Scaling of Small-Scale Dynamo Properties in the Rayleigh-Taylor Instability

Draft version June 10, 2021 Typeset using LATEX twocolumn style in AASTeX63

V. Skoutnev,1 E. R. Most,2, 3, 4 A. Bhattacharjee,1 and A. A. Philippov5

1Department of Astrophysical Sciences and Max Planck Princeton Center, Princeton University, Princeton, NJ 08544, USA 2Princeton Center for Theoretical Science, Princeton University, Princeton, NJ 08544, USA 3Princeton Gravity Initiative, Princeton University, Princeton, NJ 08544, USA 4School of Natural Sciences, Institute for Advanced Study, Princeton, NJ 08540, USA 5Center for Computational Astrophysics, Flatiron Institute, 162 Fifth Avenue, New York, NY10010, USA

(Received June 10, 2021; Revised; Accepted) Submitted to ApJ

ABSTRACT We derive scaling relations based on free-fall and isotropy assumptions for the kinematic small-scale dynamo growth rate and ampliﬁcation factor over the course of the mixing, saturation, and decay phases of the Rayleigh-Taylor instability (RTI) in a fully-ionized plasma. The scaling relations are tested using sets of three dimensional, visco-resistive MHD simulations of the RTI and found to hold in the saturation phase, but exhibit discrepancies during the mixing and decays phases, suggesting a need to relax either the free-fall or isotropy assumptions. Application of the scaling relations allows for quantitative prediction of the net ampliﬁcation of magnetic energy in the kinematic dynamo phase and therefore a determination of whether the magnetic energy either remains sub-equipartition at all velocity scales or reaches equipartition with at least some scales of the turbulent kinetic energy in laboratory and astrophysical scenarios. As an example, we consider the dynamo in RTI-unstable regions of the outer envelope of a binary neutron star merger and predict that the kinematic regime of the small-scale dynamo ends on the time scale of nanoseconds and then reaches saturation on a timescale of microseconds, which are both fast compared to the millisecond relaxation time of the post-merger.

Keywords: magnetic ﬁelds–dynamo–Rayleigh-Taylor

1. INTRODUCTION (Isobe et al. 2005; Berger et al. 2011; Hillier 2018), The Rayleigh-Taylor instability (RTI) is ubiquitous gamma-ray burst scenarios (Gull & Longair 1973; Levin- in astrophysical contexts due to the generality of the son 2009; Duffell & Macfadyen 2013; Duffell & Mac- conditions needed for its onset. The RTI operates in Fadyen 2014), and supernova explosions (Hillebrandt & regions where the density gradient is misaligned with Niemeyer 2000; Cabot & Cook 2006; Duffell & Kasen the direction of the local gravitational field or acceler- 2016). The fluid is typically a highly conducting plasma ation (Chandrasekhar 1961). The instability character- whose ability to generate and sustain magnetic fields can istically evolves with rising bubbles of lighter fluid and make the magnetohydrodynamic (MHD) version of the RTI differ from its hydrodynamic counterpart. arXiv:2106.04787v1 [physics.flu-dyn] 9 Jun 2021 sinking spikes of heavier fluid that propagate away from the unstable region, leading to mixed fluids and relax- The role of the magnetic field can be categorized by ation of the unstable density gradient. In this manner, whether the initial field strength is dynamically strong fluid mixing and transport is enhanced by the RTI in or weak. In the strong field limit, large scale magnetic many astrophysical scenarios such as the solar corona fields are able to stabilize a wide range of wavenumbers due to the restoring force of magnetic tension (Chan- drasekhar 1961; Ruderman et al. 2014). This affects Corresponding author: Valentin Skoutnev the non-linear saturation and mixing rates of the RTI [email protected] with a strong dependence on details of the initial geometry of the magnetic fields. Typically, magnetic tension 2 V. Skoutnev, E. R. Most, A. Bhattacharjee, A. A. Philippov forces suppress development of secondary shear instabil- dictions for scaling laws between the SSD growth rate, ities and can cause bubble/spike structures to rise/fall amplification factor, and parameters of the RTI (At- more rapidly than in the hydrodynamic case (Stone & wood number, gravitational acceleration, viscosity, and Gardiner 2007). This limit has application in the con- length scale) in each phase of evolution of the RTI. De- texts of solar prominences (Hillier 2018) and pulsar wind termination of the correct scaling laws is important for nebulae (Porth et al. 2014), for example. quantitatively extending results from practical simula- On the other hand, the weak field limit corresponds tion parameters to realistic experimental or astrophys- to an initial magnetic field with dynamically insignif- ical parameters, which can be separated by orders of icant strength, which allows the RTI to evolve purely magnitude. hydrodynamically at least at early times. The turbu- Section3 tests the model using sets of three- lence in the non-linear phases of the RTI has the po- dimensional (3D) visco-resistive MHD direct numerical tential to amplify the magnetic field through dynamo simulations that resolve the turbulent viscous scales, al- action. In the dynamo literature, this weak field limit lowing for results independent of numerical resolution. is also known as the kinematic dynamo regime. If the Discrepancies between the model prediction and the nu- turbulence is sufficiently vigorous and the initial mag- merical results are analyzed and the assumptions that netic energy is not too small, then the magnetic energy are likely breaking down are identified. Section4 ap- could grow to and saturate in equipartition with the ki- plies the model to the case of possible dynamo action in netic energy of the turbulence. This results in decaying RTI-unstable regions of the outer envelope of the post- MHD turbulence post-saturation of the RTI, which can merger of a binary neutron star collision. In Section5, lead to observables such as synchrotron emission. Oth- we summarize our findings, identify directions for future erwise, the magnetic energy is amplified, but remains work, and conclude. at sub-equipartition energies and results in decaying hy- 2. THEORETICAL SCALING PREDICTIONS drodynamic turbulence post-saturation of the RTI. While early, low resolution simulations of the RTI had We present scaling arguments to determine the kine- confirmed operation of the small-scale dynamo (SSD) matic small-scale dynamo growth rate and amplification (Jun et al. 1995), the quantitative scaling of dynamo factor of the magnetic energy in a Rayleigh-Taylor un- properties have been largely unstudied despite applica- stable, conducting, collisional fluid over the course of the tions in a variety of scenarios. Here we discuss two ex- growth, saturation, and decay of the instability. ample applications at opposite extremes; the fireball and Setup —Without loss of generality, we consider the stan- laser plasma experiments. Hydrodynamical simulations dard, idealized RT setup in three dimensions with a dis- of the fireball propose that the high-Reynolds number continuous density jump from ρ(z) = ρb for z < 0 to RTI-driven turbulence generates equipartition magnetic ρ(z) = ρt > ρb for z > 0 in a bounded domain of size fields that can explain the observed levels of sychnotron ∼ L and initial velocity perturbations near z = 0. Any radiation emission from these sources (Duffell & Mac- similar setup (e.g. with a continuous positive density fadyen 2013). On the other hand, several laser plasma gradient instead) can be easily related to the idealized experiments of the RTI at modest Reynolds numbers setup, and the same scaling arguments will apply. The have found magnetic energy growth and explain the ob- initial seed magnetic field is taken to be random at all servations by using the Biermann effect (Manuel et al. scales and arbitrarily weak so that the magnetic energy 2012; Gao et al. 2012; Nilson et al. 2015; Matteucci always remains much lower than the kinetic energy at all et al. 2018). However, RTI turbulence itself could be hydrodynamic scales, resulting in purely hydrodynamic a significant contributor as well, in particular, as the evolution of the velocity field. A standard model of the Reynolds number and duration of future experiments RTI in fully ionized plasmas is the set of visco-resistive increases (Galmiche & Gauthier 1996; Bott et al. 2021). MHD equations given by: A quantitative framework for either justifying the use of the equipartition argument or otherwise predicting the ∂tρ + ∇ · (ρu) = 0, (1) level of sub-equipartition magnetic energy amplification would be useful in the general case. This motivates a ∂ (ρu) + ∇ · (ρuu − BB + P ) = ρν∇2u, (2) careful study of the RTI turbulence driven dynamo. t

