wakana iwakami et al- three-dimensional simulations of standing accretion shock instability in...

8/3/2019 Wakana Iwakami et al- Three-Dimensional Simulations of Standing Accretion Shock Instability in Core-Collapse Sup

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THREE-DIMENSIONAL SIMULATIONS OF STANDING ACCRETION SHOCK INSTABILITYIN CORE-COLLAPSE SUPERNOVAE

Wakana Iwakami,1

Kei Kotake,2,3

Naofumi Ohnishi,1,4

Shoichi Yamada,5,6

and Keisuke Sawada1

Received 2007 October 10; accepted 2008 January 21

ABSTRACT

We have studied nonaxisymmetric standing accretion shock instabilities, or SASI, using three-dimensional (3D)hydrodynamical simulations. This is an extension of our previous study of axisymmetric SASI. We have prepared aspherically symmetric and steady accretion flow through a standing shock wave onto a protoYneutron star, taking intoaccount a realistic equation of state and neutrino heating and cooling. This unperturbed model is meant to represent ap-proximately the typical postbounce phase of core-collapse supernovae. We then added a small perturbation ($1%) tothe radial velocity and computed the ensuing evolutions. Both axisymmetric and nonaxisymmetric perturbations havebeen imposed. We have applied mode analysis to the nonspherical deformation of the shock surface, using sphericalharmonics. We have found that (1) the growth rates of SASI are degenerate with respect to the azimuthal index m of thespherical harmonics Yml , just as expected for a spherically symmetric background; (2) nonlinear mode couplings pro-duce only m 0 modes for axisymmetric perturbations, whereas m 6 0 modes are also generated in the nonaxisym-metric cases, according to the selection rule for quadratic couplings; (3) the nonlinear saturation level of each mode is

lower in general for 3D than for 2D, because a larger number of modes contribute to turbulence in 3D; (4) low-lmodesare dominant in the nonlinear phase; (5) equipartition is nearly established among differentm modes in the nonlinearphase; (6) spectra with respect to lobey power laws with a slope slightly steeper for 3D; and (7) althoughthese featuresare common to the models with andwithout a shock revival at the endof the simulation, the dominance of low- lmodesis more remarkable in the models with a shock revival.

Subject headinggs: hydrodynamics instabilities neutrinos supernovae: general

Online material: color figures

1. INTRODUCTION

Many efforts have been made to construct multidimensionalmodels of core-collapse supernovae (see Woosley & Janka 2005,

Kotake et al. 2006 for reviews), prompted by accumulated obser-vational evidence that core-collapse supernovae are commonlyglobally aspherical ( Wang et al. 1996, 2001, 2002). Various mech-anisms for producing the asymmetry have been discussed, includ-ing convection (e.g., Herant et al. 1994; Burrows et al. 1995; Janka& Mueller 1996), magnetic field and rapid rotation (see, e.g.,Kotake et al. 2006 for collective references), standing (stationary,spherical) accretion shock instability, or SASI (Blondin et al.2003; Scheck et al. 2004; Blondin & Mezzacappa 2006; Ohnishiet al. 2006, 2007; Foglizzo et al. 2006),and g-mode oscillationsof protoYneutron stars (Burrows et al. 2006). Most of these,however, have been investigated only with two-dimensional(2D) simulations.

Recently, a 3D study of SASI was reported by Blondin &

Mezzacappa (2007). In 2D, the shock deformation by SASI isdescribed with Legendre polynomials Pl( ), or spherical har-monics Yml (;) with m 0. Various numerical simulations have

demonstrated unequivocally that the l 1 mode is dominant andthat a bipolar sloshing of the standing shock wave occurs, withpulsational strong expansions and contractions along the sym-

metry axis (Blondin et al. 2003; Scheck et al. 2004; Blondin &Mezzacappa 2006; Ohnishi et al. 2006, 2007). In 3D, on the otherhand, Blondin & Mezzacappa (2007) perturbed a nonrotatingaccretion flow azimuthally and observed the dominance of anonaxisymmetric mode with l 1, m 1, which produces asingle-armed spiral in the later nonlinear phase. They claimedthat this spiral SASI generates a rotational flow in the accre-tion (see also Blondin & Shaw [2007] for 2D computations inthe equatorial plane), and that it may be an origin of pulsar spin.

However, many questions regarding 3D SASI still remain tobe answered. Here we are particularly interested in investigatinghow the growth of SASI differs between 3D and 2D; in particu-lar, the change in the saturation properties should be made clear.Another question is the generation of rotation in the accretion

flow by SASI (Blondin & Mezzacappa 2007); its efficiency and possible correlation with the net linear momentum should bestudied more in detail and will be the subject of a future paper(W. Iwakami et al. 2008, in preparation). We also note that neu-trino heating and cooling were neglected and the flow was as-sumed to be isentropic in Blondin & Mezzacappa (2007), butthe implementation of this physics is helpful in considering theimplications for the shock revival and nucleosynthesis (Kifonidiset al. 2006).

In this paper, we have performed 3D hydrodynamic simula-tions, employing a realistic equationof state (Shen et al. 1998) andtaking into account the heating and cooling of matter via neutrinoemission and absorption on nucleons, as done in our 2D studies(Ohnishi et al. 2006, 2007). The inclusion of neutrino heating al-

lows us to discuss how the critical luminosity for SASI-triggered

A

1Departmentof Aerospace Engineering, TohokuUniversity,6- 6- 01 Aramaki-

Aza-Aoba, Aoba-ku, Sendai 980-8579, Japan; [email protected] Division of Theoretical Astronomy, National Astronomical Observatory of

Japan, 2-21-1, Osawa, Mitaka, Tokyo 181-8588, Japan.3 Max-Planck-Institut fur Astrophysik, Karl-Schwarzshild-Strasse 1, D-85741

Garching, Germany.4

Center for Research Strategy and Support, Tohoku University, 6-6-01Aramaki-Aza-Aoba, Aoba-ku, Sendai 980-8579, Japan.

5Science and Engineering, WasedaUniversity, 3-4 -1 Okubo, Shinjuku, Tokyo

169- 8555, Japan.6

Advanced Research Institute for Science and Engineering, Waseda Univer-

sity, 3-4-1 Okubo, Shinjuku, Tokyo 169-8555, Japan.

