SPECIAL FEATURE: PERSPECTIVE

Supernova,nuclear synthesis, fluid instabilities,andinterfacial mixingSnezhana I. Abarzhia,1, Aklant K. Bhowmickb, Annie Naveha, Arun Pandianb, Nora C. Swisherb,Robert F. Stellingwerfc, and W. David Arnettd,1

Edited by William A. Goddard III, California Institute of Technology, Pasadena, CA, and approved October 3, 2018 (received for reviewDecember 31, 2017)

Supernovae and their remnants are a central problem in astrophysics due to their role in the stellarevolution and nuclear synthesis. A supernova’s explosion is driven by a blast wave causing the develop-ment of Rayleigh–Taylor and Richtmyer–Meshkov instabilities and leading to intensive interfacial mixing ofmaterials of a progenitor star. Rayleigh–Taylor and Richtmyer–Meshkov mixing breaks spherical symmetryof a star and provides conditions for synthesis of heavy mass elements in addition to light mass elementssynthesized in the star before its explosion. By focusing on hydrodynamic aspects of the problem, weapply group theory analysis to identify the properties of Rayleigh–Taylor and Richtmyer–Meshkov dynam-ics with variable acceleration, discover subdiffusive character of the blast wave-induced interfacial mixing,and reveal the mechanism of energy accumulation and transport at small scales in supernovae.

supernovae | nuclear synthesis | blast waves | Rayleigh–Taylor instabilities | Rayleigh–Taylor interfacial mixing

Supernovae are violent, disruptive explosions of stars (1).They have been a central problem in astrophysics sincetheir discovery and identification in the 1930s. The de-bris ejected from a supernova mixes with the interstellarmedium, forming a supernova remnant (SNR) (2–4).Young SNRs still retain information concerning the ex-plosion process. Explosions are initial value problems;solution requires the details of what explodes (1–5).

Astrophysics and Fluid DynamicsOur understanding of stellar evolution is based on theconvenient assumption that stars may be treated asspherically symmetric objects, at least on average (1).This approximation, which has proven to be surpris-ingly successful, allows us to evolve stars up to apresumed presupernova condition (1–3). Wemay theninvent an explosion model and compare the predic-tions with observed SNR. Unfortunately, this is a longextrapolation, with many untested choices along theway (1). The conditions in stars imply very high

Reynolds numbers, and therefore, plasma motion instars is expected to be turbulent (or at least, disor-dered) (4–7). This may break the spherical symmetry indetail but might be approximately valid on the globalscale. However, as explosion begins, spherical sym-metry is firmly broken on the global scale too, so thatconventional stellar evolution theory must fail (1, 5).

How this happens is of considerable interest (1–7).Some supernovae produce neutron stars (pulsars), andothers produce black holes. They are thought to bethe major source of galactic cosmic rays. Supernovaeare the dominant source of elements not produced bythe Big Bang (those being hydrogen and helium). Thecalcium in our bones and the iron in our blood weresynthesized in a supernova as were silver, gold, ura-nium, and thorium. The latter elements can also beproduced in neutron star mergers in the “r-process,”evidenced in the mergers’ observations and the abun-dances in dwarf galaxies (1–11). Even carbon and nitro-gen are partially produced in supernovae but more so in

aDepartment of Mathematics and Statistics, The University of Western Australia, Perth, WA 6009, Australia; bDepartment of Physics, CarnegieMellon University, Pittsburgh, PA 15213; cStellingwerf Consulting, Huntsville, AL 35803; and dThe Steward Observatory, The University of Arizona,Tucson, AZ 85721Author contributions: S.I.A. and W.D.A. designed research; S.I.A., A.K.B., A.N., A.P., N.C.S., R.F.S., and W.D.A. performed research; R.F.S.contributed new reagents/analytic tools; S.I.A., A.K.B., A.N., A.P., R.F.S., and W.D.A. analyzed data; S.I.A. guided research on fluid instabilities andmixing, including scientific and organizational aspects; A.K.B., A.N., A.P., N.C.S. studied fluid instabilities and mixing analytically and numerically;R.F.S. developed and supported the smoothed particle hydrodynamics code and provided advice on the code use; W.D.A. guided the projectconnecting the observational data, supernova, and nuclear synthesis to fluid instabilities andmixing; and S.I.A., A.K.B., A.N., A.P., R.F.S., andW.D.A.wrote the paper.The authors declare no conflict of interest.This article is a PNAS Direct Submission.Published under the PNAS license.1To whom correspondence may be addressed. Email: snezhana.abarzhi@gmail.com or wdarnett@gmail.com.This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1714502115/-/DCSupplemental.Published online November 26, 2018.

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stars that die less violently as are the “s-process” heavy elements.The details are still in debate. Supernovae are a major source ofradioactive nuclei for meteorites and for the early solar system (1–7).

Fig. 1 provides a detailed look at the Cassiopeia A (Cas A)SNRs, the youngest nearby SNR known in the Milky Way. TheSNRs of the Cas A have been produced by the explosion of amassive star. The image in Fig. 1 is combined from 18 imagestaken by NASA’s Hubble Space Telescope in 2004. It shows theCas A remnants as a broken shell of filamentary and clumpy stellarejecta glowing with the heat generated by the passage of a shockwave from the supernova blast. The various colors of the gas in-dicate differences in chemical composition. Bright green filamentsare rich in the oxygen, red and purple ones are rich in the sulfur,and blue ones are composed mostly of the hydrogen and the ni-trogen. The oxygen and the sulfur are produced by thermonuclearburning during and just before the explosion. The sulfur is formedby the oxygen nuclear burning. Some of the sulfur lies farther fromthe center than some of the oxygen, which is interpreted as due toRayleigh–Taylor instabilities (RTIs) during the explosion (12–14).

Hence, the key question is: can we work backward from the SNRtoward the underlying explosion to provide insight into the eventthat is independent of conventional stellar evolution theory?

