Supernova, Nuclear Synthesis, Fluid Instabilities, and Interfacial Mixing

SPECIAL FEATURE: PERSPECTIVE

Supernova,nuclearsynthesis,fluidinstabilities,and interfacial mixing SPECIAL FEATURE: PERSPECTIVE Snezhana 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 review December 31, 2017)

Supernovae and their remnants are a central problem in astrophysics due to their role in the stellar evolution and nuclear synthesis. A supernova’s explosion is driven by a blast wave causing the development of Rayleigh–Taylor and Richtmyer–Meshkov instabilities and leading to intensive interfacial mixing of materials of a progenitor star. Rayleigh–Taylor and Richtmyer–Meshkov mixing breaks spherical symmetry of a star and provides conditions for synthesis of heavy mass elements in addition to light mass elements synthesized in the star before its explosion. By focusing on hydrodynamic aspects of the problem, we apply group theory analysis to identify the properties of Rayleigh–Taylor and Richtmyer–Meshkov dynamics 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). Reynolds numbers, and therefore, plasma motion in They have been a central problem in astrophysics since stars is expected to be turbulent (or at least, disor- their discovery and identification in the 1930s. The de- dered) (4–7). This may break the spherical symmetry in bris ejected from a supernova mixes with the interstellar detail but might be approximately valid on the global medium, forming a supernova remnant (SNR) (2–4). scale. However, as explosion begins, spherical sym- Young SNRs still retain information concerning the ex- metry is firmly broken on the global scale too, so that plosion process. Explosions are initial value problems; conventional stellar evolution theory must fail (1, 5). solution requires the details of what explodes (1–5). How this happens is of considerable interest (1–7). Some supernovae produce neutron stars (pulsars), and Astrophysics and Fluid Dynamics others produce black holes. They are thought to be Our understanding of stellar evolution is based on the the major source of galactic cosmic rays. Supernovae convenient assumption that stars may be treated as are the dominant source of elements not produced by spherically symmetric objects, at least on average (1). the Big Bang (those being hydrogen and helium). The This approximation, which has proven to be surpris- calcium in our bones and the iron in our blood were ingly successful, allows us to evolve stars up to a synthesized in a supernova as were silver, gold, ura- presumed presupernova condition (1–3). We may then nium, and thorium. The latter elements can also be invent an explosion model and compare the predic- produced in neutron star mergers in the “r-process,” tions with observed SNR. Unfortunately, this is a long evidenced in the mergers’ observations and the abun- extrapolation, with many untested choices along the dances in dwarf galaxies (1–11). Even carbon and nitro- way (1). The conditions in stars imply very high 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, Carnegie Mellon University, Pittsburgh, PA 15213; cStellingwerf Consulting, Huntsville, AL 35803; and dThe Steward Observatory, The University of Arizona, Tucson, AZ 85721 Author 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 and mixing, 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 project connecting the observational data, supernova, and nuclear synthesis to fluid instabilities and mixing; and S.I.A., A.K.B., A.N., A.P., R.F.S., and W.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: [email protected] or [email protected]. This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1714502115/-/DCSupplemental.