1207

The Astrophysical Journal, 678:1207Y1222, 2008 May 10

# 2008. The American Astronomical Society. All rights reserved. Printed in U.S.A.


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explosion couldbe changed in 3D from those in 2D.To answerthequestions raised above, we vary the initial perturbations as well asthe neutrino luminosity, and compare the growth of SASI be-tween 2D and 3D in detail, conducting a mode analysis for boththe linear phase and the nonlinear saturation phase.

The plan of this paper is as follows. In x 2, we describe themodels and numerical methods, show the main numerical resultsin

x3, and conclude the paper in

x4.

2. NUMERICAL MODELS

The numerical methods we employ are based on the codeZEUS-MP/2 (Hayes et al. 2006), which is a computational fluiddynamics code for the simulation of astrophysical phenomena, parallelized by the MPI (message-passing) library. The ZEUS-MP/2 code employs Eulerian hydrodynamics algorithms basedon the finite-difference method with a staggered mesh. In this study,we have modified the original code substantially according to theprescriptions in our preceding 2D simulations (Ohnishi et al. 2006,2007).

We consider spherical coordinates (r; ;) with the origin atthe center of the protoYneutron star. The basic evolution equations

describing accretion flows of matter attracted by a protoY

neutronstar and irradiated by neutrinos emitted from the protoYneutronstar can be written as

d

dt : = v 0; 1

dv

dt :P : : = Q; 2

d

dt

e

P: = v Q E Q : :v; 3

dYe

dt QN; 4

GMinr

; 5

where , v, e, P, Ye, and are the density, velocity, internal en-ergy, pressure, electron fraction, and gravitational potential, re-spectively, and Gis the gravitational constant. The self-gravityof matter in the accretion flow is ignored. HereQ is the artificialviscous tensor, and Q E and QN represent the heating/cooling andelectron source/sink via neutrino absorptions and emissions byfree nucleons, respectively. The Lagrangian derivative is denotedby d/dt @/@t v = :. The tabulated realistic equation of statebased on relativistic mean field theory (Shen et al. 1998) is imple-mented according to the prescription in Kotake et al. (2003). Themass accretion rate and the mass of the central object are fixed to

be

M 1 M s1

andMin 1:4 M, respectively. The neutrinoheating is estimated under the assumptions that neutrinos are emit-ted isotropically from the central object and that the neutrino fluxis not affected by local absorptions and emissions (see Ohnishiet al. 2006). We consider only the interactions of electron-typeneutrinos and antineutrinos. Their temperatures are also assumedto be constant and are set to be Te 4 MeV and Te 5 MeV,typical values in the postbounce phase. The neutrino luminosityis varied in the range of L (6:0Y6:8) ; 10

52 ergs s1.Spherical polar coordinates are adopted. In the radial direction,

the computational mesh is nonuniform, while the grid points areequally spaced in other directions. We use 300 radial mesh pointsto coverrin r rout, where rin $ 50 km is the radius of the in-ner boundary, located roughly at the neutrino sphere, and rout

2000 km is the radius of the outer boundary, at which the flow is

supersonic. A total of 30 polar and 60 azimuthal mesh points areused to cover the whole solid angle. In order to see if this angularresolution is sufficient, we have computed a model with the 300 ;60 ; 120 mesh points and compared it to the counterpart with the300 ; 30 ; 60 mesh points. As shown in Appendix B, the resultsagree reasonably well with each other in both the linear and non-linear phases. Although the computational cost does not allow usto carry the convergence test further, we think that the resolutionof this study is good enough.

We use an artificial viscosity of tensor type, described in Ap-pendix A, instead of the von Neumann & Richtmyer type thatwas originally employedin ZEUS-MP/2. For 3D simulations witha spherical polar mesh, we find the former preferable to preventthe so-called carbuncle instability (Quirk 1994), which we ob-serve around the shock front near the symmetry axis, $ 0, .With the original artificial viscosity, an appropriate dissipation isnot obtained in the azimuthal direction for the shear flow result-ing from the converging accretion, particularly when a fine meshis used (Stone & Norman 1992). We have also applied this ar-tificial viscosity to axisymmetric 2D simulations and reproducedthe previous results (Ohnishi et al. 2006).

Figure 1 shows the radial distributions of various variables forthe unperturbed flows. The spherically symmetric steady accre-tion flow through a standing shock wave is prepared in the samemanner as in Ohnishi et al. (2006). Behind the shock wave, theelectron fraction is less than 0.5 owing to electron capture, and aregion of negative entropy gradient with positive net heatingrates is formed for the neutrino luminosities L (6:0Y6:8) ;1052 ergs s1. The values of these variables on the ghost meshpoints at the outer boundary are fixed to be constant in time, whileon the ghost mesh points at the inner boundary they are set to beidentical to those on the adjacentactive mesh points, except for theradial velocity, which is fixed to the initial value at both the innerand outer boundaries.

In order to induce nonspherical instability, we have added a

radial velocity perturbation, v

r(r; ;), to the steady sphericallysymmetric flow according to the equation

vr(r; ;) v1Dr (r) vr(; ); 6

where v1Dr (r) is the unperturbed radial velocity. In this study, weconsider three types of perturbations: (1) an axisymmetric (l 1,m 0) single-mode perturbation,

vr(;) /

ffiffiffiffiffiffi3

4

rcos v1Dr (r); 7

(2) a nonaxisymmetric perturbation with l 1,

vr(r; ;) /ffiffiffiffiffiffi

34

rcos

ffiffiffiffiffiffi3

8

rsin cos

" #v

1Dr (r); 8

and (3) a random multi-mode perturbation,

vr(; ) / rand ; v1Dr (r) (0 rand < 1); 9

where rand is a pseudorandom number. The perturbation am-plitude is set to be less than 1% of the unperturbed velocity. Wenote that there is no distinction between m 1 and 1 modeswhen the initial perturbation is added only to the radial velocity,as is the case in this paper. To put it another way, the m 1modes contribute equally. Hence, they are referred to as the

jmj 1 mode below. On theotherhand, differences do show up,

IWAKAMI ET AL.1208 Vol. 678


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for example, if random perturbations are added to the nonradialvelocity components; this case is important in discussing theorigin of pulsar spins proposed by Blondin& Mezzacappa (2007),and will be the subject of a forthcoming paper (W. Iwakamiet al. 2008). We also note that the symmetry between m 1 and

1 modes is naturally removed if the unperturbed accretion flowis rotating (see Yamasaki & Foglizzo 2008). All the models pre-sented in this paper are summarized in Table 1.

