The Astrophysical Journal, 835:170 (5pp), 2017 February 1 doi:10.3847/1538-4357/835/2/170 © 2017. The American Astronomical Society. All rights reserved. The Standing Accretion Shock Instability: Enhanced Growth in Rotating Progenitors John M. Blondin1, Emily Gipson1, Sawyer Harris1, and Anthony Mezzacappa2,3 1 Department of Physics, North Carolina State University, Raleigh, NC 27695-8202, USA Department of Physics and Astronomy, University of Tennessee, Knoxville, TN 37996-1200, USA 3 Joint Institute for Computational Sciences, Oak Ridge National Laboratory, P.O. Box 2008, Oak Ridge, TN 37831-6173, USA Received 2016 October 3; revised 2016 October 28; accepted 2016 October 31; published 2017 January 27 2 Abstract We investigate the effect of progenitor rotation on the standing accretion shock instability (SASI) using two- and three-dimensional hydrodynamic simulations. We ﬁnd that the growth rate of the SASI is a near-linearly increasing function of the speciﬁc angular momentum in the accreting gas. Both the growth rate and the angular frequency in the two-dimensional model with cylindrical geometry agree well with previous linear stability analyses. When excited by very small random perturbations, a one-armed spiral mode dominates the small rotation rates predicted by current stellar evolution models, while progressively higher-order modes are seen as the speciﬁc angular momentum increases. Key words: accretion, accretion disks – hydrodynamics – shock waves – supernovae: general Supporting material: interactive ﬁgures as a symmetry-breaking process, in which one spiral component of a sloshing mode dominates over the other. Blondin & Mezzacappa (2007) also explored the inﬂuence of progenitor rotation on the SASI spin-up scenario, and presented results from three simulations in which the infalling core material has a moderate speciﬁc angular momentum. The evolution of the SASI was markedly different in these rotating models. Rather than exhibit an initial phase characterized by the growth of the l=1 axisymmetric mode out of the initial numerical noise, the rotating models exhibited a rapid growth of the nonaxisymmetric mode right at the start of the simulation. An increased prominence of spiral modes relative to sloshing modes was also observed in 3D simulations by Iwakami et al. (2009), although their nonsteady initial conditions limited them to a comparison of the nonlinear phase in simulations with and without progenitor rotation. These qualitative numerical results are complemented by linear analyses (Yamasaki & Foglizzo 2008, hereafter YF08) that suggest the linear growth rate of nonaxisymmetric modes of the SASI increases with increasing rotation of the progenitor core. Our goal in this paper is to present 2D hydrodynamical simulations to conﬁrm the linear-stability analysis of YF08 and to extend this result to the physically relevant spherical geometry using both 2D equatorial simulations and full 3D simulations. 1. Introduction The current paradigm for core-collapse supernovae involves a relatively extended epoch, in which the nascent supernova shock stalls deep in the core. This phase, which can last for hundreds of milliseconds, can be modeled with reasonable accuracy as a steady-state accretion shock (Janka 2001). Hydrodynamic studies of steady, spherical accretion shocks have revealed the existence of a dynamical instability in this stalled epoch that has come to be know as the standing accretion shock instability, or SASI (Blondin et al. 2003). Using three-dimensional (3D) simulations, Blondin & Mezzacappa (2007) showed that although the SASI does indeed exist in 3D, the late-time evolution of a stalled accretion shock—under the idealized conditions of their model—is dominated by nonaxisymmetric modes. A surprising observation from these 3D simulations was that the nonaxisymmetric modes of the SASI could lead to the rapid accretion of a signiﬁcant angular momentum even in the case of a nonrotating progenitor. These nonaxisymmetric modes were studied by Blondin & Shaw (2007), who used 2D simulations in the equatorial plane of a spherical grid. They demonstrated that the nonaxisymmetric modes of the SASI are linearly unstable and that their growth rate is a decreasing function of the wavenumber, m. Moreover, they showed that these SASI modes correspond to a pressure wave propagating azimuthally around the accreting proto-neutron star, with the peak pressure perturbation near, but not at, the interior surface of the accretion shock. Fernández & Thompson (2009) argued