These instabilities are present in a wide variety of important circumstances involving engineering processes, geophysical and astrophysical flows, as well as in inertial confinement fusion (ICF).

Motivated by the need to better understand the instability processes that occur in astrophysical contexts^{10–14} and in ICF,^{15–19} as well the behavior in materials with strength subjected to rapid and large compressions,^{20,21} and coupled to the need to test the veracity of the computer simulations that are commonly used to model these processes, the field of high energy density physics (HEDP) has developed. HEDP experiments, used to test and validate theoretical models as well as simulation predictions, characteristically deposit tens of kilo-Joules to mega-Joules of energy in millimeter-scale targets on time scales of nanoseconds, thus directly generating plasmas of ∼1–100 megabar pressures. With the additional use of an implosion, pressures of ∼100's of gigabars can be achieved.^{22} The resulting flows emulate, by design, important processes at the plasma conditions of interest. In some cases, HEDP experiments are designed to have a certain aesthetic as well as scientific value,^{23} and result in stunning data images^{24} that may lead to new areas of study that are productive for students and researchers in HEDP.^{25–36} Beyond studying instability, applied physics problems sometimes require that instabilities be mitigated in order to achieve a practical goal.

To study the problems of interest in this paper, it is generally desirable to allow the instability to evolve to a fully developed turbulent state. This poses stringent requirements for numerical simulations, laboratory experiments, and HEDP platforms. A key objective of this paper is to highlight the necessary and sufficient conditions for the transition process that marks the onset of a fully turbulent state to occur. To make it self-contained, this manuscript is structured as follows: in Sec. II, we first characterize “fully developed turbulence” from the perspective of Kolmogorov's theory.^{37–41} In Sec. III, we turn our attention to the initiation of turbulence. In particular, we restrict ourselves to hydrodynamic instability-induced flows, as we are focused in the applications to ICF implosions (Sec. IV) and supernova explosions (Sec. V). In Sec. VI, we detail the criteria for these instability-induced flows to transition to fully developed turbulence. This is followed by discussion of two important issues: the importance of the initial conditions (Sec. VII) and the need for enhanced diagnostic measurements (Sec. VIII) in high energy density platforms. Next, we address the question whether two-dimensional (2D) turbulence resembles three-dimensional (3D) turbulence (Sec. IX). Before we conclude, some especially interesting issues and outstanding problems will be discussed.

## II. THE TURBULENT STATE

### A. Governing equations and Reynolds number

^{42,43}

*σ*is the stress tensor

_{ij}*u*is the velocity field,

_{i}*ρ*is the fluid density,

*p*is the pressure,

*μ*is the dynamic viscosity, and

*f*(

_{i}*t*) is the forcing term, which is equal to zero in a decaying flow.

^{44,45}The equation governing the transport of the density,

*ρ*, is

*Re*) characterizes the state of the flow field and is defined as

*L*and

*u*are the characteristic length scale and velocity field, respectively, and

*ν*is the kinematic viscosity given by

*ν*=

*μ*/

*ρ.*

^{46}This dimensionless parameter is obtained by comparing the nonlinear and viscous terms of the Navier-Stokes equations

^{47,48}and represents the ratio of inertial forces to viscous forces. Depending on the values of Reynolds number, the flow field is either laminar or evolving toward a turbulent flow or already turbulent. In fact, the onset of turbulence occurs when the Reynolds number is larger than a threshold value. When the viscous action is strong, the flow is laminar and any perturbations from smooth motion are damped away. When the flow is fully turbulent, however, it contains eddy motions of all sizes admissible within the flow geometry. A large part of the mechanical energy in a turbulent flow goes into the formation of these eddies, and the effects of viscosity eventually dissipate their energy as heat.

^{49}

In Van Dyke's book, “An Album of Fluid Motion,” photographs showing the flow past spheres demonstrate how dependent these flows can be on the Reynolds number. At *Re* = 100, the flow separates and a recirculating region forms. As *Re* increases to *Re* = 15 000 and *Re* > 200 000, the flow develops both the large-scales structures and small-scale motions.^{50} The same is true for the jet flows shown in Fig. 2. When *Re* is not extremely high, there are many organized large-scale features. However, when *Re* is increased significantly, the flows exhibit an extremely wide range of scale of motions.^{51}

### B. Cascade

The well-known poem by Richardson^{52} vividly illustrates some key features of the evolution of turbulent flows:

“Big whirls have little whirls

Which feed on their velocity,

And little whirls have lesser whirls

And so on to viscosity.”

The sketch in Fig. 3 is a good starting point for the study of high *Re* 3D flows. Here, the large-eddies reflect the inhomogeneities characteristic of the forces and boundary conditions which produce them. Nevertheless, since the characteristic time-scales associated with the eddies decrease geometrically with the eddy size, a large number of generations are produced during the lifetime of the large anisotropic eddies. The flow will evolve into a quasistationary self-similar hierarchy of eddies, frequently termed as the cascade process, whose wavenumbers form a geometric progression.^{40,53,54} The smallest eddies will be limited by the viscous action at the smallest length scales, the Kolmogorov dissipation scales, *η*; a precise definition of which will be given in Eq. (20).

The energy injection takes place at the largest eddy length-scales. The energy transfer process in the cascade process in 3D turbulence proceeds from large to small scales. The time required for an eddy to be distorted and, in this distortion process, generate smaller eddies is called an eddy-turnover-time. Finally, the energy is dissipated from the smallest eddies due to viscous effects.

Figure 4 provides a sense of the various length scales from a flow visualization obtained from direct numerical simulation (DNS) of homogeneous isotropic turbulence.^{55,56} The Reynolds number is very high, at *Re *=* *216 000. [Note that the Reynolds number based on the Taylor microscale (*λ*), *Re _{λ}* = 1131, is given in the original article. However,

*Re*= (20/3)

_{λ}^{1∕2}

*Re*

^{1∕2}for homogeneous and isotropic flow, see Tennekes and Lumley.

^{57}The important length scales, including the Taylor microscale, will be defined more precisely later in Subsection II C.]

Some fiducial markers are included in the image as a point of reference. The first is the length scale of the large-eddies and a precise definition, termed the integral length scale, will be provided later. Also shown is 100 times the Kolmogorov dissipation scale, *η*. This figure demonstrates that turbulence is a true multiscale problem and extremely difficult to describe numerically and analytically due to the wide range of length scales in the underlying physical problems.

### C. Statistics and length scales

Beginning with the pioneering work of Osborne Reynolds,^{58} mathematical descriptions of turbulent flows have been essentially statistical, representing, for example, the velocity, *u _{i}*, as the sum of an averaged value, $\u27e8ui\u27e9$, and a fluctuating value. In many published papers, the definition of the average is left to readers' imagination. For our case, where we are concerned with the temporal evolution of a strongly inhomogeneous field, neither temporal nor spatial averages seem appropriate, so we assume an ensemble average over a number of realizations wherein the supposedly larger-scale details of the field such as the geometry may be identical across each realization, but the finer-scale details, for example, the perturbations at an interface, may vary across the individual realizations.

It worth noting that time-averaging is inappropriate in that these flows are rapidly evolving in time. Spatial averaging may be appropriate in circumstances wherein (a) the scale of fluctuations is much smaller than the bulk motions of the fluid, or (b) when there is a homogeneous direction(s) over which to average. It must be noted that the assumption of homogeneity is not equivalent to an assumption of periodicity and that assuming homogeneity may mask effects of large scale flow features in the presumed homogeneous directions. The ensemble average provides the most generally applicable approach, but it is difficult to realize both experimentally and via direct numerical simulations. However, for the task of deriving fundamental evolution equations, and for theoretical study, the ensemble average provides the most succinct description. Hence, from a fundamental point of view, the ensemble average is the appropriate definition of the averaging process. However, for most time evolving applications, the ensemble average is approximated by averaging over any available statistically homogeneous directions, e.g., planar averaging.