Paper Outline —Section2 presents a model for the SSD 2 ∂tB − ∇ × (u × B) = η∇ B, (3) in the weak field limit based on assumptions of isotropic turbulence and free-fall scaling for the outer velocity and 2 length scale of the turbulence. The model makes pre- ∂tE + ∇ · [(E + P )u − B(B · u)] = ρκ∇ T, (4) Scaling of Small-Scale Dynamo Properties in the Rayleigh-Taylor Instability 3 where u is the velocity, B is the magnetic field, E is the net exponential amplification factor of the magnetic en- total energy density, P is the sum of the gas and mag- ergy given by: netic pressure, T is the temperature, ν is the viscosity, ME(t) Z t η is the resistivity, and κ is the thermal diffusivity. ∆ = max ln = max γ(t0)dt0, (5) We consider the subsonic limit where the fluid flows t ME(0) t 0 are nearly incompressible and therefore dynamo proper- where ME(t) = 1 R B(x, t)2d3x is the total magnetic en- ties are independent of the equation of state. Any initial 2 ergy. If u, li, and Re are enough to characterize the flow, vertical velocity perturbation with horizontal wavenum- α then dimensional constraints force γ(t) = Cγ (u/li)Re ber k triggers the RTI and grows exponentially with 1 (assuming a power law form for the Re dependence when 2 4 2 ρt−ρb rate n = (gkA + ν k ) 2 − νk , where A = ≤ 1 c ρt+ρb Re Re ), where Cγ and α are dimensionless con- is the Atwood number and effects of thermal diffusivity stants. In isotropic turbulence at high P m, the growth is are neglected. If all wavenumbers are present, the fastest known to scale with the turnover time of Kolomogorov- 2 1 growing wave number will be kc ∼ (Ag/ν ) 3 and smaller −1 scale eddies γ ∼ tν , which corresponds to α = 1/2 wavenumbers k < kc will approximately have the invis- 1 (Rincon 2019). We restrict to the high P m limit in cid growth rate n ≈ (Agk) 2 . the following analysis and assume α = 1/2; however, this assumption may need to be revisited in the mixing and decay phases where the anisotropic effects of gravity are particularly important. Not only does gravity RTI evolution and definitions —The RTI can be split affect horizontal versus vertical fluid motions differently, into four distinct phases: linear growth, mixing (or there is also asymmetry between rising and falling fluid non-linear), saturation, and decay phases. The lin- structures at increasingly higher Atwood numbers (for ear phase begins with exponential growth of any ini- a review see Zhou 2017). tial vertical velocity perturbations and ends once the A main goal of the paper is to determine how ∆ scales fluid has displaced a vertical distance comparable to with parameters of the RTI, {A, g, L, ν}. We begin by the wavelength of the initial mode. The fluid then splitting up the time integral in Eq.5 into the three rapidly becomes turbulent with outer velocity scale consecutive, turbulent phases and consider the positive contribution of each phase one at a time, u(t), integral length scale li(t), and Reynolds number Re(t) = u(t)li(t)/(2πν) evolving in time throughout the ∆ = ∆mix + ∆sat + ∆decay. (6) remaining three phases. It is convenient in what fol- √ lows to define a characteristic velocity usat = AgL, Mixing phase —In the mixing phase, a mixing region of dynamical time tdyn = L/usat, and Reynolds number vertical extent h(t) propagates away from the original Resat = usatL/(2πν). interface. A variety of models have been proposed to Following the linear phase, the initial perturbations predict the time dependence of h(t) and the results of develop in the mixing phase into the characteristic RTI simulations and experiments are on various levels of dis- bubble/spike structures that rise/fall away from the agreement that are primarily attributed to a sensitivity boundary, driven by buoyancy forcing. The mixing to initial conditions, system size effects, or effects of dif- phase ends when the upward/downward propagating fusivity. The general form of most models predict a fronts of bubbles/spikes reach the system scale L. This free-fall scaling given by begins the saturation phase of the RTI where the major- 1 2 p ity of available potential energy has been released into h(t)/L = ατ + 2 αh0/Lτ + h0/L, (7) turbulent kinetic energy. After a dynamical time scale, 2 the viscous dissipation in the turbulence becomes larger where τ = t/tdyn, α is a constant, and h0 is the length than the energy input from buoyant forcing and thus scale near when the mixing region first became non- begins the decay phase where the total kinetic energy linear (see the review Boffetta & Mazzino(2017) and ref- decreases. The fluid eventually settles into a stable state erences within). The turbulent velocity and speed of the with a negative density gradient. mixing region boundary are taken to be comparable and p The small-scale dynamo will be active when the self-consistently given by u(t)/usat = ατ + 2 αh0/L. Reynolds number Re(t) in the turbulent phases is above These scalings have been found to break down when the the critical Reynolds number Rec(P m) of the flows, size of the bubbles and spikes become comparable to the where P m = ν/η is the magnetic Prandtl number. We horizontal or vertical extent of the domain or if there characterize the dynamo with the instantaneous, expo- is a presence of a dominant mode lD, both of which d √ nential dynamo growth rate γ(t) = dt ln ME(t) and the could lead to terminal velocity scalings h(t) ∼ glDt 4 V. Skoutnev, E. R. Most, A. Bhattacharjee, A. A. Philippov √ and u(t) ∼ glD (Dimonte 2004; Banerjee & Andrews isotropic. Although the isotropy assumption is not ex- 2009; Lecoanet et al. 2012). Additionally, a larger, ab- pected to hold as the system relaxes into a stably strat- solute thermal diffusivity reduces buoyancy effects and ified state, the simple model will be useful as a point therefore slows the development of the mixing region of comparison. The total energy in isotropic turbulence (Abarzhi 2010). For instance, dimensional arguments decays as including diffusivity at low Atwood number can predict 2 2 dE(t0) 1 u(t0)3 an alternative time dependence h(t) ∼ gt / ln(gt /h0) 0 = − ρCE 0 (11) (Abarzhi et al. 2005). dt 2 li(t ) To model the dynamo growth rate, we assume that the 0 0 1 where u(t ) = (2E(t )/ρ) 2 , ρ is the average density, turbulent integral scale is comparable to the height of 0 t = 0 is the beginning of the decay phase, and CE the mixing region li(t) ≈ h(t)(Chertkov 2003; Boffetta is a dimensionless constant (Frisch 1995; Subramanian & Mazzino 2017). However, this is an assumption that et al. 2006). The problem lies in determining the re- may also need to be revisited since the turbulence is not lation between the evolving integral scale and energy in steady state and there may be a delay between energy 0 −s (li(t ) ∼ E ). Typical choices for s lie in the range generation at large scales and dissipation at small scales 1 1 5 ≤ s ≤ 3 with s = 1/5 corresponding to Batchelor- (Livescu et al. 2009). For simplicity, we also assume the type and s = 1/3 to Saffman-type turbulence, depend- free-fall scalings hold. Then the dynamo exponentially ing on properties and initial conditions of the turbulence grows with rate: (Ishida et al. 2006). However, if the integral scale is already at the system scale and the system is bounded ( 1 1 i.e. in a numerical simulation), then l cannot grow and 2 p 3 ! 2 i Cγ Resat (ατ + 2 αh0/L) 0 γmix(t) ≈ thus remains at the system scale (li(t ) ≈ L) indepen- t 1 2 p dyn 2 ατ + 2 αh0/Lτ + h0/L dent of E (corresponding to s → 0) (Skrbek & Stalp (8) 2000; Touil et al. 2002). Solving this system for u(t0) p 0 After a transient time τ ≈ h0/(αL), the growth will and li(t ), the growth rate is given by: 1 1 −1 2 asymptotically scale as γmix(t) ∼ t Re τ 2 . Depend- dyn sat 3 1 2 −3−2s ing on the geometry, the mixing phase will last for a time 0 Cγ (2α) 4 Resat 1 0 2(1+2s) γdec(t ) ≈ 1+(s+ )CEτ (12) interval ∆tmix ∼ tdyn when the mixing region reaches tdyn 2 the vertical system scale h(t) ≈ L. Thus, with free-fall scalings, ∆mix is asymptotically found to be The dynamo growth rate will eventually become zero when Re(t0) ≈ Rec(P m). Assuming the dynamo is ac- √ 3 0 0 2 tive in the decay phase from τ = 0 to τ = ∆t /t 2 2αCγ 1 ∆tmix dec dyn ∆mix ≈ Re 2 (9) decay sat 1/CE, ∆ is given by: 3 tdyn