www.pnas.org/cgi/doi/10.1073/pnas.1714502115 PNAS Latest Articles | 1of9 Downloaded by guest on September 24, 2021 stars that die less violently as are the “s-process” heavy elements. RTI/RMI and RT/RM mixing occur at a broad range of astro- The details are still in debate. Supernovae are a major source of physical phenomena in low- and high-energy density regimes (1, 14). radioactive nuclei for meteorites and for the early solar system (1–7). Examples include the appearance of stiff light years-long structures Fig. 1 provides a detailed look at the Cassiopeia A (Cas A) in molecular hydrogen clouds, the formation of accretion disks and SNRs, the youngest nearby SNR known in the Milky Way. The black holes, and the processes of stellar evolution (1–7). The latter SNRs of the Cas A have been produced by the explosion of a ranges from a birth of a star due to the interstellar gas collapse to life massive star. The image in Fig. 1 is combined from 18 images of a star with the extensive material mixing in the stellar interior and taken by NASA’s Hubble Space Telescope in 2004. It shows the to death of a star in the supernova (1–7, 14, 15). In supernovae, the Cas A remnants as a broken shell of filamentary and clumpy stellar blast wave-driven RT/RM mixing of the outer and inner layers of the ejecta glowing with the heat generated by the passage of a shock progenitor star creates conditions for synthesis of heavy and in- wave from the supernova blast. The various colors of the gas in- termediate mass elements in addition to light mass elements syn- – – dicate differences in chemical composition. Bright green filaments thesized in the star before its explosion (1 7, 14, 15, 19 25). are rich in the oxygen, red and purple ones are rich in the sulfur, In everyday life and in extreme astrophysics environments, RT and blue ones are composed mostly of the hydrogen and the ni- flows are observed to have similar qualitative features of their – trogen. The oxygen and the sulfur are produced by thermonuclear evolution (12 16). RTI starts to develop when the fluid interface is – burning during and just before the explosion. The sulfur is formed slightly perturbed near its equilibrium state (12 14). The flow transits from an initial stage, where the perturbation amplitude by the oxygen nuclear burning. Some of the sulfur lies farther from grows quickly, to a nonlinear stage, where the growth rate slows the center than some of the oxygen, which is interpreted as due to and the interface is transformed into a composition of small-scale Rayleigh–Taylor instabilities (RTIs) during the explosion (12–14). shear-driven vortical structures and a large-scale coherent structure Hence, the key question is: can we work backward from the SNR of bubbles and spikes [with the bubble (spike) being the portion of toward the underlying explosion to provide insight into the event the light (heavy) fluid penetrating the heavy (light) fluid]. RT flows that is independent of conventional stellar evolution theory? are usually three-dimensional (3D) and have two macroscopic- To do so, we have to better understand fluid dynamic aspects length scales—the amplitude in the acceleration direction and of the multiphysics problem of supernovae, particularly Rayleigh- the spatial period or the wavelength in the normal plane. The flow Taylor instability, Richtmyer–Meshkov instability (RMI), and Rayleigh– eventual stage is the self-similar interfacial mixing (12–15). Taylor (RT)/Richtmyer–Meshkov (RM) interfacial mixing that are While RT and RM flows are similar in many regards, there are ’ – caused by supernova sblast(1,12 16). RTI develops at the in- also important distinctions (12–18, 26–29). For instance, postshock terface of the fluids with distinct densities that are accelerated RM dynamics is a superposition of two motions (17, 18, 26–29). – against their density gradients (12 14). RMI develops when the These are the background motion of the fluid bulk and the growth acceleration is induced by a shock refracting a perturbed interface of interface perturbations (26–29). In the background motion, both of the fluids with distinct acoustic impedances (17, 18). Intense in- fluids and their interface move as whole unit in the transmitted – terfacial RT/RM mixing of the fluids ensues with time (12 18). shock direction; this motion occurs even for an ideally planar interface, and is supersonic for strong shocks. The growth of the interface perturbations is due to impulsive acceleration by the shock; it develops only for a perturbed interface (26–29). The rate of this growth is subsonic, and the associated motion is incompressible. The growth rate is constant initially and decays with time later. RM unstable interface is transformed to a composition of a large-scale structure of bubbles and spikes and small-scale shear-driven vortical structures. Small-scale nonuniform structures also appear in the bulk (26–29). Self-similar RM mixing develops, and the energy supplied initially by the shock gradually dissipates (17, 18, 26–29). In RT mixing with constant acceleration, the length scale, the velocity scale, and the Reynolds number increase with time (14– 16). At a first glance, such flow should quickly proceed to a fully disordered turbulent state (30). Turbulence is a stochastic process insensitive to deterministic conditions with intense energy transport from the large to the small scales (30–32). In supernova environments, such transport may supply to small