*N*is the number of elements in the ensemble and the superscript (

_{e}*n*) represents the

*n*th realization in the ensemble. The fluctuational part, denoted with a prime, is then simply

^{59–61}are constructed by applying this decomposition (or the related mass-averaged statistical decomposition) to the Navier-Stokes equations and deriving an exact equation for the mean. Unfortunately, the mean flow equation will have second-order moments which must either be modeled or for which an additional evolution equation must be derived. For our current purpose—describing the Kolmogorov cascade process—we will assume an incompressible, constant density flow. Under these restrictions, the evolution equation will have a quadratic nonlinearity in the fluctuating velocity

*K*, by

^{38}then the two-point Reynolds stress tensor is independent of $xc$ and can be written as

*E*(

_{i}*k*) is the component energy spectrum.

### D. Kolmogorov phenomenology

Kolmogorov's 1941 “Phenomenology,” which is basically a dimensional argument, is of particular significance owing to limitations of computational resources in the foreseeable future. Figure 5 (Ref. 63) shows the relevant features of the Kolmogorov 1941 theory. Essentially, a flow field always has an energy-containing scale, which corresponds to the large-eddies discussed in the previous subsection II B. The “energy spectrum,” *E*(*k*), defined above, provides a useful mathematical object for understanding Kolmogorov's theory. The contributions to the energy spectrum, *E*(*k*), from the energy-containing scale, are located in the vicinity of the low wavenumbers. At the other end of the energy spectrum is the so-called dissipative scale, where *E*(*k*) decreases exponentially as a result of the viscous action. Moreover, Kolmogorov predicted that if the Reynolds number is sufficiently large, the energy-containing scale will be separated from the dissipation scale by an inertial range in which *E*(*k*) scales as *k*^{−5∕3}. The larger the Reynolds number, the wider the inertial range will be.

The phenomenology makes the following assumptions:

The turbulence is locally isotropic, i.e., at small scales, the turbulence statistics are invariant under translation and rotation.

The turbulence is near “equilibrium,” i.e., the energy injection (the forcing on the turbulence), $F$, that drives the turbulence is approximately matched by the rate, $E$, at which the viscous effects dissipate the energy, i.e., $F\u2248E$.

The wave number characterizing the energy injection,

*k*, is much smaller than the wave number_{F}*k*characterizing where the energy is dissipated, $kF\u226akd$._{d}The cascade process through

*k*-space is “local,” i.e., the transfer through wavenumber*k*depends on the dynamics in the neighborhood near*k.*^{64–66}

It is also essential for the analysis that the energy transfer in Fourier space is conservative due to the nature of the nonlinearities of the Navier-Stokes equations.

*C*is the so-called Kolmogorov constant, and the exponents

_{K}*p*and

*q*are determined by dimensional arguments. Since

*E*(

*k*) has units of $(Length3/Time2),\u2009E$ has units of $(Length2/Time3)$, and

*k*has units of (1/Length), the time scaling of

*E*can only be reconciled if the exponent of $E$ is

*p*=

*2/3, where now $E2/3$ has units of $(Length4/3/Time2)$. Thus, the exponent of*

*k*must be

*q*= −5/3 to produce the correct length scaling for

*E*. The result is the Kolmogorov scaling law

*C*is a universal constant, but it appears to be approximately

_{K}*C*≈ 1.5.

_{K}^{67–70}In addition, Kolmogorov deduced that the length scale, $\eta \u223c1/kd$ (the “Kolmogorov microscale”) which characterizes the range of scales where energy is dissipated by viscous processes should be independent of the large-scales, and a function only of the active variables at fine scales, e.g., viscosity,

*ν*, and the dissipation rate itself, $E$. Dimensional analysis then gives the result that

Figure 6 is a compilation of experimental measurements of the energy spectrum.^{71} The longest inertial ranges are especially noteworthy: Grant *et al.* collected their data from the Discovery Passage, a tidal channel of the west coast of Canada.^{72} Praskovsky *et al.* obtained their results from a return channel in the large wind tunnel of the Central Aerohydrodynamic Institute.^{73} Saddoughi and Veeravalli carried out their experiments with the 80 × 120-foot Full-Scale Aerodynamics Facility at NASA Ames Research Center.^{71} Clearly, the energy-containing and dissipation scales in these and other high *Re* flows are very well separated by an extended inertial range. Moreover, the measured energy spectrum confirms the *k*^{−5∕3} scaling predicted by Kolmogorov. For contrast, a number of low *Re* flows energy spectra are also included in Fig. 6. For these experiments, the energy-containing scales are closely linked to the dissipation scales. As a result, no inertial range can be discerned.

^{74}and Corrsin

^{75}applied/extended the Kolmogorov 1941 theory to the case of a passive scalar, e.g., weak temperature fluctuations, and deduced that the scalar spectrum,

*E*

_{T}(

*k*), behaves as

*C*is a constant. The scalar microscale,

_{T}*η*, is then equal to

_{T}*Sc*∼ 600 for the brine/water experiments of Dalziel

*et al.*

^{76}and the follow-up experimental work by Banerjee and Mutnuri

^{77}has

*Sc*∼ 10

^{3}. Molecular mixing measurements were reported

^{78}for a high-Schmidt-number (

*Sc*∼ 10

^{3}) buoyancy-driven turbulent Rayleigh-Taylor (RT) mixing layer in a water channel facility with a weakly buoyant flow.

*L*of the turbulence, are typically independent of viscous effects, and are a function of the large-scale variables, i.e., the energy injection rate, $F$, which, in turn, is usually related to the overall geometry of the flow domain. Again dimensional analysis produces

^{47}

*λ*, where

^{57}for more details.) This definition is particularly convenient for application to simulation data.

^{62}$R(r)$. Note that $R(r)$ is an even function, so that it is symmetric $R(r)=R(\u2212r)$. Performing the Taylor expansion, the leading terms are

^{79,80}

### E. Astrophysical flows, turbulence, and the masterpiece

“The Starry Night” (Fig. 7) was painted by van Gogh in 1889 during his periods of prolonged psychotic agitation.^{81} The tie between the distant star observed by Hubble (Fig. 8) and this famous artwork was noted by NASA. The Museum of Modern Art (MoMA), New York City also noted that “the whirling forms in the sky match published astronomical observations of clouds of dust and gas known as nebulae.” While highly suggestive, can the masterpiece be linked to the fluid flows of turbulence?

Aragón *et al.* used digital images of these paintings of van Gogh to show that the statistics of luminance contains the characteristic fingerprint of turbulent flow.^{82} Indeed, these authors showed that Starry Night, and other impassioned van Gogh paintings, transmitted the essence of turbulence with high realism. This was shown by comparing the “structure functions” deduced from the paintings (analogous to those for computed for turbulence^{83}) shown in Fig. 9 to actual measurements of structure functions from a wind tunnel.^{84}

*et al.*

^{85}and Woosley

*et al.*

^{86}found

*et al.*

^{87}determined that

^{88,89}

### F. Turbulence and mixing

Why do we care about turbulence? Usually it is due to its effect on mixing. In turbulent flows, the mixing of the mass, momentum, and energy occurs at a significantly more rapid rate than for laminar flows.

Turbulent mixing can be grouped into three categories,^{90} based on increased role of mixing. The first category is composed of those cases where turbulent flow drives mixing, but the mixing itself plays essentially no role in the dynamics of the turbulent velocity field. A good example is mixing of a dye in a liquid, where the scalar is the dye concentration,^{91} which does not influence flow dynamics for this type of mixing.