that in addition to the lateral acoustic waves, the growth of the SASI involves radial advection similar to the original vortical-acoustic cycle described by Foglizzo (2002). The axisymmetric “sloshing” modes originally observed by Blondin et al. (2003) can be constructed by combining two equal and opposite nonaxisymmetric “spiral” modes (Blondin & Shaw 2007). Equivalently, Fernández (2010) interpreted spiral modes as the superposition of two sloshing modes. Kazeroni et al. (2016) interpreted the presence of a spiral mode 2. Numerical Model of Standing Accretion Shock The idealized model of a standing accretion shock presented in this paper is similar to previously reported simulations (Blondin & Mezzacappa 2006; Blondin & Shaw 2007; Fernández & Thompson 2009), but with the addition of angular momentum in the accreting gas. We also include the option of using a 2D cylindrical geometry to facilitate a direct comparison with YF08. Speciﬁcally, the unperturbed model is the steady-state accretion of an ideal gas, with an adiabatic index of g = 4 3, onto a spherical surface of radius r* subject to a Newtonian potential of U = -GM r . We use the steadystate solution for a spherical accretion shock given by Houck & Chevalier (1992) using cooling parameters of a = 3 2 and 1 The Astrophysical Journal, 835:170 (5pp), 2017 February 1 Blondin et al. b = 5 2 along with its analog in a cylindrical geometry. In both cases, the amplitude of the cooling function is adjusted to produce a nominal value for the shock stand-off distance of rs r* = 5. We scale this model to the radius of the accretion shock, rs, and the freefall velocity at the accretion shock, vf (in the absence of rotation). Note that the post-shock density varies as r -3 nears the shock in both the cylindrical and spherical cases. The primary difference is a faster deceleration, v∝r2, in the cylindrical case compared to v µ r in the spherical case (Kazeroni et al. 2016). Since the radial proﬁle of the sound speed is similar, this means that the interior Mach number is lower in the cylindrical case compared to the spherical case. We added rotation to these models in the same manner as YF08, assuming a constant speciﬁc angular momentum, h = rvf , in the supersonic, infalling gas. The effective gravitational acceleration in this scaled model is given by ge = h2 0.5 - 2 , r3 r Figure 1. Post-shock entropy proﬁles with increasing speciﬁc angular momentum. The height above the surface of the accreting star is normalized to the shock height. The two bold lines correspond to h=0.16, above which the entropy at the base begins to increase with an increasing rotation; and h=0.25, which is the fastest rotator for which we could generate a steadystate model. (1 ) the accretion shock, which in our scaled model is given by hence the effective gravity at the accretion surface is zero when h=0.316. Using the ﬁducial supernova parameters listed in YF08 (M = 1.3M, r* = 50 km ), h is in units of rs vf = 2GMrs = 9.3 ´ 1016 cm2 s-1 for rs r* = 5. This gives a factor of 14820 difference in the scaling of a speciﬁc angular momentum used in this paper relative to that used in YF08. For comparison, Heger et al. (2005) ﬁnd a speciﬁc angular momentum in a supernova progenitor core of the order ~3 ´ 1016 cm2 s-1 (close to the upper limit of h=0.3 in our scaled units), without magnetic braking and 1015 cm2 s-1 (a scaled h ~ 0.01) when including magnetic torques (Spruit 2002). We used the time-dependent hydrodynamics code VH-1 to evolve this accretion shock model in 1D, 2D, and 3D. Because we are interested in the linear growth, during most of the evolution, the spherical shock is virtually stationary with respect to the numerical grid. Nonetheless, we included strong shock dissipation in the form of strong ﬂattening parameters and grid wiggling in the angular direction (Colella & Woodward 1984) to minimize the carbuncle instability (Quirk 1994) at the shock and produce a quiet, post-shock gas even when the shock begins to deviate from spherical. The ability to measure the growth of speciﬁc SASI modes in the linear regime requires a very quiet numerical steady state that minimizes random numerical perturbations seeding multiple SASI modes. To create such quiescent initial conditions for the multi-dimensional simulations, we ran a series of 1D models that vary the speciﬁc angular momentum from zero up to the maximum value that would produce a steady solution. In each model, we evolved past a time of 100 rs vf to ensure a steady state. In order to