The saturation of SASI by parasitic instabilities
aa r X i v : . [ a s t r o - ph . S R ] A p r Draft version July 25, 2018
Preprint typeset using L A TEX style emulateapj v. 11/10/09
THE SATURATION OF SASI BY PARASITIC INSTABILITIES
J´erˆome Guilet, Jun’ichi Sato & Thierry foglizzo
Laboratoire AIM, CEA/DSM-CNRS-Universit´e Paris Diderot, IRFU/Service d’Astrophysique,CEA-Saclay F-91191 Gif-sur-Yvette, France.
Draft version July 25, 2018
ABSTRACTThe Standing Accretion Shock Instability (SASI) is commonly believed to be responsible for largeamplitude dipolar oscillations of the stalled shock during core collapse, potentially leading to anasymmetric supernovae explosion. The degree of asymmetry depends on the amplitude of SASI, butthe nonlinear saturation mechanism has never been elucidated. We investigate the role of parasiticinstabilities as a possible cause of nonlinear SASI saturation. As the shock oscillations create bothvorticity and entropy gradients, we show that both Kelvin-Helmholtz and Rayleigh-Taylor types ofinstabilities are able to grow on a SASI mode if its amplitude is large enough. We obtain simpleestimates of their growth rates, taking into account the effects of advection and entropy stratification.In the context of the advective-acoustic cycle, we use numerical simulations to demonstrate how theacoustic feedback can be decreased if a parasitic instability distorts the advected structure. Theamplitude of the shock deformation is estimated analytically in this scenario. When applied to theset up of Fern´andez & Thompson (2009a), this saturation mechanism is able to explain the dramaticdecrease of the SASI power when both the nuclear dissociation energy and the cooling rate are varied.Our results open new perspectives for anticipating the effect, on the SASI amplitude, of the physicalingredients involved in the modeling of the collapsing star.
Subject headings: hydrodynamics — instabilities — shock waves — supernovae: general INTRODUCTION
Despite decades of active research (Colgate & White1966; Bethe & Wilson 1985), the core collapse super-novae mechanism remains elusive. The failure of themost sophisticated 1D models to explode the majority ofmassive progenitors (Liebend¨orfer et al. 2001) suggeststhat multidimensional effects are essential for a success-ful explosion. Understanding the hydrodynamical in-stabilities responsible for this symmetry breaking, andmore specifically their nonlinear dynamics, is thereforerequired to understand the explosion mechanism.The region between the neutrinosphere and theshock deserves particular attention because two in-stabilities take place there: neutrino-driven convec-tion (Herant et al. 1992, 1994; Burrows et al. 1995;Janka & M¨uller 1996; Foglizzo et al. 2006), and thenewly discovered Standing Accretion Shock Instabil-ity (SASI) (Blondin et al. 2003; Ohnishi et al. 2006;Foglizzo et al. 2007; Scheck et al. 2008). 2D simu-lations suggest that the complex fluid motions trig-gered by these instabilities could lead to a success-ful explosion either by helping the classical neutrino-driven mechanism (Buras et al. 2006; Marek & Janka2009; Murphy & Burrows 2008) or by a new mecha-nism based on the emission of acoustic waves fromthe proto-neutron star (Burrows et al. (2006, 2007),see however Weinberg & Quataert (2008)). The largescale ( l = 1 −
2) induced asymmetry could also ex-plain the high kick velocities of newly formed neu-tron stars (Scheck et al. 2004, 2006) and may affectsignificantly their spin (Blondin & Mezzacappa 2007;Yamasaki & Foglizzo 2008).The linear phase of the two instabilities hasbeen described in details by Foglizzo et al. (2006);Blondin & Mezzacappa (2006); Foglizzo et al. (2007); Yamasaki & Yamada (2007); Fern´andez & Thompson(2009a). Neutrino-driven convective modes with a largeangular scale can be stabilized by a fast advection ofmatter through the gain region, whereas SASI is alwaysdominated by large scale modes. This linear argumentfavors SASI as the cause of the prominent l = 1 − METHOD
Estimating the growth rate of the secondaryinstabilities
We estimate the local growth rate of the parasitic in-stability by using a simplified description of the linearSASI mode that keeps only the features that are essen-tial for the physics of the instability.A SASI mode has a complex structure of entropy,vorticity and pressure perturbations which can becomputed by a linear analysis (Foglizzo et al. 2007;Yamasaki & Yamada 2007). We focus on the advectedstructure of the SASI mode, where the parasitic insta-bilities operate. As we seek a local description of theparasites, we assume a planar geometry. Denoting by z the vertical direction, and x the transverse direction,the structure of the advected entropy/vorticity wave (de-noted as S , w respectively) is approximated by a sinusoidwith a vertical wave number K z = ω/v , where ω is theSASI frequency and v is the flow velocity: S ( z, t ) = S ( z ) + ∆ S × cos ( ωt − K z z ) (1) w ( z, t ) = ∆ w × cos ( ωt − K z z ) , (2)where S is the entropy profile of the radial flow, and∆ S , ∆ w are the amplitudes of entropy and vorticityperturbations associated with the SASI mode. The ad-vected entropy/vorticity wave in a SASI mode is actu-ally tilted with respect to the horizontal direction, butwe neglect this tilt for the sake of simplicity ( K z isa factor & K x ∼ p l ( l + 1) /r in a typical SASI mode (Foglizzo et al.2007)). In what follows we omit the index z and simplynote K = ω/v the wave number of advected SASI per-turbations. The horizontal wave number of each para-sitic instability is denoted as k . The notation A refers to a quantity ( A ) in the stationary flow, unperturbedby SASI. Its perturbation by the SASI mode is denotedas ∆ A , and δA refers to a parasitic perturbation. Thegrowth rates of the KHi and RTi are directly related tothe amplitude of the SASI mode through its profile ofvorticity and entropy gradient. Non adiabatic cooling istaken into account in the shape of SASI eigenfunctions,but is neglected in the dynamical evolution of the para-sites. We choose the usual adiabatic index of a relativisticgas γ = 4 / v ,second by assessing the effects of a uniform entropy gra-dient ∇ S , and third by taking into account advection.In the first two steps, a standard linear mode analysiscan be used because the flow is stationary. The equationsdetermining the evolution of the perturbations (e.g. Ap-pendix A of Foglizzo & Ruffert (1999)) are solved