scales some activation energy required for the synthesis of heavy mass ele- Fig. 1. An image taken with NASA’s Hubble Space Telescope provides ments (1). Recent advances in the theory and experiment of RTI a detailed look at the remains of a supernova explosion known as Cas and RT mixing have found, however, that, in the high- and low- A. It is the youngest such remnant known in the Milky Way. The image energy density regimes, the properties of heterogeneous, aniso- is made from 18 separate images taken in December 2004 by using tropic, nonlocal, and statistically unsteady RT mixing depart from Hubble’s Advanced Camera for Surveys. It shows the Cas A remnant as a broken ring of bright filamentary and clumpy stellar ejecta. These those of homogeneous, isotropic, local, and statistically steady huge swirls of debris glow with the heat generated by the passage of a canonical turbulence (14–16, 30–34). High Reynolds number, shock wave from the supernova blast. The various colors of the gas while necessary, is not a sufficient condition for turbulence to indicate differences in chemical composition. Image courtesy of NASA/ occur. RT mixing exhibits more order and has stronger correla- ESA/Hubble Heritage (STScI/AURA)-ESA/Hubble Collaboration, Frank tions, weaker fluctuations, and stronger sensitivity to deterministic Summers (Space Telescope Science Institute, Baltimore), Robert Fesen – (Dartmouth College, Hanover, NH), and J. Long (ESA/Hubble, conditions compared with canonical turbulence (14 16). For RM Garching bei München, Germany). mixing, strong sensitivity to deterministic conditions has also been

2of9 | www.pnas.org/cgi/doi/10.1073/pnas.1714502115 Abarzhi et al. Downloaded by guest on September 24, 2021 found along with its nominally large Reynolds number and small- scale interfacial vortical structures (26–29). In supernovae in a hydrodynamic approximation, accelerations are induced by blast waves (1). Blast waves can be viewed as strong variable shocks (1, 19–25, 33). Blast wave dynamics is self-similar, and the blast wave-induced acceleration is a power law function of time (spatial coordinate) (1, 19–25, 33). Several questions thus appear. What are the properties of RTI with variable acceleration, and how do they differ from those of RTI with constant acceleration and from those of shock-driven RMI? How can these properties be applied for interpretation of astrophysical data? What are their AB potential outcomes for stellar evolution and nuclear synthesis? Fig. 3. Qualitative velocity field in laboratory reference frame In this work, we consider hydrodynamic aspects on the multi- near the bubble tip for incompressible immiscible ideal fluids at physics problem of supernovae and their remnants. We focus on some Atwood number for a nonlinear solution in RT/RM family in the dynamics of RTI and RT mixing with variable acceleration in a (A)thevolumeand(B) the plane, with the interface marked by a broad parameter regime (Figs. 2–7). The acceleration is a power dashed curve. law function of time (spatial coordinate). We find that, for variable acceleration, RT/RM dynamics is multiscale and has two illustrated by numerical simulations of strong shock-driven RMI — macroscopic-length scales the amplitude in the acceleration (17, 18, 26–29, 36). Such effects should be considered in the in- direction and the spatial period in the normal plane (14–16, 34). terpretation of observational data (1). Technical details of this Depending on the exponent of the acceleration power law, the work are given in SI Appendix for corresponding sections. dynamics can be RT type or RM type. For RT type, the acceleration sets the timescale at early stage and defines the nonlinear dy- Fluid Instabilities and Interfacial Mixing namics and the interfacial mixing at later stages. For RM type, the Governing Equations. Dynamics of ideal fluids is governed by initial growth rate sets the timescale at early stage; at late stages, the conservation of mass, momentum, and energy: the drag defines the nonlinear dynamics and the interfacial mix- X 3 ing, and the initially supplied energy gradually dissipates (33, 35). ∂ρ=∂t + ∂ρv =∂x = 0, ∂ρv =∂t + ∂ρv v ∂x + ∂P=∂x = 0, i i i j=1 i j j i The critical values of the exponent at which the transition occurs ∂E=∂t + ∂ðE + PÞv =∂x = 0, from RT- to RM-type dynamics are distinct for the linear, nonlinear, i i and mixing regimes. Particularly for blast wave-induced acceler- = ations, the linear and nonlinear dynamics are RT type, and the where xi are the spatial coordinates with ðx1, x2, x3Þ ðx, y, zÞ; t ρ ρ mixing is RM type, but RM-type mixing develops quicker than the is time; ð , v, P, EÞ are the fields of density , velocity v, pres- = ρ + 2= acceleration prescribes (33, 35). While for subdiffusive mixing sure P, and energy E ðe v 2Þ; and e is the specific internal dynamics, superdiffusive canonical turbulence may be a challenge energy (25). The latter refers to energy per unit mass con- to occur, other mechanisms are possible for energy accumulation tained within a system, excluding the kinetic and the potential at small scales. They are due to small-scale nonuniform structures energy of the system as a whole (25). For immiscible fluids, the developing in the fluid bulk and including cumulative jets, hot and fluxes of mass, momentum, and energy obey the boundary conditions at the interface cold spots, high- and low-pressure regions, and localizations as ½v · n = 0, ½P = 0, ½v · τ = any, ½W = any,