keep the accretion shock at a steadystate radius of unity for all models, we derived an empirical function for the cooling amplitude as a function of the speciﬁc angular momentum. The dependence of the steady, symmetric solutions on an speciﬁc angular momentum is shown in Figure 1 for the spherical geometry case. For values of h below about 0.1, the post-shock proﬁle is only slightly changed by the presence of rotation, with a uniformly lower entropy throughout the postshock region as the rotation rate increases. This entropy shift is consistent with the change of the entropy immediately behind s = P rg = 0.064 (1 - 2h2)7 6. (2 ) At around h » 0.16, the entropy at the base of the settling ﬂow begins to increase with an increasing h rather than decreasing. By h » 0.25, the post-shock proﬁle has become signiﬁcantly ﬂattened, with signiﬁcant energy loss and deceleration occurring in the last numerical zone above the accreting surface. We are not able to adequately resolve the inner boundary at these high rotation rates and it may be that a steady-state solution does not exist in this regime of rapid rotation. A similar behavior is seen in the case of the cylindrical geometry, although the transition to ﬂatter proﬁles begins at a slightly larger value of h » 0.18 3. SASI Growth in Two-Dimensions The 2D simulations reported herein use a numerical grid with 1024 evenly spaced zones in angles from 0 to 2p using periodic boundary conditions, and 384 logarithmically spaced zones in a radius from r* = 0.2 to 2.0, which provides a roughly constant value of Dr r that is also comparable to Df . Note that this nonuniform spacing in the radius provides higher spatial resolution near the steep gradients created by cooling in the vicinity of the accreting stellar surface. The strength of the SASI was measured by computing the power in the m=1 Fourier components of the deviation of angular momentum from the initial value: 2p a (r ) = ò0 b (r ) = ò0 2p (h - rvf) cos fdf , (3 ) (h - rvf) sin fdf. (4 ) The amplitude of the SASI is then measured by the radial integral of the full amplitude over the region of shocked gas, C= rs òr* a2 (r ) + b2 (r ) dr. (5 ) An exponential function is ﬁtted to each growth curve of the amplitude to extract the growth rate, Im(ω). The phase angle, given by tan (d ) = a b , is computed at a radius slightly inside 2 The Astrophysical Journal, 835:170 (5pp), 2017 February 1 Blondin et al. Figure 2. Example evolution illustrating the growth of the m=1 spiral mode in the spherical geometry for h=0.035. The images show the deviation of the angular momentum from the initial value, h - vf r , with an arbitrary normalization so that the pattern can be seen while the amplitude changes by several orders of magnitude. Red is positive values, blue is negative, and white is zero. The center of each image corresponds to roughly the time each image was recorded. Figure 3. Different SASI modes are excited at different levels of rotation. The values of the speciﬁc angular momentum in the spherical accreting ﬂow are, from left to right, h=0.025, 0.115, 0.205, and 0.245. the accretion shock. The time evolution of the phase data is corrected for shifts of π, and values near p 2 are dropped because of the large scatter. The resulting phase curve is ﬁtted with a straight line, the slope of which gives the angular speed of the SASI wave corresponding to the Re(ω). We ﬁrst present a series of runs using spherical divergence, without any initial perturbation other than the inherent numerical noise of the simulation. This approach allows us to follow the growth from a very small amplitude, but it does not provide control over the mode that is excited. An example evolution that is dominated by m=1 is shown in Figure 2 for a model with h=0.035. In this case, the amplitude can be followed over six orders of magnitude. Note that the maximum amplitude represented by the color scale in the ﬁrst image is low enough that the acoustic noise can be seen overlaid on the m=1 spiral mode of the SASI. We ﬁnd that higher-order angular modes dominate at higher rotation rates, as illustrated in Figure 3. Below h ~ 0.08, the evolution is dominated by m=1. Above this value, m=2 is the dominant mode, with higher values of m appearing at higher rotation rates. This behavior is consistent with YF08, who found that m=2 becomes the dominant mode for h > 0.04 in the case of cylindrical geometry. We ﬁnd roughly this same transition value in our cylindrical models. Note that YF08 did not show results for rotations