3 1 4 Saturation phase —In the saturation phase, the mixing decay 2 4(2α) Cγ ∆ ≈ Resat (13) region encompasses the entire domain h(t) = L ≈ li(t) CE(1 − 2s) and the turbulent kinetic energy is maximal. The veloc- √ Result —Combining our results, each phase in Eq6 con- ity scale u(t) ≈ 2αusat can be solved either from the 1 2 free-fall scaling in the mixing phase or by equating con- tributes a term proportional to Resat giving 4 verted initial potential energy PE ∼ (ρt − ρn)αgL to 1 2 3 turbulent kinetic energy KE ∼ (ρt +ρb)u L . The dy- √ 3 2 1 2 3 1 2 2 2α ∆tmix −1 2 ∆ ≈ Re C namo growth rate is then γsat(t) ≈ Cγ (2α) 4 t Re . sat γ dyn sat 3 tdyn We expect the velocity scale and integral length scale to 3 ! remain roughly constant for a time interval ∆tsat ∼ tdyn 3 ∆tsat 4(2α) 4 Cγ + (2α) 4 + , (14) comparable to the dynamical time before the turbulence tdyn CE(1 − 2s) enters the decay phase. The exponential ampliﬁcation from this phase should scale as Since both ∆tmix ∼ tdyn from free-fall scaling and ∆t ∼ t from dimensional constraints, all the terms sat dyn 3 1 ∆tsat sat 4 2 in the parenthesis are constants and thus: ∆ ≈ Cγ (2α) Resat (10) tdyn 1 1 3 1 4 4 4 2 A g L Decay phase —From this point, the turbulence decays ∆ ∼ Resat ∼ 1 , (15) ν 2 and the dynamo growth rate decreases. Again for simplicity, we assume that the free decaying turbulence is independent of constants in the models. Scaling of Small-Scale Dynamo Properties in the Rayleigh-Taylor Instability 5

In summary, the clean result comes from the choice of Thus, our results for ∆ are an upper bound since free-fall velocity (u ∼ usat) and length (li ∼ L) scales the SSD will leave the kinematic regime during some 1 −1 2 phase of the RTI if ∆ is too large. As a rough esti- leading to a dynamo with growth rate γ ∼ tdynResat mate, any sub-equipartition, seed magnetic field con- acting over dynamical timescales tdyn and resulting in 1 2 figuration with magnetic energy greater than ME(t = amplification of ∆ ∼ γtdyn ∼ Resat. 1 − 2 −∆ 0) & Resat KEsate will reach the dynamical regime Low P m case —The RTI is also applicable to astrophysi- at some point in the RTI evolution and possibly satu- cal scenarios where the magnetic Prandtl number P m = rate before the end of the decay phase of the RTI, which Rm/Re can be less than one, such as in stellar interiors would result in MHD turbulence in the remaining RTI during supernova explosions. The only modification is evolution. We briefly study this regime in Section 3.3. the estimate of the growth rate. The resistive scale at While in principle one can build on our kinematic dy- −3/4 low P m is inside the inertial range (lη ∼ Rm L > lν ) namo model and include the non-linear dynamo growth and the dynamo is thought to be driven by resistive- phase by using a time dependent (t), we do not pur- 1 scale eddies whose turnover times are tη ∼ L/(URm 2 ) sue this idea in further detail because 1) the non-linear (Iskakov et al. 2007). For Re Rec(P m), where growth phase in simulations is short and difficult to test Rec(P m 1) = O(102), this corresponds to a reduced and 2) it is unclear how the feedback from the strong 1 growth rate given by γ ∼ URm 2 /L and an exponential Lorentz forces across a growing range of velocity scales amplification factor that instead scales as will affect the time evolution of the RTI velocity field.