numer-ically. A simple fitting formula for the maximum growthrate is proposed, as a function of the SASI amplitude andthe background stratification.The third step is related to the concept of global ver-sus local instability (Huerre & Monkewitz 1990), but thegradients in the direction of advection preclude the use ofstandard analytical techniques. If the fluid is advectedtoo fast, the instability may be able to grow in a la-grangian way but would actually decay at a fixed radiusas the perturbations are advected away. We use numeri-cal simulations to measure the propagation speed of theparasitic instability in the z -direction, by perturbing theSASI mode over a limited region. Adjusting the mea-sured speeds with a physical but approximate descriptionof the propagation, we obtain a simple analytical esti-mate of the growth rate σ parasite of each parasitic insta-bility, taking into account the SASI amplitude, the back-ground stratification, and the advection speed (Eqs. 12and 18). Estimating the saturation amplitude
The growth rate σ parasite is an increasing function ofthe SASI amplitude. A parasitic instability can affect thedynamics of SASI if its amplitude δA becomes compa-rable to the SASI amplitude ∆ A . The growth of theratio δA/ ∆ A requires σ parasite > σ sasi . We use thiscriterion to estimate the saturation amplitude of SASI.Pessah & Goodman (2009) use a similar criterion to es-timate the saturation amplitude of the MRI.The criterion σ parasite = σ sasi defines the minimumamplitude ∆ A min ( r ) of SASI above which parasites cancompete with SASI at a given radius r , despite advec-tion and cooling. The parasitic instabilities can alterthe growth of SASI only if their growth takes place ina region which is vital to the mechanism of SASI. Forexample, if the mechanism of SASI is interpreted as anadvective-acoustic cycle, this cycle is most sensitive tothe region between the shock and the deceleration regionwhere most of the acoustic feedback is produced. Fortu-nately, as will be shown in Sect. 6, the local saturationamplitude ∆ A min ( r ) displays a broad minimum aroundthe radius ( r ∗ + r sh ) /
2, which defines a global saturationamplitude ∆ A min without much sensitivity on the detailsof the SASI mechanism.he saturation of SASI 3 Limitations
By focussing on the growth of parasitic instabilities,we ignore other nonlinear processes which could playa role in saturating the amplitude of SASI. Amongthem, the steepening of acoustic waves into shocks (e.g.Fern´andez & Thompson (2009b)), the decoherence of themode due to the finite displacement or velocity of theshock, or the exchange of energy by resonant mode cou-pling, could be important in some parameter range.When compared to published simulations (Sect. 6), ourestimate of the saturation amplitude of SASI based onparasitic instabilities is encouraging in view of the manysimplifications inherent to our method: our descriptionof the parasitic growth neglects the spherical geometryof the flow, neglects the tilt of the SASI wave with re-spect to the horizontal direction, and assumes an adia-batic evolution of the parasites. We further assume thatSASI is dominated by a single
SASI mode of finite am-plitude, which we describe using a linear approximation.Steepened advected waves, induced by steepened acous-tic waves reaching the shock, could affect the growth ofparasitic instabilities by introducing larger vorticity andsharper entropy gradients. We also neglect the produc-tion of entropy by acoustic waves steepening into shocks,and the creation of vorticity by the baroclinic interactionof entropy gradients with pressure waves. THE KELVIN-HELMHOLTZ INSTABILITY (KHI)
The KHi in a sinusoidal velocity profile
The KHi feeds on the kinetic energy available in shearflows. A necessary condition for its growth is the pres-ence of a maximum in the absolute value of vorticity(Drazin & Reid 1981). The linear growth of SASI cre-ates a sinusoidal velocity profile v x ( z ) = ∆ v sin ( Kz ) , (3)with two such maxima per wavelength.Heyvaerts & Priest (1983) have demonstrated thatthis profile is indeed unstable to perturbations with asmall horizontal wave number k < K , with a growthrate σ < k ∆ v .In order to estimate the maximum growth rate and thecorresponding wavelength, we complement their analyti-cal study by a numerical mode calculation where the flowprofile is described by Eq. (3), with periodic boundaryconditions in z = 0 and z = 1. The results (Fig. 1) showgood qualitative agreement with the expectation fromHeyvaerts & Priest (1983): the maximum growth rate σ = 0 . K ∆ v is reached for a wave number k = 0 . K ,and the KHi is stable for k > K . The maximum KHigrowth rate is thus a fraction of the maximum vortic-ity in the flow (see also Foglizzo & Ruffert (1999)). TheMach number M has little effect on the instability unlessit approaches unity (we used M = 0 . Effect of stratification on the KHi
The buoyancy force in a stably stratified atmosphere isable to stabilize the KHi if Ri > /
4, where the Richard-son number Ri characterizes the relative strengths of
Fig. 1.—
Growth rate σ of the KHi as a function of the transversewave number k . The dashed line is the eigenvalue for a sinusoidalvelocity profile and periodic boundary conditions. The full linesshow the effect of stratification with the Richardson number Riranging from 0 to 0.2. The triangles and circles are measuredfrom simulations where perturbations are localized (triangles) orextended (circles). σ is normalized by the maximum vorticity w ,and k is normalized by the vertical wavenumber K of the SASImode. buoyancy and shear (e.g. Chandrasekhar (1961)):Ri ≡ N w , (4)where w ≡ ~ ∇ × ~v is the vorticity. In the absence of acomposition gradient, the Brunt-V¨ais¨al¨a frequency N isdefined by: N ≡ − γ − γ g ∇ S. (5)The entropy S is here measured in dimensionless units S ≡ log( P/ρ γ ) / ( γ − k ∼ . K to k ∼ . K (Fig. 1). The marginalstability is reached for a critical value of the Richardsonnumber Ri = 0 .
24 in close agreement with the expectedvalue Ri = 1 / Effect of advection on the KHi
In the context of core collapse, the SASI vorticity waveis advected toward the neutron star. Using a physicalargument, we first give a naive estimate of the speed atwhich an unstable perturbation can propagate againstthe stream, and evaluate the reduced growth rate in thepresence of advection. Numerical simulations are used inorder to obtain more accurate estimates.