above h=0.07 in our units, but if the transition for m=2 to m=3 is as abrupt as the transition from m=1 to m=2, then their results are consistent with such a transition occurring at a value of h > 0.07. In order to explicitly excite the m=1 spiral mode of the SASI, we added to the gas above the shock a slight (0.1%) angular velocity perturbation with a cos (f ) dependence, but with an f offset at different radii to account for the wave speed of the SASI. Speciﬁcally, we added a phase offset given by t · Re (w ), where we use a linear ﬁt to the data shown in Figure 5 to ﬁnd Re(ω)≈1.8h+0.45 (for the cylindrical Figure 4. Growth curves of the m=1 mode of the SASI in the spherical model showing a systematic increase in the growth rate with increasing values of speciﬁc angular momentum in the infalling gas. These growth curves span values of h from 0.01–0.158. geometry: Re(ω)≈2.3h+0.40) and τ is the freefall time to reach the shock (r = 1) from a given radius, t= ò1 r dr vf . (6 ) Other forms of perturbation were tried, including density, pressure, and velocity above and below the shock. Most of these gave similar results for small values of h, but appeared to excite more complicated modes (e.g., multiple radial nodes) at a high rotation. We ran two identical sets of models with varying speciﬁc angular momentums, one set with a radial geometry corresponding to a cylindrical divergence and one corresponding to a spherical divergence. Figure 4 shows a series of growth curves for the spherical model covering a range of h from 0.01–0.146. The resulting growth rates as a function of rotation are plotted in Figure 5. We ﬁnd that the growth rate of the m=1 mode is a roughly linearly increasing function of the speciﬁc angular 3 The Astrophysical Journal, 835:170 (5pp), 2017 February 1 Blondin et al. Figure 5. Growth rate and frequency of the m=1 mode of the SASI as a function of the speciﬁc angular momentum in the progenitor core for the spherical model. Also shown are linear ﬁts at low rotation rates: Im(ω)≈1.5h+0.15, Re(ω)≈1.8h+0.45. Figure 6. Growth rate and frequency of the m=1 SASI wave as a function of the speciﬁc angular momentum in the progenitor core for the cylindrical model. The solid lines are from the linear stability analysis of YF08. 1D to provide quiescent initial conditions for studying the SASI in 2D means evolving the ﬂow in 2D to achieve such a steady state for 2D. This 2D ﬂow, however, is subject to the SASI. For our choice of rs r* = 5, the accretion shock is just at the edge of instability for l=2, but unstable for l=1 (Foglizzo et al. 2007). To prevent the growth of the unstable l=1 mode, we evolve only one quadrant for the initial conditions, assuming the equator’s reﬂection symmetry. The l=2 mode is observed, but gradually decays away. This 2D solution is then mapped onto the 3D Yin–Yang grid. The resulting 3D structure of the spiral SASI near the end of the linear stage is illustrated in Figure 7, and the wave pattern in the equatorial plane is compared with the corresponding 2D simulation in Figure 8. The m=1 spiral SASI is excited in the same manner as the 2D simulations, with the perturbation in vf dependent on f and the radius (to match the spiral pattern of the SASI) but independent of θ. The amplitude is computed without any assumption of the functional dependence on θ. The evolutions of the amplitudes for the 3D simulation and the corresponding 2D model are shown in Figure 9. The time axis is shifted in such a way that both simulations reach the transition to nonlinear evolution at roughly the same time. The growth curve for the 3D simulation shows some oscillation relative to the 2D results, which suggests the presence of power in the sloshing (l = 1) mode. Nonetheless, the overall growth rate is consistent with the growth rate found in the equivalent 2D simulation of the equatorial wedge. momentum in the infalling gas up until h ~ 0.15. This is also the value of h where rotation starts to signiﬁcantly change the radial structure of the post-shock ﬂow as shown in Figure 1. The rotational frequency of the spiral SASI wave increases roughly linearly with the speciﬁc angular momentum of the infalling gas. If this frequency corresponds to a wave traveling in a circular path around the accreting star, one would expect a Doppler shift of the wave speed (YF08), w = w0 + W (r ), where W (r ) = h rp2 is the rotational frequency of the gas at a characteristic propagation radius, rp. Note that this deﬁnition of rp is not the same as the co-rotation radius deﬁned