1 1 3 However, we do note a useful estimate for the time it 1 A 4 g 4 L 4 2 takes for the dynamo to saturate during the dynamical ∆ ∼ Rmsat ∼ 1 . (16) η 2 regime ∆td.r.. If the magnetic energy at the start of 1 − 2 Saturation of the SSD —We discuss the extension of our the dynamical regime is ME ∼ Re KE, at satura- model of the dynamo in the kinematic regime to the tion is ME = f · KE, and the transfer rate of kinetic dynamical regime and saturation of the SSD at high energy is roughly constant ∼ KE/tdyn, then we sim- P m. Our results above are only applicable in the kine- ply find that ∆td.r. is comparable to the dynamical time − 1 matic regime of the dynamo where Lorentz forces are ∆td.r. ∼ tdyn(f − Re 2 )/ζ. unimportant and the induction equation is a linear func- tion of the velocity field, which leads to exponential growth of magnetic energy. However, when the magnetic energy becomes comparable to the kinetic energy 3. SIMULATION SCALING ANALYSIS 2 − 1 2 at viscous scales (h|B| i ∼ Re 2 h|u| i), the dynamo In this section, we use three dimensional direct nu- enters the dynamical regime where feedback on fluid merical simulations of the RTI to test the theoretical motions from Lorentz forces become important and the scaling predictions presented in Section2. We show and dynamo switches to polynomial-order growth (Rincon discuss the results of parameters scans with the Atwood 2019). The dynamical regime is also known as the number, gravitational acceleration, and viscosity. non-linear growth phase, for which there are several models in the literature. In one possible model, the growth rate decreases due to sequential suppression of dynamo-generating motions by the Lorentz forces start- ing from the viscous scales and ending at the forcing 3.1. Numerical Setup scales (Schekochihin et al. 2002). Assuming the growth We use the Athena++ code (Stone et al. 2019) to solve rate is set by the turn over time of the smallest, un- the visco-resistive MHD equations in a 3D Cartesian do- quenched velocity scale and that the magnetic energy is main with a similar setup to previous works (Stone & in equipartition with all smaller velocity scales, one can Gardiner 2007). The horizontal −L/2 ≤ x, y ≤ L/2 easily show that the magnetic energy will grow linearly directions have periodic boundary conditions while the 3 in time ME(t) ≈ ζt, where ∼ u /li is the trans- vertical direction −L ≤ z ≤ L has reflecting boundary fer rate of kinetic energy and ζ is a dimensionless con- conditions, where L = 0.1. All simulations use a resolu- 2 stant. When the Lorentz forces become strong at the tion of NxNyNz = 512 ×1024, RK3 for the timestepper, forcing scales, the dynamo will fully saturate in near- and HLLD for the Riemann solver (Miyoshi & Kusano equipartition between the total magnetic and kinetic en- 2005). ergy with a ratio ME/KE = f = O(10−1), where f is The initial hydrodynamic conditions set the fluid den- a dimensionless constant . sity ρ(z) = ρt in the top half of the domain, ρ(z) = ρb in 6 V. Skoutnev, E. R. Most, A. Bhattacharjee, A. A. Philippov the lower half of the domain, and a seed vertical velocity perturbation given by:

  X anx,ny i 2π (n x+n y) πz u (t = 0) = < e√ e L x y cos z  n 2L  nx,ny (17) where a is a random complex number, n = enx,ny q 2 2 nx + ny, and 0 < n ≤ nmax with nmax = 32. The magnetic field is initialized with an isotropic spectrum M(k) ≈ const. in the wavenumber range 0 < k ≤ kmax = 2πnmax/L. The total initial magnetic energy R EM (t = 0) = M(k)dk is set to a dynamically in- −19 significant value EM (t = 0) ≈ 10 when studying the kinematic regime in Section 3.2, and set higher when studying of the saturation regime. Lastly, we fix the magnetic Prandtl number to P m = 3 and the thermal Figure 1. Evolution of the magnetic energy (top left), in- Prandtl number to P r = ν/κ = 1 for all runs. stantaneous dynamo growth rate (top right), kinetic energy Asymptotic scaling laws of hydrodynamic quantities (bottom left), and approximate Reynolds number (bottom in the mixing phase of the RTI are known to be sensi- right) of the fiducial simulation. The background shading de- tive to initial conditions (IC) and the box aspect ratio notes the linear (blue), mixing (green), saturation (orange), (Dimonte 2004; Boffetta & Mazzino 2017). Thus, pa- and decay (red) phases. rameter scans with different choices of IC (e.g. varying nmax) could lead to slightly different scaling laws. It is bility. We expect neither edge case to be applicable in not clear which choice of IC is generally most physically a realistic astrophysical system. applicable since the RTI setup is already quite ideal- The parameter scans are based on a fiducial simula- ized. We discuss throughout this article how our dy- tion with parameters {A, g, ν−1} = {0.67, 0.65, 3 × 105} namo results may vary for choices of IC different from which we use below to describe a typical simulation and ours. One common alternative option is mode perturba- explain our method of analysis. Figure1 shows the tions with wavelengths that go down to the grid scale, time evolution of the kinetic energy, magnetic energy, while another is restricting to a shell of modes in Fourier dynamo growth rate, and approximate Reynolds num- space (Dimonte 2004). Our choice of IC was motivated ber of the fiducial simulation. When the Reynolds num- to allow us to study the effect of changing viscosity on ber rises above the critical Reynolds number Re(t) & the dynamo while retaining a similar hydrodynamic RTI Rec(P m > 1) ≈ 60, the dynamo growth rate sharply evolution by having our fastest growing mode to always increases. The dynamo is triggered near the beginning be fixed at kmax < kc for our chosen range of viscosities. of the mixing phase (green shaded region in Figure1), In order to expect our results to be generalizeable, it which we define as the time when KE(t) = 0.01PEmix, mix 4 is important to at least check that a minor variations where PE = 0.5gL (ρt − ρb) is the potential en- of ICs for a fixed set of parameters has a minimal ef- ergy released from complete mixing of the fluids. The fect on ∆. We have found this to generally be the case kinetic energy rises until a maximum in the middle of (n) for a fiducial set of parameters where we widely varied the saturation phase at the saturation time tsat given (n) nmax ≥ 8 for both the initial velocity and magnetic field by KE(tsat) = max(KE(t)), at which point we also spectra. The two exceptions we found are the edge cases (n) (n) (n) have the numerical values for usat = u(tsat) and tdyn = of a single mode RTI (n = 1) or a purely uniform (n) max L/u . weak initial magnetic field. The single mode RTI be- sat We define the mixing phase to end and the saturation comes non-linear near the system scale unlike the multi- (n) phase (orange shaded region) to begin at time t = t − mode RTI, leading to negligible contribution from ∆mix sat t(n) . Similarly, the saturation phase we define to end and a reduced ∆. In the uniform magnetic field case, dyn and the decay phase (red shaded region) to begin at time the dynamo additionally has a short and intense tran- (n) (n) sient growth at the beginning of the mixing phase due t = tsat + tdyn. At some point in the decay phase the to coherent alignment of the field with the secondary magnetic energy will reach its maximum and we define shear layers that undergo the Kelvin-Helmholtz insta- the decay phase to end for the purposes of the dynamo. Scaling of Small-Scale Dynamo Properties in the Rayleigh-Taylor Instability 7

scales visible in the density field. The saturation phase (middle column) has large eddies on the system scale with the magnetic field developing on small scales, as would be qualitatively expected of isotropic turbulence. Lastly, the decay phase (right column) shows a negative mean density gradient in which residual kinetic energy sloshes fluid around while the magnetic fields still appear on small but slightly larger scales than in the saturation phase.