Physical argument
The simplest illustration of the KHi (e.g.Drazin & Reid (1981)) considers an incompressiblefluid with a discontinuity of horizontal velocity: v x = − ∆ v for z <
0, and v x = ∆ v for z >
0. Unstablemodes exist for any horizontal wave number with agrowth rate σ = k x ∆ v , and their structure on bothsides of the discontinuity is a decreasing exponential: δA ∝ e − k x | z | . Viewed in a frame moving with a verticalvelocity v z , the time dependence of the perturbationis: δA ∝ e ( σ − k x v z ) t at a given height z above thediscontinuity. A condition for the perturbation to begrowing at a given radius can be deduced: σ > k x v z ,which may be interpreted as a propagation speed v prop of the KHi equal to: v prop = σ k x . (6)The local growth rate in the frame moving with respectto the discontinuity is thus decreased as follows: σ = σ (cid:18) − v z v prop (cid:19) . (7)Of course this physical argument is too simple to bedirectly applicable to the case of a sinusoidal shear waveadvected downward, where each unstable shear layer isfollowed by another one. Could the propagation be fasterif these adjacent shear layers cooperate ? Numerical simulations of the KHi
Using the code RAMSES (Teyssier 2002;Fromang et al. 2006) we performed numerical sim-ulations of a sinusoidal velocity profile described byEq. (3) (with K = 2 π , ∆ v = 1, c = 5). The compu-tational domain was a box − < z <
8, 0 < x < z = − . z = 0 . − ∆ v . The runs presentedhere have a resolution of 1024 × ∼ . k/K ∼ . .In the two bottom plots in Fig. 2 the KHi mode has al-ready propagated downward and upward from the initial The growth rates (triangles) are slightly smaller than the linearvalues with periodic boundary condition, but are close to those ofthe modes whose spatial extent is restricted to three wavelengths(Sect. 3.2). Note that they are not expected to match exactly, butthe fact that they have similar values suggests a common physicalorigin of the reduced growth rate, namely the restricted extent ofthe region where the KHi grows. For comparison we also ran asimulation where the whole flow was perturbed instead of the re-gion z = [ − . , . Fig. 2.—
Different stages in the evolution of the KHi on a sinu-soidal velocity profile. The upper plot is the initial condition: asinusoidal transverse velocity wave with random perturbations lo-calized between − . < z < . middle plot shows the time when the KHi just reacheda nonlinear amplitude, the mode structure is still clear. Finallythe bottom plot shows a more developed nonlinear stage of theinstability. In all plots, the grayscale represents the vorticity. Fig. 3.—
Propagation of the KHi along the z direction. Thethree curves represent different modes with a number of horizontalwavelengths ranging from 2 (upper curve) to 6 (lower curve). Theplotted quantity, z + ( t ), corresponds to the spatial extent in whichthe KHi has reached an amplitude of δv = 10 − ∆ v . The curveswere shifted in time so that they start to deviate from 0 at t = 0. he saturation of SASI 5 Fig. 4.—
Upper plot:
Propagation speed v prop of the KHi as afunction of the wave number k , measured in the numerical simu-lations (triangles). The full line shows the quantity 2 . σ/k , where σ is the growth rate computed in the linear analysis of Sect. 3.1.The factor 2 . Bottom plot : maximum growth rate (full line,left axis) and associated wave number (dashed line, right axis) asa function of the advection velocity. The thin straight line witharrows is a linear approximation of the growth rate (Eq. (8)) . perturbations. For a quantitative study of the propaga-tion speed, we measured the spatial range [ − z + , z + ] inwhich the KHi has reached an arbitrary amplitude (say δv = 10 − ∆ v ), as a function of time for different wavenumbers. As expected from 3.3.1, the large wavelengthperturbations propagate faster than the short wavelengthones (Fig. 3).The propagation velocity measured in the simulationsis typically a factor 2 . σ/k suggested in 3.3.1, but the dependence on k is similar(upper plot in Fig. 4). We interpret this high propagationspeed as the sign that adjacent shear layers do cooperate.Approximating the propagation speed by v prop = 2 . × σ/k , the wavelength dependence of the growth rate fora given advection velocity is estimated using Eq. (7).The maximum growth rate is smaller, and obtained fora longer wavelength than without advection, becauseshorter wavelengths propagate more slowly (Fig. 4). Themaximum growth rate σ KHmax decreases almost linearlywith the advection velocity v z . We approximate thiscurve linearly as follows: σ KHmax = σ (cid:18) − v z v eff0 (cid:19) , (8)where σ = 0 . w is the maximum KHi growthrate without advection, and v eff0 is an effective propa- gation speed which we estimate to be (in the absence ofstratification) v eff0 ∼ . × ∆ v ≃ . w/K . Note that v prop in Eq. (7) depends on the wavelength k of the per-turbation, whereas v eff in Eq. (8) is independent of k ,because σ KHmax is the growth rate maximized over allwavelengths. v eff can be interpreted as an average prop-agation speed of the modes growing fastest at differentadvection speeds. Analytical estimate of the KHi growth rate withboth advection and stratification
In the presence of stratification, Eq. (8) becomes : σ KH = σ KHstrat (cid:18) − v z v eff (cid:19) , (9)where σ KHstrat = σ (1 − Ri / Ri ) is the maximumgrowth rate of the KHi in the presence of stratificationbut in the absence of advection, and v eff is the effec-tive propagation velocity in the presence of stratification. Eq. (6) suggests that the propagation speed is propor-tional to the growth rate, giving : v eff = σ KHstrat σ v eff0 , (10)Injecting Eq. (10) into Eq. (9) then gives the followingformula : σ KH = σ (cid:18) − RiRi − v z v eff0 (cid:19) , (11)where σ = 0 . w is the growth rate in the absenceof stabilizing effect, Ri = 0 .
24, and v eff0 = 1 . w/K .Equation (11) can be rewritten as a function of the SASIamplitude: σ KH = 0 . w − . N ∆ w − Kv z . (12)In a SASI eigenmode, the vorticity ∆ w at a given po-sition in the flow is directly proportional to the relativeshock displacement ∆ r/r sh . The factor w sasi ( r ) definedby ∆ w ≡ w sasi ∆ r/r sh depends on the radius, and is aresult of the linear mode analysis of SASI.A local saturation amplitude of SASI due to the par-asitic growth of the KHi, at a given radius, is deducedfrom Eq. (12) and the criterion σ KH = σ sasi :∆ rr sh = h(cid:0) Kv z + 2 σ sasi (cid:1) + 4 . N i / + Kv z + 2 σ sasi w sasi . (13)The discussion of a global saturation amplitude is post-poned to Sect. 6. THE RAYLEIGH-TAYLOR INSTABILITY (RTI)
Simple RTi in a sinusoidal entropy profile
The RTi feeds on the potential energy available whena low entropy fluid is sitting on top of a higher entropyone, in a gravitational acceleration g . Buoyancy is char-acterized by the Brunt-V¨ais¨al¨a frequency N (Eq. (5)).The flow is unstable if N is negative, and the typicalgrowth rate σ RT of short wavelengths perturbations is: σ RT ≡ (cid:0) − N (cid:1) = (cid:18) γ − γ g ∇ S (cid:19) . (14) Fig. 5.—
Growth rate of the RTi as a function of the transversewave number. The full lines are the result of the linear analysis of asinusoidal entropy profile embedded in a stable background entropygradient and a gravity field. The different curves show differentvalues of the background entropy gradient ranging from ∇ S = 0(upper curve) to ∇ S = 0 . ∇ ∆ S (lower curve). The triangles showthe growth rate measured in the numerical simulations withoutbackground entropy gradient. In the sinusoidal entropy profile created by SASI, thesign of the entropy gradient ∇ S changes every half wave-length: the entropy wave is made of adjacent layers ofstably stratified and Rayleigh-Taylor unstable fluid. TheRTi is expected to grow fastest where the entropy gradi-ent is most negative.By computing the eigenmodes of a sinusoidal entropyprofile with a vertical wavenumber K in a constant grav-ity field, we verified that the growth rate σ ∼ σ RT atshort wavelength (Fig. 5), and σ ∼ . σ RT if k ∼ K . Inthis calculation, as in Sect. 3.2, the entropy gradient andgravity profiles have a constant amplitude over a lim-ited region of space (three vertical wavelengths), and aresmoothly connected to zero outside this region. Effect of stratification on the RTi
Just as KHi, the RTi can be stabilized by the presenceof a stable entropy gradient in the stationary flow ∇ S .The flow is stable where the stationary gradient ∇ S isstronger than the SASI gradient ∇ ∆ S . The maximumgrowth rate is expected to scale as the Brunt-V¨ais¨al¨a fre-quency associated with the most negative entropy gradi-ent: σ RT = (cid:20) γ − γ g ∇ (∆ S + S ) (cid:21) . (15)This was checked by adding a background positive en-tropy gradient to the previous mode analysis. The re-sulting growth rate, shown in Fig. 5, follows Eq. (15) atshort wavelengths. At longer wavelength, the RTi is sta-bilized slightly faster. The cause may be that these largerscale modes, in addition to grow on less intense negativeentropy gradient, are also sensitive to the higher positiveentropy gradient (which are stable). Effect of advection on the RTi
Physical argument
The simple physical argument applied to the KHi inSect. 3.3.1 can be adapted to the case of the RTi by con-sidering an incompressible fluid with a discontinuity ofdensity with ρ = ρ − ∆ ρ for z < ρ = ρ + ∆ ρ for Fig. 6.—
Different stages in the evolution of the RTi on a si-nusoidal entropy profile.
