Some general properties of the advective-acoustic instability are described and understood using a toy model, which is simple enough to allow for analytical estimates of the eigenfrequencies. The ...essential ingredients of this model, in the unperturbed regime, are a stationary shock and a subsonic region of deceleration. For the sake of analytical simplicity, the two-dimensional unperturbed flow is parallel and the deceleration is produced adiabatically by an external potential. The instability mechanism is determined unambiguously as the consequence of a cycle between advected and acoustic perturbations. The purely acoustic cycle, considered alone, is proven to be stable in this flow. Its contribution to the instability can be either constructive or destructive. A frequency cutoff is associated with the advection time through the region of deceleration. This cutoff frequency explains why the instability favors eigenmodes with a low frequency and a large horizontal wavelength. The relation between the instability occurring in this highly simplified toy model and the properties of standing accretion shock instability observed in the numerical simulations of stellar core collapse is discussed. This simple setup is proposed as a benchmark test to evaluate the accuracy, in the linear regime, of numerical simulations involving this instability. We illustrate such benchmark simulations in a companion paper.
ABSTRACT
The gravitational collapse of rapidly rotating massive stars can lead to the onset of the low T/|W| instability within the central proto-neutron star (PNS), which leaves strong signatures in ...both the gravitational wave (GW) and neutrino emission. Strong large-scale magnetic fields are usually invoked to explain outstanding stellar explosions of rapidly rotating progenitors, but their impact on the growth of such instability has not yet been cleared. We analyse a series of three-dimensional magnetohydrodynamic models to characterize the effects of different magnetic configurations on the development of the low T/|W| and the related multimessenger features. In the absence of magnetic fields, we observe the growth on dynamical time-scales of the low T/|W|, associated with a strong burst of GW and a correlated modulation of the neutrino emission. However, models with a strong magnetic field show a quenching of the low T/|W|, due to a flattening of the rotation profile in the first ∼100 ms after shock formation caused by the magnetic transport of angular momentum. The associated GW emission is weakened by an order of magnitude, exhibits a broader spectral shape, and has no dominant feature associated with the PNS large-scale oscillation modes. Neutrino luminosities are damped along the equatorial plane due to a more oblate PNS, and the only clear modulation in the signal is due to Standing Accretion Shock Instability activity. Finally, magnetized models produce lower luminosities for νe than for $\bar{\nu }_e$, which is connected to a higher concentration of neutron-rich material in the PNS surroundings.
ABSTRACT
We study the impact of rotation on the hydrodynamic evolution of convective vortices during stellar collapse. Using linear hydrodynamics equations, we study the evolution of the vortices ...from their initial radii in convective shells down to smaller radii where they are expected to encounter the supernova shock. We find that the evolution of vortices is mainly governed by two effects: the acceleration of infall and the accompanying speed up of rotation. The former effect leads to the radial stretching of vortices, which limits the vortex velocities. The latter effect leads to the angular deformation of vortices in the direction of rotation, amplifying their non-radial velocity. We show that the radial velocities of the vortices are not significantly affected by rotation. We study acoustic wave emission and find that it is not sensitive to rotation. Finally, we analyse the impact of the corotation point and find that it has a small impact on the overall acoustic wave emission.
The linear stability of isothermal Bondi accretion with a shock is studied analytically in the asymptotic limit of high incident Mach number ${\cal M}_{1}$. The flow is unstable with respect to ...radial perturbations as expected by Nakayama (1993), due to post-shock acceleration. Its growth-time scales like the advection time from the shock ${r}_{\rm sh}$ to the sonic point rson. The growth rate of non-radial perturbations $l=1$ is higher by a factor ${\cal M}_{1}^{2/3}$, and is therefore intermediate between the advection and acoustic frequencies. Besides these instabilities based on post-shock acceleration, our study revealed another generic mechanism based on the cycle of acoustic and vortical perturbations between the shock and the sonic radius, independently of the sign of post-shock acceleration. The vortical-acoustic instability is fundamentally non-radial. It is fed by the efficient excitation of vorticity waves by the isothermal shock perturbed by acoustic waves. The growth rate exceeds the advection frequency by a factor $\log{\cal M}_{1}$. Unstable modes cover a wide range of frequencies from the fundamental acoustic frequency ~$ c/{r}_{\rm sh}$ up to a cut-off ∼ $c/r_{\rm son}$ associated with the sonic radius. The highest growth rate is reached for $l=1$ modes near the cut-off. The additional cycle of acoustic waves between the shock and the sonic radius is responsible for variations of the growth rate by a factor up to 3 depending on its phase relative to the vortical-acoustic cycle. The instability also exists, with a similar growth rate, below the fundamental acoustic frequency down to the advection frequency, as vorticity waves are efficiently coupled to the region of pseudosound. These results open new perspectives to address the stability of shocked accretion flows.