3.2. Scaling Results To test the scaling relations of the dynamo model proposed in Section2, we preform a series of parameter scans of the Atwood number, gravitational acceleration, and viscosity. We choose our fiducial simulation with values of {A, g, ν−1} = {0.67, 0.65, 3e5} (corresponding to max(Re(t)) ≈ 300) and then run a parameter scan across a maximum range of each parameter, while keeping all others Figure 2. Two-dimensional slices in the X −Z plane of the density (top row) and vertical magnetic field (bottom row) fixed. For the gravitation acceleration parameter, in the mixing phase (left column), saturation phase (mid- we scan g ∈ {0.25, 0.35, 0.5, 0.65, 0.8, 1.0, 1.2, 1.4}, dle column), and decay phase (right column) of the fiducial for the Atwood number, we scan A ∈ simulation. {0.2, 0.33, 0.43, 0.5, 0.6, 0.67, 0.71, 0.76, 0.82, 0.88, 0.9}, and for the viscosity, we scan ν−1 ∈ Note, while the exact definitions of the start and end {2, 2.5, 3, 3.5, 4, 4.5, 5} × 105. The lower bound for each c time of each phase are somewhat arbitrary, we find that parameter is constrained by needing Resat Re in our results are not sensitive to reasonable variations of order for the dynamo growth rate to be approximated 1 the definitions. by it’s asymptotic power law γ ∼ Re 2 , while the upper With the above definitions, we obtain the numerical bound is constrained by requiring the Kolmogorov scale values of the mean dynamo growth rate and amplifica- to be larger than the grid scale. In particular, the lowest phase tion factor in each phase (γphase and ∆ ) with the value of the explicit viscosity is chosen to be a factor following expressions: of two above the numerical viscosity for the numerical setup, which is estimated based on the decay rate of Z tend 1 Alfven waves as described in AppendixA. Note, it is γphase = γ(t)dt, (18) ∆tphase tbeg critical to at least marginally resolve the Kolmogorov scale, because otherwise the arbitrary simulation grid phase ME(tend) scale introduces another length scale that breaks the ∆ = ln = γphase∆tphase, (19) ME(tbeg) scaling arguments. First we examine the dynamo amplification factor

∆tphase = tend − tbeg (20) across the entire RTI instability. A linear fit in log-log space between ∆ and each RTI parameter gives: where tbeg and tend are the beginning and ending times of each phase. The total amplification factor is then ∆ ∼ A0.40±0.03g0.35±0.03ν−0.51±0.09, (22) very closely given by: For reference, the predicted free-fall scaling relations are max(ME(t)) ∆ = ∆mix + ∆sat + ∆dec ≈ ln . (21) given by: min(ME(t)) ∆ ∼ A0.25g0.25ν−0.5. (23) For completeness, two-dimensional slices from the While the viscosity exponent is in good agreement fiducial simulation of the density ρ and vertical mag- with the model (within one standard deviation), the netic field Bz during each of the three turbulent phases exponents of A and g are close in magnitude to the are shown in Figure2. The amplified magnetic field in predicted value of 0.25, but are different by a statisti- the mixing phase (left column) closely follows the ris- cally significantly amount (roughly five and three stan- ing bubbles and falling spike structures at intermediate dard deviations). An alternative way to view the result 8 V. Skoutnev, E. R. Most, A. Bhattacharjee, A. A. Philippov

A g ν

∆ff 0.25 0.25 -0.5

∆mix 0.68±0.16 0.43±0.10 -0.79±0.23

∆sat 0.25±0.03 0.24±0.02 -0.38±0.04

∆dec 0.28±0.23 0.37±0.13 -0.35±0.33

γff 0.75 0.75 -0.5

γmix 1.01±0.05 0.77±0.03 -0.54± 0.07

γsat 0.81±0.07 0.81±0.05 -0.60±0.10

γdec 0.17±0.19 1.0±0.27 -1.3±0.36

∆tff -0.5 -0.5 0

∆tmix -0.33±0.12 -0.34±0.08 -0.24±0.17

∆tsat -0.56±0.06 -0.57±0.04 0.21±0.10

∆tdec 0.1±0.29 -0.62±0.38 0.96±0.41 Table 1. Table of the scaling exponents between the dynamo ampliﬁcation factor, mean dynamo growth rate, and duration of each phase (rows) and the RTI parameters (columns). For example, the entry for ∆mix and A means 0.68±0.16 ∆mix ∼ A . For reference, ∆ff , γff , ∆tff denote the prediction for the exponent based on the free-fall model.

0.5 the predicted ∆ ∼ Resat. Breaking up ∆ further into the contribution of each phase ∆phase in the bottom panel of Figure3 clearly shows that the mixing phase is in disagreement while the saturation and decay phases are in good agreement with the model (although the decay phase data is highly scattered). To better understand the discrepancy, it is necessary to examine the scaling relations of the mean growth rate and duration of each phase with the RTI parameters. Following a linear fit in log-log space between phase {∆ , γphase, ∆tphase} and {A, g, ν} in each phase Figure 3. Top: The total magnetic energy amplification separately, Table1 shows the resulting scaling expo- 1 2 nents. Overall, the free-fall model predictions are again factor ∆ versus Resat for each simulation on a log − log scale 1 2 in good agreement for the saturation phase (all roughly is shown along with a linear fit ln ∆ = m ln Resat + b. The parameter scans are represented with red circles for the vis- within two standard deviations), while in several dis- cosity scan, cyan squares for the gravitational acceleration agreements for the mixing and decay phases. The devi- scan, and magenta triangles for the Atwood scan. Bottom: ation of the m = 1.29±0.08 result for ∆ from the model The total magnetic energy amplification factor ∆ is split up prediction (m = 1) can thus be explained by a likely into the contributions from the mixing (green circles), sat- failure of model assumptions in the mixing and decay uration (orange squares), and decay phases (red triangles) 1 phases. The value of m = 1.29 is still close to 1, how- 2 and plotted versus Resat. ever, because ∆sat provides the dominant contribution to ∆. We now present a more detailed analysis of the is shown in the top panel of Figure3 where the ∆ is mixing and decay phases. 1 2 plotted versus Resat on a log-log scale, so a linear fit 1 Mixing phase —The goal is to find which assumptions ln ∆ = m ln Re 2 + b with a slope of m = 1 means per- sat in the model of Section2 are violated in the mixing fect agreement. The data does appear to qualitatively phase. We begin by comparing the time dependence of follow a power law; however, the fit also shows a minor KE(t) and γ (t) between free-fall model predictions but statistically significant disagreement with the model mix and re-scaled simulation time series averaged over all quantified by the measured value of m = 1.29 ± 0.08. In runs, shown in Figure4. The free-fall model with the other words, with a measured value of b = −0.90, the 0.65±0.04 isotropy assumption asymptotically predicts a time de- simulations have a scaling ∆ ≈ 0.4Resat instead of 1.5 0.5 0.5 pendence γmix ∼ u(t) /li(t) ∼ t for the dynamo Scaling of Small-Scale Dynamo Properties in the Rayleigh-Taylor Instability 9

