11institutetext: 1 Université Paris-Saclay, Université Paris Cité, CEA, CNRS,
AIM, 91191, Gif-sur-Yvette, France
2Department of Physics and Energetic Cosmos Laboratory, Nazarbayev University,
Astana 010000, Kazakhstan
11email<EMAIL_ADDRESS>
# Effect of stellar rotation on the development of post-shock instabilities
during core-collapse supernovae

A.-C. Buellet1 T. Foglizzo1 J. Guilet1 E. Abdikamalov2

(Received ??, 2022; accepted ??,2022)

###### Abstract

Context. The growth of hydrodynamical instabilities is key to trigger 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. In this paper, we assess the effect of stellar rotation on SASI when
neutrino heating is taken into account, and the effect of rotation on the
convective instability when neutrino heating is higher. 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 use 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 explore rotation effects on the relative strength of SASI
and convection by considering a large range of specific angular momenta,
neutrino luminosities and mass accretion rates.

Results. The nature of the dominant non-axisymmetric instability developing in
the equatorial postshock region depends on both the convection parameter
$\chi$ and the rotation rate. Equatorial convective modes with $\chi\gtrsim 5$
are hampered by differential rotation. At smaller $\chi$, 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 $\sim 30\%$ of the
Keplerian rotation at the PNS surface, the growth rate of the dominant mode
depends weakly on the rate of neutrino heating which highlights the arising of
a new instability regime. Similarly, the frequency of this mode is
surprisingly independent of the heating rate, with a strong prograde spiral
$m=2$ dominating over a large parameter range which is favourable to the
production of gravitational waves. For any heating rate, 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 postshock instabilities can be
distinguished depending on the rotation rate. For small rotation rates (less
than $10\%$ of the Keplerian rotation at the PNS surface), differential
rotation has a quadratic stabilising or destabilising effect on equatorial
purely convective modes and a linear destabilising effect on SASI.
Intermediate rotation rates ($10$ to $30\%$ of the Keplerian rotation) lead to
the emergence of mixed SASI/convection/rotation modes involving large angular
scales. Finally, strong rotation erases the influence of the buoyancy and
heating rate on the instability. This independency allows for a reduction of
the parameter space, which is valuable for gravitational wave analysis.

###### Key Words.:

supernova – convection – rotation – hydrodynamics

##  1 Introduction

The death of massive stars begins with the collapse of their inner core, which
forms a proto-neutron star (PNS). The outer core of the star bounces off and
creates a shock wave that propagates outward, gradually losing energy,
dissociating iron atoms until it stalls. The development of multidimensional
instabilities during this phase of stalled shock impacts both the
multimessenger signature and the revival of the shock (Herant et al. 1994;
Janka & Mueller 1996; Couch & O’Connor 2014; Takiwaki et al. 2016; Janka et
al. 2016; Müller 2020; Burrows & Vartanyan 2021). Among them, the stationary
accretion shock instability (SASI) (Blondin et al. 2003; Blondin & Mezzacappa
2006) can generate shock oscillations and contribute to pushing the shock
further up (Scheck et al. 2008; Marek & Janka 2009; Hanke et al. 2013). The
magnitude of this effect depends on the concurrent growth of the neutrino-
driven convection, which can generate turbulence in the post-shock region
(Abdikamalov et al. 2015; Radice et al. 2016). Different paths to explosions
dominated either by neutrino-driven convection or SASI may occur depending on
the precise physical conditions determined by the progenitor structure (Müller
et al. 2012; Murphy et al. 2013; Fernández et al. 2014) and the magnitude of
pre-collapse turbulence asymmetries (Müller et al. 2017).

The development of neutrino-driven convection and/or SASI leaves clear
signatures in gravitational waves (Murphy et al. 2009; Kuroda et al. 2016;
Andresen et al. 2017). The eigenfrequencies deduced from the perturbative
study of the accretion flow can be recognised in the gravitational wave signal
despite the nonlinear development of instabilities (Torres-Forné et al. 2018,
2019b) and can be used to reconstruct the flow structure (Torres-Forné et al.
2019a; Sotani & Takiwaki 2020; Sotani et al. 2021). These asteroseismic
properties can help constrain the equation of state at nuclear densities
(Kuroda et al. 2016; Sotani et al. 2017).

The neutrino signal is also a precious messenger carrying direct information
on the frequency of large-scale shock oscillations induced by SASI, currently
detectable for a galactic supernova (Tamborra et al. 2013).

The competition between SASI and convection seems to vary from one numerical
code to another (Hanke et al. 2013; Ott et al. 2018; O’Connor et al. 2018;
Glas et al. 2019). A detailed understanding of the instabilities mechanisms
can help interpret the results of numerical simulations and guide the
exploration of the parameter space. SASI results from the interaction of
pressure and advected perturbations of entropy and vorticity between the shock
with the surface of the PNS (Foglizzo et al. 2007; Foglizzo 2009; Fernández &
Thompson 2009; Guilet & Foglizzo 2012). The convective instability in the
post-shock region is driven by neutrinos emitted by the cooling PNS: in the
region where the absorption of neutrino energy exceeds the losses by neutrino
emission, the heating by neutrino absorption creates a negative entropy
gradient favourable to convection (Bethe 1990). The growth of this neutrino-
driven convection requires a strong enough neutrino heating such that the
buoyancy timescale is shorter than one third of the advection timescale across
this region (Foglizzo et al. 2006). Above a critical heating rate, the
fundamental oscillatory mode of SASI becomes the purely growing convective
instability that dominates the dynamics of the flow (Yamasaki & Yamada 2007;
Fernández et al. 2014).