ABwhere ½... denotes the jump of functions across the interface; n and τ are the normal and tangential unit vectors of the interface with n = ∇θ=j∇θj and ðn · τÞ = 0; and θ = θðx, y, z, tÞ is a local 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 spatially extended system, the flow can be periodic in the plane ðx, yÞ normal to the z direction of gravity g and has no mass sources:

v = 0, v = 0. CD jz→+∞ jz→−∞

Acceleration g, jgj = g, is directed from the heavy to the light fluid. Initial conditions include initial perturbations of the flow fields (12–18, 25–29, 34). For ideal fluids, the initial conditions set the length-scale λ and the timescale τ. Here,pffiffiffiffiffiffiffiffiffiffiλ is the perturbation wavelength (spatial period), and τ ∼ λ=g0 in case of constant acceleration g0. In realistic fluids, small scales are usually stabilized, and a characteristic scale λm Fig. 2. One parameter family of regular asymptotic solution for 3D 2 1=3 corresponds to a fastest growing mode [i.e., λm ∼ ðν =g0Þ , flow with group p6mm at some Atwood numbers. RT/RM-type ν – ’ nonlinear dynamics: the bubble velocity vs. (A and C) the bubble where is the kinematic viscosity] (37 39). The ratio of the fluids curvature and (B and D) the interfacial shear. densities and the density jump at the interface is parameterized

Abarzhi et al. PNAS Latest Articles | 3of9 Downloaded by guest on September 24, 2021 interface [i.e., the bubble tip ð0, 0, z0ðtÞÞ]. Governing equations are then reduced to a dynamical system in terms of surface vari- ables and moments, each of which is an infinite sum of weighted Fourier amplitudes; the system solution is sought (34, 46, 48–50). For group p6mm, to the first order N = 1, the interface is 2 2 z* − z0ðtÞ = ζðx + y Þ, and the dynamical system is À Á ~ M = −M = −v, ρ ζ_ − 2ζM − M =4 = 0, AB 0 0 .h 1 2 ρ ζ_ − ζ ~ + ~ = − ~ = l 2 M1 M2 4 0, M1 M1 any, Fig. 4. Asymptotic solutions for RT mixing with variable acceleration. 0 1 , The mixing is (A) RT type and (B) RM type with time- and space- À Á _ _ varying accelerations. ρ _ + ζ _ − 2 + ζ = ρ @ ~ − ζ ~ − ~ 2 + ζ A h M1 4 M0 M1 8 g l M1 4 M0 M1 8 g .

= ρ − ρ = ρ + ρ < < → by the Atwood number A ð h lÞ ð h lÞ,0 A 0; A 1 ρ =ρ → ∞ → ρ =ρ → – for h l and A 0for h l 1(12 18). Here, v ≥ 0andζ ≤ 0 are the bubble velocity and curvature; ~ To rigorously describe RT dynamics, one has to solve the MðMÞ are the heavy (light) fluid moments. Group theory is fur- problem of extreme complexity: solve a system of nonlinear partial ther applied to solve the closure problem, find regular asymp- differential equations in 4D space–time, solve the boundary value totic solutions forming a continuous family, study the solutions’ problems for a subset of nonlinear partial differential equations at a stability, elaborate properties of nonlinear RTI (14, 34, 48–50). nonlinear freely evolving interface and at the outside boundaries Group theory for the mixing dynamics finds that, in RT mixing, and also solve the ill-posed initial value problem, with account for the momentum and energy are gained and lost at any scale; the singularities and secondary instabilities developing in a finite time dynamics of a parcel of fluid is governed by a balance per unit (14, 34). A complete theory of RTI applicable at all scales and all mass the rates of momentum gain, μ~, and momentum loss, μ,as times has yet to be developed. Rigorous theories have successfully handled the problem in well-defined approximations; empirical _ h = v, v_ = μ~ − μ, models have repeatedly described a broad set of data with nearly the same set of parameters. The reader is referred to review and where h is the length scale along the acceleration g, v is the research papers (12–18, 26–30, 33–54) for details of theoretical and corresponding velocity, and μ~ðμÞ is the magnitude of the rate numerical studies of RT dynamics. of gain (loss) of specific momentum in the acceleration direc- It is worth noting that, despite complexity and noisiness tion (14, 16, 54). The rate of gain (loss) of specific momentum is resulting from interactions of all of the scales, RT dynamics is μ~ = ~«=v (μ = «=v), with ~«ð«Þ being the rate of gain (loss) of spe- observed to have certain features of universality and order, and is cific energy. The rate of energy gain is ~« = fgv, f = f ðAÞ, thus eligible to first principle considerations, such as group theory rescaled g f → g hereafter. The rate of energy dissipation is (14, 15, 34, 55, 56). For linear and nonlinear RTI, group theory « = Cv3=L, with a length scale L and a drag C, C ∈ ð0, ∞Þ (14– analysis uses theory of discrete groups to solve the boundary 16, 35, 54). Momentum model has the same symmetries and value and initial value problems (34, 48–50). For RT mixing, group scaling transformations as the governing equations (14, 16, 34, theory is implemented in the momentum model with equations 54). It can be solved by applying the Lie groups. The cases that have the same symmetries and scaling transformations as the L ∼ λ and L ∼ h correspond to the nonlinear dynamics and the governing equations (14, 16, 34, 40, 54). Some principal results of self-similar mixing (14–16, 34, 54). In each case, asymptotically, group theory analysis—the multiscale character of nonlinear dy- there is the particular solution of RT type ðha, vaÞ and the namics, to which both the spatial period λ and the amplitude h contribute; the tendency to keep isotropy in the plane normal to the 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 nonlinear dynamics studies RT flows that consist of large-scale coherent = ∇Φ B structures in which dynamics is potential, vhðlÞ hðlÞ, and shear- driven interfacial vortical structures are small (34). For a spatially extended system, coherent structure is periodic in the plane normal to the acceleration direction. It is invariant with respect to a discrete group G with generators that are translations in the plane, rotations, and reflections (34). To be structurally stable, coherent dynamics must be invariant under one of the spatial groups with A C the inversion in the plane, such as the groups of hexagon p6mm, square p4mm, rectangle p2mm in 3D, and group pm11 in 2D (34, Fig. 5. Evolution of strong shock-driven RMI. (A) Snapshots of the 48). By applying irreducible representations of the group, the flow regions at some time instances. Dependence of RM initial Φ growth rate on the initial perturbation amplitude: (B) the growth rate potential hðlÞ is expanded as a Fourier series, and spatial ex- data and (C) the data fit by the model. Reprinted from ref. 29, with pansion is further made in a vicinity of a regular point of the the permission of AIP Publishing.