In the next section, we perform the mode analysis as follows.The deformation of the shock surface can be expanded as a linearcombination of the spherical harmonic components Yml (;):

RS(;) X1l0

Xlml

c ml Ym

l (;); 10

where Yml is expressed by the associated Legendre polynomialPml and a constant K

ml , given as

Yml Km

l Pml ( cos )e

im; 11

Kml ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2l 14

(l m)!(l m)!

s: 12

The expansion coefficients can be obtained as

c ml

Z20

d

Z0

d sin RS(; ) Ym

l (; ); 13

where the superscript asterisk () denotes complex conjugation.

3. RESULTS AND DISCUSSIONS

3.1. Axisymmetric Single-Mode Perturbation

Before presenting the results of 3D simulations, we first dem-onstrate the validity of our newly developed 3D code. We com-pare the axisymmetric flows obtained by 2D and 3D simulationsfor the axisymmetric (l 1, m 0) single-mode perturbation.By 2D simulations we mean that axisymmetry is assumed andcomputations are done in the meridian section with all deriva-tives dropped, whereas in 3D simulations we retain all these de-rivatives,and a 3D mesh is employed. This validation is importantbecause numerical errors may induce azimuthal motions evenfor the axisymmetric initial conditions. Hence, we have to con-firmthat azimuthalerrors do notappear in the nonaxisymmetricsimulation.

Figure 2 shows the time evolutions of the average shock ra-dius RS for 2D (Fig. 2a) and 3D (Fig. 2b) results. The average

shock radius RS is obtained from the expansion coefficientc0

0 in

TABLE 1

Summary of All the Models

Model Perturbation

Neutrino

Luminosity L(1052 ergs s1)

I ................... single-mode, l 1, m 0 6.0

II.................. multi-mode, l 1, m 0 and l 1, jmj 1 6.0III................. single-mode, l 1, m 0 with

random perturbation at t 400 ms6.0

IV ................ random perturbation 6.0

V.................. random perturbation 6.4

VI ................ random perturbation 6.8

VII.. ... ... ... .. .. axisymmetric random perturbation 6.0

VIII.. .. ... ... ... . axisymmetric random perturbation 6.8

IX ................ none 6.0

X.................. none 6.8

Fig. 1.Profiles of various variables for the unperturbed spherically symmetric accretion flows with different neutrino luminosities.

3D SIMULATIONS OF SASI IN SUPERNOVAE 1209No. 2, 2008


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equation (10) multiplied by K00 . The solid, dashed, and dotted linescorrespond to the neutrino luminosities L of 6.0, 6.4, and 6:8 ;1052 ergs s1, respectively. One cannot expect a perfect agreementbetween two computations of the exponential growth of the insta-bility followed by turbulent motions through mode coupling, asconsidered here; however, it is still obvious that the results of the2D and 3D simulations agree in essential features. In particular, inboth the 2D and 3D results, we see that the RS is settled to an al-mostconstant value forL 6:0and6:4 ; 10

52 ergs s1, whereasit continues to increase forL 6:8 ; 10

52 ergs s1.

Figure 3 presents the normalized amplitudes jc ml /c00j as a func-tion of time for model I (L 6:0 ; 1052 ergs s

1) for 2D (Fig. 3a)

and 3D (Fig. 3b) results. The red, blue, black, and gray solid linescorrespond to the modes of (l; m) (1; 0), (2,0), (3,0), and (4,0),respectively. As can be seen, the time evolution can be divided intotwo phases. First is the linear growth phase, in which the amplitudeof the initially imposed mode with (l; m) (1; 0) grows exponen-tially. This lasts for$100 ms. Higher modes are also generated bymode couplings and grow exponentially during this phase. Thenstarts the second phase, which is characterized by the saturation ofamplitudes of the order of 0.1. In this phase, the accretion flow be-comes turbulent. It is interesting to note that in this nonlinear satura-

tion phase the amplitude of the mode with (l; m) (2; 0) is almostof the same order as the initially imposed mode with (l; m) (1; 0)

Fig. 2.Time evolutions of the average shock radius RS in 2D (left) and 3D (right) for the axisymmetric l 1, m 0 single-mode perturbation.

Fig. 3.Time evolutions of the normalized amplitudes jcml /c00j for model I with the axisymmetric l 1, m 0 single-mode perturbation in 2D (left) and 3D (right).

[See the electronic edition of the Journal for a color version of this figure.]



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and is dominant over other modes, which fact was observed inOhnishi et al. (2006).

Most important of all, however, is the fact that none of them 6 0 modes are produced, implying that the perturbed flow re-tains axisymmetry, which is a necessary condition for the numer-ical study of nonaxisymmetric instability. Although the resultsof the 2D and 3D simulations are not identical, the essential fea-tures such as the linear growth rate of the initial perturbation(l 1, m 0) and the production of other modes via nonlinearmode coupling, as well as the saturation levels in the nonlinearphase, are in good agreement for the two cases. The quantita-tive differences between the 2D and 3D results mainly originatefrom the difference in the time steps, which depend on the mini-mum grid width. Note that in addition tothe 300 radial and 30 polarmesh points used for the 2D computations, 60 azimuthal meshpoints are employed in the 3D simulations and, as a result, smallertime steps are usually taken for the 3D computations.

3.2. Nonaxisymmetric Perturbation with l 1

Now we discuss the results of genuinely 3D simulations, inwhich the nonaxisymmetric perturbation with (l 1, jmj 1)isadded to the axisymmetric (l 1, m 0) perturbation.

Figure 4 shows the time evolutions of some of the normalized

amplitudes jcm

l /c0

0j for model II with the neutrinoluminosityL 6:0 ; 1052 ergs s1. The red, yellow, blue, light blue, green, black,brown, violet, and pink solid lines denote the amplitudes of themodes with (l; jmj) (1; 0), (1,1), (2,0), (2,1), (2,2), (3,0), (3,1),(3,2), and (3,3), respectively. In the linear phase, the initially im-posed modes with (l; jmj) (1; 0) and (1,1) grow exponentially,just as in the axisymmetric single-mode perturbation. It should benoted that the growth rate of the (l; jmj) (1; 1) mode is identicalto that of the (l; m) (1; 0) mode. This is just as expected for thespherically symmetric background, for which modes with thesame l but differentm are degenerate in the linear eigenfrequency.