by YF08. The constant slope found for Re(ω) implies a value of rp independent of h for low to moderate values of rotation. Using a slope of 1.8 measured from the data in Figure 5, we infer a characteristic propagation radius of rp » 0.75, which is relatively close to the accretion shock. Similarly, if this SASI wave is traveling at the speed of sound, cs, we can use the wave speed in the absence of rotation, together with the post-shock proﬁle of the sound speed, to ﬁnd the radius at which w0 = cs r . This implies an effective propagation radius of 0.85. Our numerical results for the cylindrical model are in good agreement with the linear results of YF08, as shown in Figure 6. Values for Re(ω) from YF08 were computed from their deﬁnition of the corotation radius, rco. The linear dependence on h of both the wave speed and the growth rate fades beyond h ~ 0.07. Moreover, at higher values of h, it is difﬁcult to ﬁnd a consistent way of always exciting the lowest radial harmonic of the m=1 mode. Nonetheless, for the range of h reported in YF08, our results are in good agreement. 5. Conclusion We have used time-dependent hydrodynamic simulations of an idealized standing accretion shock to investigate nonaxisymmetric modes of the SASI in the presence of a progenitor rotation. We speciﬁcally address moderate rotation rates with a speciﬁc angular momentum in the range of 1015 - 1016 cm2 s-1, consistent with current stellar evolution models (Heger et al. 2005). We have shown that: 4. SASI Growth in Three-Dimensions We extended our numerical model to 3D using “Yin–Yang” spherical overset grids (Kageyama & Sato 2004). We made every effort to keep the numerical details of the 2D and 3D simulations the same, which includes using the same radial resolution. The angular resolution, however, was cut in half to reduce the computational time. We assume the speciﬁc angular momentum of the infalling gas varies as h (q ) = h 0 sin2 q , such that the angular velocity, vf µ sin q , approaches zero at the rotation axis (Iwakami et al. 2009). In the presence of an angular momentum, the steady-state unperturbed ﬂow outside of the orbital plane is necessarily 2D. Following our procedure of evolving the ﬂow to a steady state in 1. Both the growth rate and the angular frequency of the m=1 spiral mode of the SASI increase linearly with increasing speciﬁc angular momentum of the accreting gas. The linear increase begins to break down at high rotation rates where the radial proﬁle of the unperturbed accretion ﬂow shows signiﬁcant changes. 2. In the cylindrical geometry, these results are in good agreement with the linear stability analysis of YF08. 4 The Astrophysical Journal, 835:170 (5pp), 2017 February 1 Blondin et al. Figure 7. Three-dimensional structure of the m=1 mode of the SASI is illustrated with the deviation of speciﬁc angular momentum from the axisymmetric initial conditions. The left image is looking down the rotation (z) axis. The right image is looking along the orbital plane. The accretion shock is represented by the transparent gray spherical surface and the accreting star represents the inner solid spherical surface. An interactive version of this ﬁgure is available online. Figure 8. The two-dimensional simulations of the SASI (left image), illustrated here by h - vf r , provide a close approximation of the ﬂow in the equatorial plane of full 3D simulations (right image). E.G. and S.H. were supported by NSF REU award AST1062736. Computer simulations were run at the Texas Advanced Computing Center (TACC) at The University of Texas at Austin using an allocation from the Extreme Science and Engineering Discovery Environment (XSEDE), which is supported by National Science Foundation grant number ACI1053575. References Blondin, J. M., & Mezzacappa, A. 2006, ApJ, 642, 401 Blondin, J. M., & Mezzacappa, A. 2007, Natur, 445, 58 Blondin, J. M., Mezzacappa, A., & DeMarino, C. 2003, ApJ, 584, 971 Blondin, J. M., & Shaw, S. 2007, ApJ, 656, 366 Colella, P., & Woodward, P. 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If the linear increase in angular speed of the SASI wave is interpreted as a Doppler boost, these results imply that the SASI perturbation pattern are propagating in the angular direction at an effective radius of roughly 75% of the accretion shock radius. 4. In the one 3D example presented herein, the growth rate and the angular speed of the SASI is the same in 3D as those found in 2D simulations of the equatorial wedge. 5

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