To do so, we have to better understand fluid dynamic aspectsof the multiphysics problem of supernovae, particularly Rayleigh-Taylor instability, Richtmyer–Meshkov instability (RMI), and Rayleigh–Taylor (RT)/Richtmyer–Meshkov (RM) interfacial mixing that arecaused by supernova’s blast (1, 12–16). RTI develops at the in-terface of the fluids with distinct densities that are acceleratedagainst their density gradients (12–14). RMI develops when theacceleration is induced by a shock refracting a perturbed interfaceof the fluids with distinct acoustic impedances (17, 18). Intense in-terfacial RT/RM mixing of the fluids ensues with time (12–18).

RTI/RMI and RT/RM mixing occur at a broad range of astro-physical phenomena in low- and high-energy density regimes (1, 14).Examples include the appearance of stiff light years-long structuresin molecular hydrogen clouds, the formation of accretion disks andblack holes, and the processes of stellar evolution (1–7). The latterranges from a birth of a star due to the interstellar gas collapse to lifeof a star with the extensive material mixing in the stellar interior andto death of a star in the supernova (1–7, 14, 15). In supernovae, theblast wave-driven RT/RM mixing of the outer and inner layers of theprogenitor star creates conditions for synthesis of heavy and in-termediate mass elements in addition to light mass elements syn-thesized in the star before its explosion (1–7, 14, 15, 19–25).

In everyday life and in extreme astrophysics environments, RTflows are observed to have similar qualitative features of theirevolution (12–16). RTI starts to develop when the fluid interface isslightly perturbed near its equilibrium state (12–14). The flowtransits from an initial stage, where the perturbation amplitudegrows quickly, to a nonlinear stage, where the growth rate slowsand the interface is transformed into a composition of small-scaleshear-driven vortical structures and a large-scale coherent structureof bubbles and spikes [with the bubble (spike) being the portion ofthe light (heavy) fluid penetrating the heavy (light) fluid]. RT flowsare usually three-dimensional (3D) and have two macroscopic-length scales—the amplitude in the acceleration direction andthe spatial period or the wavelength in the normal plane. The floweventual stage is the self-similar interfacial mixing (12–15).

While RT and RM flows are similar in many regards, there arealso important distinctions (12–18, 26–29). For instance, postshockRM dynamics is a superposition of two motions (17, 18, 26–29).These are the background motion of the fluid bulk and the growthof interface perturbations (26–29). In the background motion, bothfluids and their interface move as whole unit in the transmittedshock direction; this motion occurs even for an ideally planar in-terface, and is supersonic for strong shocks. The growth of the in-terface perturbations is due to impulsive acceleration by the shock;it develops only for a perturbed interface (26–29). The rate of thisgrowth is subsonic, and the associated motion is incompressible.The growth rate is constant initially and decays with time later. RMunstable interface is transformed to a composition of a large-scalestructure of bubbles and spikes and small-scale shear-driven vor-tical structures. Small-scale nonuniform structures also appear inthe bulk (26–29). Self-similar RM mixing develops, and the energysupplied initially by the shock gradually dissipates (17, 18, 26–29).

In RT mixing with constant acceleration, the length scale, thevelocity scale, and the Reynolds number increase with time (14–16). At a first glance, such flow should quickly proceed to a fullydisordered turbulent state (30). Turbulence is a stochastic processinsensitive to deterministic conditions with intense energy trans-port from the large to the small scales (30–32). In supernova en-vironments, such transport may supply to small scales someactivation energy required for the synthesis of heavy mass ele-ments (1). Recent advances in the theory and experiment of RTIand RT mixing have found, however, that, in the high- and low-energy density regimes, the properties of heterogeneous, aniso-tropic, nonlocal, and statistically unsteady RT mixing depart fromthose of homogeneous, isotropic, local, and statistically steadycanonical turbulence (14–16, 30–34). High Reynolds number,while necessary, is not a sufficient condition for turbulence tooccur. RT mixing exhibits more order and has stronger correla-tions, weaker fluctuations, and stronger sensitivity to deterministicconditions compared with canonical turbulence (14–16). For RMmixing, strong sensitivity to deterministic conditions has also been

Fig. 1. An image takenwith NASA’s Hubble Space Telescope providesa detailed look at the remains of a supernova explosion known as CasA. It is the youngest such remnant known in the Milky Way. The imageis made from 18 separate images taken in December 2004 by usingHubble’s Advanced Camera for Surveys. It shows the Cas A remnant asa broken ring of bright filamentary and clumpy stellar ejecta. Thesehuge swirls of debris glowwith the heat generated by the passage of ashock wave from the supernova blast. The various colors of the gasindicate differences in chemical composition. Image courtesy ofNASA/ESA/Hubble Heritage (STScI/AURA)-ESA/Hubble Collaboration, FrankSummers (Space Telescope Science Institute, Baltimore), Robert Fesen(Dartmouth College, Hanover, NH), and J. Long (ESA/Hubble,Garching bei München, Germany).

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found along with its nominally large Reynolds number and small-scale interfacial vortical structures (26–29).

In supernovae in a hydrodynamic approximation, accelerationsare induced by blast waves (1). Blast waves can be viewed as strongvariable shocks (1, 19–25, 33). Blast wave dynamics is self-similar,and the blast wave-induced acceleration is a power law function oftime (spatial coordinate) (1, 19–25, 33). Several questions thus ap-pear. What are the properties of RTI with variable acceleration, andhow do they differ from those of RTI with constant acceleration andfrom those of shock-driven RMI? How can these properties beapplied for interpretation of astrophysical data? What are theirpotential outcomes for stellar evolution and nuclear synthesis?