Itcan also beseen fromFigure 4 thatthe modes with(l; jmj) (2; 0), (2,1), (2,2) are first produced by the nonlinear mode cou-plings of (l; jmj) (1; 0), (1,1). Then the modes with (l; jmj)

(3; 0), (3,1), (3,2), (3,3) can be produced by the couplings of the

first-order modes with (l; jmj) (1; 0), (1,1) and the second-order modes with (l; jmj) (2; 0), (2,1), (2,2). Even higher ordermodes are produced subsequently toward the nonlinear satura-tion, but they are not shown here. Although the branching ratiosshould be investigated in more detail before coming to any con-

clusions, the above sequence of mode generation strongly sug-gests that the nonlinear coupling is mainly of quadratic nature.

In the nonlinear phase that begins att$ 200 ms, these modeamplitudes are saturated and settled into a quasi-steady state. Asin the axisymmetric case, the l 1 modes both with m 0 andjmj 1 are dominant over other modes in this stage for the non-axisymmetric case. One thing to be noted here, however, is thefact that the saturated amplitudes for model II are lower thanthose for model I in general (see Fig. 3b). This is, we think, be-cause the turbulent energy is shared by a larger number of modesin the nonaxisymmetric case than in the axisymmetric case. Todemonstrate this more clearly, we have added a random pertur-bation to the radial velocity in the axisymmetric model I att 400 ms, by which time the axisymmetric nonlinear turbulence

has fully grown. We refer to this model as model III. The timeevolutions of the normalized amplitudes jc ml /c

00j for model III are

shown in Figure 5. It is clear that the axisymmetric m 0 modeamplitudes are reduced after the additional perturbation is given,and the nonaxisymmetric m 6 0 modes grow to saturation levelat t$ 450 ms. The power spectra of the turbulent motions willbe discussed in more detail later.

3.3. Random Multi-Mode Perturbations

3.3.1. Dynamical Features and Critical Luminosity

Having understood the elementary processes of the lineargrowth and nonlinear mode couplings in the previous section,we now discuss the 3D SASI induced by random multi-mode

perturbations, which are supposed to be closer to reality. In thissubsection, we show the typical dynamics, paying attention to thetime evolution of the shock wave, and hence to the critical lumi-nosity, at which the stalled shock is revived.

Figure 6 shows the time variations of the average shock radiusRS for models IV, V, and VI, with neutrino luminosities L 6:0, 6.4, and 6:8 ; 1052 ergs s1, respectively. It is found that forL 6:0 and 6:4 ; 10

52 ergs s1, the shock settles to a quasi-steady state after the linear growth, whereas it continues to expandforL 6:8 ; 10

52 ergs s1, which is also true for the axisym-metric counterpart displayed in the right panel of Figure 6 forcomparison. This implies that the critical neutrino luminosity isnot much affected by the change from 2D to 3D. In the follow-ing discussion we refer to the models with L 6:0 and 6:4 ;

1052

ergs s1

as the non-explosion models, and the model with

Fig. 4.Timevariations of the normalized amplitudes jcml /c00j formodel II,inwhich the modes with (l; jmj) (1; 0); (1; 1) are initially imposed. [See the elec-tronic edition of the Journal for a color version of this figure.]

Fig. 5.Time evolutionsof the normalizedamplitudes jcml /c00j formodelIII, in

which we have added a random perturbation to the radial velocity at t 400 ms.[See the electronic edition of the Journal for a color version of this figure.]



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L 6:8 ; 1052 ergs s1 as the explosion model, and we look

into their dynamical features in turn. The mode analyses will bedone in xx 3.3.2 and 3.3.3.

Figure 7 shows the side views of the iso-entropy surfaces to-gether with the velocity vectors in the meridian section at fourdifferent times for the non-explosion model with L 6:0 ;1052 ergs s1 (model IV). The hemispheres (0 ) of eightiso-entropy surfaces are superimposed on one another, and theoutermost surface almost corresponds to the shock front. The ini-tial perturbations grow exponentially in the linear phase ( Fig. 7a;t 40 ms). At the end of the linear phase, a blob of high entropyis formed, which subsequently grows (Fig. 7b; t 93 ms). High-entropy blobs are continuously generated, and the nonlinear phasebegins (Fig. 7c; t 100 ms). Circulating flows occur inside theblobs, while high-velocity accretions onto a protoYneutron starsurround the blobs. These blobs expand and shrink repeatedly,being distorted, split, and merged with each other inside the shockwave (Fig. 7d; t 350 ms). Reflecting these complex motions,the shock surface oscillates in all directions but remains almostspherical for the nonaxisymmetric model. This is in sharp contrastto the axisymmetric case, in which large-amplitude oscillationsoccur mainly in the direction of the symmetry axis ( Blondin et al.2003; Ohnishi et al. 2006).

Figure 8 displays snapshots of the iso-entropy surfaces andthe velocity vectors for the explosion model with L 6:8 ;1052 ergs s1 (model VI ). As in the non-explosion model, the se-quence of events starts with the linear growth of the initial pertur-bation inside the shock wave (Fig. 8a; t 40 ms). In the explosionmodel, however, many high-entropy blobs emerge simultaneouslymuch earlier on (Fig. 8b; t 70 ms) than in the non-explosionmodel. These blobs then repeatedly expand and contract, beingdistorted, split, and merged together just as in the non-explosioncounterpart (Fig. 8c; t 80 ms). After that, some of the blobsget bigger, absorbing other blobs (Fig. 8d; t 350 ms) and, as aresult, the shock radius increases almost continuously up to theend of the computation (t 400 ms), as already demonstrated inFigure 6a. At this point, the shock surface looks ellipsoidal ratherthan spherical. However, the major axis is not necessarily aligned

with the symmetry axis, nor is the flow is symmetric with respectto this major axis.

3.3.2. Mode Spectra

Next we look into the spectral intensity in more detail. As astandard case, we take model IV with L 6:0 ; 10

52 ergs s1.Figure 9 shows the time evolutions of the normalized amplitudesjc ml /c

00j with m 0 and compares them with the axisymmetric

counterparts in model VII. Note that the m 6 0 modes also existin the nonaxisymmetric model; these are not shown in the figurebut will be discussed in the following paragraphs. As can beclearly seen, the amplitude of each mode grows exponentiallyuntil $100 ms, which is the linear phase. Note in particular thatthe growth rate and oscillation frequency for the l 1 mode arethe same as those obtained for the model with the single-modeperturbation. After$100 ms, the mode amplitudes are saturatedand the evolution enters the nonlinear phase, with a clear domi-nance of the modes with l 1; 2. It is also evident in the figure thatthe saturation level is lower in general for the nonaxisymmetric casethan for the axisymmetric one, which confirms the claim in the pre-vious section based on the results for the single-mode perturbation.