The upper plot is the initial condition:a sinusoidal entropy wave embedded in a constant gravity field,with random perturbations of the vertical velocity localized be-tween − . < z < . The middle plot shows the time when theRTi just reached a nonlinear amplitude, the mode structure is stillclear. Finally the bottom plot shows a more developed nonlinearstage of the instability. In all plots, the grayscale represents theentropy. z >
0. Perturbations with a horizontal wavenumber k x are unstable with a growth rate σ = ( k x g ∆ ρ/ρ ) / (e.g.Chandrasekhar (1961)). The vertical structure of theRTi mode on both sides of the discontinuity is a decreas-ing exponential: δA ∝ e − k x | z | due to the incompressiblenature of the flow ( k z + k x = 0). Using the same argu-ment as in Sect. 3.3, the estimated propagation speed is v prop = σ /k x . In an entropy wave, each unstably strat-ified layer is followed by a stably stratified one. Unlikethe instability of a vorticity wave, adjacent layers are notexpected to cooperate. The vertical propagation of theRTi is expected to be accelerated by unstable layers anddecelerated by stable ones. Numerical simulations of the RTi
In order to measure the propagation speed of the RTi,we performed numerical simulations of a sinusoidal en-tropy profile S ( z ) described by: S ( z ) = ∆ S sin ( Kz ) . (16)In units of the vertical wavelength 2 π/K ≡
1, we chose acomputational domain − < z < < x <
8, withperiodic boundary conditions in the x direction, and aresolution of 1024 × Fig. 7.—
Propagation of the RTi along the z direction. Thethree curves represent modes with a wave number ranging from k = 0 . K (upper curve) to k = 2 K (lower curve). The plottedquantity, z + ( t ), corresponds to the spatial extent in which theRTi has reached an amplitude of δv = 10 − c . The curves havebeen shifted in time so that they start to deviate from 0 at t = 0. restricted to − < z <
5, the medium being uniform for | z | >
5. At | z | = 8 we imposed a zero gradient boundarycondition. The different phases of the RTi growth areillustrated by three snapshots in Fig. 6. The RTi growsat all scales resolved by the grid. Numerical convergencehas been checked for the modes studied in the following( k/K & . − . < z < . .
01% of the sound speed c in order to follow their propagation.As expected in the previous Section, the propagationis less regular than for the KHi: it is slow in the sta-bly stratified regions and faster in the unstable ones(Fig. 7). It is however possible to measure a globalpropagation speed, which is remarkably close to the es-timate σ/k deduced from the mode analysis (upper plotin Fig. 8). The propagation speed is well approximatedby v prop = 1 . σ/k .Using Eq. (7), the RTi growth rate for a given advec-tion speed is expressed as a function of the wave num-ber. It is maximum at a longer wavelength than in theabsence of advection, because shorter wavelengths prop-agate more slowly. We approximate linearly the decreaseof the maximum RTi growth rate σ RTmax when the ad-vection velocity v z increases (bottom plot in Fig. 8): σ RTmax = σ (cid:18) − v z v eff0 (cid:19) , (17)where σ = 0 . N is the RTi growth rate with-out advection and v eff0 is an effective propagation speedwhich we estimate to be v eff0 ≃ . N/K . Analytical estimate of the RTi growth rate withboth advection and stratification
By a similar reasoning as in Sec. 3.4, the RTi growthin the presence of both advection and stratification is
Fig. 8.—
Upper plot: propagation speed of the RTi as a functionof its wave number k . The triangles are measured in the simu-lations. The black line is the quantity σ/k estimated from thelinear mode calculation. Bottom plot: maximum growth rate ofthe RTi as a function of the advection velocity (full thick line) andcorresponding wave number (dashed line). The thin line with ar-rows illustrates the linear approximation of the growth rate usedin Eq. (17). approximated using the results of Sects. 4.2 and 4.3: σ RT = 0 . (cid:20) γ − γ g ∇ (∆ S + S ) (cid:21) − . Kv z . (18)The entropy gradient ∇ (∆ S ) in a SASI mode is propor-tional to the relative shock displacement ∆ r/r sh . Defin-ing ∇ S sasi by ∇ (∆ S ) ≡ ∇ S sasi ∆ r/r sh , we use the cri-terion σ RT = σ sasi to obtain an explicit estimate of thelocal saturation amplitude of SASI due to the parasiticgrowth of the RTi:∆ rr sh = ∇ S ∇ S sasi + γ .