In this work, we consider hydrodynamic aspects on the multi-physics problem of supernovae and their remnants. We focus onthe dynamics of RTI and RT mixing with variable acceleration in abroad parameter regime (Figs. 2–7). The acceleration is a powerlaw function of time (spatial coordinate). We find that, for variableacceleration, RT/RM dynamics is multiscale and has twomacroscopic-length scales—the amplitude in the accelerationdirection and the spatial period in the normal plane (14–16, 34).Depending on the exponent of the acceleration power law, thedynamics can be RT type or RM type. For RT type, the accelerationsets the timescale at early stage and defines the nonlinear dy-namics and the interfacial mixing at later stages. For RM type, theinitial growth rate sets the timescale at early stage; at late stages,the drag defines the nonlinear dynamics and the interfacial mix-ing, and the initially supplied energy gradually dissipates (33, 35).The critical values of the exponent at which the transition occursfrom RT- to RM-type dynamics are distinct for the linear, nonlinear,and mixing regimes. Particularly for blast wave-induced acceler-ations, the linear and nonlinear dynamics are RT type, and themixing is RM type, but RM-type mixing develops quicker than theacceleration prescribes (33, 35). While for subdiffusive mixingdynamics, superdiffusive canonical turbulence may be a challengeto occur, other mechanisms are possible for energy accumulationat small scales. They are due to small-scale nonuniform structuresdeveloping in the fluid bulk and including cumulative jets, hot andcold spots, high- and low-pressure regions, and localizations as

illustrated by numerical simulations of strong shock-driven RMI(17, 18, 26–29, 36). Such effects should be considered in the in-terpretation of observational data (1). Technical details of thiswork are given in SI Appendix for corresponding sections.

Fluid Instabilities and Interfacial MixingGoverning Equations. Dynamics of ideal fluids is governed bythe conservation of mass, momentum, and energy:

∂ρ=∂t + ∂ρvi=∂xi = 0,  ∂ρvi=∂t +X3

j=1∂ρvivj

�∂xj + ∂P=∂xi = 0,

 ∂E=∂t + ∂ðE +PÞvi=∂xi = 0,

where xi are the spatial coordinates with ðx1, x2, x3Þ= ðx, y, zÞ; tis time; ðρ, v,P,EÞ are the fields of density ρ, velocity v, pres-sure P, and energy E = ρðe+ v2=2Þ; and e is the specific internalenergy (25). The latter refers to energy per unit mass con-tained within a system, excluding the kinetic and the potentialenergy of the system as a whole (25). For immiscible fluids, thefluxes of mass, momentum, and energy obey the boundaryconditions at the interface

½v · n�= 0,  ½P�= 0,  ½v · τ�= any,  ½W �= any,

where ½...�denotes the jump of functions across the interface; nand τ are the normal and tangential unit vectors of the inter-face with n=∇θ=j∇θj and ðn · τÞ=0; and θ= θðx, y, z, tÞ is alocal scalar function, with θ= 0 at the interface and with θ> 0(θ<0) in the bulk of the heavy (light) fluid marked with sub-script hðlÞ. The specific enthalpy is W = e+P=ρ. In a spatiallyextended system, the flow can be periodic in the plane ðx, yÞnormal to the z direction of gravity g and has no mass sources:

vjz→+∞ = 0,  vjz→−∞ = 0.

Acceleration g, jgj=g, is directed from the heavy to the lightfluid. Initial conditions include initial perturbations of the flowfields (12–18, 25–29, 34). For ideal fluids, the initial condi-tions set the length-scale λ and the timescale τ. Here, λ is theperturbation wavelength (spatial period), and τ∼

ffiffiffiffiffiffiffiffiffiffiλ=g0

pin

case of constant acceleration g0. In realistic fluids, smallscales are usually stabilized, and a characteristic scale λmcorresponds to a fastest growing mode [i.e., λm ∼ ðν2=g0Þ1=3,where ν is the kinematic viscosity] (37–39). The ratio of the fluids’densities and the density jump at the interface is parameterized

A B

C D

Fig. 2. One parameter family of regular asymptotic solution for 3Dflow with group p6mm at some Atwood numbers. RT/RM-typenonlinear dynamics: the bubble velocity vs. (A and C) the bubblecurvature and (B and D) the interfacial shear.

A BFig. 3. Qualitative velocity field in laboratory reference framenear the bubble tip for incompressible immiscible ideal fluids atsome Atwood number for a nonlinear solution in RT/RM family in(A) the volume and (B) the plane, with the interface marked by adashed curve.

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by the Atwood number A= ðρh − ρlÞ=ðρh + ρlÞ, 0<A< 0; A→1for ρh=ρl →∞ and A→0 for ρh=ρl → 1 (12–18).

To rigorously describe RT dynamics, one has to solve theproblem of extreme complexity: solve a system of nonlinear partialdifferential equations in 4D space–time, solve the boundary valueproblems for a subset of nonlinear partial differential equations at anonlinear freely evolving interface and at the outside boundariesand also solve the ill-posed initial value problem, with account forsingularities and secondary instabilities developing in a finite time(14, 34). A complete theory of RTI applicable at all scales and alltimes has yet to be developed. Rigorous theories have successfullyhandled the problem in well-defined approximations; empiricalmodels have repeatedly described a broad set of data with nearlythe same set of parameters. The reader is referred to review andresearch papers (12–18, 26–30, 33–54) for details of theoretical andnumerical studies of RT dynamics.

It is worth noting that, despite complexity and noisinessresulting from interactions of all of the scales, RT dynamics isobserved to have certain features of universality and order, and isthus eligible to first principle considerations, such as group theory(14, 15, 34, 55, 56). For linear and nonlinear RTI, group theoryanalysis uses theory of discrete groups to solve the boundaryvalue and initial value problems (34, 48–50). For RT mixing, grouptheory is implemented in the momentum model with equationsthat have the same symmetries and scaling transformations as thegoverning equations (14, 16, 34, 40, 54). Some principal results ofgroup theory analysis—the multiscale character of nonlinear dy-namics, to which both the spatial period λ and the amplitude hcontribute; the tendency to keep isotropy in the plane normal tothe acceleration and the discontinuous dimensional cross-over;the order in RT mixing that may be coexistent with a quasiturbu-lent state—self-consistently explain the observations (14, 15, 26–29, 34, 48–50, 54–58).