In Figure 10, we display snapshots at four different timesof the spectral distributions for both the nonaxisymmetric (leftpanels) and axisymmetric (right panels) cases. The top panelscorrespond to the linear phase, in which the intensity is distributedrather uniformly over all modes. As time passes, however, the am-plitudes of low-lmodes grow much more rapidly than those withhigher l(second and third rows). One can see a similarity in thetime evolutions between the nonaxisymmetric and axisymmetricmodels. It should be noted again that there is no superiority in thelinear growth rate among differentm modes in the nonaxisymmet-ric case. Since the amplitudes of different modes are oscillating intime with some phase lags, the mode with (l; jmj) (1; 0) is larg-est at one instance (Fig. 10b; t 30 ms), whereas the mode with(l; jmj) (1; 1) has the greatest amplitude at another instance(Fig. 10b; t 60 ms). On average, however, none of them issuperior to others. This is also true of the explosion modelsdescribed in the next section. After the nonlinear phase starts

Fig. 6.Time evolutions of the average shock radius RS for models IV, V and VI, in which a random multi-mode perturbation is imposed in 3D (left) and 2D (right).



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(bottom panels), the growths of all modes are saturated, and thespectral distributions are settled to be quasi-steady. It is againobvious in these panels that the low-l modes are dominant inthe nonlinear phase, and the saturation level is lower in the non-axisymmetric case.

Now we turn our attention to the quasi-steady turbulence inthe nonlinearphase. Shown as a function oflin Figure 11 are thepower spectrajc ml /c

00j

2averaged over the interval from t 150

to 400 ms. The nonaxisymmetric and axisymmetric cases areshown by open and filled circles, respectively. In the left panel,

modes with differentm are plotted separately, whereas they aresummed in the right panel. We find that the time-averaged powerspectra are not very different among the modes with the samel but different m. This implies that the equipartition of the tur-bulence energy is nearly established among modes with the samel. This will have important ramifications for the origin of pulsarspin, and will be discussed in our forthcoming paper (W. Iwakamiet al. 2008, in preparation).

The right panel of Figure 11demonstrates that the time-averagedpower spectrum summed overm for the nonaxisymmetric case is

Fig. 7.Iso-entropysurfacesand velocity vectors in the meridian section for model IV. The hemispheres (0 ) of eight iso-entropy surfaces are superimposed,with the outermost surface nearly corresponding to the shock front. [See the electronic edition of the Journal for a color version of this figure.]



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not very different from that for the axisymmetric counterpart. Thismeans that the total turbulence energy does not differ very muchbetween the axisymmetric and nonaxisymmetric cases. As a re-sult, the power of each mode in the nonaxisymmetric case issmaller by roughly a factor of 2l 1 than thatof the same lmodein the axisymmetric case. This is the reason why we have ob-served that the saturation level of the nonlinear SASI is smallerin the nonaxisymmetric case than in the axisymmetric case andthe shock surface oscillates in all directions with smaller ampli-

tudes in the nonaxisymmetric flow, whereas it sloshes in the di-

rection of the symmetry axis with larger amplitudes in the axi-symmetric case. The difference in the fluctuations of the averageshock radius seen in Figures 6a and 6b can also be explained inthe same manner.

Another interesting feature observed in Figure 11 is the factthat the time-averaged power spectra obey a power law atlk 10for both the nonaxisymmetric and axisymmetric models. Twostraight lines in Figure 11a are the fits to the data forl ! 10, eachobtained for the axisymmetric and nonaxisymmetric models. The

powers are found to be 5.7 and 4.3 for the nonaxisymmetric

Fig. 8.Iso-entropy surfaces and velocity vectors in the meridian section for the explosion model (model VI ). Note that the displayed region is 2.5 times larger forpanel d. [See the electronic edition of the Journal for a color version of this figure.]



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and axisymmetric cases, respectively. Although the difference isalmost unity, which originates from the difference in multiplicityof the modes with the same l, the slope excluding this effect is stilla little bit steeper in the nonaxisymmetric case, as seen in Fig-ure 11b. At the moment we do not know if this difference in slopeis real or not, and the power itself also remains to be explainedtheoretically.

3.3.3. The Explosion Models

So far, we have been discussing the non-explosion models, inwhich the SASI is saturated in the nonlinear phase and is settledto a quasi-steady state. In considering the possible consequencesof SASI after a supernova explosion occurs, however, it is alsointeresting to investigate the explosion models, in which a shockrevival appears as a result of SASI.

Figure 12 shows the time evolutions of the normalized ampli-tudes jc ml /c

00j for the explosion model (model VI) and compares

them with the axisymmetric counterpart (model VIII). One fea-ture shared by boththe axisymmetric and nonaxisymmetric explo-sion models is that the nonlinear stage is divided into two phases.The earlier phase is quite similar to the nonlinear phase that wehave seen so far for the non-explosion models. The later phase, onthe other hand, has the distinctive feature that the l 1 mode be-

comes much more remarkable than other modes, and the oscil-lation period tends to be longer as the shock radius gets larger andthe advection-acoustic cycle, supposedly an underlying mecha-nism of the instability, takes longer. The later prominence of thel 1 mode is intriguing, and will be important to anyquantitativediscussion of the pulsar kick velocity. A theoretical account, how-ever, is yet to be given. The differences between the nonaxisym-metric and axisymmetric models, such as the saturation level, arecarried over to the explosion models.