4of9 | www.pnas.org/cgi/doi/10.1073/pnas.1714502115 Abarzhi et al. Downloaded by guest on September 24, 2021 prefactor, G > 0, with dimensions ½G = m=s2+a and ½a = 1. For a given wavelength (period) λ and for a ∈ ð−∞, −2Þ ∪ ð−2, +∞Þ, − = + − there are two timescales τ = ðkGÞ 1 ða 2Þ and τ = ðkv Þ 1, where k G pffiffiffi 0 0 is the wave vector with k = 4π=λ 3 for 3D flow with group p6mm and v0 is the initial growth rate set by the initial conditions and/or by −1 the impulsive acceleration. At a = −2, the timescale is τ0 = ðkv0Þ , and value Gk parameterizes the acceleration strength. Time is t > t0 > 0, t0 >> fτG, τ0g. At early stage, for a > − 2, the timescale is τG, and the linear dynamics 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 −1 growth rate and is RM type. At a = −2, the timescale is τ0 = ðkv0Þ , and the linear dynamics changes its character from RT to RM type Fig. 6. Wave interference and order–disorder in RM flow. Snapshots with the decrease of Gk. of the flow regions at some time instances for two-wave initial At late stage, for a > − 2, the nonlinear dynamics is RT type; perturbations, with the same waves being in (Left) antiphase and ζ ∼ (Right) random phase. regular asymptotic solutions depend on time as k and v, M, ~ a=2 M ∼ t for t >> τG. For a < − 2, the nonlinear dynamics is RM type; regular asymptotic solutions depend on time as ζ ∼ k and v, M, ~ −1 homogeneous solution of RM type ðhd, vdÞ (35). These solutions M ∼ t for t >> τ0.Ata = −2, regular asymptotic solutions are ζ ∼ k are effectively decoupled due to their distinct symmetries. ~ −1 and v, M, M ∼ t for t >> τ0, and they are RT (RM) type for Gk >> 1 (Gk << 1). These regular asymptotic solutions form a family (34). Smoothed Particle Hydrodynamics Simulations. Supernova For a > − 2 and at a = −2 with Gk >> 1, for regular asymptotic environments are characterized by the conditions of high energy solutions of RT-type nonlinear dynamics, the bubble velocity v ≥ 0 density, strong shocks, sharp changes of flow fields, large per- depends on its curvature ζ, ζ < 0, as turbations, and small effects of dissipation and diffusion. Nu- .pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi merical modeling of these extreme regimes is a challenge. This is v taG=k = ð−2Aðζ=kÞÞ 9 − 64ðζ=kÞ2 because numerical methods should satisfy numerous competing rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi −1 requirements to capture shocks, track interfaces, and accurately 2 × −48ðζ=kÞ + A 9 + 64ðζ=kÞ . account for dissipative processes. An efficient approach for modeling strong shock-driven dynamics is the smoothed particle ζ ∈ ζ ζ = − = hydrodynamics (SPH) implemented in the smoothed particle hy- For every A, this function domain is ð cr ,0Þ, cr ð3 8Þk, drodynamics code (SPHC) (36). and the range is v ∈ ð0, vmaxÞ, with v = 0 achieved at ζ = 0 and ζ = ζ = ζ = ζ ζ ∈ ζ The SPH is a Lagrangian method representing a continuous at cr and with v vmax achieved at max, max ð cr ,0Þ. fluid by means of fixed mass SPH particles and thus, reducing the The multiplicity of the nonlinear solutions is associated with the governing partial differential equations to ordinary differential nonlocal and singular character of the interfacial dynamics (34, 45, equations (26–29, 36). SPHC has been originated in astrophysics, 46, 48–50). The solutions exist and converge with increase in ap- and in addition to astrophysical gravitational problems, it has proximation order. The number of the family parameters is identified been successfully applied, tested, and validated in multiphysics by symmetry of the global flow (34). For group p6mm, the dynamics 2 2 problems in fluids, plasmas, materials for modeling the strong is highly isotropic, z* − z0 ∼ ζðx + y Þ, and the interface morphology shock-driven RMI, the Noh problem, and other flow problems as is captured by the principal curvature ζ (Fig. 2) (34, 48–50). well as reactive and supercritical fluids, material transformation The multiplicity is also due to the presence of shear at the under impact, ablation process in hypersonic flows, and charge interface (58). Defining the shear as the spatial derivative of imbalance in plasmas (36). When applied to strong shock-driven RMI, it has achieved an excellent agreement with the experiments and with the rigorous zero-order, linear, and group theories. The latter includes the multiscale character of the nonlinear dynamics and is evinced, for instance, in the flattening of RM bubbles, flow fields structure, and sensitivity to deterministic conditions (26–29, 36). Here, we apply group theory to study RTI/RMI and RT/RM mixing with variable acceleration in a broad parameter regime. We further conduct SPHC simulations of strong shocks-driven RM flows to study their sensitivity to deterministic conditions and their small-scale dynamics at the interface and in the bulk. We find the properties of blast wave-induced RT/RM dynamics and propose the mechanisms for energy accumulation and transport at small scales in supernovae.