Group Theory Approach. Group theory for linear and nonlineardynamics studies RT flows that consist of large-scale coherentstructures in which dynamics is potential, vhðlÞ =∇ΦhðlÞ, and shear-driven interfacial vortical structures are small (34). For a spatiallyextended system, coherent structure is periodic in the planenormal to the acceleration direction. It is invariant with respect to adiscrete groupGwith generators that are translations in the plane,rotations, and reflections (34). To be structurally stable, coherentdynamics must be invariant under one of the spatial groups withthe inversion in the plane, such as the groups of hexagon p6mm,square p4mm, rectangle p2mm in 3D, and group pm11 in 2D (34,48). By applying irreducible representations of the group, thepotential ΦhðlÞ is expanded as a Fourier series, and spatial ex-pansion is further made in a vicinity of a regular point of the

interface [i.e., the bubble tip ð0, 0, z0ðtÞÞ]. Governing equationsare then reduced to a dynamical system in terms of surface vari-ables and moments, each of which is an infinite sum of weightedFourier amplitudes; the system solution is sought (34, 46, 48–50).For group p6mm, to the first order N= 1, the interface isz* − z0ðtÞ= ζðx2 + y2Þ, and the dynamical system is

M0 =− ~M0 =−v,  ρh �_ζ− 2ζM1 −M2=4

�= 0,  

ρl

�_ζ− 2ζ ~M1 + ~M2

.4�= 0,M1 − ~M1 = any,  

ρh  �_M1�4+ ζ _M0 −M2

1

�8+ ζg

�= ρl

0@ _~M1

,4− ζ _~M0 − ~M

21

�8+ ζg

1A.

Here, v ≥ 0 and ζ≤ 0 are the bubble velocity and curvature;Mð ~MÞ are the heavy (light) fluid moments. Group theory is fur-ther applied to solve the closure problem, find regular asymp-totic solutions forming a continuous family, study the solutions’stability, elaborate properties of nonlinear RTI (14, 34, 48–50).

Group theory for the mixing dynamics finds that, in RT mixing,the momentum and energy are gained and lost at any scale; thedynamics of a parcel of fluid is governed by a balance per unitmass the rates of momentum gain, ~μ, and momentum loss, μ, as

_h= v,   _v = ~μ− μ,

where h is the length scale along the acceleration g, v is thecorresponding velocity, and ~μðμÞ is the magnitude of the rateof gain (loss) of specific momentum in the acceleration direc-tion (14, 16, 54). The rate of gain (loss) of specific momentum is~μ= ~«=v (μ= «=v), with ~«ð«Þ being the rate of gain (loss) of spe-cific energy. The rate of energy gain is ~«= fgv, f = f ðAÞ,rescaled g  f →g hereafter. The rate of energy dissipation is«=Cv3=L, with a length scale L and a drag C, C ∈ ð0,∞Þ (14–16, 35, 54). Momentum model has the same symmetries andscaling transformations as the governing equations (14, 16, 34,54). It can be solved by applying the Lie groups. The casesL∼ λ and L∼h correspond to the nonlinear dynamics and theself-similar mixing (14–16, 34, 54). In each case, asymptotically,there is the particular solution of RT type ðha, vaÞ and the

A BFig. 4. Asymptotic solutions for RT mixing with variable acceleration.The mixing is (A) RT type and (B) RM type with time- and space-varying accelerations.

CA

B

Fig. 5. Evolution of strong shock-driven RMI. (A) Snapshots of theflow regions at some time instances. Dependence of RM initialgrowth rate on the initial perturbation amplitude: (B) the growth ratedata and (C) the data fit by the model. Reprinted from ref. 29, withthe permission of AIP Publishing.

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homogeneous solution of RM type ðhd , vdÞ (35). These solutionsare effectively decoupled due to their distinct symmetries.

Smoothed Particle Hydrodynamics Simulations. Supernovaenvironments are characterized by the conditions of high energydensity, strong shocks, sharp changes of flow fields, large per-turbations, and small effects of dissipation and diffusion. Nu-merical modeling of these extreme regimes is a challenge. This isbecause numerical methods should satisfy numerous competingrequirements to capture shocks, track interfaces, and accuratelyaccount for dissipative processes. An efficient approach formodeling strong shock-driven dynamics is the smoothed particlehydrodynamics (SPH) implemented in the smoothed particle hy-drodynamics code (SPHC) (36).

The SPH is a Lagrangian method representing a continuousfluid by means of fixed mass SPH particles and thus, reducing thegoverning partial differential equations to ordinary differentialequations (26–29, 36). SPHC has been originated in astrophysics,and in addition to astrophysical gravitational problems, it hasbeen successfully applied, tested, and validated in multiphysicsproblems in fluids, plasmas, materials for modeling the strongshock-driven RMI, the Noh problem, and other flow problems aswell as reactive and supercritical fluids, material transformationunder impact, ablation process in hypersonic flows, and chargeimbalance in plasmas (36). When applied to strong shock-drivenRMI, it has achieved an excellent agreement with the experimentsand with the rigorous zero-order, linear, and group theories. Thelatter includes themultiscale character of the nonlinear dynamics andis evinced, for instance, in the flattening of RM bubbles, flow fieldsstructure, and sensitivity to deterministic conditions (26–29, 36).

Here, we apply group theory to study RTI/RMI and RT/RMmixing with variable acceleration in a broad parameter regime.We further conduct SPHC simulations of strong shocks-driven RMflows to study their sensitivity to deterministic conditions and theirsmall-scale dynamics at the interface and in the bulk. We find theproperties of blast wave-induced RT/RM dynamics and proposethe mechanisms for energy accumulation and transport at smallscales in supernovae.

RTI with Variable AccelerationWe consider RT dynamics with time-varying acceleration g=Gta,where a is the acceleration exponent, a∈ ð−∞,+∞Þ, and G is

prefactor, G> 0, with dimensions ½G�=m=s2+a and ½a�= 1. For agiven wavelength (period) λ and for a∈ ð−∞,−2Þ∪ ð−2,+∞Þ,there are two timescales τG = ðkGÞ−1=ða+2Þ and τ0 = ðkv0Þ−1, where k

is the wave vector with k = 4π=λffiffiffi3

pfor 3D flow with group p6mm

and v0 is the initial growth rate set by the initial conditions and/or by

the impulsive acceleration. At a=−2, the timescale is τ0 = ðkv0Þ−1,and value Gk parameterizes the acceleration strength. Time ist > t0 > 0, t0 >> fτG, τ0g.

At early stage, for a> − 2, the timescale is τG, and the lineardynamics is driven by the acceleration and is RT type. For a< − 2,the timescale is τ0, and the linear dynamics is driven by the initial

growth rate and is RM type. At a=−2, the timescale is τ0 = ðkv0Þ−1,and the linear dynamics changes its character from RT to RM typewith the decrease of Gk.