The second category is characterized by the situation where the scalar being mixed plays an intrinsic role in driving the dynamics of the velocity field. Flows induced by the RTI, wherein the scalar is the density field and the forcing on the momentum field is due to the coupling of the scalar to the acceleration vector, are an example of this category of mixing. For the third and last category, the scalar being mixed has the dominant active role in the behavior of the overall flow field. Examples for this category include flows driven by combustion^{92,93} and/or by detonation and deflagration phenomena.^{94–96}

## III. TIME-DEPENDENT FLOWS INDUCED BY HYDRODYNAMIC INSTABILITIES

Because of our interest in supernova explosions and in ICF, we will restrict our attention to turbulent flows induced by hydrodynamic instabilities with perturbed interfaces. The three “classical” instabilities that play a significant role in these applications are the RTI, RMI, and KHI. Each of these instabilities involves a perturbed interface between two fluid regions. For the case of the RTI, the density varies across the interface, whereas for the RMI, the acoustic speed varies across the interface although this variation in acoustic speed is typically associated with a density difference as well. The KHI typically involves an interface across which the fluid velocity varies—in the classical case, the variation in velocity is discontinuous. The KHI interface may also be associated with a density difference across the interface. The nature of the interface varies with the circumstances of the problem. The interface may be discontinuous in the fluid properties or it may be diffuse. Molecular interactions such as surface tension, diffusion-induced fluid stresses, or reactions may be involved. These complications may play a substantial role in the evolution of the layer. Also, in most practical situations, an accurate and precise description of the interface may be difficult to obtain. Additionally, these instabilities can appear in a number of circumstances and can occur in convergent or divergent geometries in addition to the usual planar cases. Finally, it is typical in many astrophysics and HEDP circumstances to have all three instabilities concurrently.

*q*represents a heat flux. It should be noted that

_{i}*q*consists of a conduction flux ∼−∇

*T*and an enthalpy flux (this has not always been correctly treated, see Cook

^{97}for example).

*Y*is the mass-fraction of the

_{i}*i*th-species and we assume a Fickian diffusion model wherein $Di$ is an effective binary species diffusion coefficient.

^{98}The diffusional fluxes are given by

^{97,99}

*Ma*, is small ( $Ma\u21920$), the divergence of the velocity field becomes

^{100–102}

^{103}The typical approach is a perturbative expansion. However, if the perturbations of the interface between the two fluids continue to grow, they quickly move beyond the regime in which instability theory is applicable. Numerous theories have been proposed to describe the nonlinear regime, with varying degrees of success. However, in finite time, these instabilities typically end up as an inhomogeneous turbulent mixing layer where in the core of the mixing layer is in a “fully turbulent” state.

### A. The Rayleigh-Taylor instability

^{104}This instability was first studied in detail by Lord Rayleigh (Strutt),

^{1}and later by Taylor,

^{2}and continues to be subject to much theoretical, experimental, and computational scrutiny (for an example of a rigorous mathematical treatment, see Refs. 105–107). The classical case concerns an incompressible fluid, and an interface between the heavier and lighter fluids is represented by a two-dimensional surface described by a single two-dimensional Fourier mode, i.e.,

*z*-axis,

*k*and

_{x}*k*are the Fourier wave numbers in the

_{y}*x*and

*y*directions, respectively, and $\u211c$ denotes the real part of its argument. The perturbed interface represented by Eq. (39) can be viewed as a simple one-dimensional single-mode perturbation with a wavenumber of $k=kx2+ky2$. In general, the RTI occurs whenever

^{108}

David Alfaro Siqueiros, a well-known Mexican muralist (Fig. 11), used this method of absorption of two or more superimposed colors which infiltrate one into another (Figs. 12 and 13). The technique is referred to as “accidental painting.”^{129,130}

Chandrasekhar^{103} and Drazin and Reid^{109} presented detailed derivations of the RTI linear stability analyses for the case of a sinusoidal perturbation at the interface, assuming no diffusion. Chandrasekhar included analyses for the cases with and without surface tension, and for inviscid fluids as well as viscous fluids and fluids with differing viscosities. However, he only presented results for inviscid fluids and fluids with identical kinematic viscosities. In addition, he extended the analysis to include hydromagnetic effects.

*a*(

*t*), of the Fourier mode describing the interface initially grows as

*a*

_{0}is the initial Fourier amplitude, and is equivalent to half the width,

*h*(

*t*), of the interface (peak-to-valley),

*g*is the acceleration applied to the interface,

*k*is the wavenumber of the sinusoidal perturbation, and $A$ is the Atwood number

*ρ*

_{2}>

*ρ*

_{1},

*ρ*

_{2}and

*ρ*

_{1}are the densities of the two fluids. The linear stability analyses are limited to very small times. However, numerous attempts have been made to extend perturbative-type models to a higher order in the perturbative expansions, and to include nonlinear effects, as well as account for a greater variety of initial perturbations—examples include (but are by no means limited to) Refs. 110–122. These models have been extended in various ways, including nonplanar geometries and nonconstant accelerations. In addition, they show reasonable congruence to experiments up to some point in time (a more extensive discussion can be found in Ref. 7). It is also interesting to note that Zhang and co-workers

^{123,124}have advanced quantitative theories for RTI and RMI for all density ratios, using the methods of asymptotic matching

^{125}and two-point Padé approximations.

^{126–128}

An obvious difficulty in applying some of these models to practical circumstances is in characterizing the initial interface perturbation with sufficient accuracy to reliably apply these analyses. Also, these analyses are typically limited to the evolution of a small number of modes. This is problematic if the mixing layer transitions to a fully developed turbulent state. In a circumstance with a multimodal initial condition, the RTI quickly tends toward an apparently chaotic turbulent behavior. At late times (beyond the linear stability theory regime), the lighter fluid “rises” up through the heavier fluid, which in turn “drops” through the lighter fluid. The characteristic structures of the lighter fluid penetrating the heavier fluid are usually referred to as “bubbles” and the heavier fluid forms “spikes” as it penetrates the lighter fluid (Fig. 14). The evolution of the mixing layer width can be obtained from both numerical simulations and experimental measurements.

To clarify further why the term spikes is used, we recall that at very high density ratios, e.g., water/air, as used in the early RT experiments, the dense fluid forms spikes as it penetrates the lighter fluid. Most notably, “mushroom” shapes form on the spike tips at lower density ratios and for small density differences there is little asymmetry. However, using the terms bubble and spike at all density ratios has persisted in the RT/Richtmyer-Meshkov (RM) literature.

Figure 15 presents images from an experiment^{131} demonstrating the emergence of the spikes and bubbles from an initial perturbation, and an apparent transition toward a turbulent state. One feature of RTI seen in this figure is the emergence of the mushroom shape at the tips of the bubbles and spikes.^{132} This feature is also observed in many RMI simulations and experiments, as well as in some instances of the KHI. For the case of the RTI and RMI, these mushroom features are due to the shears induced at the tips of the bubbles and spikes as the fluid being penetrated must move laterally as well as in the direction of the penetrating fluid to accommodate the penetrating fluid.

The Rayleigh-Taylor instability and the subsequent transition to a turbulent Rayleigh-Taylor mixing layer (RTML) have been the subject of intense experimental scrutiny. A typical laboratory RTI experiment consists of a test chamber (or “ampoule”) with two liquids of differing densities in a stably stratified configuration. This ampoule is then driven toward the floor of the test facility with an acceleration of greater than 1 *g*, thus rendering the fluid configuration unstable. The first notable RTI experiment was due to Lewis^{133} and used compressed air to drive the test ampoule. Various mechanisms have been used to drive the ampoule, including rocket motors,^{134} linear electric motors,^{135,136} gas guns,^{137} and weights and pulleys.^{131}

Experiments have also been conducted to examine the behavior if the acceleration is not strictly normal to the fluid interface (e.g., Ref. 134) and the behavior under variable acceleration histories, e.g., Ref. 136. In this context, the RMI is frequently treated as an impulsively driven RTI. Note that many of the experiments do not attempt to “characterize” the initial perturbative state of the interface, but instead initialize the RTI with mutually immiscible fluids, or with weakly diffusive fluids that are essentially quiescent at the moment the acceleration is applied. However, experiments to investigate the behavior of single mode perturbations have also been conducted, e.g., Refs. 131 and 138.