The numerical dynamo growth rate in the simulation (n) 0.71±0.17 data is slightly steeper γmix ∼ t (within two standard deviations from the prediction) and the kinetic energy growth significantly shallower KE(n) ∼ t3.11±0.23 (four standard deviations from the prediction). The discrepancy suggests a slightly alternative scaling of either u(t), h(t), or li(t) in our simulations. Fitting the time power laws to u(t) and h(t) independently for the fiducial simulation (not shown), we find the numerical time dependence u(n) ∼ t0.8±0.025 and h(n) ∼ t1.55±0.06, which suggests an intermediate scaling between the free- fall and terminal velocity scaling (u ∼ t0, h ∼ t1) and has also been observed in previous simulations (Lecoanet et al. 2012). One possible reason we do not obtain free- fall scaling may be our choice of initial conditions which do not set kmax = kc, while another reason may be that the vertical dimension of the box is not asymptotically Figure 4. Time evolution of the dynamo growth rate (top) large enough to reduce the effect of the linear term in and kinetic energy (bottom) of every simulation. For each Equation7. The effects of thermal diffusion may also simulation, the time is re-scaled by the dynamical time, tdyn, play a role, despite the moderately high resolution of 1 −1 2 our simulations. The deviation of ∆t from free-fall the dynamo growth rate is re-scaled by tdynResat, and the ki- mix netic energy is re-scaled by PEmix. The background shading predictions in Table1 is likely tied to the above reasons is the same as in Figure1. as well. Nonetheless, the time dependence in the simulations (n)1.5 (n)0.5 0.45±0.05 predict γmix ∼ u /h ∼ t , which is even less steep than the free-fall prediction. This motivates checking which of the remaining major assumptions break down: 1) isotropy of the dynamo-generating scales 2) whether the integral scale and mixing height are directly proportional. The assumption of isotropy of turbulence at dynamo-generating scales appears to be sup-

ported by simulation data since α = d ln γmix/(d ln ν) ≈ 0.5 in Table1. Additionally, while the anisotropy of the total kinetic energy (defined by 2KEz(t)/KEh(t)) is large in the mixing phase as shown in the top panel of Figure5, the anisotropy in the total magnetic energy (defined by 2MEz(t)/MEh(t)) is much smaller and ap- proaches unity by the end of the mixing phase (bottom panel of Figure5). We expect the anisotropy of the magnetic field in the mixing phase to be even lower for larger Reynolds numbers since higher resolution simulations of the RTI than ours find strong evidence for isotropy at Kolmogorov scales (Zingale et al. 2005; Cabot & Cook 2006). Figure 5. Time evolution of the anisotropy of total ki- We check the second assumption by computing li(t) = netic (top) and magnetic (bottom) energy of every simula- (R k−1E(k, t)dk)/(R E(k, t)dk) for the fiducial simulation. The background shading is the same as in Figure1. (n) 1.1±0.04 tion and find li ∼ t , which is statistically significantly different than the time dependence for h(n) ∼ growth rate and KE(t) ∼ u(t)2h(t) ∼ t4 for the total 1.55±0.06 t . Notably, the direct substitution γmix ∼ 2 kinetic energy with u ∼ t and l (t) ∼ h(t) ∼ t , where 3 1 i (n) 2 (n) 2 0.65±0.04 u /li ∼ t has a better agreement with R h(t) R L/2 R L/2 2 3 (n) 0.71±0.17 u(x, t) d x the observed γmix(t) ∼ t . However, the large u(t)2 = −h(t) −L/2 −L/2 (24) 2h(t) 10 V. Skoutnev, E. R. Most, A. Bhattacharjee, A. A. Philippov

tra maintains roughly the same level of quasi-isotropy at all scales. Both of these patterns are observed in stably stratified turbulence with the only difference that the horizontal velocity components dominate at large scales instead (Skoutnev et al. 2021). Thus, the effect of the large-scale anisotropy on the dynamo in the mixing phase can perhaps be modeled as a reduction in the effective Reynolds number through an effective Froude number, F r < 1, similar to the way the buoyancy Reynolds number, Rb = F r2Re, controls the dynamo growth rate in stably stratified turbulence (Sk- outnev et al. 2021). This extension may help explain

the signiﬁcant discrepancy of the exponent relating γmix and the Atwood number in Table1, for instance.

Decay Phase —The scatter in the data and uncertainty in the scaling exponents is unfortunately large in the decay phase. The uncertainty for some exponents in Table1 is Figure 6. Normalized component energy kinetic (blue), comparable to the mean values, making it not possible EK,i(k), and magnetic (red), EM,i(k), spectra in the mixing to meaningfully compare with asymptotic predictions of phase (at t = 2) of the fiducial simulation. The kinetic and the model. We attribute this to intermittency in the magnetic spectra are normalized by the total kinetic and RTI turbulence where occasionally a intermediate-scale, magnetic energy at t = 2 , respectively. heavy parcel remains suspended for longer than usual and causes residual large-scale forcing, which can be seen (n) uncertainty of the time dependence of γmix makes any as brief increases of KE(t) and 2KEz(t)/KEh(t) in the conclusive determination difficult. decay phase of some runs in Figures4 and5, respec- Overall, a detailed understanding of the dynamo in the tively. The main assumption that the turbulence in the mixing phase of the RTI requires a further systematic decay phase is freely decaying may perhaps be violated. study of the effects of initial conditions, box aspect ratio, It is not obvious whether residual forcing effects will per- and diffusivities, which we leave for future work. sist in the decay phase at even higher Reynolds numbers.

We would like to note two speculatively important Additionally, the large uncertainty in γdec may also be modifications to the model not examined in this study: a finite Reynolds number effect since the rapid decay 1) accounting for the non-steady-state nature of the mix- of Re(t) from the moderate value of Re in our simula- ing phase turbulence 2) incorporating effects of large- tions may violate the assumptions that Re Rec and scale anisotropy in the velocity field into the dynamo ∆tdec tdyn/CE. growth rate. The first modification requires including The asymptotic time dependence of the dynamo the time delay between the buoyancy forcing at the scale growth rate and kinetic energy similarly has large uncer- 0 li(t) and the dissipation rate (t), which is dissipat- tainties. We find that li(t) ≈ t with li ≈ 0.3L through- ing cascading energy due to forcing from earlier times out the entire decay phase, suggesting s = 0 as expected (Livescu et al. 2009). This is important because the since the turbulence has reached the box scale. This pre- dynamo operates at the dissipation scale and can more dicts an asymptotic scaling u(t) ∼ t0−1, KE(t) ∼ t0−2, 1 0−1.5 0 accurately be expressed as γ(t) ∼ ((t)/ν) 2 (Beresnyak and γ(t) ∼ t , where t = 0 is the beginning of the de- 3 2012), with (t) ∼ u(t) /li(t) a good approximation only cay phase. The numerical power law fit (Figure4) gives if the cascade rate is faster than the time rate of change KE(n) ∼ t0−4.36±1.35, and γ(n) ∼ t0−4.74±1.59, which of li(t) and u(t). are both significantly steeper than the model prediction. The second modification may require incorporating The behavior of decaying RTI turbulence and associated the effect of large-scale velocity anisotropy in the dy- dynamo is clearly poorly approximated by the model of namo growth rate. As shown in Figure6, the velocity freely decaying, isotropic turbulence. spectra in the mixing phase is highly anisotropic at large Fortunately, the decay phase has a smaller contribu- scales with a dominant vertical component and becomes tion to the total dynamo amplification factor (see bot- quasi-isotropic at intermediate and smaller scales (for tom panel of Figure3) than either the mixing or sat- kL/2π & 10) while the magnetic field component spec- uration phases, which mitigates the effects of the large uncertainties. Reducing the uncertainties to better un- Scaling of Small-Scale Dynamo Properties in the Rayleigh-Taylor Instability 11 derstand the decay phase will likely require simulations with much higher Reynolds numbers.