At late stage, for a> − 2, the nonlinear dynamics is RT type;regular asymptotic solutions depend on time as ζ∼ k and v,M,~M∼ ta=2 for t >> τG. For a< − 2, the nonlinear dynamics is RM type;regular asymptotic solutions depend on time as ζ∼ k and v,M,~M∼ t−1 for t >> τ0. At a=−2, regular asymptotic solutions are ζ∼ kand v,M, ~M∼ t−1 for t >> τ0, and they are RT (RM) type for Gk >> 1(Gk << 1). These regular asymptotic solutions form a family (34).

For a> − 2 and at a=−2 with Gk >> 1, for regular asymptoticsolutions of RT-type nonlinear dynamics, the bubble velocity v ≥ 0depends on its curvature ζ,  ζ< 0, as

v. ffiffiffiffiffiffiffiffiffiffiffiffiffi

taG=kp

=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið−2Aðζ=kÞÞ

p �9− 64ðζ=kÞ2

� 

× ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

−48ðζ=kÞ+A�9+ 64ðζ=kÞ2

�r −1

.

For every A, this function domain is ζ∈ ðζcr , 0Þ, ζcr =−ð3=8Þk,and the range is v ∈ ð0, vmaxÞ, with v = 0 achieved at ζ=0 andat ζ= ζcr and with v = vmax achieved at ζ= ζmax,  ζmax ∈ ðζcr , 0Þ.

The multiplicity of the nonlinear solutions is associated with thenonlocal and singular character of the interfacial dynamics (34, 45,46, 48–50). The solutions exist and converge with increase in ap-proximation order. The number of the family parameters is identifiedby symmetry of the global flow (34). For group p6mm, the dynamicsis highly isotropic, z* − z0 ∼ ζðx2 + y2Þ, and the interfacemorphologyis captured by the principal curvature ζ (Fig. 2) (34, 48–50).

The multiplicity is also due to the presence of shear at theinterface (58). Defining the shear as the spatial derivative of

Fig. 6. Wave interference and order–disorder in RM flow. Snapshotsof the flow regions at some time instances for two-wave initialperturbations, with the same waves being in (Left) antiphase and(Right) random phase.

Fig. 7. Small-scale nonuniform structures in the fields of temperature(Left) and pressure (Right) in RM flow, including bulk-immersedcumulative jets, hot (cold) spots, and high- (low-) pressure regions.Reprinted from ref. 27, with permission of AIP Publishing.

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the jump of tangential velocity at the interface, Γ=ΓxðyÞ, with

ΓxðyÞ = ∂�vxðyÞ

��∂xðyÞ, we find shear Γ= ~M1 −M1 near the bubble

tip and its dependence on the interface curvature ζ. Forζ∈ ð−ζcr , 0Þ, shear Γ is 1–1 function on ζ, Γ∈ ðΓmin,ΓmaxÞ achievingvalue Γmin at ζ= 0 and value Γmax at ζ= ζcr.

The fastest stable solution is the physically significant solution (Fig.2) (14, 34, 48–50, 58). For the fastest stable solutions, the de-pendence of curvature and velocity ðζmax, vmaxÞ on the Atwoodnumber A is complex: vmax = vmaxðGta, k,AÞ,  ζmax = ζmaxðk,AÞ (Fig.2). However, there is the invariant v2max=ððtaG=kÞð8jζmaxj=kÞ3Þ= 1,implying that the nonlinear dynamics is characterized by the contri-bution of the wavelength λ, the amplitude h, and their derivatives andis, thus, multiscale (34, 48–50).

Regular asymptotic solutions have important physics proper-ties. For these solutions, there is effectively no motion of the fluidsin the bulk and away from the interface, there is intense motionnear the interface, and shear is present at the interface leading toformation of interfacial vortical structures, in agreement with ob-servations (Fig. 3) (34, 48–50).

Regular asymptotic solutions have important global proper-ties. The flow tends to conserve isotropy in the plane (34, 48–50).The 3D highly symmetric dynamics is universal (34, 48). That is, onthe substitution k = 2π=λ, the nonlinear solutions describe thedynamics of 3D flow with group p4mm. For 3D low-symmetricflows with group p2mm, there is a two-parameter family of reg-ular asymptotic solutions; among the family solutions, only nearlyisotropic bubbles are stable. The dimensional 3D–2D cross-over isdiscontinuous (34, 48).

At a=−2 with Gk << 1 and for a> − 2, for regular asymptoticsolutions of RM-type nonlinear dynamics, the bubble velocityv ≥ 0 depends on the bubble curvature ζ, ζ< 0, as

vðktÞ=�3− 2Aðζ=kÞ

�−5+ 64ðζ=kÞ2

���9− 64ðζ=kÞ2

�

×�−48ðζ=kÞ+A

�9+ 64ðζ=kÞ2

��−1.

The function domain is ζ∈ ðζcr , 0Þ,  ζcr =−ð3=8Þk, and therange is v ∈ ðvmin, vmaxÞ, with v = vmin achieved at ζ= ζcr andv = vmax achieved at ζ=0. For ζ∈ ðζmin, 0Þ, ζmin ≥ ζcr, shear Γis 1–1 function on ζ, Γ∈ ðΓmin,ΓmaxÞ, with Γ=Γmin at ζ= ζminand Γ=Γmax at ζ= 0. The fastest stable solution is the physi-cally significant solution (Fig. 2) (14, 34, 48–50, 58).

RT and RM families have similar physical, mathematical, andglobal properties (Figs. 2 and 3). These include the structure of thevelocity fields, the existence of the family of solutions, their de-pendence on the flow symmetry and the interfacial shear, thetendency to keep isotropy in the plane normal to the acceleration,and the discontinuity of the 3D–2D cross-over. However, theirlocal properties are distinct (Fig. 2) (34, 50). Particularly, for RMbubbles, the velocity is a monotone function on the curvature, andthe fastest stable solution corresponds to a flat bubble withvmaxðAktÞ=3= 1,  ζmax = 0. The quasi invariant of this solution

ð4=3Þtv2max= ðdv=dζÞjζ=ζmax

= ð1+ ð5=2ÞðA=2Þ2Þ−1 ≈ 1 implies that

the wavelength and the amplitude both contribute to the non-linear dynamics. A steep dependence of the velocity on the shearfor physically significant solutions in RT/RM families suggests theuse of highly accurate methods with the interface tracking fornumerical modeling RT/RM dynamics (Figs. 2 and 3).