56 ( γ − g ∇ S sasi ( σ sasi + 0 . k sasi v z ) . (19) ACOUSTIC FEEDBACK IN THE PRESENCE OFPARASITIC INSTABILITIES
The analytical estimates of the saturation ampli-tude obtained in Eq. (13) and (19) from the criterion σ parasite = σ sasi are directly compared to the numeri-cal simulations of SASI in Sect. 6. Before that, we usethe simplified toy-model of Sato et al. (2009) to evalu-ate the nonlinear effect of the parasitic instabilities onthe advective-acoustic cycle. The distortion of the SASImode by growing parasites, illustrated by the bottomplots in Fig. 2 and 6, is expected to induce a decrease inthe acoustic feedback and stabilize the advective-acoustic Fig. 9.—
Efficiency of the acoustic feedback as a function of theamplitude of the advected wave. The dashed line corresponds toan entropy vorticity wave (case i), the full line to an entropy wave(case ii) and the dotted line to a vorticity wave (case iii). The twovertical lines represent the estimate of the amplitude cutoff wherethe corresponding efficiency has decreased by 50%: dashed for theentropy-vorticity wave, full for the entropy wave. cycle responsible for the growth of SASI.The “problem 1” studied by Sato et al. (2009) dealswith the deceleration of an advected wave through anexternal potential, in a planar toy-model. This deceler-ation region of size ∆ z ∇ generates an acoustic feedback,measured at a distance z meas above it. The advectedwave of amplitude ǫ S is perturbed by a random noiseacting as a seed for the parasitic instabilities. We choose∆ z ∇ = 0 . M = 5, c /c = 0 . ωτ aac / π = 2.The numerical technique based on a AUSMDV schemeis described in Sato et al. (2009).The effects of the KHi and RTi on the acoustic feed-back are studied together and separately by performingthree sets of simulations: i) with the same mixture ofentropy and vorticity as produced by a perturbed shock(Eq. (9-13) of Sato et al. (2009)), ii) with the same en-tropy structure but no vorticity, iii) with the vorticitystructure of (i) but no entropy.The pressure measured at z meas = 3 is Fourier trans-formed in the x -direction and in time, in order to esti-mate the part of the acoustic feedback which is coherentwith the initial advected wave. This coherent feedback isresponsible for the closure the advective-acoustic cycle.Above a certain amplitude threshold, the acoustic feed-back efficiency decreases from the value predicted by thelinear analysis to a small fraction of this value (Fig. 9).This threshold is measured as the amplitude ǫ S at whichthe acoustic feedback efficiency is 50% of its linear value.We find a value of ǫ S = 0 .
68 for case (i) and ǫ S = 1 . n = 1 component is decreased sig-nificantly in the region where the parasites have grown(Fig. 11). The vertical structure of the feedback is alsodistorted.In Fig. 9, the small amplitude of the acoustic feed-back in the case (iii) decreases by 30% for ǫ S = 0 .
8, andincreases again for ǫ S = 1: this increase is due to the Fig. 10.—
Acoustic feedback from an entropy-vorticity wave (casei) in the presence of the KHi parasites. The left column representsa wave of linear amplitude ( ǫ S = 0 .
1, below the cutoff), while theright column represents a wave of non-linear amplitude ( ǫ S = 1,above the cutoff). The three rows show entropy (upper), vorticity(middle), and pressure (bottom) perturbations. The horizontaldashed lines represent the extent of the potential jump. The KHiis able to grow only on the non-linear wave (right column). pressure associated with the KHi, propagating againstthe flow.Can the linear description of the KHi and RTi (Sect. 3and 4) predict the value of the threshold? We use theanalytical estimates of the KHi and RTi growth rates(Eqs. (12) and (18)) at marginal stability to estimatethe threshold amplitude ǫ S ( z ) above which the KHi orRTi can grow despite the stabilizing effect of advection.The ǫ S -threshold for neutral stability is expected to bea lower bound for the threshold measured in the simula-tions (Fig. 13).The RTi can grow only in the region − . < z < . z ∼ ǫ S = 0 .
78, which appears to be agood estimate of the amplitude above which the RTi cangrow and damp the acoustic feedback.In the case (ii) where the upstream advected wave con-tains no vorticity, the growth of the KHi is subdominantand does not affect the acoustic feedback because it takesplace below the region of deceleration (Fig. 13). The RTihe saturation of SASI 9
Fig. 11.—
Same as Fig. 10 but filtered to keep only the n = 1component of the horizontal structure. In the entropy and vorticityprofile a clear decrease in amplitude is visible where the KHi hasgrown. In the right column, the pressure perturbations are slightlysmaller and the coherence of the vertical structure is lost. is thus the dominant instability. The linear threshold( ǫ S = 0 .
78) deduced from Fig. 13 is about 40% smallerthan the value of ǫ S = 1 . ǫ S ∈ [0 . , . .
68 versus 1 . ∼ . ǫ S =0 . −
40% smaller thanthe threshold at half efficiency measured in the simula-tions. Let us emphasize that in case (i), although thelinear acoustic feedback is essentially generated by theentropy wave, vorticity plays an essential role in deter-mining the saturation threshold through the KHi. Thisillustrates the non-trivial interplay of vorticity and en-
Fig. 12.—
Acoustic feedback from an entropy wave (case ii) inthe presence of RTi parasites. The left column represents a wave oflinear amplitude ( ǫ S = 0 .