The power spectra averaged over time intervals of 100 t 200 ms and 250 t 400 ms are presented in Figure 13a and13b, respectively. As in the non-explosion models, the equiparti-tion among differentm modes is again established for the explo-sion models. This is true even in the later nonlinear phase, as seen

in the right panel of the figure. As a result, the saturation level is

smaller for the nonaxisymmetric case than for the axisymmetriccase, as mentioned above. The power spectra in the earlier non-linear phase look very similar to those found in the non-explosionmodels, with the power law being satisfied forlk 10. In the laternonlinear phase, on the other hand, the power law is extended tomuch lowerl. This is related to the late-time prominence of thel 1 mode, as mentioned above. In fact, the amplitudes in thelowerlportion of the power spectrum are enhanced in 250 t 400. The mechanism of this amplification remains to be under-stood, but it might be related to the volume-filling thermal con-vection advocated for the late stage of convective instability in asupernova core by Kifonidis et al. (2006). As a result of this evo-lution of the spectrum, the shock tends to be ellipsoidal as theshock expands.

3.4. Neutrino Heating

The SASI is supposed to be an important ingredient not onlyfor a pulsars proper motion and spin, but also for the explosionmechanism itself. In this section, we look into neutrino heatingin nonaxisymmetric SASI.

In Figure 14 we show the color contours of the net heating ratein the meridian section for the non-explosion model with L 6:0 ; 1052 ergs s1 (nonaxisymmetric model IV and axisymmet-

ric model VII ), as well as for the nonaxisymmetric explosionmodel with L 6:8 ; 1052 ergs s1 (model VI). In the early

phase, a cooling region with negative net heating rates existsaround the protoYneutron star, while the heating region extendsover the cooling region up to the stalled shock wave in all cases.As time passes and the SASI grows, a pocket of regions withpositive but relatively low net heating rates appear. These regionscorrespond to the high-entropy blobs (high-entropy rings for theaxisymmetric case), where the circulating flow exists, as observedin Figures 7b, 7c, and 7d. Since the neutrino emission in these re-gions is more efficient than in the surroundings, the net heatingrate is rather low.

Althoughthe criticalneutrino luminositiesare not verydifferentbetween the nonaxisymmetric and axisymmetric cases, the spatial

distributions of neutrino heating are different. In the axisymmetric

Fig. 9.Time evolutions of the normalized amplitudes jc ml /c00j formodelsIV (left) andVII(right). Note that them 6 0 modes alsoexist in the nonaxisymmetric model,

but are not shown in this figure. [See the electronic edition of the Journal for a color version of this figure.]



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Fig. 10.Normalized amplitudes jc ml /c00j for model IV (left panels) and model VII (right panels) at different times. Note that the time-averaged values are plotted in

panels dand h. [See the electronic edition of the Journal for a color version of this figure.]


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case, the shock wave oscillates up and down, whereas it moves inall directions in the nonaxisymmetric case. The oscillation am-plitudes are larger in the axisymmetric than in the nonaxisym-metric case in general, as repeatedly mentioned. Reflecting thisdifference in the shock motions, the neutrino heating is enhancedmainly in the polar regions in the axisymmetric case, while in thenonaxisymmetric case the heating rate is affected by SASI chieflyaround the high-entropy blobs. These effects are clearly seen inFigure 14 (see also Fig. 7).

As described inx

3.3.1, the generation of the high-entropyblobs starts around the end of the linear phase and continues dur-ing the nonlinear phase. Although the blobs repeatedly expand

and contract, sometimes merging or splitting, the turbulent mo-tions together with the neutrino heating finally settle to a quasi-steady state in the non-explosion model. For the explosion model,on the other hand, the number and volume of high-entropy blobsincrease much more quickly, and as a result, the heating regionprevails, pushing the shock waveoutward and narrowing the cool-ing region near the protoYneutron star.

Figure 15 shows the time evolutions of the net heating ratesintegratedover the gain region inside theshock wave.For the non-explosion model (model IV; solid line), the volume-integratedheating rate grows until t$ 150 ms, but is saturated there-after, whereas it increases continuously for the explosion model

Fig. 11.Time-averaged power spectra for models IV and VII. The average is taken over t 150Y400 ms. In the left panel, modes with different m are plottedseparately; they are summed up in the right panel.

Fig. 12.Time evolutions of the normalized amplitudes jcml /c00j for the explosion models model VI (left) and model VIII (right). [See the electronic edition of the

Journal for a colorversion of this figure.]

3D SIMULATIONS OF SASI IN SUPERNOVAE 1217


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(model VI; dash-dotted line). These behaviors of the volume-integrated heating rate are in accord with the time evolutions of theaverage shock radius, as shown in Figure 6a. It is clear from thefigure that the heating rates are mainly affected by SASI during thenonlinear phase. For comparison,the spherically symmetric coun-terparts, models IX with L 6:0 ; 10

52 ergs s1 and X withL 6:8 ; 10

52 ergs s1, are also presented as dashed and dottedlines, respectively. Note that the last model does not lead to shockrevival even with this high neutrino luminosity. It is found that theSASI enhances the neutrino heating for both the explosion and

non-explosion models.It is understandably a difficult task to make a clean comparisonwith more realistic simulations, since they are highly dynamical,and the extraction of an unperturbed background is not easy. TheSASI they observe is more often than not nonlinear from the be-ginning. We do, however, point out that the heating rate of ourmodel IVis similarto what was observed in the nonrotating modelof Marek & Janka (2007; see the line for HW-2D in the middlepanel of Fig. 7) aftert $ 200 ms. Hence, our results with simpli-fied physics are consistent with their results taking more detailedphysics into account, at least for the nonlinear phase. The linearphase is more difficult to compare, since their models probablyhave no linear phase at all. This clearly demonstrates the com- plimentary roles of the idealized toy models and detailed

simulations.4. CONCLUSIONS

In this paper we have studied the nonaxisymmetric SASI by3D hydrodynamical simulations, taking into account a realisticEOS and neutrino heating and cooling. We have added variousnonaxisymmetric perturbations to spherically symmetric steadyflows that accrete through a standing shock wave onto a protoYneutron star, being irradiated by neutrinos emitted from the protoYneutron star. Mode analysis has been done for the deformation ofthe shock surface by the spherical harmonics expansion. Afterconfirming that our 3D code is able to reproduce for the axisym-metric perturbations the previous results for 2D SASI that weobtained in Ohnishi et al. (2006), we have done genuinely 3D

simulations and found the followings.