Mixing DynamicsRT/RM Dynamics for Time-Varying Acceleration. In the non-linear regime with L∼ λ, asymptotic solutions for the momentummodel are consistent with group theory results. In the mixing re-gime with L∼ h, asymptotic solutions for the momentum modelare ha =Bat2+a and hd =Bdt2+acr , where the critical exponent is

acr =−2+ ð1+CÞ−1 with acr → − 1 for C→ 0 and acr → − 2 forC→∞ (33). For solution ðha, vaÞ, the exponent is set by the ac-celeration’s exponent, ð2+ aÞ, and the prefactor Ba is set by theacceleration parameters and the drag. This mixing is RT type; it isdriven by the acceleration. For solution ðhd , vdÞ, the exponent is

set by the drag, ð2+ acr Þ= ð1+CÞ−1, and the prefactor Bd is set bythe initial conditions. This mixing is RM type; it is driven by thedrag (the dissipation, since μ= «=v) (35). For a> acr the mixing isRT type. For a< acr, the mixing is RM type. At a∼ acr, a transitionoccurs from RT- to RM-type mixing (33, 35).

RT-Type Mixing. Properties of the asymptotic solutions indicatethat, for a> acr, when the dynamics is the acceleration driven inthe nonlinear and mixing regimes, two states are possible. Onestate is achieved for L∼ λ; it is the state with asymptotic balance ofthe rates of momentum, jμj∼ j~μj∼ ta, and with j _v=μj→ 0 (14, 33,54). The other is achieved for L∼ h; it is the state withj~μj∼ jμj∼ j _vj∼ ta and with an algebraic imbalance of the rates ofmomentum, ~μ≠ μ (14, 33). Per observations, at a= 0, the imbal-ance is small, ð~μ− μÞ=~μ<< 1 (51–53). Hence, RT mixing may de-velop when the amplitude h is the scale for energy dissipation,L∼ h (14, 33, 35). It may also develop due to the growth of periodλ∼Gta+2 in the nonlinear regime. Because the dynamics is mul-tiscale, the growth of the period λ is possible and is not a neces-sary condition for the mixing to occur (14, 29, 33, 35, 54).

For a> acr, in RT-type mixing, the length scales with time asL∼ ta+2, and the velocity scales as v ∼ ta+1. The length scale in-creases with time for a> acr. The velocity scale increases fora> − 1, is constant for a=−1, and decreases for a< − 1. Recallthat diffusion scaling law is L∼ t1=2, whereas canonical turbulenceis superdiffusive with L∼ t3=2 (25, 31, 32, 59). By comparing theseexponents, we find that, in RT-type mixing with a> acr, the dy-namics L∼ ta+2 is super ballistics for a> 0, ballistics at a= 0, steadyat the flex point a=−1, superdiffusion for a> − 3=2, quasidiffu-sion at a=−3=2, and subdiffusion for −3=2> a> acr . Large ve-locities correspond to large (small) length scales for a> − 1(acr < a< − 1).

RM-Type Mixing. RM-type mixing develops for a< acr; its rates ofgain and loss of momentum are asymptotically imbalancedj~μ=μj→ 0, whereas jμj∼ j _vj∼ tacr (33, 35). RM-type mixing maydevelop when the amplitude h is the scale for energy dissipation,L∼ h. It may also develop when the period λ grows with time,as λ∼ tðacr+2Þ+ðacr−aÞ ðλ∼ tðacr+2ÞÞ for −2< a< acr ða< − 2Þ. In RM-type mixing with a< acr, the length scale increases with time,L∼ tðacr+2Þ, whereas the velocity scale v ∼ tðacr+1Þ decreases withtime, and large velocities correspond to small length scales, sinceðacr + 2Þ∈ ð−1,0Þ for C > 0.

Space-Varying Acceleration. Similar analyses can be conductedwhen the acceleration is a power law function on the spatial co-ordinate, g∼ hn. Particularly, the nonlinear dynamics is RT type forn∈ ð−∞, 2Þwith v ∼ tn=ð2−nÞ, and RM type for n→ −∞with v ∼ t−1.For n→ −∞, the nonlinear dynamics changes from RT to RM type.

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Solutions for the nonlinear dynamics with time- and space-varying ac-celerations can be transformed one into another with α→ 2n=ð2  −   nÞ.

The mixing dynamics is RT type for n∈ ðncr , 1Þwith h∼ t2=ð1−nÞ,and RM type for n∈ ð−∞,ncr Þ with h∼ t2=ð1−ncr Þ. The critical ex-ponent is ncr =   − 2C   −   1, with ncr → − 1 for C→0 and ncr → −∞for C→ +∞. Solutions for the mixing dynamics with time- andspace-varying accelerations can be transformed one into anotherwith α→ 2n=ð1  −   nÞ.

Fig. 4 illustrates RT/RM-type mixing for various values of theexponents a and n. Solutions are derived for C being a stochasticprocess with log-normal distribution with the mean hCi= 3.6 andthe SD σ = hCi=2 leading to acr ≈−1.78 ðncr ≈−8.2Þ. Mean valuesof quantities h, v, j _vj,g are plotted in red, blue, green, and black,respectively, in Fig. 4 (35).

Blast Wave-Induced Mixing. Consider now RT/RM mixing in-duced by blast waves with the first kind (Sedov–Taylor) self-similarity and the second kind (Guderley–Stanyukovich–Landau)self-similarity (19–25). Note that exponents acr ðncr Þ have valuestypical for blast waves.