An alternate approach to generating the RTI and RTML is to use coflowing streams of two fluids with different densities that pass a “splitter” plate.^{139–143} This approach permits the combining of the RTI with the KHI and also allows the formation of long-time averages of the mixing layer statistics. The disadvantage is that the applied acceleration is limited to 1 *g*, and mixing layers are spatially evolving, rather than temporally evolving; thus, the statistics only roughly approximates the axisymmetry present in the accelerated ampoules.

Another method of experimentally producing RTI was described by Dalziel *et al.*^{76} and Banerjee and Mutnuri,^{77} in which the heavy and light fluids are kept apart by a slider-plate, with the heavy fluid above the plate and the light fluid below. When the plate is removed, the RTI is initiated. Dalziel *et al.*^{76} discussed and used numerical simulations to assess the effects of the vorticity layer deposited by the plate as it is removed. Again, as with the approach with the coflowing streams, the splitter plate approach is limited to accelerations of 1 *g*.

Other novel experiments of RTI and RTML have been conducted in cylindrical geometries using pressurized gas-driven gelatin layers.^{144,145} The use of laser driven fusion facilities has allowed the design and execution of three-dimensional spherical and cylindrical and spherical RTI and RTML experiments as well albeit introducing complexities in terms of the physical models required to describe the events.

The RTI has also been the subject of investigation via numerical simulation. The RTI problem presents significant numerical challenges beyond those seen in simulations of single-fluid incompressible fluids. Unless the Boussinesq limit of small density fluctuations is enforced, or the fluids are deemed immisicible, the velocity field is no longer solenoidal [see Eq. (38)]. In addition, as the fluids interpenetrate each other and the edges of the mixing layer begin to accelerate, the density gradients at the tips of the spikes and bubbles increase, in spite of the effects of diffusion, placing enormous demands on the ability of the numerical algorithms to resolve the gradients. A pioneering effort to simulate the RTI and subsequent RTML was presented by Youngs.^{146,147} Examples include those of large-eddy simulations (LES) of Ref. 148 and the direct numerical simulations (DNS) of Refs. 101, 102 and 149–151. The DNS studies require the inclusion of molecular viscosity and molecular diffusion to regularize the field and permit resolution of the boundaries between the fluids. Simulations of these flows with immiscible fluids have been presented by, for example, in Refs. 152 and 153. The immiscible case generally involves a “front-tracking” method^{153} or a “volume-of-fluid” method.^{146} Another approach to simulating these flows is the so-called implicit large eddy simulation (ILES) methodology,^{154–156} which relies on the implicit damping in certain classes of finite difference methods to treat the sharp gradients and discontinuities present in the underlying flow field.^{157} The interesting reader is referred to Youngs^{158} for a comparison between ILES and DNS for four test cases.

### B. The Richtmyer-Meshkov instability

The RMI shares many significant features with the RTI. The RMI occurs at a perturbed interface between two fluids with different densities. The instability is then initiated by the abrupt and temporary application of a pressure gradient, typically a shock-wave traversing the interface in a direction normal to the interface, or in some cases, an impulsive application of acceleration, in essence, and impulsively driven RTI. Note that if the incident shock-wave is weak, i.e., low Mach Number, and the perturbations are small, then the compressible case is well-matched by the impulsive RTI case. However, unlike the RTI, the RMI perturbations will grow for both the case of $\u2207p\xb7\u2207\rho >0$ as well as for $\u2207p\xb7\u2207\rho <0$. For the case where $\u2207p\xb7\u2207\rho >0$, the interface will undergo a phase inversion prior to the growth of the amplitude of the perturbation. Unlike the RTI, since the pressure gradient in the RMI is not sustained, the RMI-induced mixing layer is not a driven turbulent field. Consequently, if the initial shock-wave or impulsive acceleration is not sufficiently large, the RMI may not develop into a fully developed turbulent state.

^{3}and the first successful experimental reproduction of the instability was due to Meshkov in 1969.

^{4}Richtmyer's analysis of the RMI used Taylor's linear theory

^{2}and assumed an interface perturbation like that given by Eq. (39). His analysis treated only the case of the shock-wave moving from the lighter to the heavier fluid, and also treated the shock as an impulsive acceleration. It is important to note that, for the RMI, the linear theory only applies to the case where $ka0\u226a0$, where

*k*is again the wavenumber of the perturbation and

*a*

_{0}is the initial amplitude. Richtmyer modeled the impulse as $g=\Delta V\delta (t)$, where Δ

*V*is the change in velocity of the interface as the shock passes through, and

*δ*(

*t*) is the Dirac delta function. The equation governing the growth of the amplitude is then

Various modifications to this formulation have been proposed, and the interested reader is referred to the reviews.^{7,159} Richtmyer only considered the instance of the shock moving from lighter to the heavier fluid, which turns out to be the case most relevant to ICF. The case of the shock moving from the lighter to the heavier fluid produces a reflected shock-wave in addition to the forward propagating wave. For the case of the shock moving from the heavier to the lighter fluid, the reflected wave is a rarefaction wave. Also, these two cases (lighter/heavier vs heavier/lighter) result in the deposition of vorticity of opposite signs. This is illustrated in Fig. 16, showing the lighter/heavier on the left and the heavier/lighter case on the right. The case wherein the shock moves from the heavier gas to the lighter gas causes a phase-reversal of the interface to occur before the interface begins to grow, whereas the lighter-heavier case causes a direct growth of the interface.^{160}

Other RMI analyses have been based on an initial velocity or vorticity distribution at the interface, for examples, see Refs. 161–163. Indeed, the linear theory gives modes $exp\u2009(\xb1nt)$. So there is a choice of an $a=a0cosh(nt)$ (initial amplitude) mode or an $a=a0sinh(nt)$ mode [which assumes an initial velocity distribution, see, for example, Eq. (11) of Mikaelian^{161}]. Historically, before RM was considered, most researchers took $a\u0307(0)=0$, a natural choice. For example, Taylor explicitly said that if the initial amplitude is finite and if “the initial velocity is zero,” then the solution is $cosh(nt)$. Indeed, one had no incentive to consider a finite initial $a\u0307(0)$ when RTI is the only consideration.^{163} It should be noted that when both an initial vorticity and an initial velocity (in the sense of $a\u0307$) are given, the two can be related as was done by Jacobs and Sheeley^{162} [see their Eq. (17)]. If initially both the amplitude (*a*) and velocity ( $a\u0307$) are provided, then the two may be degenerate and cancel each other out at late times as Mikaelian^{163} discussed recently.

The most obvious way to experimentally generate the RMI and RMML (Richtmyer-Meshkov mixing layer) is to use a shock tube. The first experiments to demonstrate the RMI were due to Meshkov.^{4} Meshkov used an extremely thin nitrocellulose membrane with an imposed sinusoidal perturbation to separate the light and heavy fluids in a shock tube prior to initiating the shock-wave. When the shock-wave encounters the membrane, the membrane is destroyed, and the RMI behavior is observed. Meshkov examined both the light/heavy configurations and the heavy/light configurations. Although the membranes are reduced to small fragments, these fragments have some effect on the flows. Prasad *et al.*^{164} used a polymeric membrane supported by a wire mesh to separate the gases and studied (among other things) the effect of the membrane and mesh on the evolution of the mixing layer (see also Erez *et al.*^{165}).