First, the model of the initial perturbation with (l; jmj) (1; 0)and (1,1) has demonstrated that the nonaxisymmetric SASI pro-ceeds much in the same manner as the axisymmetric SASI: thelinear phase, in which the initial perturbation grows exponen-tially, lasts for about 100 ms and is followed by a nonlinear phase,in which various modes are produced by nonlinear mode cou-pling, and their amplitudes are saturated. It has been found thatthe criticalneutrino luminosity, above which shock revival occurs,is not very different between 2D and 3D. For neutrinoluminositieslower than the critical value, the SASI settles to a quasi-steady tur-

bulence. We have found that the saturation level of each mode inthe nonaxisymmetric SASI is lower in general than that of its axi-symmetriccounterpart. This is mainly due to the fact that the num-ber of the modes contributing to the turbulence is larger for thenonaxisymmetric case. The sequence of mode generation, on theother hand, strongly suggests that nonlinear mode coupling ischiefly quadratic in nature.

Second, the simulations with random multi-mode perturba-tions imposed initially have shown that the dynamics in the lin-ear phase is essentially a superposition of those of single modes.Toward the end of the linear phase, high-entropy blobs are gen-erated continuously and grow, starting the nonlinear phase. Wehave observed that these blobs repeatedly expand and contract,merging and splitting from time to time. In the non-explosion

models, these nonlinear dynamics lead to the saturation of modeamplitudes and quasi-steady turbulence. For the explosion mod-els, on the other hand, the production of blobs proceeds muchmore quickly, followed by an oligarchic evolution, with a rela-tively small number of large blobs absorbing smaller ones, andas a result the shock radius increases almost monotonically. Thespectral analysis has clearly demonstrated that low-lmodes arepredominant in the nonlinear phase, just as in the axisymmetriccase. We have also shown that equipartition is nearly establishedamong differentm modes in the quasi-steady turbulence, and thatthe spectrum summed over m in the nonaxisymmetric case isquite similar to the axisymmetric counterpart. This implies thatthe larger number of modes in the nonaxisymmetric case is themain reason why the amplitude of each mode is smaller in 3D

than in 2D. The power spectrum is approximated by a power law

Fig. 13.Time-averaged power spectra for the explosion models, model VI and VIII. The averages are taken for (a) t 100Y200 ms and (b) t 250Y400 ms.

IWAKAMI ET AL.1218


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Fig. 14.Color contours of thenet heating rate inthe meridiansectionfor modelIV (top panels),modelVII(middle panels),andmodelVI (bottom panels) at differenttimes. All panels have the same spatial scale. [ See the electronic edition of the Journal for a color version of this figure.]


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forlk 10. Although the slope is slightly steeper for the nonaxi-symmetric models, whether the difference is significant or not isunknown at present.

We have seen in the explosion models, on the other hand, thatthe oscillation period of each mode becomes longer in the latenonlinear phase, as the shock radius gets larger and the advec-tion-acoustic cycle becomes longer. What is more interesting isthe fact that in this late phase the dominance of low l modesbecomes even more remarkable. Although this may be related tothe volume-filling thermal convection, the mechanism has yet to

be revealed. This spectral evolution leads to the global deforma-tion of the shock surface to an ellipsoidal shape, whose majoraxis is not, however, necessarily aligned with the symmetry axis.

Finally, we have presented the neutrino heating in 3D SASI. Ithas been shown that the volume-integrated heating rate is affectedmainly in the nonlinear phase. A comparisonwith spherically sym-metric counterparts has confirmed that the SASI also enhancesneutrino heating in the nonaxisymmetric case. Although the crit-ical neutrinoluminosity in the nonaxisymmetric SASI is not muchchanged from that for the axisymmetric case, the spatial distri-bution of neutrino heating is different in the nonlinear phase.Relatively narrow regions surrounding high-entropy blobs areefficiently heated for the nonaxisymmetric case, while wider re-gions near the symmetry axis are heated strongly, in accord with

the sloshing motion of shock wave along the symmetry axis, forthe axisymmetric case. For the non-explosion models, the high-entropy blobs produced by neutrino heating occupy the insideof shock wave, repeatedly expanding and contracting and being

intermittently split and merged, but the flow finally settles to aquasi-static state. For the explosion models, on the other hand, thehigh-entropy blobs are generated much more quickly and extendthe heating region, pushing the shock outward and narrowing thecooling region near the protoYneutron star.

In this paper we have not discussed the instability mechanism,which is still controversial at the moment. Based on our previouswork (Ohnishi et al. 2006), we prefer the advection-acoustic cycleproposed by Foglizzo (2001, 2002) to the purely acoustic cyclediscussed by Blondin & Mezzacappa (2006). It is, however, fair tomention that most of our models have ratios of the shock radius totheinner boundaryradius for which the recent analysis by Laming(2007) predicts the operation of a pure acoustic cycle. Since hisanalysis includes some approximations, it is certainly not the finalverdict in our opinion, and further investigations are needed.

In the present study we have not observed a persistent segre-gation of angular momentum in the accretion flow, such as foundby Blondin & Mezzacappa (2007) and Blondin & Shaw (2007),although instantaneous spiral flows are frequently seen. As dis-cussed above, equipartition is nearly established among differentm modes in our models. It should be emphasized here again, how-

ever, that we have added the initial perturbations only to the radialvelocity in this study, and as a result, modes with m 1 areequally contributing. We defer the analysis of models with initialnonaxisymmetric perturbations added also to the azimuthal com-ponent of the velocity to a forthcoming paper ( W. Iwakami et al.2008, in preparation), in which we will also discuss a possiblecorrelation between the kick velocity and spin of neutron stars ifthey are indeed produced by the 3D SASI. Many questions re-garding the SASI still remain to be studied; in particular, the influ-ences of rotation and magnetic field are among the top priorities,and will be addressed elsewhere in the near future.

W. I. expresses her sincere gratitude to Kazuyuki Ueno,

Michiko Furudate, and the members of the Sawada and Uenolaboratory for continuing encouragement and advice. She wouldalso like to thank Yoshitaka Nakashima of Tohoku Universityfor his advice on visualization by AVS. K. K. expresses his thanksto Katsuhiko Sato for his continuous encouragement. Numericalcomputations were performed on the Altix3700Bx2 at the Insti-tute of Fluid Science, Tohoku University, and on VPP5000 andthe general common use computer system at the Center for Com-putational Astrophysics (CfCA), National Astronomical Observa-tory of Japan. This study was partially supported by the Programfor Improvement of Research Environment for Young Researchersfrom Special Coordination Funds for Promoting Science and Tech-nology (SCF), Grants-in-Aid for Scientific Research (S19104006,S14102004, 14079202, 14740166), and a Grant-in-Aid forthe 21st

century COE program Holistic Research and Education Centerfor Physics of Self-organizing Systems of Waseda Universityby the Ministry of Education, Culture, Sports, Science, and Tech-nology (MEXT) of Japan.