For Sedov–Taylor self-similar dynamics, the scaling dependence

of the solution is set by the energy releaseΕ, ½Ε�= kgðm=sÞ2, and thefluid density ρ, ½ρ�= kg=m3ð2,1Þ, in 3Dð2D,1DÞ in case of point (line,plane) energy source (19, 20, 24). The invariance ofenergy density ðΕ=ρÞ leads to scaling laws for the length∼ t2=5ðt1=2, t2=3Þ, velocity ∼ t−3=5ðt−1=2, t−1=3Þ, and acceleration∼ t−8=5ðt−3=2, t−4=3Þ. This suggests that the blast wave dynamics issubstantially slower than canonical turbulence (31, 32). By com-paring the blast wave acceleration exponent with the value acr, wefind that this mixing can be RT type for small drag C < 3=2ð1, 1=2Þand RM type for large drag C > 3=2ð1, 1=2Þ.

For Guderley–Stanyukovich–Landau self-similar dynamics, theload history and momentum transport should be accounted for.The corresponding invariant FðΕ, P, ρÞ is a function of the energyrelease Ε, pressure P, and fluid density ρ, with ½F�=m=sθ and0< θ< 1 (21–24). In this case, the solution is a power law with thelength ∼ tθ, velocity v ∼ tθ−1, and acceleration v ∼ tθ−2, so that theacceleration exponent is −2< ðθ− 2Þ< − 1. By comparing these

values with acr =−2+ ð1+CÞ−1, we find that this mixing can be RTtype for small drag C < ðθ−1 − 1Þ and RM type for large dragC > ðθ−1 − 1Þ.

Small-Scale Nonuniform Structures of the Flow Fields. Hence,depending on the drag value, the blast-wave induced mixing canbe RT type with a ∼ acr or RM type with a < acx, acr ∈ (−2,−1); ineither case, larger velocities (velocity fluctuations) correspond tosmall length scales, and the dynamics is sensitive to deterministicconditions. In canonical turbulence, large velocities (velocityfluctuations) correspond to large length scales, and the dynamicsis independent of deterministic conditions (31, 32). In the blastwave-induced (subdiffusive) mixing, the canonical (super-diffusive) turbulence may be a challenge to develop (unless thereis a source, other than gravity, supplying turbulent energy to thefluid system). If so, what are other possible mechanisms for energyaccumulation at small scales, which is necessary for nuclear syn-thesis in supernovae (1)?

Such mechanisms may exist due to small-scale nonuniformstructures of the flow fields (26–29, 59). Particularly, subdiffusiveprocesses are often characterized by localizations. If a spot withhigh energy fluctuation appears in the flow, it can be trapped due

to the slow dynamics. If the spot is hot enough, it may initiate anuclear reaction accompanied by some energy release and mayfurther induce a chain of other reactions and processes (1–3). Blastwaves are the special strong shocks (24). In strong shock-drivenRMI, the shock–interface interaction may lead to intense pro-duction of small-scale nonuniform structures in the bulk in addi-tion to small-scale shear-driven vortical structure at the interface(26–29). These nonuniform structures may include cumulative jets,hot and cold spots, high- and low-pressure regions, and mayenable strong energy fluctuations at small scales.

Strong Shock-Driven RM FlowsSPH Simulations. To illustrate scale coupling in strong shock-driven RMI, we use the SPHC in a hydrodynamic approximationfor ideal monoatomic gases with adiabatic indexes γhðlÞ = 5=3 andthe Atwood number A= f0.3, 0.6, 0.7, 0. 8, 0. 95g. They have highenergy per atom. The shock Mach number is M = {3,5,7,10} de-fined relatively to the light fluid with the speed of sound cl (26–29).The interface is normal to the shock; the initial perturbationwavelength and amplitude are λ and a0. In SPHC, we scale thelength, velocity, and time with λ, v∞, λ=v∞, where v∞ðM,A, γhðlÞÞ isthe background motion velocity value; it is supersonic. RM initialgrowth rate v0ðM,A, γhðlÞ, λ, a0Þ is subsonic.

Fig. 5A illustrates SPHC simulations of the postshock dynamicsof strong shock-driven RMI. The superposition of the growth ofthe interface perturbation with the background motion of thefluids, the formation of large-scale coherent structure of bubblesand spikes, the bubble flattening at late times, and the occurrenceat small scales of interfacial vortical structures and bulk-immersedcumulative jets are seen in Fig. 5A. The flow regions are shown,with red (blue) for the light (heavy) fluid particles and green for thelight fluid interfacial particles in Fig. 5A.

Sensitivity of RMDynamics to Deterministic Conditions. In RMIwith a single-wave initial perturbation, the initial growth rate v0 de-pends on λ and a0, whereas the nonlinear dynamics retains memoryof the initial conditions (26–29). Fig. 5B shows the dependence ofthe initial growth rate on initial perturbation amplitude for givenðM,A, γhðlÞ, λÞ and for a0=λ∈ ½0.1, 1�. The data are confidently de-

scribed by the model ðv0=v∞Þ=A=C1ða0=λÞe−C1ða0=λÞ with C1 ≈ 4.26and C2 ≈ 2.63 (Fig. 5 B and C), suggesting that, in addition towavelength λ, RM dynamics has the characteristic amplitude scalea0 max ≈ 0.38λ,  ka0 max ≈ 2.4, at which the maximum initial growthrate is achieved, v0 max=v∞ ≈ 0.6A. The ratio of the RMI initial andlinear growth rates decays exponentially with the initial amplitude,v0=½v0�linear = e−ða0=a0 maxÞ (Fig. 5) (28).

For a multimode initial perturbation, the order and disorder inRM flow are highly sensitive to deterministic conditions, includingthe wavelengths, the amplitudes, and the relative phases of theinitial perturbation waves (29). Fig. 6 show snapshots of the flowregions of late stages of the RMI, with the same correspondingvalues of the Mach and Atwood numbers and the observationaltime. The initial perturbations have two waves with the sameamplitudes and wavelengths ðλ1ð2Þ, a1ð2ÞÞ. However, the flowkeeps order in some cases and is disordered in the others. Thisdifference is due to the relative phase φ and the interference ofwaves constituting the initial perturbation, as group theory finds(29). In Fig. 6, Left, the waves are antiphase, φ= π, and the flowsymmetry group is pm11; this flow keeps order (29, 34). In Fig. 6,Right, the waves have random phase, φ= π=2, and the flow sym-metry group is p1; this flow is disordered (29, 34). While at a first

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glance, the disordered RM flow might appear turbulent, a morecautious consideration is required. The disorder in the randomphase case in Fig. 6 is induced by initial conditions, suggestingdeterministic chaotic (rather than stochastic turbulent) dynamics(29, 31).