3.3. Saturation In this section we study the case where the dynamo is able to saturate before the RTI fully relaxes. We simulate this regime by initializing the magnetic energy with ME(t = 0) ≈ 10−5PEmix for the fiducial run, 1 − 2 −∆ which satisfies the ME(t = 0) ≥ Resat KEsate = O(10−7) · PEmix criterion for the dynamo to reach the dynamical regime as discussed in Section2. The energy evolution is shown in the inset of Figure7 where the dynamical regime is observed to begin around t ≈ 4 when the slope of the magnetic energy growth sharply drops. The following non-linear growth phase appears to last only briefly until t ≈ 5 (tdyn ≈ 0.5 for the fiducial 1 − 2 simulation). This is expected since the Resat ∼ 0.03 for the fiducial simulation and the final ratio of magnetic to kinetic energy is f ≈ 0.1, leading to a short predicted time to saturation in the dynamical regime of ∆td.r. ∼ Figure 7. Main figure shows the kinetic energy spectrum 1 − 2 tdyn(f − Resat ). (solid lines) and magnetic energy spectrum (dashed lines) To qualitatively test the phenomenological model of right after the saturation phase at t = 4 (red) and in the the dynamo in the dynamical regime discussed in Sec- decay phase (blue) at t = 10. The spectra are normalized by the potential energy that would be released from com- tion2, we plot the isotropic energy spectra of a cubic plete mixing PEmix. The inset shows the time evolution of volume in the region −L/2 ≤ z ≤ L/2 at two repre- the kinetic energy (solid lines) and magnetic energy (dashed sentative times t ∈ {4, 10} in Figure7. At t = 4 (red lines), both also normalized by PEmix. curves) just after the dynamo has enter the non-linear growth phase, the magnetic energy is larger on a scale Kelvin-Helmholtz instability (KHI) active at the con- by scale basis below an intermediate scale kL/2π ∼ 30. tact region of the two stars in contributing to the mag- In the fully saturated phase of the dynamo at t = 10 netic field amplification observed in the core of the post- (blue curves), the magnetic energy is larger than the ki- merger (Kiuchi et al. 2015; Aguilera-Miret et al. 2020). netic energy at basically all but the largest scales. Both It is generally assumed that the Reynolds number of the of these observations are in accord with the predictions KHI turbulence is high enough that the dynamo will sat- from the current understanding of the dynamical dy- urate in near equipartition with the kinetic turbulence, namo regime in steady-state turbulence as discussed in although it is unclear how fast that will happen (Ki- Section2. uchi et al. 2018). The KHI, however, cannot explain Overall, these results show that the RTI is capa- the large amount of dynamo action that is observed in ble of generating small-scale magnetic fields in near- the surface layer during the merger (Kiuchi et al. 2015). equipartition (ME/KEdec = f = O(10−1)) with the We propose that the RTI is likely responsible for dy- decaying turbulent kinetic energy, KEdec, which is a namo action in the outer parts of the merger where cen- fraction on the order of KEdec/P Emix = O(10−2) of trifuged denser matter can be seen falling down onto the initially available potential energy. These magnetic less dense matter at adjacent longitudes. Rapid satura- fields then can act as seed fields for further processes tion of the dynamo and the associated amplification of driven by larger scales in the astrophysical object. the magnetic field to near-equipartition with the RTI- 4. RAYLEIGH-TAYLOR DRIVEN DYNAMO IN driven turbulence might critically affect the prospects NEUTRON STAR MERGERS of long-term mass ejection (Metzger et al. 2018; Ciolfi & Kalinani 2020), as well as jet launching (Ciolfi 2020; One open question in recent studies of binary neu- Möstaet al. 2020) from a stable magnetar remnant. Us- tron star mergers is the source, efficiency, and time scale ing our model, we verify that the RTI-driven dynamo of magnetic field amplification in the post-merger star does saturate and provide a quantitative estimate for (Price & Rosswog 2006; Kiuchi et al. 2015; Giacomazzo et al. 2015). Several studies have proposed the role of the 12 V. Skoutnev, E. R. Most, A. Bhattacharjee, A. A. Philippov the timescale for saturation in the conditions of the post time. The estimate is as follows: merger neutron star envelope. ! We first need an estimate for the Reynolds number 1 KE ∆tk.r. ∼ ln 1 γ 2 Resat. The viscosity of neutron star matter is strongly ResatME(0) dependent on the temperature and density conditions. ! KE −7 −3 Assuming low temperatures of T ' 1 MeV and den- ∼2 log10 1 × 10 − 10 µs 14 3 2 sities around nuclear saturation, i.e. 2 × 10 g/cm , ResatME(0) −3 2 the kinematic shear viscosity ν ∼ 3 × 10 m /s in tdyn. (25) the electron scattering dominated regime (Shternin & Yakovlev 2008). On the other hand, for temperatures After the kinematic regime, the dynamo will continue T ' 10 MeV, neutrino-emitting Urca processes (e.g. to grow in the dynamical regime and fully saturate on a − − n → p e ν¯e and p e → n νe ) might be the domi- timescale of 4 2 nant contribution, leading to ν ' 10 m /s at densities 1 − 2 around saturation (Alford et al. 2018). The resistivity ∆td.r. ∼ tdyn(f − Resat )/ζ for a warm (T ' 1 MeV) neutron star crust is set by ∼ tdynf/ζ electron scattering of correlated nuclei (Harutyunyan & ∼ O(10−1)µs (26) Sedrakian 2016) and has a value of η ' 5 × 10−7m2/s (Harutyunyan et al. 2018), which puts neutron star mat- using the model of the non-linear dynamo growth phase ter well into the high P m regime with P m = O(104−11). described in Section2 and assuming f = O(10−1) − 1 The extreme density gradients at the surface of a neu- 2 Resat . tron star, imply that the Atwood number A ≈ 1. We Since the duration of the kinematic dynamo regime consider a thin layer exhibiting these gradients and be- (upper bound of nanoseconds) and duration of the dy- coming RTI unstable on a length scale that is a frac- namical dynamo regime (upper bound of microseconds) tion of a scale height L ∼ 0.1H (H ∼ 1 km). We of the RTI-driven turbulence in the envelope are both approximate the gravitational acceleration by means much smaller than the relaxation time of the merger of the Newtonian expression for surface gravity, i.e. (order of milliseconds), we expect magnetic energies to GM 16 2 g ' R2 ≈ 3 × 10 m/s . Although technically, usat be in near-equipartition with the kinetic energy of the as obtained from the expressions above would be su- RTI-driven turbulence in the envelope across essentially perluminal (owing to the simplified Newtonian assump- the entire of duration the post-merger evolution. tions made here), we take this as an indication that usat ' c. Finally, all these parameters correspond to 5. SUMMARY AND CONCLUSIONS 6 13 an estimate Resat ∼ O(10 − 10 ). If we assume our 1 We present a model for the kinematic small-scale dy- 2 model holds (∆ ≈ C∆Resat with C∆ = O(1)), then in namo in the mixing, saturation, and decay phases of a RTI unstable region the expected amplification would the Rayleigh Taylor instability of an ionized, collisional be ∆ ∼ O(103 − 106). The assumption that the dy- plasma with free-fall and isotropy assumptions for the namo will reach equipartition with the kinetic energy turbulence in each of the phases. The model quantita- 1 of the viscous scales, KE/Re 2 is easily justified since tively predicts scaling relations between the properties ∆ 1 e KE/(ME(0)Re 2 ) even for generous estimates of of the dynamo (growth rate and total magnetic energy order KE/ME(0) ∼ 10O(10), where ME(0) is the ini- exponential amplification factor) and parameters of the tial magnetic energy density near the surface of one of RTI (Atwood number, gravitational acceleration, length the initial neutron stars and KE is the characteristic scale, and viscosity). The model predictions are tested turbulent kinetic energy in post-merger envelope. with sets of three dimensional direct numerical simula- We can now estimate how long it will take the dynamo tions that solve the visco-resistive MHD equations using to leave the kinematic regime (∆tk.r.) using the dynamo the Athena++ code. The main results are itemized be- 1 −1 2 3 7 −1 low: growth rate γ ∼ tdynResat ∼ O(10 − 10 )µs , where we have used tdyn = L/usat ∼ 0.3 µs for the dynamical • We find that the total magnetic energy exponential amplification factor, ∆, based on simulation data scales as α ∆ ≈ C∆Resat (27)