APPENDIX A

TENSOR ARTIFICIAL VISCOSITY

Here we present the tensor-type artificial viscosity, Q, that we use in this paper. It is a direct extension to 3D of that employed byStone & Norman (1992) for 2D simulations. Following the notations of Stone & Norman (1992), Q can be written as

Q l2: = v :v

1

3: = v e

!if : = v < 0;

0 otherwise;

8


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where ldenotes a constant with a dimension of length, :v (vi; j vj; i)/2 is the symmetrized velocity-gradient tensor, and e is the unittensor. Dropping the off-diagonal components, we utilize only the diagonal components of Q in this study, which are written in finitedifference form as

Q11i; j; k l2i; j; kdi; j; k : = v i; j; k :v(11)

i; j; k

1

3: = v i ;j; k

!;

Q22i; j; k l2

i; j; kdi; j; k : = v i; j; k :v(22)

i; j; k

1

3 : = v i; j; k !

;

Q33i; j; k l2i; j; kdi; j; k : = v i; j; k :v(33)

i; j; k

1

3: = v i; j; k

!; A2

where di; j; k stands for the density at the site specified by (i;j; k), and li; j; k is written as

li; j; k max (C2 dx1ai; C2 dx2aj; C2 dx3ak): A3

Here C2 is a dimensionless constant controlling the number of grid points, over which a shock is spread, and dx1ai, dx2aj, and dx3ak arethe grid widths of the a-mesh at the i, j, and kth grid points in the r, , and directions, respectively. The a-mesh and b-mesh are dis-tinguished in ZEUS-MP/2 and are defined on the cell edge and cell center, respectively (see Stone & Norman 1992 for more details).

The artificial viscous stress : =Q and artificial dissipation Q : :v in the momentum equation (2) and energy equation (3) are

given as

: = Q (1) 1

g22 g31

@

@x1g22 g31Q11

;

: = Q (2) 1

g2g232

@

@x2g232Q22

Q11

g2g32

@g32@x2

;

: = Q (3) 1

g2g32

@Q33

@x3; A4

Q : :v l2: = v1

3:v(11) :v(22)

2 :v(11) :v(33)

2 :v(33) :v(22)

2 !; A5

where g2 r, g31 r, and g32 sin are the metric components for the spherical coordinates, (x1;x2;x3) (r; ;) (see again Stone& Norman 1992). These equations are discretized as

: = Q 1; i; j; kg2b 2i g31biQ11i; j; k g2b

2i1g31bi1Q11i1; j; k

g2a 2i g31aidx1bi;

: =Q 2;i; j; kg32b 2j Q22i; j; k g32b

2j1Q22i; j1; k

g2big32a2i dx2bj

Q11i; j; k Q11i; j1; k

2g2big32aj

@g32aj

@x2

;

: =Q 3;i; j; kQ33i; j; k Q33i; j; k1g2bi; g32bjdx3bk

; A6

(Q : :v)i; j; k l2

i; j; kdi; j; k( : = v)i; j; k1

3

n( :v(11))i; j; k ( :v(22))i; j; k

h i2 ( :v(11))i; j; k ( :v(33))i; j; kh i2 ( :v(33))i; j; k ( :v(22))i; j; kh i2o; A7

where g2a, g31a, and g32a are g2, g31, and g32 defined on the a-mesh, and g2b, g31b, and g32b are those defined on the b-mesh. Theterms dx1bi, dx2bj, and dx3bk represent the width of the b mesh at the i, j, and kth grid points in the r, , and directions, respectively.Finally, the velocity-gradient tensor ( :v(11))i; j; k; ( :v(22))i; j; k, and ( :v(33))i; j; k are given by

:v(11)

i; j; k

v1i1; j; k v1i; j; k

dx1ai;

:v(22)

i; j; k

v2i; j1; k v2i; j; k

g2bidx2aj

v1i; j; k v1i1; j; k2g2bi

@g2bi

@x1

;

:v(33)

i; j; k

v3i; j; k1 v3i; j; kg2big32bjdx3ak

v2i; j; k v2i; j1; k

2g31big32bj

@g32bj

@x2

v1i; j; k v1i1; j; k2g31bi

@g31bi

@x1

: A8



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APPENDIX B

NUMERICAL CONVERGENCE TESTS

In orderto see if the numerical resolution employed in themain body is sufficient, we increase the number of angular grid points andcompare the results. Figure 16a shows the time evolutions of the normalized amplitudes jc ml /c

00j in the linear phase. In this comparison,

we impose the l 1, jmj 1 perturbation initially. We refer to the model with 300 ; 30 ; 60 mess points as MESH0, and to that with300 ; 60 ; 120 grid points as MESH1 in the figure. We find that the linear growth rates agree with each other reasonably well, al-

though the coarser mesh slightly overestimates the growth time. Figure 16bpresents the power spectrajcml /c

00j

2

that are time-averagedover the nonlinear phase. The random perturbation is imposed in this case. It is again clear that the results for MESH0 are in goodagreement with those for MESH1.

It should be mentioned that for MESH0 it takes 32 parallel processors about 1.5 days to compute the evolution up to t 400 ms,while MESH1 needs 22 days even for 128 parallel processors. This is partly because of the difference in the Courant numbers, whichare set to 0.5 for MESH0 but 0.1 for MESH1 to better use the tensor-type artificial viscosity. Although this severe limit of CPU timedoes notallow us to do more thoroughconvergence tests, we think, based on the results of the tests shown above, that ourresults givenin this paper are credible.

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Fig. 16.Convergence tests. (a) Time evolutions of the normalized amplitudes jcml /c00j in the linear phase. The single-mode perturbation with l 1, jmj 1 is im-

posed initially. The models with 300 ; 30 ; 60 and 300 ; 60 ; 120 mess points are referred to as MESH0 and MESH1, respectively. (b) Time-averaged power spectrajcml /c

00j

2. The average is takenover 150 t 400 ms. In thiscomparison, the random multi-mode perturbation is imposed.[ Seethe electronic edition of theJournalfor a

colorversion of this figure.]

IWAKAMI ET AL.1222

wakana iwakami et al- three-dimensional simulations of standing accretion shock instability in...

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