We analyze the linear stability of a stalled accretion shock in a perfect gas with a parameterized cooling function 8P super(b-a)P super(a) 8 r super(b-a) P super(a). The instability is dominated by ...the l = 1 mode if the shock radius exceeds 2-3 times the accretor radius, depending on the parameters of the cooling function. The growth rate and oscillation period are comparable to those observed in the numerical simulations of Blondin & Mezzacappa. The instability mechanism is analyzed by separately measuring the efficiencies of the purely acoustic cycle and the advective-acoustic cycle. These efficiencies are estimated directly from the eigenspectrum and also through a WKB analysis in the high-frequency limit. Both methods prove that the advective-acoustic cycle is unstable and that the purely acoustic cycle is stable. Extrapolating these results to low frequency leads us to interpret the dominant mode as an advective-acoustic instability, different from the purely acoustic interpretation of Blondin & Mezzacappa. A simplified characterization of the instability is proposed, based on an advective-acoustic cycle between the shock and the radius rv where the velocity gradients of the stationary flow are strongest. The importance of the coupling region in this mechanism calls for a better understanding of the conditions for an efficient advective-acoustic coupling in a decelerated, nonadiabatic flow, in order to extend these results to core-collapse supernovae.
Performing two-dimensional hydrodynamic simulations including a detailed treatment of the equation of state of the stellar plasma and for the neutrino transport and interactions, we investigate here ...the interplay between different kinds of non-radial hydrodynamic instabilities that can play a role during the postbounce accretion phase of collapsing stellar cores. The convective mode of instability, which is driven by the negative entropy gradients caused by neutrino heating or by variations in the shock strength in transient phases of shock expansion and contraction, can be identified clearly by the development of typical Rayleigh-Taylor mushrooms. However, in those cases where the gas in the postshock region is rapidly advected towards the gain radius, the growth of such a buoyancy instability can be suppressed. In this situation the shock and postshock flow can nevertheless develop non- radial asymmetry with an oscillatory growth in the amplitude. This phenomenon has been termed "standing (or spherical) accretion shock instability" (SASI). It is shown here that the SASI oscillations can trigger convective instability, and like the latter, they lead to an increase in the average shock radius and in the mass of the gain layer. Both hydrodynamic instabilities in combination stretch the advection time of matter accreted through the neutrino-heating layer and thus enhance the neutrino energy deposition in support of the neutrino-driven explosion mechanism. A rapidly contracting and more compact nascent neutron star turns out to be favorable for explosions, because the accretion luminosity and neutrino heating are greater and the growth rate of the SASI is higher. Moreover, we show that the oscillation period of the SASI observed in our simulations agrees with the one estimated for the advective-acoustic cycle (AAC), in which perturbations are carried by the accretion flow from the shock to the neutron star and pressure waves close an amplifying global feedback loop. A variety of other features in our models, as well as differences in their behavior, can also be understood on the basis of the AAC hypothesis. The interpretation of the SASI in our simulations as a purely acoustic phenomenon, however, appears difficult.