Jones and Jacobs^{166} generated a perturbed interface in a vertical shock tube without a membrane by flowing the heavy gas through the bottom of the vertical shock tube, and the light gas through the top, and removing them through slots in the test section, thus maintaining a smooth flat interface. A sinusoidal perturbation is then induced in this flat interface by oscillating the shock tube. However, molecular diffusion tends to “smear” the interface between the two-fluids. For multimode interfaces RMI, Fig. 17 shows a sequence of planar laser-induced fluorescence (PLIF) images for a He/Ar interface.^{167,168}

Another approach to avoid the use of membranes in shock-tubes is to use a “slider plate” to separate the two fluids and then remove the slider plate immediately before initiating the shock-wave, see, for example, Brouillette and Sturtevant^{169} and Puranik *et al.*^{170}

The third approach to generating the RMI is to explicitly generate an impulsive RTI. This approach allows certain types of RTI experimental facilities to be re-purposed for RMI experiments. For example, Ref. 162 vertically accelerated an ampoule containing the test fluids by “bouncing” it off a fixed coil spring. Prior to dropping it on the spring, a sinusoidal standing wave was induced at the interface between the heavy and light fluids by oscillating the ampoule. Likewise Dimonte and Schneider^{136} used their linear electric motor to impulsively drive the ampoule. The reader is referred to Ref. 175 for recent discussions of experimental measurements of RMI and RMML.

Explosively driven experiments of RMI and RMML in metals have been studied by Buttler *et al.*,^{171} Prime *et al.,*^{172} among others.^{173} As in the case of the RTI and RTML, laser-driven experiments provide the ability to conduct experiments of the RMI and the RMML,^{174} but additional physical processes, such as equations of state, are introduced.

Numerical methods have proven to be a useful tool for studying the RMI and RMML. However, in addition to all the numerical challenges presented by the RT problem, the RM cases now compound these issues with the necessity of treating compressibility and the presence of shock waves and rarefaction waves. Recently, Wong *et al.* carried out high-resolution 2D and 3D shock-capturing adaptive mesh refinement simulations of multispecies mixing driven by RMI.^{176} Direct resolution of the shock wave is generally extremely computationally costly in three dimensions. As a result, front-tracking methods or ILES approaches are frequently used in two-dimensional simulations, e.g., Ref. 177, whereas in three-dimensions the useful tools appear to be LES^{99,178,179} and ILES, e.g., Refs. 157 and 180–183. It should be noted that the ability to track shock waves and fronts can be highly useful for studying internal shock wave interactions in the mixing layers at high Mach number.^{177} Yet, for highly turbulent scenarios, the dynamics of the turbulence are intrinsically three-dimensional [the two-dimensional case is in some sense pathological (see Sec. IX)], and thus three-dimensional simulations are required to capture the phenomena with physical fidelity. The ILES approach has also been applied to the shock/reshock problem and to similar problems regarding the effects of compressibility on the development of RMI in the case where $ka$ is not much smaller than unity.^{183} Note that again numerical simulations afford the ability to easily simulate arbitrary, e.g., convergent geometries, and provide access to all the field variables.^{184–188}

One additional phenomenon frequently encountered in experiments,^{189–191} modeling,^{192} and simulations^{193–196} of RMI and ICF is the so-called “reshock” event. A reshock event occurs when a shock wave traverses a RM-induced mixing layer. This can occur, e.g., when a shock-wave reflects off the end of the shock-tube and strikes the mixing layer again. Reshock events typically result in a rapid growth of the turbulence within the mixing layer—much more than produced during the initial RTI initiation. Taylor's linear theory (as applied by Richtmyer and others) generally does not apply to a reshock event since, at the time of reshock, the requirement that $ka1\u226a0$ (where *a*_{1} is the amplitude of the perturbation when the reshock event occurs) is usually not true. The production mechanism is due in part to Kelvin-Helmholtz-induced shear layers induced by the shock crossing the mixing layer. Figure 18 shows the development of the RMI and RMML, with a subsequent reshock event from a shock tube experiment by Tomkins *et al.*^{197} The initial incident shock had a Mach number of *Ma* = 1.2. After the first shock crosses the interface, the emergence of the characteristic mushroom shapes is observed. The reshock event occurs in the figure, and the mixing layer subsequently evolves rapidly toward a chaotic, turbulent state.

### C. The Kelvin-Helmholtz instability

^{6}and subsequently by Lord Kelvin.

^{5}As in the case of the RTI and RMI, the classical instability analysis assumes a sinusoidal perturbation and an initially exponential growth rate for the unstable mode(s). The analyses presented by Chandrasekhar

^{103}and by Drazin and Reid

^{109}include the effects of buoyancy, surface tension, rotation, and a diffuse, rather than discontinuous interface. For the case of buoyancy, and with a discontinuous interface, the growth rate of the interface is given by

*g*is the acceleration normal to the interface,

*V*

_{1}and

*V*

_{2}are the fluid velocities across the interface (parallel to the interface), $A$ is the Atwood number,

*k*is the wavenumber of the disturbance, and

*ρ*

_{1}and

*ρ*

_{2}are the fluid densities on each side of the interface.

It is generally assumed that the KHI will evolve toward an approximately self-similar turbulent state.^{198} The mechanisms of the transition process have been studied using both DNS, for example by Metcalfe *et al.*,^{199} and experimentally, for example, by Winant and Browand.^{200} When the KHI achieves a chaotic state, it is typically referred to as a “temporally” evolving shear layer. This terminology distinguishes it from the closely related “spatially” evolving shear layer. A temporally evolving shear layer is difficult to realize in experiments, but easier to simulate than the spatially evolving case. The spatially evolving shear layer is usually experimentally generated by conjoining two fluids with different speeds downstream from a splitter plate, as shown in Fig. 19.^{201}

It should also be noted that the KHI and Kelvin-Helmholtz (KH) mixing layer (KHML) can be initiated by a shock wave passing through a density interface at an angle that is oblique to the interface.^{202} In this case, the difference in acoustic velocities between the two fluids gives rise to different rates at which the shock travels through the fluids, resulting in a velocity difference across the interface, thus inducing a shear instability. As mentioned in the Introduction, many authors have performed experiments in which shock-induced shear layers have been generated on laser-driven ICF platform.

An early and pioneering experimental study of the turbulent shear layer was due to Liepmann and Laufer.^{203} More recent significant experiments have been published in Refs. 204–206 among others. Several experiments on spatially evolving shear layers have been designed to be able to study the coupled effects of buoyancy and shear, for example, Refs. 207 and 208, as well as the effects of compressibility on the evolution of the free shear layers.^{201,209} Recently, San and Maulik^{210} and Rahman and San^{211} studied the stratified KH turbulence of compressible shear flows, while Gan *et al.*^{212} and Lin *et al.*^{213} carried out simulations of nonequilibrium KHI.

Explosively driven shock-induced shears have been examined by Silver *et al.*,^{214} which represented a mix of RT and KH instabilities and mixing. The experiment was conducted by using explosively lined cylinders and employed radiography for diagnostics.

Computations of shear layers have been conducted using a variety of algorithms. In fact, shear layers have been an important topic of turbulence research and CFD (computational fluid dynamics) studies for decades (The series of biennial Turbulence and Shear Flow Phenomena Symposia is the main venue for disseminating recent and ongoing research. http://www.tsfp-conference.org/index.html.)