RTI with Variable Acceleration Fig. 7. Small-scale nonuniform structures in the fields of temperature (Left) and pressure (Right) in RM flow, including bulk-immersed = a We consider RT dynamics with time-varying acceleration g Gt , cumulative jets, hot (cold) spots, and high- (low-) pressure regions. where a is the acceleration exponent, a ∈ ð−∞, +∞Þ, and G is Reprinted from ref. 27, with permission of AIP Publishing.

Abarzhi et al. PNAS Latest Articles | 5of9 Downloaded by guest on September 24, 2021 the jump of tangential velocity at the interface, Γ = Γ ,with Mixing Dynamics Ä Å xðyÞ ~ RT/RM Dynamics for Time-Varying Acceleration. In the non- Γ = ∂ v ∂xðyÞ, we find shear Γ = M1 − M1 near the bubble xðyÞ xðyÞ ∼ λ tip and its dependence on the interface curvature ζ. For linear regime with L , asymptotic solutions for the momentum model are consistent with group theory results. In the mixing re- ζ ∈ ð−ζ ,0Þ, shear Γ is 1–1 function on ζ, Γ ∈ ðΓ , Γ Þ achieving cr min max ∼ Γ ζ = Γ ζ = ζ gime with L h, asymptotic solutions for the momentum model value min at 0 and value max at cr. + + = 2 a = 2 acr The fastest stable solution is the physically significant solution (Fig. are ha Bat and hd Bd t , where the critical exponent is = − + + −1 → − → → − 2) (14, 34, 48–50, 58). For the fastest stable solutions, the de- acr 2 ð1 CÞ with acr 1 for C 0 and acr 2 for ζ C → ∞ (33). For solution ðh , v Þ, the exponent is set by the ac- pendence of curvature and velocity ð max, vmaxÞ on the Atwood a a = a ζ = ζ celeration’s exponent, ð2 + aÞ, and the prefactor B is set by the number A is complex: vmax vmaxðGt , k, AÞ, max maxðk, AÞ (Fig. a 2 = a = ζ = 3 = acceleration parameters and the drag. This mixing is RT type; it is 2). However, there is the invariant v ððt G kÞð8j maxj kÞ Þ 1, max driven by the acceleration. For solution ðh , v Þ, the exponent is implying that the nonlinear dynamics is characterized by the contri- d d + = + −1 bution of the wavelength λ, the amplitude h, and their derivatives and set by the drag, ð2 acr Þ ð1 CÞ , and the prefactor Bd is set by is, thus, multiscale (34, 48–50). the initial conditions. This mixing is RM type; it is driven by the μ = «= > Regular asymptotic solutions have important physics proper- drag (the dissipation, since v) (35). For a acr the mixing is < ∼ ties. For these solutions, there is effectively no motion of the fluids RT type. For a acr, the mixing is RM type. At a acr, a transition in the bulk and away from the interface, there is intense motion occurs from RT- to RM-type mixing (33, 35). near the interface, and shear is present at the interface leading to RT-Type Mixing. Properties of the asymptotic solutions indicate formation of interfacial vortical structures, in agreement with ob- that, for a > a , when the dynamics is the acceleration driven in servations (Fig. 3) (34, 48–50). cr the nonlinear and mixing regimes, two states are possible. One Regular asymptotic solutions have important global proper- state is achieved for L ∼ λ; it is the state with asymptotic balance of ties. The flow tends to conserve isotropy in the plane (34, 48–50). the rates of momentum, jμj ∼ jμ~j ∼ ta, and with jv_=μj → 0 (14, 33, The 3D highly symmetric dynamics is universal (34, 48). That is, on 54). The other is achieved for L ∼ h; it is the state with the substitution k = 2π=λ, the nonlinear solutions describe the jμ~j ∼ jμj ∼ jv_j ∼ ta and with an algebraic imbalance of the rates of dynamics of 3D flow with group p4mm. For 3D low-symmetric momentum, μ~ ≠ μ (14, 33). Per observations, at a = 0, the imbal- flows with group p2mm, there is a two-parameter family of reg- ance is small, ðμ~ − μÞ=μ~ << 1 (51–53). Hence, RT mixing may de- ular asymptotic solutions; among the family solutions, only nearly velop when the amplitude h is the scale for energy dissipation, isotropic bubbles are stable. The dimensional 3D–2D cross-over is L ∼ h (14, 33, 35). It may also develop due to the growth of period discontinuous (34, 48). λ ∼ Gta+2 in the nonlinear regime. Because the dynamics is mul- At a = −2 with Gk << 1 and for a > − 2, for regular asymptotic tiscale, the growth of the period λ is possible and is not a neces- solutions of RM-type nonlinear dynamics, the bubble velocity sary condition for the mixing to occur (14, 29, 33, 35, 54). v ≥ 0 depends on the bubble curvature ζ , ζ < 0, as For a > a , in RT-type mixing, the length scales with time as cr a+2 a+1 2 2 L ∼ t , and the velocity scales as v ∼ t . The length scale in- vðktÞ = 3 − 2Aðζ=kÞ −5 + 64ðζ=kÞ 9 − 64ðζ=kÞ creases with time for a > acr. The velocity scale increases for > − = − < − a 1, is constant for a 1, and decreases for a 1. Recall −1 2 ∼ 1=2 × −48ðζ=kÞ + A 9 + 64ðζ=kÞ . that diffusion scaling law is L t , whereas canonical turbulence is superdiffusive with L ∼ t3=2 (25, 31, 32, 59). By comparing these exponents, we find that, in RT-type mixing with a > a , the dy- The function domain is ζ ∈ ðζ ,0Þ, ζ = −ð3=8Þk, and the cr cr cr ∼ a+2 > = range is v ∈ ðv , v Þ, with v = v achieved at ζ = ζ and namics L t is super ballistics for a 0, ballistics at a 0, steady min max min cr = − > − = v = v achieved at ζ = 0. For ζ ∈ ðζ ,0Þ, ζ ≥ ζ , shear Γ at the flex point a 1, superdiffusion for a 3 2, quasidiffu- max min min cr = − = − = > > is 1–1 function on ζ, Γ ∈ ðΓ , Γ Þ, with Γ = Γ at ζ = ζ sion at a 3 2, and subdiffusion for 3 2 a acr. Large ve- min max min min > − and Γ = Γ at ζ = 0. The fastest stable solution is the physi- locities correspond to large (small) length scales for a 1 max < < − cally significant solution (Fig. 2) (14, 34, 48–50, 58). (acr a 1). RT and RM families have similar physical, mathematical, and < global properties (Figs. 2 and 3). These include the structure of the RM-Type Mixing. RM-type mixing develops for a acr; its rates of gain and loss of momentum are asymptotically imbalanced velocity fields, the existence of the family of solutions, their de- jμ~=μj → 0, whereas jμj ∼ jv_j ∼ tacr (33, 35). RM-type mixing may pendence on the flow symmetry and the interfacial shear, the develop when the amplitude h is the scale for energy dissipation, tendency to keep isotropy in the plane normal to the acceleration, ∼ λ – L h. It may also develop when the period grows with time, and the discontinuity of the 3D 2D cross-over. However, their + + − + as λ ∼ tðacr 2Þ ðacr aÞ λ ∼ tðacr 2Þ for −2 < a < a a < − 2 .InRM- local properties are distinct (Fig. 2) (34, 50). Particularly, for RM ð Þ cr ð Þ type mixing with a < acr, the length scale increases with time, bubbles, the velocity is a monotone function on the curvature, and + + ∼ ðacr 2Þ ∼ ðacr 1Þ the fastest stable solution corresponds to a flat bubble with L t ,whereasthevelocityscalev t decreases with = = ζ = time, and large velocities correspond to small length scales, since vmaxðAktÞ 3 1, max 0. The quasi invariant of this solution ðacr + 2Þ ∈ ð−1,0Þ for C > 0. 2 2 −1 ð4=3Þtv = ðdv=dζÞjζ=ζ = ð1 + ð5=2ÞðA=2Þ Þ ≈ 1 implies that max max the wavelength and the amplitude both contribute to the non- Space-Varying Acceleration. Similar analyses can be conducted linear dynamics. A steep dependence of the velocity on the shear when the acceleration is a power law function on the spatial co- for physically significant solutions in RT/RM families suggests the ordinate, g ∼ hn. Particularly, the nonlinear dynamics is RT type for use of highly accurate methods with the interface tracking for n ∈ ð−∞,2Þ with v ∼ tn=ð2−nÞ, and RM type for n → −∞with v ∼ t−1. numerical modeling RT/RM dynamics (Figs. 2 and 3). For n → −∞, the nonlinear dynamics changes from RT to RM type.