1, below the cutoff), while the right col-umn represents a wave of non-linear amplitude ( ǫ S = 2, above thecutoff). The two rows show entropy (upper) and pressure (bottom)perturbations. The horizontal dashed lines represent the extent ofthe potential jump. The RTi is able to grow only on the non-linearwave (right column). Fig. 13.—
Amplitude ǫ S corresponding to the marginal stabilityof secondary instabilities, as a function of z in the problem 1 ofSato et al. (2009). Full lines show the KHi marginal stability: theentropy-vorticity wave (thick line, case i), the entropy wave (thinline, case ii). The dashed line shows the RTi marginal stability inboth cases. The two vertical dotted lines show the vertical extentof the potential jump. tropy in the advective-acoustic cycle. COMPARISON WITH (MORE) REALISTICSIMULATIONS: THE EFFECT OF NUCLEARDISSOCIATION
In order to test the parasitic scenario against resultsfrom numerical simulations, we apply the above esti-mates to the set up of Fern´andez & Thompson (2009a),0
Fig. 14.— “Local saturation amplitude” as a function of the ra-dius for the KHi ( upper plot ) and RTi ( bottom plot ) instabilities.The thick lines show the fundamental mode for different dissocia-tion energies (upper curves correspond to lower energies). The thinlines show higher harmonics with the dissociation energy ǫ = 0 . v . where the energy loss at the shock ( ǫ in their notations)due to the dissociation of iron is varied in a parameter-ized way from ǫ = 0 (the setup of Blondin & Mezzacappa(2006)) to ǫ = 0 . v . Here v ff is the free fall velocityat the shock. The shock radius is kept constant by ad-justing the cooling function, which is varied by a factorup to 127 (the shock radius and the adiabatic index are r sh /r star = 2 . γ = 4 / ǫ .By solving the radial structure of eigenmodes in thesetup of Fern´andez & Thompson (2009a), we calculatethe parameters w sasi and ∇ S sasi and use Eqs. (13) and(19) to estimate the “local saturation amplitude” of SASIoscillations above which the parasites grow faster thanSASI at a given radius. As indicated by Fig. 14, the sat-uration amplitude of SASI associated with each parasiticinstability decreases strongly when ǫ increases.The global saturation amplitude can be estimated asthe minimum of the local saturation amplitude, at leastif this minimum is sufficiently broad and above the cou-pling radius. The curves in Fig. 14 show a minimum at an Fig. 15.—
Roles of the advection, the entropy stratification, andthe SASI growth rate in the “local saturation amplitude”. Thisis illustrated with the fundamental SASI mode at ǫ = 0 and theRTi, but it is qualitatively the same if one considers other SASImodes, dissociation energies, or the KHi. The thick line shows the“local saturation amplitude”, the thin line the same quantity if oneneglects the growth rate of SASI, the dotted and the dashed linesthe contributions of the advection and the entropy stratification. intermediate radius between the proto-neutron star andthe shock, approximately at r mini ∼ ( r sh + r ∗ ) /
2. Thiscan be understood by the fact that higher up the shockparasites are efficiently stabilized by advection, whileclose to the proto-neutron star they are strongly stabi-lized by the entropy stratification (Fig. 15). The most ef-ficient growth of the parasites therefore takes place whereneither advection nor stratification is strong. As the min-ima of the curves in Fig. 14 are quite flat, the parasitesshould be able to grow in a large region of the flow aroundthe radius r mini when SASI saturates.The saturation amplitude predicted by our analy-sis of the KHi and RTi is compared with the resultsof the simulations by Fern´andez & Thompson (2009a)in Fig. 16. The RTi (thick full line) is expected tobe the dominant parasite because it grows at smallerSASI amplitudes than the KHi. We note that someRTi structures are clearly visible in the simulations ofFern´andez & Thompson (2009a) (online edition) for ǫ =0 . v and ǫ = 0 . v , in agreement with our conclusionthat the RTi is the dominant secondary instability. How-ever, these RTi structures are less obvious when ǫ = 0.The amplitude in the simulations decreases by a factor ∼
25 between ǫ = 0 and ǫ = 0 . v , while our estimatedecreases by a factor 15. In addition to reproducing cor-rectly the trend, the saturation amplitude given by thissaturation mechanism is 15 −
50% smaller than the simu-lated value for all ǫ . This is consistent with Sect. 5, wherewe found that the stability threshold of the parasites was ∼ −
40% smaller than the amplitude at which theireffect is important. Given the many approximations in-volved in our analytic description of the parasites, andthe many other nonlinear effects we neglected, the com-parison in Fig. 16 is considered very encouraging.One of our assumptions is that the background sta-tionary flow is unchanged, which is justified for low sat-uration amplitudes but is less justified if the saturationamplitude is very nonlinear. The uncertainty of our an-alytical estimate for large saturation amplitudes is illus-he saturation of SASI 11
Fig. 16.—
Saturation amplitude of SASI as a function ofthe dissociation energy ǫ : Comparison of the simulations byFern´andez & Thompson (2009a) with the parasitic instabilities sce-nario. The black diamonds show the amplitude of the saturated l = 1 mode in the simulations of Fern´andez & Thompson (2009a)(rms fluctuation of the l = 1 Legendre coefficient averaged oververy long timescales, after the flow has settled to a quasi-steadystate). The empty diamonds show the saturation amplitude nor-malized by r sh − r ∗ , where r sh is the average shock radius duringthe nonlinear phase of SASI, instead of the shock radius in the sta-tionary flow for the black diamonds (R. Fernandez, private commu-nication). The thick lines are the saturation amplitude predictedfor the most unstable mode of the KHi (gray line), and the RTi(black). The dotted (respectively dashed) line shows the RTi sat-uration amplitude when σ sasi is set to 0 in Eqs. (19) (respectivelywhen one considers only the fundamental mode of SASI). trated in Fig. 16 by the empty diamonds, where thesaturation amplitude is normalized using the averagedshock radius during the nonlinear phase of SASI insteadof its value in the stationary flow. This new normaliza-tion brings the values from the simulations closer to thepredicted ones (R. Fernandez, private communication).What is the dominant effect causing the dramatic de-crease of the SASI amplitude when ǫ is increased? Us-ing Eq. (19) allows us to identify the contributions ofthe SASI growth rate σ sasi , the relative amplitude of theSASI entropy wave ∇ S sasi , the background entropy strat-ification ∇ S and the background advection velocity v z .According to our linear stability analysis, the most un-stable SASI mode is the fundamental one if ǫ is small, thefirst harmonics if ǫ > . v and the second harmonicsif ǫ > . v . The comparison between the dashed lineand thick full line in Fig. 16, separated by about ∼ ∇ S sasi is partly due to the factthat entropy gradients in a SASI mode increase with fre-quency. It is striking that the sharp drop in the SASIamplitude that is observed by Fern´andez & Thompson(2009a) coincides with the shift from the fundamental tothe first radial overtone. Our analysis suggests howeverthat this shift may be the cause of only a small fractionof the decrease, and should not be considered as a gen-eral feature of the saturation of SASI with dissociation.Indeed, repeating our analysis for other aspect ratios ofthe shock to star radius indicates that this shift can alsohappen much earlier (e.g. for r sh /r ∗ = 0 .
2) or not at all(e.g. for r sh /r ∗ = 0 . σ RT = 0) can be compared to the saturationamplitude ( σ RT = σ sasi ). From Fig. 16 (dotted and thicklines), we estimate that the lower growth rate of SASIfor high dissociation energy is responsible for a ∼ v z . As the RTidevelops more easily in a slow flow, the saturation am-plitude of SASI naturally decreases when dissociation isincreased. This major effect contributes to a factor 4 . σ = 0in order to distinguish it from the effect of the change inthe SASI growth rate).An additional factor 2 is due to the change in the flowprofile, in particular to the decrease of the entropy strat-ification ∇ S which favors the growth of parasitic insta-bilities. DISCUSSION
SASI or neutrino-driven convection?
Fern´andez & Thompson (2009b) argued that the largeamplitude l = 1 oscillations appearing in their numer-ical simulations including iron dissociation and a heat-ing function is due to neutrino-driven convection ratherthan SASI, since SASI is stabilized by iron dissociationaccording to Fern´andez & Thompson (2009a). Does irondissociation at the shock really prevent SASI from grow-ing to large amplitudes in a realistic core collapse? Inrealistic simulations the compression factor at the shockcan reach ∼
10, which corresponds to ǫ = 0 .