### D. Nonlinear behavior and transition to self-similarity

Historically, the initial growth of these instabilities has been treated as a linear (or weakly nonlinear) process. However, as the velocities grow and the mixing widths increase, the nonlinearities inherent in the Navier-Stokes equations begin to grow and eventually play a dominant role in the mixing process. When the flow fields achieve a state of fully developed turbulence (the precise meaning of which will be discussed later), the flows tend to exhibit approximately self-similar behaviors. These self-similar behaviors can generally be deduced from fairly simple dimensional arguments—much like those employed by Kolmogorov. However, the transition mechanisms that lead to the approximately self-similar behavior are subtle. We will now discuss the characteristics of the self-similar states and describe some metrics that indicate when the transition to self-similarity will occur.

^{217,218}The appropriate scaling groups are

^{198,224}

*h*is a characteristic length-scale of the mixing layer, e.g., the bubble or spike size, or the mixing layer width. The two parameter group (

*τ*and

*ξ*) is assumed to be restricted to a simple power-law subgroup, $\xi =\tau \zeta $, thus yielding

For the KH mixing layer at late times *ζ* = 1. For the RTI mixing layer at late times, *ζ* = 2, but the growth will look almost linear when $0<t\u2248t0$, and only when $t\u226bt0$ will the growth be obviously quadratic. This alone might account for some variations in reported measurements. The same is true for the late-time, RMI induced mixing layer, where *ζ* = *θ*. The *t ^{θ}* scaling is only valid when $t\u226bt0$. The time required for the late-time similarity scaling laws to be observed can be easily met if the temporal transition criteria are satisfied (see Subsection VI C).

*b*refers to the bubble-side, and the subscript

*s*refers to the spike side.

^{104,146,148,152,219–223}This is precisely the form suggested by Eq. (50) when

*ζ*= 2 and $\alpha b,sAg=hb/s,0/tb/s,02$. Under this limit, one obtains

*τ*must be nondimensional for the values of $h\u0303b,0,\u2009h\u0303s,0$,

*θ*, and

_{b}*θ*to be physically meaningful. Clark and Zhou

_{s}^{198}represented the dimensionless time scale shown in Eq. (50), i.e., as $\tau =(t+tb/s,0)/tb/s,0$. Note that if $\theta b=\theta s=\theta $, then we can identify

*ζ*=

*θ*in Eq. (50), and the layer is self-similar. The values of the exponents range from 0.17 to 0.667. Table 6.4 of Zhou

^{7}provides a rather complete picture of the variation in the estimates of the coefficients.

^{136}that

*θ*>

_{s}*θ*implies that the ratio of the spike and bubble widths grows without bound in time implying non-self-similar internal structure. On the other hand, the viewpoint of Refs. 181, 224, and 225, where a single value of

_{b}*θ*for the layer as a whole is assumed, indicates ( $t\u226btb/s,0$)

It is worth mentioning three recent publications relevant to this issue. Thornber *et al.*^{225} compared and contrasted the RMI scaling exponent from the numerical simulation databases from several independently developed codes. Elbaz and Shvarts^{226} evaluated a new formulation based on the well-known mode-coupling models of Haan,^{110,111} Shvarts *et al.*,^{114} and Ofer *et al.*^{115} In addition to providing a self-consistent extension of previous RTI theoretical studies, Elbaz and Shvarts found the results of RMI scaling exponent, *θ* (2/5 in 2D and 1/3 in 3D). Using the Eddy Damped Quasi-Normal Markovian theory (EDQNM)^{227–230} of turbulence, Soulard *et al.* found that *θ* = 1/4 at low Atwood number, $A\u21920$.^{231} It would be interesting to further confirm that the values of *θ* from these publications will be valid at late-time.

It has been observed experimentally and in simulations that in both the temporal and spatial KHI mixing cases, the turbulent shear layers transition toward an approximately self-similar state. If compressibility and buoyancy effects are negligible, the temporally evolving shear layer width grows linearly in time,^{198} i.e., *ζ* = 1 in Eq. (50), and the spatially evolving shear layer width grows linearly with position downstream from the splitter plate.^{201} The turbulent kinetic energy (defined in Sec. II) on the centerline of the layer becomes constant with respect to time for the temporally evolving layer, and constant with respect to downstream position for the spatially evolving case. Like the RTML, the KH shear layer represents an anisotropic inhomogeneous driven turbulent field, where the driver is the differential velocity across the layer.

The Kelvin-Helmholtz shear layer plays a significant role in the RTML and RMML cases as well, where the characteristic mushroom-shapes of the tips of the spikes and bubbles are induced by a shear instability between the penetrating and penetrated fluids. As the bubbles and spikes continue to grow, the shear between the light fluid and the heavy fluids within the mixing layer will give rise to secondary KHI growth and mixing. The mushroom shape is, in fact, physically associated with the shear process itself, rather than the RTI or RMI processes—see Fig. 20.^{208} The mushroom features appear also in other sheared flows, e.g., a round jet, shown in Fig. 21, where the fluids are the same in both the inflowing jet and the fluid into which the jet flows.^{232} The fluids are distinguished in the experiment via a dye.

In particular, if viscous effects are small, short wavelength perturbations grow quickly and then saturate. Long wavelength perturbations grow at a slower rate and saturate at a larger amplitude. This implies that the dominant length scale should increase for the RT, RM and KH cases. If the dominant length scale increased in proportion to the width of the mixing zone, the process tends to become self-similar. A number of empirical models have been proposed, and the interested reader is referred to Refs. 7, 215, and 216 for details.

^{149}

The corresponding behavior of the RMML observed in experiments and simulations is generally represented by the form given in Eq. (53), where *θ _{b}* < 1 and

*θ*< 1, and generally $\theta s\u2260\theta b$. Typically, the experimentalist or simulationist will assign different power-law exponents and prefactors to the bubble-side and spike-side of the RMML, but the reader should be warned this matter has some subtlety—if $\theta b\u2260\theta s$, the behavior is not truly self-similar, as noted above.

_{s}Summarizing for RTI and RMI at onset of self-similarity, it is reasonable to expect the following:

For RTI, the virtual starting thickness, $hb/s,0$, depends on how long it takes for the flow to become self-similar, which in turn depends on the spectrum of initial perturbations.

^{149}For RMI, it is difficult to run simulations or experiments far enough to achieve self-similarity (Youngs, see pages 38–39 of Ref. 8) and in many cases, the time origin, $tb/s,0$, must be incorporated for the influence of initial conditions.

^{198}

The value of *θ* is sensitive to the choice of the point where self-similarity is assumed to be established, as illustrated by Thornber.^{233} The value of *α* obtained in Eq. (55) for a turbulent mixing layer is dependent to some degree on the definition of the edge of the mixing layer, so caution should be exercised when comparing various estimates of the mixing layer growth rate.^{234} Indeed, it is important to note that there is no universally accepted definition of the “edge” of a turbulent mixing layer—different authors may use different metrics to define the edge of a mixing layer. For example, the edge might be defined as that position (in a planar layer) where the penetrating liquid occupies some small specified volume-fraction, or perhaps mass-fraction of the overall fluid.

^{235}

*Y*

_{1}and

*Y*

_{2}are the mass fractions of the two fluids, which could be viewed as a more direct marker of the evolution of the mixing layers due to hydrodynamic instabilities than the mixing layer width. A specifically attractive feature of the mixed mass $M$ is that it is a conserved quantity.

The importance of the mixed mass is evident from recent measurements using the high-energy-density-physics platform.^{18,236–238} Figure 22 shows the measured total neutron yield across the ensemble of cryogenic low-foot deuterium-tritium (DT) experiments against their inferred CH(Si) mix mass. While there is large variability in the amount of measured mix, there is clearly a trend of neutron yield dropping with increasing mix.^{18}

### E. Other interesting plasma instabilities

The RTI, RMI, and KHI are hydrodynamic instabilities in that they arise in purely hydrodynamic descriptions and do not fundamentally depend on additional plasma properties (e.g., electric or magnetic fields) or on kinetic effects. While this article focuses on situations in ICF/HEDP and astrophysics where these instabilities play a key role, there are also many cases where instabilities and turbulence that are not purely hydrodynamic occur in plasmas. For example, kinetic instabilities also arise in the above-mentioned settings, and there is a whole “ecosystem” of instabilities that arises in magnetized plasmas, in which the “background” magnetic field plays a key role. A detailed discussion of these topics is beyond the scope of current article, and we will only provide a very concise sketch here for completeness.