6of9 | www.pnas.org/cgi/doi/10.1073/pnas.1714502115 Abarzhi et al. Downloaded by guest on September 24, 2021 Solutions for the nonlinear dynamics with time- and space-varying ac- to the slow dynamics. If the spot is hot enough, it may initiate a celerations can be transformed one into another with α → 2n=ð2 − nÞ. nuclear reaction accompanied by some energy release and may 2=ð1−nÞ – The mixing dynamics is RT type for n ∈ ðncr ,1Þ with h ∼ t , further induce a chain of other reactions and processes (1 3). Blast = − 2 ð1 ncr Þ waves are the special strong shocks (24). In strong shock-driven and RM type for n ∈ ð−∞, ncr Þ with h ∼ t . The critical ex- RMI, the shock–interface interaction may lead to intense pro- ponent is ncr = − 2C − 1, with ncr → − 1 for C → 0 and ncr → −∞ for C → + ∞. Solutions for the mixing dynamics with time- and duction of small-scale nonuniform structures in the bulk in addi- space-varying accelerations can be transformed one into another tion to small-scale shear-driven vortical structure at the interface – with α → 2n=ð1 − nÞ. (26 29). These nonuniform structures may include cumulative jets, Fig. 4 illustrates RT/RM-type mixing for various values of the hot and cold spots, high- and low-pressure regions, and may exponents a and n. Solutions are derived for C being a stochastic enable strong energy fluctuations at small scales. process with log-normal distribution with the mean hCi = 3.6 and Strong Shock-Driven RM Flows the SD σ = hCi=2 leading to acr ≈−1.78 ðncr ≈−8.2Þ. Mean values of quantities h, v, jv_j, g are plotted in red, blue, green, and black, SPH Simulations. To illustrate scale coupling in strong shock- respectively, in Fig. 4 (35). driven RMI, we use the SPHC in a hydrodynamic approximation γ = = for ideal monoatomic gases with adiabatic indexes hðlÞ 5 3 and = Blast Wave-Induced Mixing. Consider now RT/RM mixing in- the Atwood number A f0.3, 0.6, 0.7, 0. 8, 0. 95g. They have high = duced by blast waves with the first kind (Sedov–Taylor) self- energy per atom. The shock Mach number is M {3,5,7,10} de- – similarity and the second kind (Guderley–Stanyukovich–Landau) fined relatively to the light fluid with the speed of sound cl (26 29). The interface is normal to the shock; the initial perturbation self-similarity (19–25). Note that exponents acr ðncr Þ have values λ typical for blast waves. wavelength and amplitude are and a0. In SPHC, we scale the λ λ= γ For Sedov–Taylor self-similar dynamics, the scaling dependence length, velocity, and time with , v∞, v∞, where v∞ðM, A, hðlÞÞ is of the solution is set by the energy release Ε, ½Ε = kgðm=sÞ2,andthe the background motion velocity value; it is supersonic. RM initial growth rate v ðM, A, γ , λ, a Þ is subsonic. fluid density ρ, ½ρ = kg=m3ð2,1Þ,in3Dð2D,1DÞ in case of point (line, 0 hðlÞ 0 plane) energy source (19, 20, 24). The invariance of Fig. 5A illustrates SPHC simulations of the postshock dynamics energy density ðΕ=ρÞ leads to scaling laws for the length of strong shock-driven RMI. The superposition of the growth of = = = − = − = − = the interface perturbation with the background motion of the ∼ t2 5ðt1 2, t2 3Þ, velocity ∼ t 3 5ðt 1 2, t 1 3Þ, and acceleration fluids, the formation of large-scale coherent structure of bubbles ∼ t−8=5ðt−3=2, t−4=3Þ. This suggests that the blast wave dynamics is and spikes, the bubble flattening at late times, and the occurrence substantially slower than canonical turbulence (31, 32). By com- at small scales of interfacial vortical structures and bulk-immersed paring the blast wave acceleration exponent with the value a ,we cr cumulative jets are seen in Fig. 5A. The flow regions are shown, find that this mixing can be RT type for small drag C < 3=2 1, 1=2 ð Þ with red (blue) for the light (heavy) fluid particles and green for the > = = and RM type for large drag C 3 2ð1, 1 2Þ. light fluid interfacial particles in Fig. 5A. For Guderley–Stanyukovich–Landau self-similar dynamics, the load history and momentum transport should be accounted for. Sensitivity of RM Dynamics to Deterministic Conditions. In RMI Ε ρ The corresponding invariant Fð , P, Þ is a function of the energy with a single-wave initial perturbation, the initial growth rate v de- Ε ρ = = θ 0 release , pressure P, and fluid density , with ½F m s and pends on λ and a , whereas the nonlinear dynamics retains memory < θ < – 0 0 1 (21 24). In this case, the solution is a power law with the of the initial conditions (26–29). Fig. 5B shows the dependence of θ θ− θ− length ∼ t , velocity v ∼ t 1, and acceleration v ∼ t 2, so that the the initial growth rate on initial perturbation amplitude for given − < θ − < − γ λ =λ ∈ acceleration exponent is 2 ð 2Þ 1. By comparing these ðM, A, hðlÞ, Þ and for a0 ½0.1, 1. The data are confidently de- −1 = − + + −C1ða =λÞ values with acr 2 ð1 CÞ , we find that this mixing can be RT scribed by the model ðv0=v∞Þ=A = C1ða0=λÞe 0 with C1 ≈ 4.26 < θ−1 − type for small drag C ð 1Þ and RM type for large drag and C2 ≈ 2.63 (Fig. 5 B and C), suggesting that, in addition to − C > ðθ 1 − 1Þ. wavelength λ, RM dynamics has the characteristic amplitude scale a0 max ≈ 0.38λ, ka0 max ≈ 2.4, at which the maximum initial growth Small-Scale Nonuniform Structures of the Flow Fields. Hence, rate is achieved, v0 max=v∞ ≈ 0.6A. The ratio of the RMI initial and depending on the drag value, the blast-wave induced mixing can linear growth rates decays exponentially with the initial amplitude, − = ∼ < ∈ − − = = ða0 a0 max Þ be RT type with a acr or RM type with a acx, acr ( 2, 1); in v0 ½v0linear e (Fig. 5) (28). either case, larger velocities (velocity fluctuations) correspond to For a multimode initial perturbation, the order and disorder in small length scales, and the dynamics is sensitive to deterministic RM flow are highly sensitive to deterministic conditions, including conditions. In canonical turbulence, large velocities (velocity the wavelengths, the amplitudes, and the relative phases of the fluctuations) correspond to large length scales, and the dynamics initial perturbation waves (29). Fig. 6 show snapshots of the flow is independent of deterministic conditions (31, 32). In the blast regions of late stages of the RMI, with the same corresponding wave-induced (subdiffusive) mixing, the canonical (super- values of the Mach and Atwood numbers and the observational diffusive) turbulence may be a challenge to develop (unless there time. The initial perturbations have two waves with the same λ is a source, other than gravity, supplying turbulent energy to the amplitudes and wavelengths ð 1ð2Þ, a1ð2ÞÞ. However, the flow fluid system). If so, what are other possible mechanisms for energy keeps order in some cases and is disordered in the others. This accumulation at small scales, which is necessary for nuclear syn- difference is due to the relative phase φ and the interference of thesis in supernovae (1)? waves constituting the initial perturbation, as group theory finds Such mechanisms may exist due to small-scale nonuniform (29). In Fig. 6, Left, the waves are antiphase, φ = π, and the flow structures of the flow fields (26–29, 59). Particularly, subdiffusive symmetry group is pm11; this flow keeps order (29, 34). In Fig. 6, processes are often characterized by localizations. If a spot with Right, the waves have random phase, φ = π=2, and the flow sym- high energy fluctuation appears in the flow, it can be trapped due metry group is p1; this flow is disordered (29, 34). While at a first