14 in thepresent study and a SASI amplitude of 6% of the shockdistance ( r sh − r ∗ ), quite smaller than without dissocia-tion (40%). We point out however that a significant frac-tion of this amplitude decrease may be an artifact of theparameterization, which changed the cooling function bya factor 127 in order to keep the ratio r sh /r ∗ constant.This parameterization has the great advantage of beinginsensitive to geometrical effects that may arise if the as-pect ratio between the shock and the cooling surface ischanged. However cooling is then artificially low whendissociation is taken into account without heating. As aconsequence, the resulting flow profile may not be morerealistic than the flow profile without dissociation. Asour analysis suggests that entropy gradients play an im-portant role in the saturation of SASI, we investigatedthe effect of keeping the cooling function constant whendissociation is varied, resulting in a change of the shockradius (from 2 . r ∗ to 1 . r ∗ , for 0 < ǫ < . v ). Byperforming the same analysis as in Section 6, we thenfind that the saturation amplitude of SASI should de-crease significantly less than when the shock radius iskept constant : ∆ r/ ( r sh − r ∗ ) decreases by a factor 1 . r/r sh and ∆ r/r ∗ are decreased bya factor 3 . r sh is constant). Althoughthe geometrical effects make a direct comparison difficult,the fact that all these numbers are significantly smallerthan the former variation by a factor 15 suggests thatthe decrease of the cooling function, necessary to keepthe shock radius constant, plays a key role in decreas-ing the saturation amplitude. More insight on this issue2may be gained by including the effect of neutrino heatingin a parameterized manner, such that dissociation couldbe varied while both the cooling function and the shockradius are constant. This calculation is left for a futurestudy.The numerical simulations by Scheck et al. (2008) sug-gest that SASI is able to grow to large amplitudes evenin the presence of dissociation. These simulations aresignificantly more realistic than the set up studied here,since they include a realistic equation of state where dis-sociation is taken into account in a physical way, anda simplified treatment of neutrino heating and cooling.They also differed from those by Fern´andez & Thompson(2009a) by their choice of a moving inner boundary mim-icking the proto neutron star contraction. In some of themodels of this article (e.g. W00), neutrino-driven convec-tion was artificially suppressed but still SASI oscillationscould grow to non negligible amplitudes. It is howeverdifficult to determine which difference between the twosetups affects most importantly the saturation amplitudeof SASI.Incidentally, it is worth noting that RTi mushroomshave been identified growing on the SASI entropy gradi-ents in Fig. 7 of Scheck et al. (2008) and were interpretedin their Sect. 6.1 as secondary convection. Although inthat article convection was not recognized as an agent ofSASI saturation, the fact that the RTi appears at a SASIamplitude close to the saturation amplitude is consistentwith a parasitic mechanism of saturation.The interaction between SASI and neutrino-drivenconvection is complex and still poorly understood. Couldneutrino-driven convection prevent the growth of SASIby breaking its mode structure ? One may arguethat neutrino-driven convection does not feed upon theSASI mode energy, but rather converts free energy fromthe stationary gradients into vorticity. Could neutrino-driven convection feed SASI, either by creating vortic-ity which would enter the advective-acoustic cycle, or bycreating sound waves (Fern´andez & Thompson 2009b)?These difficult questions are beyond the scope of ourstudy. Distinguishing RTi from KHi in the simulations
The RTi is often characterized by finger-like ormushroom-like structures as in Fig. 6, while the KHi ischaracterized by vortices as in Fig. 2 and 10. However, ina complex flow containing both entropy gradients, shearand advection, RTi mushrooms may look like vortices(Fig. 12).The following criterion may be more useful: the RTishould occur preferentially where the entropy perturba-tion is maximum, while the KHi occurs where the shearis maximum. In a sloshing mode these two maxima arevery distinct: the entropy oscillation is maximum at thepole (where the shock speed is maximum), while vorticityis maximum at the equator (where the inclination of theshock is maximum). The parasitic structures visible inthe simulations by Scheck et al. (2008) and in the moviespublished online by Fern´andez & Thompson (2009a) aremore vigorous near the pole, in agreement with our anal-ysis.Furthermore, the RTi structure should grow preferen-tially on the half wavelength of a SASI mode where theentropy gradient is negative. By contrast, the KHi should grow on the whole extent of the SASI wavelength. Aninspection of Fig. 7 of Scheck et al. (2008) and of themovies by Fern´andez & Thompson (2009a) confirms thisdistinct feature of the RTi.
Numerical resolution needed to resolve the parasites
An interesting concern raised by this saturation mech-anism is that simulations should be able to resolve theparasitic instabilities properly in order to give reliableresults on the nonlinear behavior of SASI. The RTi is ashort wavelength instability, but as is shown in Sect. 4.3advection tends to stabilize the small scales and makesthe RTi dominated by large scales. Entropy stratifica-tion on the other hand favors small scales. As in theset up studied here the RTi is found to develop whereboth stabilizing effects are important, it is hard to makeany prediction for its dominant wavelength. Any conver-gence study should verify that the grid size allows for thegrowth of parasites.As an example, Scheck et al. (2008) witnessed thegrowth of Rayleigh-Taylor mushrooms with a typical an-gular scale of l ∼ −
30. They were able to cap-ture this small angular scale by using 360 angular zonesfor 180 ◦ . Most 2D simulations use a resolution with > −
200 angular zones, and would probably resolvethese scales (200 zones in Murphy & Burrows (2008),121 in Burrows et al. (2006), 128-192 in Marek & Janka(2009), and 60-120 zones in Ohnishi et al. (2006)). How-ever 3D simulations may not be able to resolve such smallscales. For example Iwakami et al. (2008, 2009) mostlyuse a resolution of 30 angular zones for 180 ◦ , which maybe too coarse to capture such a small scale behavior. Wenote that the saturation amplitude of the low- l modes( l = 1 −
3) in Fig. 16 of Iwakami et al. (2008) is slightlysmaller at “high resolution” (60 zones) than at “low res-olution” (30 zones). While this may be explained by asuppression of parasitic instabilities at low resolution, wecannot exclude that a different saturation process maytake place in 3D, as discussed below.