Kinetic streaming instabilities can arise in ICF and HEDP experiments and can result in rapid (much faster than collisional) transfer of flow energy into heat.^{239} The Weibel instability is an electromagnetic streaming instability that has been postulated as a dissipation mechanism important in collisionless astrophysical shocks,^{240} and has been produced and observed in specially designed HEDP experiments.^{241} A variety of plasma instabilities can cause unwanted scattering of laser energy^{242,243} or (sometimes desirable) conversion into different waves.^{244}

In the case of tokamaks, as an important example of laboratory magnetized plasmas, various categorizations of the instabilities and turbulence are of value. Denoting n and m as the toroidal and poloidal mode numbers, respectively, there are “macro” or global instabilities for which the modes are large scale and have low n and m, and small-scale “micrometer”-instabilities, for which the modes have the high n and m. The simplest extension of hydrodynamics that allows for significant electromagnetic effects is the “magnetohydrodynamic” (MHD) description,^{245–248} which allows for self-consistent evolution of the magnetic field along with the usual fluid fields (e.g., velocity, pressure, mass density). A relevant and useful (if singular) limit of MHD is “ideal” MHD, in which dissipative effects such as resistivity and viscosity are neglected. Large-scale ideal MHD instabilities can be particularly virulent and can result in large rapid losses of tokamak energy or plasma up to the complete loss of the plasma. When the plasma is ideal-MHD stable, it may still be unstable to “nonideal” modes, i.e., those fundamentally dependent on resistivity or other dissipation mechanisms. These act on slower time scales but can nevertheless also lead to loss of plasma or significant degradation in plasma confinement.

Plasma instabilities can also be classified according to the driving source of free energy. Current-driven instabilities, i.e., those driven by equilibrium electrical plasma currents, include ideal and resistive “kink” modes. The latter are also known as “tearing” modes^{249} as they involve a tearing of magnetic flux surfaces, resulting in a rearrangement of their topology. There is a variant known as “neoclassical” tearing modes, in which the “bootstrap” current associated with magnetically trapped particle orbits and collisions plays a significant role. Finer scale current driven modes in the edge pressure “pedestal” region are known as “peeling” modes. Pressure driven modes, include RT-like interchange modes. Here, if the pressure gradient is in the same direction as the mean curvature or magnetic-field gradient, then an “internal” RT-like instability can result, in which whole magnetic “flux tubes” of plasma will interchange. If this “bad” magnetic field gradient and curvature are localized poloidally, i.e., in the outside region of the toroidal tokamak plasma, then “ballooning” modes result, in which the flux-tube interchange is also poloidally localized. An additional potentially important source of free energy that can cause instabilities is energetic ions, which may result (via resonant charge exchange) from neutral beams used for plasma heating or arise as fusion products.

The MHD modes are “electromagnetic” in that they involve coupled evolution of the flows and the (solenoidal) magnetic field, and the (irrotational) electric field is unimportant. There are also electrostatic modes in which the electric field evolution is dominant over that of the magnetic field. These tend to arise at fine to very fine spatial scales but can degrade the plasma (particle and energy) confinement in plasmas that are stable to large scale modes. The toroidal ion^{250} and electron temperature gradient modes^{251} are electrostatic internal RT-like instabilities driven by the combination of temperature gradients and the magnetic field gradient and curvature. In the absence of magnetic field gradients and curvature, temperature gradients can still drive an instability by creating an effective negative compressibility.^{252,253} Pressure, temperature, and density gradients can drive various (diamagnetic) drift instabilities through a variety of effective dissipation mechanisms such as collisional resistivity^{254} and Landau damping.^{255} Trapped-particle instabilities are typically drift modes in which collisional or collisionless responses of trapped particles provide effective dissipation.^{256} Radial shear in the parallel velocity gradient, due to toroidal rotation shear driven typically by the momentum imparted by neutral beams, can also be an important source of instability drive.

Good pedagogical references on ideal MHD instabilities include Bateman^{245} and Freidberg,^{246} and a more general treatment of instabilities (especially global/macroinstabilities) can be found in Wesson.^{247}

Some of the instabilities discussed above have been shown to have major consequences. “Disruption” events,^{257,258} which can result in complete rapid loss of plasma and current, large populations of highly energetic “runaway” electrons, large inductive forces, and material damage, are due to various causes including phase locking of multiple large-scale MHD modes. “Mirnov oscillations”^{259} seen in magnetic signals result from tearing modes and reconnection near the outer surface with n = 1, m = *q _{a}*, the number of toroidal circuits a field line makes for each poloidal circuit near the edge of the plasma. Large “sawtooth” oscillations in the central temperature and density

^{260}can result from internal m = n = 1 reconnection. Magnetically trapped energetic ions can drive “fishbone” oscillations

^{261}via resonance between their precession and m = n = 1 MHD perturbation. Fine scale ballooning modes act in combination with the peeling modes at the pressure pedestals that arise in high-confinement (“H-”) mode plasmas and result in a phenomenon known as “edge-localized modes”

^{262}in which the pedestal confinement cyclically collapses or degrades. A key way to limit power flow to material surfaces in a tokamak is to create a “detached” plasma region near material plates in which a layer of cold plasma radiates away the much energy before it can reach the plates. The creation of this detached layer is tricky because a “MARFE” instability

^{263}can arise in which the cold plasma expands into the core region. Finally, even in the absence of such large-scale events, the finer scale turbulence results in transport of heat and particles that set the limits on fusion plasma performance.

Nowadays, there is a vast body of work using kinetic and fluid simulations of both large-scale instabilities, “events,” small-scale instabilities and the associated turbulence. In addition to quantitative calculation and parameterization of transport and comparison of predicted turbulence signatures with experiment,^{264} interesting fundamental phenomena have been discovered and characterized. For example, there is a nonlinear upshift in the gradients needed to drive significant turbulence relative to the thresholds for the underlying linear modes^{265–268} and the saturation by stable modes at wavenumbers similar to those of unstable modes^{269,270} as opposed to the wavenumber “cascades” familiar in hydrodynamic turbulence.

Before closing this section, it is worth mentioning some other instabilities and turbulence found in space plasma and geophysics (see, for example, Seropian *et al.*^{271}). The magneto-rotational instability is an instability that arises in accretion disks due to the decrease in angular velocity as a function of distance, which results in a turbulent viscosity (transport of angular momentum) in the disk. The reader is referred to classical references by Velikhov^{272} and Chandrasekhar.^{273} In the magnetospheres of the Earth, planets and stars, reconnection, shocks, and plasma jets are important issues (see, for example, Collinson *et al.*^{274}). The link between shocks, turbulence, and magnetic reconnection in collisionless plasmas has been addressed, for instance, by Karimabadi *et al.*^{275} Finally, the Sun and stars exhibit a variety of observable phenomena, including flares, eruptions, prominences, sunspots, and turbulent dynamo (Mackay *et al.*^{276}).