Small-Scale Structures in RM Flow. SPHC simulations accuratelycapture small-scale dynamics in strong shock-driven RMI. At theinterface, we observe small-scale shear-driven vortical structuresdue to the Kelvin–Helmholtz instability (26–29). In the bulk, be-tween the transmitted shock and the interface, we observe small-scale nonuniform localized structures of the flow fields (Fig. 7) (26).These include the “reverse” cumulative jets, which are short andenergetic and develop due to collisions of converging fluid flowsin the bulk (56, 60); the hot (cold) spots, which are the localizedregions with the temperature much higher (lower) than that of theambient; and the regions with high (low) pressure (26–29). Thesewell-pronounced small-scale structures are volumetric in nature(Fig. 7). They develop, since the flow dynamics is adjusted to themotion of the interface.

Hence, the strong shock-driven RM dynamics is sensitive todeterministic conditions, may keep order at large and smallscales, and may have localized nonuniform structures at smallscales in the bulk in addition to shear-driven vortical structures atthe interface.

DiscussionSupernovae and their remnants are a central problem in astro-physics due to their role in the formation of neutron stars and blackholes in the processes of stellar evolution and nuclear synthesis (Fig.1) (1). We have considered this multiphysics problem in a hydro-dynamic approximation, with supernova blast causing the devel-opment of RTI, RMI, and RT/RM interfacial mixing with variableacceleration (1–7). We have applied group theory (14, 34) to studyRTI/RMI and RT/RM mixing with variable acceleration, and identi-fied properties of linear, nonlinear, and mixing RT/RM dynamicsthat have not been discussed before (Figs. 2–7).

We have found that, for g∼ ta, the linear and nonlinear dy-namics is RT type and is acceleration driven for a> − 2; for a< − 2,the dynamics is RM type and is driven by the initial growth rate atearly stage and by the drag (dissipation) at late stages. RT–RMtransition occurs at a=−2 with varying of the accelerationstrength. For any a, the nonlinear regular asymptotic solutionsform a continuous family; this multiplicity is due to the interfacialshear, and the fastest stable solution is physically significant (Figs.2 and 3). The principal result of the group theory analysis is themultiscale character of the nonlinear RT/RM dynamics, to whichtwo macroscopic-length scales—spatial period and amplitude—contribute. This further leads to two distinct mechanisms of thedevelopment of RT/RMmixing: the growth of the wavelength andthe dominance of the amplitude, each resulting in the imbalanceof gain and loss of the rates of specific momentum (14, 34).

The mixing is RT type for a> acr and is RM type for a< acr. RT–RM transition occurs for a∼ acr by varying the acceleration ex-

ponent; the critical exponent acr =−2+ ð1+CÞ−1,  acr ∈ ð−2,−1Þ

depends on the flow drag (Fig. 4). For −2< a< acr, the asymptoticdynamics is RT type in the linear and nonlinear regimes and RMtype in the mixing regime: The acceleration triggers the instabilityand defines the linear and nonlinear dynamics, but it plays ef-fectively no role in the mixing regime. The mixing dynamics isnevertheless faster than the acceleration prescribes. The criticalexponent acr has the exponent values typical for blast waves.

For RT-type mixing with a> acr and length-scale L∼ ta+2, with thedecrease of the acceleration exponent, the dynamics changes itscharacter from superballistics to subdiffusion. For RM-type mixingwith a< acr and length-scale L∼ tacr+2, the dynamics is faster thanthe acceleration prescribes and is subdiffusive for C > 1. Fora> − 1 ða< − 1Þ, large velocities correspond to large (small) scales.

In blast wave-driven dynamics, the acceleration exponent canbe a∼ acr or a< acr (19–25). The blast wave-driven mixing can thusbe RT type with a∼ acr or RM type with a< acr. In either case, themixing has large velocities (velocity fluctuations) at small scales,and the dynamics is essentially subdiffusive. Subdiffusive pro-cesses are known to depend on deterministic conditions and havesmall-scale localizations (Figs. 5–7). RM dynamics is indeed sen-sitive to deterministic conditions, including the wavelengths, theamplitudes, the relative phase, and the interference of wavesconstituting the initial perturbation. In RM flows, localized non-uniform structures develop at small scales in the bulk in addition toshear-driven vortical structures at the interface. These volumetricstructures may include cumulative jets, hot and cold spots, and high-and low-pressure regions (26–29). They are well-pronounced andcause strong fluctuations of flow fields. Depending on deterministicconditions, RM dynamics may keep order at large and small scales.

What are potential outcomes of this hydrodynamics for su-pernova and nuclear synthesis (1)?

In blast wave-driven RT/RM mixing in supernovae, canonical tur-bulence is traditionally considered as the mechanism for energyaccumulation at small scales. According to our results, for theacceleration parameters typical for blast waves (19–25), superdiffusiveturbulence (31, 32) may be a challenge to implement in subdiffusiveRT/RMmixing. However, the conditions of heterogeneity, nonlocality,anisotropy, and statistical unsteadiness that are common for RT/RMflows (14, 15, 33, 34) may lead to appearance of small-scale non-uniform structures in the bulk of the blast wave-driven mixing flows,whereas the slow subdiffusive transport may result in energy locali-zation and trapping at small scales (26–29, 59). These effects areconsistent with and may explain the richness of structures observed insupernovae, including the Cas A (Fig. 1) (1). We suggest that sucheffects be considered in interpretation of observational data.

Our work focuses on hydrodynamic aspects of the multiphysicsproblem of supernovae and their remnants and serves for sys-tematic studies of the problem in perspective.

AcknowledgmentsWe thank the University of Western Australia, the National Science Foundation,and the Theoretical Astrophysics Program and the Steward Observatory at theUniversity of Arizona for support.

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