Effects of other physical ingredients
The saturation mechanism described in this paper canbe used to anticipate the effects of many other physicalingredients of the core collapse model (e.g. 3D versus 2D,the rotation rate, the magnetic field) although a detailedanalysis is beyond the scope of this paper. •
3D versus 2D : 3D simulations allow for non-axisymmetric modes that are artificially forbiddenin axisymmetric simulations, thus a greater num-ber of modes is available to the linear developmentof SASI. A single mode l = 1, m = 0 often dom-inates in 2D, whereas 3 modes l = 1, m = 0 , ± m . Wecannot exclude that nonlinear processes associatedwith the coupling between different mode, ignoredin our analysis, are more important in 3D thanin 2D. Contrary to Iwakami et al. (2008) however,Blondin & Mezzacappa (2007) found that one spi-ral mode dominates the 3D dynamics.he saturation of SASI 13Assuming the parasitic growth of instabilities isthe dominant saturation mechanism, our analy-sis based on a linear description of the parasiteswould predict the same saturation amplitudes ofSASI in 2D or 3D. However the nonlinear behav-ior of the RTi is known to differ in 3D and 2D(e.g. Goncharov (2002), Cabot (2006)), and thismay affect the saturation of SASI. Iwakami et al.(2008) reported a smaller saturation amplitude ofeach individual SASI mode in 3D as compared to2D, although the numerical convergence of this re-sult should be further checked (Section 7.3). Ifconfirmed, it would raise the following questions: is this difference in amplitude a consequence ofthe different non linear Rayleigh-Taylor behavior in3D? Or is this the signature of a different satura-tion mechanism based on the interaction of m = 0modes ? A more systematic parametric study, sim-ilar to Fern´andez & Thompson (2009a) but in 3D,could help check the relevance of parasitic instabil-ities in 3D. • Rotation rate : Yamasaki & Foglizzo (2008) haveshown that rotation increases the growth rate ofthe spiral modes rotating in the same directionas the steady flow, while stabilizing the counter-rotating ones. If the rotation is strong enough,a single spiral mode dominates the evolution ofSASI (Blondin & Mezzacappa 2007; Iwakami et al.2008). According to our analysis (Eq. (19)), thelarger growth rate of the spiral mode could lead toa larger saturation amplitude of SASI. Neverthe-less, a detailed calculation using the exact entropyand vorticity profiles of the SASI eigenmodes in arotating flow is required in order to make an accu-rate prediction. • Magnetic field strength : The effect of the mag-netic field on the linear phase of SASI is yet tobe understood (Guilet & Foglizzo 2010), but its ef-fect on parasitic instabilities can already be antic-ipated from the point of view of the magnetic ten-sion which tends to prevent motions that distortthe magnetic field lines. This effect is stabilizingfor the perturbations with a wave vector parallel tothe magnetic field, but does not affect those whosewave vector is perpendicular. One would then ex-pect that the magnetic field does not change themaximum RTi growth rate, but selects RTi modeswith a wave vector perpendicular to the field lines.In contrast, the KHi can be suppressed if the mag-netic field along the direction of the transverse ve-locity is strong enough. In a situation where theKHi were the dominant parasitic instability, a mag-netic field could potentially allow for a larger sat-uration amplitude. SUMMARY
In this article we have developed for the first time a pre-dictive mechanism for the saturation of SASI. In this sce-nario the saturation happens when a parasitic instabilityis able to grow fast enough to compete with SASI. Twotypes of instabilities are of potential importance: the RTigrowing on the entropy gradients created by SASI, and the KHi growing on the vorticity involved in the SASImode. For each of these parasites, two stabilizing effectswere found to be crucial: the entropy stratification inthe stationary flow and the advection of matter towardthe neutron star. An estimate of the growth rates takinginto account these effects has been obtained in Sect. 3and 4 for the KHi and RTi respectively. The satura-tion amplitude of a given SASI mode has been evaluatedby comparing its growth rate with that of the parasiticinstabilities.Using numerical simulations, we studied the effect ofparasitic instabilities on the acoustic feedback in the sim-plified context of the toy model introduced by Foglizzo(2009) and Sato et al. (2009). This confirmed the idea ofa threshold in amplitude above which the acoustic feed-back is reduced, that is determined by the ability of para-sitic instabilities to grow despite advection. A reasonableestimate of this threshold has been obtained by measur-ing the threshold of marginal stability, which is foundto be 25 −
40% lower than the amplitude at which thefeedback is decreased by 50%.The saturation mechanism by parasitic instabilitiescan reproduce the decrease of the SASI power withdissociation energy observed in the simulations ofFern´andez & Thompson (2009a). Our amplitude esti-mate based on linear growth rates remains 15 −50%lower than the saturation amplitude observed in theirsimulations, which is consistent with our simulationof the acoustic feedback in the toy-model. Further-more our analysis suggests that the RTi is the dom-inant secondary instability. This is consistent withthe presence of RTi mushrooms in the simulations ofFern´andez & Thompson (2009a) as well as Scheck et al.(2008).The strong decrease of the SASI power, when both thedissociation energy is increased and the cooling is de-creased, can be traced back to 4 different effects thathelp the parasitic growth of the RTi: (i) the slowerthe advection velocity in the postshock flow, the fasterthe propagation of the RTi against the flow, (ii) thesofter the negative entropy profile in the backgroundflow, the easier its destabilization by the SASI entropywave, (iii) the slower the growth of SASI, the lower thethreshold for competing parasites, (iv) the steeper theentropy wave in the SASI mode, the faster the RTi.The first two effects are the dominant ones. We pointout that other studies (Scheck et al. 2008; Ohnishi et al.2006), which included the effect of nuclear dissocia-tion, witnessed powerful SASI oscillations, contrary toFern´andez & Thompson (2009a). These studies differedfrom that of Fern´andez & Thompson (2009a) by the useof a realistic equation of state, the inclusion of some heat-ing, and in the case of Scheck et al. (2008) the contrac-tion of the inner boundary. The dominant cause of thediffering simulations has not been identified yet. Furtherinvestigations could shed more light on this question byapplying the analysis described in this article to set upsthat include one or several of these additional ingredi-ents (heating, realistic EOS, contraction of the innnerboundary).Although the saturation mechanism proposed inthis paper compares favorably with the results ofFern´andez & Thompson (2009a), it should be testedwith results from other setups in order to confirm its va-4lidity, in particular in the case of 3D simulations. We pro-pose that future simulations should look for signs of theparasitic instabilities, and check that the angular resolu-tion is sufficient to resolve them. If confirmed, our resultswould open new perspectives for anticipating the effecton the SASI amplitude of other physical ingredients suchas the equation of state, the heating rate, the rotation,and magnetic field of the progenitor star. They couldalso be useful as an input for analytical models studyingthe possible consequences of SASI, such as the model forgravitational wave emission proposed by Murphy et al.(2009) (see also Marek et al. (2009)). The authors are grateful to S. Fromang for stimulatingdiscussions and his help with the code RAMSES. Inter-esting discussions with J. Griffond and B.-J. Grea arealso acknowledged. R. Fernandez is thanked for shar-ing data from his simulations, as well as for insightfulcomments on an early draft. The anonymous referee isalso thanked for his remarks that signicantly improvedthis article. This work has been partially funded by theVortexplosion project ANR-06-JCJC-0119.