## IV. INERTIAL CONFINEMENT FUSION IMPLOSION

### A. Spherically converging implosion

ICF aims to achieve thermonuclear fusion by compression of deuterium-tritium (DT) fuel to extreme densities and temperatures.^{277} ICF requires both the heating the fuel to sufficiently high temperatures to initiate fusion burn (i.e., reach a sufficiently large DT fusion reaction cross section) and also to compress the fuel to high densities so that the inertia of the fuel itself can confine the burning plasma long enough to achieve significant fuel burn up. Experiments are under way to demonstrate the first steps of ICF ignition at the National Ignition Facility (NIF) at Lawrence Livermore National Laboratory (LLNL).^{278} In these experiments, the indirect drive approach to ICF is being pursued where NIF's 1.8 MJ of laser energy is first converted to soft x-rays inside of a high-Z enclosure or hohlraum.^{16} These soft x-rays then ablate the outer surface of a spherical low-Z capsule suspended in the middle of the hohlraum, driving strong shocks into the capsule to implode and heat the fuel. Experiments where the capsule is driven directly by laser beams (direct drive) are also being studied, principally at the University of Rochester's Laboratory for Laser Energetics (LLE).^{279}

In the typical ignition scheme, a spherically converging implosion is arranged to heat a low-density but high temperature central hot spot, while a surrounding colder fuel layer is compressed to high density but at much lower temperatures to achieve the necessary confinement.^{280} This separation into a high-temperature hot spot (∼10 keV) and a cold but dense main fuel (>500 g/cm^{3}) minimizes the overall energy investment necessary to reach ignition conditions. Ideally, if the central hot spot is large enough that the fusion product *α*-particles slow down within and self-heat the hot spot, run away fusion ignition can occur, propagating a detonation wave into the surrounding dense fuel. At a hot spot density of ∼100 g/cm^{3}, this requires a hot spot radius of ∼30 *μ*m and an energy investment of ∼40 kJ. With ∼100 mg of surrounding DT fuel compressed to > 500 g/cm^{3}, a burn duration of ∼50 ps is possible resulting in a burn-up fraction of ∼1/3 and a yield of ∼10 MJ. Assuming a total coupling efficiency of ∼10%, this idealized scaling suggests that substantial energy gain is possible with only ∼400 kJ of absorbed energy. Decades of experimental and theoretical ICF research have shown this scaling to be highly optimistic, however, in large part due to the impact of hydrodynamic instabilities during the implosion.

### B. Various phases of hydrodynamic instabilities

The high compression necessary to achieve the relevant densities for ICF inevitably implies a high convergence and high velocity implosion. In turn, this high convergence and acceleration to high velocity provide ample opportunity for hydrodynamic instabilities to develop, predominantly the well-known RT and RM instabilities.

Details of a typical ICF implosion are shown in Fig. 23. The figure shows radius vs time of an imploding ICF capsule from a 1-D Lagrangian simulation. The different colored traces show the different materials making up the shell: the interior volume is filled with low-density DT gas, the main fuel is a thin layer of DT ice shown in green, and the outer shell is composed of various layers of doped plastic (CH) that are ablated by the incident x-ray flux. The time dependence of the hohlraum radiation temperature (or x-ray flux) that drives the implosion is shown by the thick black curve with the scale on the right. Each step in the radiation temperature launches a shock into the capsule and ultimately accelerates the shell to a velocity of more than 350 km/s by the peak of the x-ray drive. Stagnation, and hopefully fusion ignition, occurs in the last few hundred picoseconds of the implosion.

Line-outs of the shell density from the three representative times marked in Fig. 23 are shown in Fig. 24. During the first time shown in Fig. 24, several strong shocks can be seen propagating through the DT fuel and plastic ablator shell. As the shocks cross the various interfaces in the capsule, including the ablation front (or outer surface of the imploding capsule), RM instabilities can develop. While the RM modulation amplitudes are generally quite small due to the very smooth surfaces of the initial capsule and grow only slowly with time, RM growth is important in setting the initial amplitudes (and phases) for subsequent RT growth.^{281} After the shocks break out from the inside of the shell, the shell begins to accelerate, and a subsequent RT unstable phase of the implosion begins, as represented by the second time in the figure. This phase again involves instability growth at both the ablation front and any interior interfaces with an unstable density gradient.

At the ablation front, the ablation of the outer capsule surface exerts a strong stabilizing effect on RT growth.^{282–284} This stabilization is essentially due to the “polishing away” of the perturbations that occur under ablation: the steepened temperature gradients in the neighborhood of a perturbation maximum cause enhanced heat flux and enhanced drive pressure near that maximum, both of which tend toward the suppression of the perturbation maximum and an effective reduction in its growth. Sufficiently short wavelengths are in fact completely stabilized and only a finite spectrum of modes can grow in the presence of ablation, unlike in the classical case (see also Subsection IV C).

At the interior interfaces, however, particularly the interface between the DT fuel and plastic ablator, there is no ablative stabilization, and classical RT instability growth can occur.^{285,286} The only means of controlling this instability growth is by managing the amplitude of the unstable Atwood number at this interface throughout the implosion. It is precisely for this reason that the dopant layers shown in Fig. 23 are added to the plastic ablator. By doping the plastic and increasing its opacity, the interior layers of the ablator are subject to less x-ray heating from the driving x-ray radiation. This causes the interior ablator layers to remain relatively cool and hence dense if they are compressed to the same pressure. In turn, this reduces the unstable Atwood number and subsequent instability growth between the DT fuel and ablator. This interior stabilization comes at a cost; however, in that adding dopant to the ablator also steepens the outside ablation front during the implosion causing that surface to become more RT unstable. ICF implosion design, and indeed the success of ICF, depends on properly managing these competing instability demands.

Finally, in the last few hundred picoseconds of the implosion, the shell stagnates to form a low density but high temperature and high pressure hot spot. In this phase, labeled as three in Figs. 23 and 24, the low density hot spot now decelerates the denser shell, and again RM and RT instabilities can develop at the interface between them. If the perturbation seeds are large enough or the instability growth is fast enough during this phase, perturbations of the hot spot-main fuel boundary can grow to amplitudes large enough to quench the desired hot spot ignition.

Note that none of the various phases of instability growth shown in Figs. 23 and 24 occurs in isolation. Indeed, a principle complicating factor of ICF is that instabilities that develop at the ablation front during the acceleration phase of the implosion can feed through the finite thickness shell and act as seeds for instability growth during deceleration around the edge of the hot spot. Moreover, there are multiple initial imperfections in real ICF targets that can act as initial seeds for RM and RT growth. These include the roughnesses inherent to each surface in the capsule at the start of the implosion, both the external ablator surface and all of the internal surfaces between the ablator layers, ablator and DT fuel, and finally the inner surface of the DT ice layer. Bulk inhomogeneities, either in density or opacity, can also seed instabilities, as can asymmetries in how the implosion is driven, either by x-rays or directly by lasers. Finally, all of the discrete features inherent in any practical target are also very important seeds to instabilities, the 45 nm thick plastic membrane or “tent” used to support the capsule at shot time and the micrometer-scale fill tube used to fuel the capsule with DT prior to the shot are the leading examples of these.

The interactions of all of these multiple effects and the reality that instabilities in ICF implosions typically evolve into a weakly nonlinear phase make a detailed understanding of hydrodynamic instability evolution in ICF extremely complicated. Naturally, heavy reliance must then be made on numerical simulation for a quantitative understanding. Over the previous decades, instability modeling in the ICF context has become quite sophisticated relying on highly developed multiphysics codes that couple radiation transport, hydrodynamics, thermonuclear burn, and detailed material properties (equations of state and opacities) in realistic 3D geometry.^{287–289} Indeed, quantitative comparisons between simulation and experimental data from high-convergence NIF implosions have only proven feasible with these fully 3D, multieffect simulations.^{290–293}

### C. Ablative Rayleigh-Taylor instability

^{282,284,294–302}ARTI was responsible for a significant fraction of the degraded implosion performance observed during the National Ignition Campaign (NIC) on the National Ignition Facility (NIF).

^{303–306}The subsequent high-foot implosion,

^{307–313}which was designed to mitigate ARTI performed much better. The essen