with constants C∆ ≈ 0.4 and α ≈ 0.65 ± 0.04. This is in fairly close agreement with the model Scaling of Small-Scale Dynamo Properties in the Rayleigh-Taylor Instability 13

prediction of α = 0.5, but the difference is sta- magnetic energy amplification in the outer regions tistically significant. An analysis of the dynamo of the post-merger in global simulations, comple- in each phase reveals that the model correctly mentary to amplification by saturation of the the predicts scaling relations in the saturation phase, Kelvin-Helmholtz instability observed in the core. while having several discrepancies in the mixing Applying the scaling relations to the parameter and decays phases. regime of RTI-unstable regions of the outer en- velopes of binary neutrons star mergers, the model • An analysis of the dynamo scaling relations and predicts that the kinematic regime of the small- time dependence in the mixing phase shows that scale dynamo will end on the time scale of nanosec- the free-fall and isotropy assumptions are fairly onds and then reach saturation on a timescale well supported with a few minor discrepancies, of microseconds, which are both fast in compar- which are attributed to deviations of the evolution ison to the millisecond relaxation timescale of the of the hydrodynamic turbulence from free-fall pre- merger. dictions. For example, we do not find strong agreement with the free-fall predictions of quadratic The flexibility of the model allows for easy extensions scaling of mixing height with time, linear scaling of in future studies of the dynamo in RTI turbulence. The the root mean square velocity with time, nor con- model primarily prescribes a time dependence for the stant proportionality between mixing height and instantaneous integral length and outer velocity scale in the instantaneous integral scale. These discrepan- each phase of the RTI, which are then substituted into cies are well known in the literature and primarily an equation for the dynamo growth rate. Using alter- attributed to the choice of initial conditions and native time dependencies based on the choice of initial the effects of finite diffusivities in simulations. We conditions, or allowing a time delay between forcing and leave a more detailed analysis of the SSD in the dissipation could improve understanding of the SSD in mixing phase for future studies. the mixing phase, for example. We leave such extensions for future work. • In the decay phase, the dynamo scaling relations and time dependencies have large uncertainties and our model assumption of freely, decay- ACKNOWLEDGMENTS ing isotropic turbulence is not a good fit. We attribute this to residual buoyant forcing and pos- We acknowledge the Flatiron’s Center for Computa- sibly finite Reynolds number effects. Fortunately, tional Astrophysics (CCA) and the Princeton Plasma Physics Laboratory (PPPL) for the support of collabora- the magnetic amplification in the decay phase is small compared to the contributions from the mix- tive CCA-PPPL meetings on plasma-astrophysics where ing and saturation phases. insightful comments and discussions contributed to this work. Research at the Flatiron Institute is supported by • In the saturation regime of the small-scale dy- the Simons Foundation. Simulations were carried out on namo, the magnetic field is found to reach near- the Frontera cluster with NSF Frontera grant number equipartition with the large scales and super- AST20008. A. B. was supported by the DOE Grant equipartition with the intermediate and small for the Max Planck Princeton Center (MPPC). A.P. scales of the decaying velocity field of the RTI. We acknowledges support by the National Science Foun- study this regime by running a single simulation dation under Grant No. AST-1909458. E.R.M. grate- with a moderate initial magnetic field so that the fully acknowledges support from a joint fellowship at the dynamo reaches saturation before the RTI fully Princeton Center for Theoretical Science, the Princeton relaxes. Gravity Initiative and the Institute for Advanced Study. V. S. was supported by Max-Planck/Princeton Center • We propose that the small-scale dynamo driven for Plasma Physics (NSF grant PHY-1804048). by RTI turbulence helps explain observations of

APPENDIX 14 V. Skoutnev, E. R. Most, A. Bhattacharjee, A. A. Philippov

Figure 8. Main figure shows the effective numerical viscosity versus the grid resolution for our numerical setup. The inset plot shows the the exponentially decaying kinetic energy (solid lines) of the Alfven wave at different resolutions and fits (dashed black lines) that provide estimates for the effective numerical viscosities.

A. DECAYING ALFVEN WAVE In a grid-based code like Athena++, numerical diffusivity acts as a resolution-dependent and algorithm-dependent effective viscosity that regularizes the turbulent cascade in a hydrodynamical simulation without an explicit viscosity. An explicit viscosity will only be meaningful if it is larger than the effective viscosity. We estimate the effective numerical viscosity of our numerical setup (RK3 for the timestepper and HLLD for the Riemann solver) by launching an Alfven wave along the main diagonal in a cubical domain with zero explicit viscosity and√ measuring the decay rate. The Alfven wave has wave number k = (1, 1, 1) · 2π/L in a background field B0 = (1, 2, 0.5) and the domain has 3 triply-periodic boundary conditions with a resolution Ngrid. The kinetic energy of the wave will decay approximately −2ν k2t exponentially KE(t) ∼ e eff . We measure νeff with a linear fit on a plot of log(KE(t)) versus time for four grid −1.51 resolutions as shown in the inset of Figure8. The main plot in Figure8 shows a clean power-law fit νeff ∼ Ngrid . The −6 value of the numerical viscosity is approximately νeff ≈ 10 for the resolution Ngrid = 512 used in our simulations of the RTI. This informs our choice of the explicit viscosity ν ≈ 3 · 10−6 for the fiducial simulation and ν = 2 · 10−6 for our lowest choice of viscosity in the viscosity parameter scan in Section3.

Software: Athena++ (Stone et al. 2019)

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