A toy model is analyzed in order to evaluate the linear stability of the gain region immediately behind a stalled accretion shock, after core bounce. This model demonstrates that a negative entropy ...gradient is not sufficient to warrant linear instability. The stability criterion is governed by the ratio j of the advection time through the gain region divided by the local timescale of buoyancy. The gain region is linearly stable if j < 3. The classical convective instability is recovered in the limit j >> 3. For j > 3, perturbations are unstable in a limited range of horizontal wavelengths centered around twice the vertical size H of the gain region. The threshold horizontal wavenumbers k sub(min) and k sub(max) follow simple scaling laws such that Hk sub(min) a 1/j and Hk sub(max) a j. The convective stability of the l = 1 mode in spherical accretion is discussed, in relation with the asymmetric explosion of core-collapse supernovae. The advective stabilization of long-wavelength perturbations weakens the possible influence of convection alone on a global l = 1 mode.
The instability of Bondi-Hoyle-Lyttleton accretion, observed in numerical simulations, is analyzed through known physical mechanisms and possible numerical artefacts. The mechanisms of the ...longitudinal and transverse instabilities, established within the accretion line model, are clarified. They cannot account for the instability of BHL accretion at moderate Mach number when the pressure forces within the shock cone are taken into account. The advective-acoustic instability is considered in the context of BHL accretion when the shock is detached from the accretor. This mechanism naturally explains the stability of the flow when the shock is weak, and the instability when the accretor is small. In particular, it is a robust proof of the instability of 3D accretion when $\gamma=5/3$ if the accretor is small enough, even for moderate shock strength (${\cal M}\sim 3$). The numerical artefacts that may be present in existing numerical simulations are reviewed, with particular attention paid to the advection of entropy/vorticity perturbations and the artificial acoustic feedback from the accretor boundary condition. Several numerical tests are proposed to test these mechanisms.
Context.
The growth of hydrodynamical instabilities is key to triggering a core-collapse supernova explosion during the phase of stalled accretion shock, immediately after the birth of a ...proto-neutron star (PNS). Stellar rotation is known to affect the standing accretion shock instability (SASI) even for small rotation rates, but its effect on the onset of neutrino-driven convection is still poorly known.
Aims.
We assess the effect of stellar rotation on SASI when neutrino heating is taken into account as well as the effect of rotation on neutrino-driven convection. The interplay of rotation with these two instabilities affects the frequency of the mode
m
= 2, which can be detected with gravitational waves at the onset of a supernova explosion.
Methods.
We used a linear stability analysis to study the dynamics of the accreting gas in the equatorial plane between the surface of the PNS and the stationary shock. We explored rotation effects on the relative strength of SASI and convection by considering a large range of specific angular momenta and neutrino luminosities.
Results.
The nature of the dominant non-axisymmetric instability developing in the equatorial post-shock region depends on both the convection parameter,
χ
, and the rotation rate. Equatorial convective modes with
χ
≳ 5 are hampered by differential rotation. At smaller
χ
, however, mixed SASI-convective modes with a large angular scale,
m
= 1, 2, 3, can take advantage of rotation and become dominant for relatively low rotation rates, at which centrifugal effects are small. For rotation rates exceeding ∼30% of the Keplerian rotation at the PNS surface, a new instability regime is characterised by a frequency that, when measured in units of the post-shock velocity and radius,
v
sh
/
r
sh
, is nearly independent of the convection parameter,
χ
. A strong prograde
m
= 2 spiral dominates over a large parameter range and is favorable to the production of gravitational waves. In this regime, a simple linear relation exists between the oscillation frequency of the dominant mode and the specific angular momentum of the accreted gas.
Conclusions.
Three different regimes of post-shock instabilities can be distinguished depending on the rotation rate. For low rotation rates (less than 10% of the Keplerian rotation at the PNS surface), differential rotation has a linear destabilising effect on SASI and a quadratic stabilising or destabilising effect on the purely convective equatorial modes depending on their azimuthal wavenumber. Intermediate rotation rates (10% to 30% of the Keplerian rotation) lead to the emergence of mixed SASI-convection-rotation modes that involve large angular scales. Finally, strong rotation erases the influence of the buoyancy and heating rate on the instability. This independence allows for a reduction in the parameter space, which can be helpful for gravitational wave analysis.
PDF