Abarzhi et al. PNAS Latest Articles | 7of9 Downloaded by guest on September 24, 2021 glance, the disordered RM flow might appear turbulent, a more depends on the flow drag (Fig. 4). For −2 < a < acr, the asymptotic cautious consideration is required. The disorder in the random dynamics is RT type in the linear and nonlinear regimes and RM phase case in Fig. 6 is induced by initial conditions, suggesting type in the mixing regime: The acceleration triggers the instability deterministic chaotic (rather than stochastic turbulent) dynamics and defines the linear and nonlinear dynamics, but it plays ef- (29, 31). fectively no role in the mixing regime. The mixing dynamics is nevertheless faster than the acceleration prescribes. The critical Small-Scale Structures in RM Flow. SPHC simulations accurately exponent acr has the exponent values typical for blast waves. capture small-scale dynamics in strong shock-driven RMI. At the + For RT-type mixing with a > a and length-scale L ∼ ta 2,withthe interface, we observe small-scale shear-driven vortical structures cr decrease of the acceleration exponent, the dynamics changes its due to the Kelvin–Helmholtz instability (26–29). In the bulk, be- character from superballistics to subdiffusion. For RM-type mixing tween the transmitted shock and the interface, we observe small- + < ∼ acr 2 scale nonuniform localized structures of the flow fields (Fig. 7) (26). with a acr and length-scale L t , the dynamics is faster than > These include the “reverse” cumulative jets, which are short and the acceleration prescribes and is subdiffusive for C 1. For energetic and develop due to collisions of converging fluid flows a > − 1 ða < − 1Þ, large velocities correspond to large (small) scales. in the bulk (56, 60); the hot (cold) spots, which are the localized In blast wave-driven dynamics, the acceleration exponent can regions with the temperature much higher (lower) than that of the be a ∼ acr or a < acr (19–25). The blast wave-driven mixing can thus ambient; and the regions with high (low) pressure (26–29). These be RT type with a ∼ acr or RM type with a < acr. In either case, the well-pronounced small-scale structures are volumetric in nature mixing has large velocities (velocity fluctuations) at small scales, (Fig. 7). They develop, since the flow dynamics is adjusted to the and the dynamics is essentially subdiffusive. Subdiffusive pro- motion of the interface. cesses are known to depend on deterministic conditions and have Hence, the strong shock-driven RM dynamics is sensitive to small-scale localizations (Figs. 5–7). RM dynamics is indeed sen- deterministic conditions, may keep order at large and small sitive to deterministic conditions, including the wavelengths, the scales, and may have localized nonuniform structures at small amplitudes, the relative phase, and the interference of waves scales in the bulk in addition to shear-driven vortical structures at constituting the initial perturbation. In RM flows, localized non- the interface. uniform structures develop at small scalesinthebulkinadditionto shear-driven vortical structures at the interface. These volumetric Discussion structures may include cumulative jets, hot and cold spots, and high- Supernovae and their remnants are a central problem in astro- – physics due to their role in the formation of neutron stars and black and low-pressure regions (26 29). They are well-pronounced and holes in the processes of stellar evolution and nuclear synthesis (Fig. cause strong fluctuations of flow fields. Depending on deterministic 1) (1). We have considered this multiphysics problem in a hydro- conditions, RM dynamics may keep order at large and small scales. dynamic approximation, with supernova blast causing the devel- What are potential outcomes of this hydrodynamics for su- opment of RTI, RMI, and RT/RM interfacial mixing with variable pernova and nuclear synthesis (1)? acceleration (1–7). We have applied group theory (14, 34) to study In blast wave-driven RT/RM mixing in supernovae, canonical tur- RTI/RMI and RT/RM mixing with variable acceleration, and identi- bulence is traditionally considered as the mechanism for energy fied properties of linear, nonlinear, and mixing RT/RM dynamics accumulation at small scales. According to our results, for the that have not been discussed before (Figs. 2–7). acceleration parameters typical for blast waves (19–25), superdiffusive We have found that, for g ∼ ta, the linear and nonlinear dy- turbulence (31, 32) may be a challenge to implement in subdiffusive namics is RT type and is acceleration driven for a > − 2; for a < − 2, RT/RM mixing. However, the conditions of heterogeneity, nonlocality, the dynamics is RM type and is driven by the initial growth rate at anisotropy, and statistical unsteadiness that are common for RT/RM – early stage and by the drag (dissipation) at late stages. RT RM flows (14, 15, 33, 34) may lead to appearance of small-scale non- = − transition occurs at a 2 with varying of the acceleration uniform structures in the bulk of the blast wave-driven mixing flows, strength. For any a, the nonlinear regular asymptotic solutions whereas the slow subdiffusive transport may result in energy locali- form a continuous family; this multiplicity is due to the interfacial zation and trapping at small scales (26–29, 59). These effects are shear, and the fastest stable solution is physically significant (Figs. consistent with and may explain the richness of structures observed in 2 and 3). The principal result of the group theory analysis is the supernovae, including the Cas A (Fig. 1) (1). We suggest that such multiscale character of the nonlinear RT/RM dynamics, to which effects be considered in interpretation of observational data. two macroscopic-length scales—spatial period and amplitude— Our work focuses on hydrodynamic aspects of the multiphysics contribute. This further leads to two distinct mechanisms of the development of RT/RM mixing: the growth of the wavelength and problem of supernovae and their remnants and serves for sys- the dominance of the amplitude, each resulting in the imbalance tematic studies of the problem in perspective. of gain and loss of the rates of specific momentum (14, 34). Acknowledgments The mixing is RT type for a > acr and is RM type for a < acr.RT– ∼ We thank the University of Western Australia, the National Science Foundation, RM transition occurs for a acr by varying the acceleration ex- and the Theoretical Astrophysics Program and the Steward Observatory at the −1 ponent; the critical exponent acr = −2 + ð1 + CÞ , acr ∈ ð−2, −1Þ University of Arizona for support.

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