Most core-collapse simulations and theoretical work neglect the impact of
rotation, which is therefore not well known (Suwa et al. 2010; Chatzopoulos et
al. 2016; Fujisawa et al. 2019; Powell & Müller 2020). Rotation is however
thought to play an important role in at least a small fraction of core-
collapse supernovae with more extreme properties, such as superluminous
supernovae (Woosley 2010; Inserra et al. 2013) or hypernovae and long gamma-
ray bursts (Woosley 1993; Metzger et al. 2011). It is furthermore possible
that rotation plays a less extreme role in a larger fraction of core-collapse
supernovae. Theoretical models of stellar evolution constrained by the
efficient transport of angular momentum inferred from asteroseismic
observations of red giants (Cantiello et al. 2014), and the observations of
pulsar spins (Popov & Turolla 2012) suggest that the majority of supernova
explosions originate from slowly rotating stellar cores ($<0.5$ rad/s at
$1000$km).

Rotation rates up to 2 rad/s are more exceptional but commonly considered to
explain extreme events such as hypernovae, superluminous supernovae and GRBs.
In this regime, the convective dynamo (Thompson & Duncan 1993; Raynaud et al.
2020, 2022; Masada et al. 2022; White et al. 2022) and the magnetorotational
instability (Akiyama et al. 2003; Obergaulinger et al. 2009; Guilet & Müller
2015; Guilet et al. 2022; Reboul-Salze et al. 2021, 2022) are expected to
amplify efficiently the magnetic field of the PNS. Raynaud et al. (2020)
predicted that magnetar-like magnetic fields can be generated by the strong
field branch of the convective dynamo, which takes place for specific angular
momentum larger than $4\times 10^{15}\,\mathrm{cm^{2}/s}$. This threshold
corresponds to an angular frequency as low as $0.4$ rad/s at $1000$ km in the
progenitor. The extraction of rotational energy with such a strong magnetic
field can lead to strong magnetorotational explosions (e.g. Burrows et al.
2007; Takiwaki et al. 2009; Kuroda et al. 2020; Obergaulinger & Aloy 2021;
Bugli et al. 2021).

Beside the generation of magnetic fields, rotation can have several other
effects: the centrifugal force affects the shape of the neutrinosphere; the
differential rotation modifies the development of SASI and convective
instabilities; differential rotation can also generate a new instability.
These multiple effects are often studied in the hydrodynamical approximation
because the dynamo timescale is uncertain.

The centrifugal force diminishes the action of gravity and results in a larger
radius of the neutrinosphere in the equatorial plane. The equatorial decrease
in neutrino luminosity and neutrino mean energy is not favourable to the
explosion according to axisymmetric simulations (Marek & Janka 2009). The
lower equatorial temperature favours neutrino heating in the polar region and
a bipolar explosion (Suwa et al. 2010).

Stellar rotation can be favourable to the development of a vigorous, prograde,
spiral SASI mode dominated by large angular scales $l=1,2$ when neutrino
absorption is neglected (Blondin & Mezzacappa 2007). This destabilising effect
of differential rotation exists even for slow rotation with a negligible
centrifugal contribution, as shown by perturbative analyses (Yamasaki &
Foglizzo 2008; Walk et al. 2022) and confirmed by numerical simulations
(Kazeroni et al. 2017; Blondin et al. 2017). However, the driving mechanism of
this rotational destabilisation is not understood yet (Walk et al. 2022).

Classical studies of the effect of rotation on convection considered a
rotation axis aligned with gravity. In a viscous fluid with thermal diffusion,
the critical Rayleigh number defining the onset of thermal convection is
increased by rotation (Chandrasekhar 1961; Rossby 1969; Wedi et al. 2021). The
few studies that considered the impact of rotation in the equatorial plane did
not involve radial advection, which is crucial in the supernova case (Feudel &
Feudel 2021). According to the Solberg-Høiland criterion, the development of
axisymmetric convection can be stabilised by rotation if the specific angular
momentum increases outward (Endal & Sofia 1978). In axisymmetric simulations
of stellar core-collapse, this effect produces less vigorous convective
motions in the equatorial plane, and results in later-time explosions which
are weaker at the equator than at the poles (Fryer & Heger 2000). These 2D
results seemed to be confirmed in 3D for the fastest spinning progenitors
(Fryer & Warren 2004) ($\Omega\sim 4.1$ rad/s at $1000$ km), in a regime where
centrifugal effects can be dominant at reducing both the effective gravity and
the neutrino luminosity in the equatorial plane, and thus a reduced buoyancy
compared to the polar region.

Estimating the effect of modest rotation in the equatorial region of post-
shock convection is less obvious when centrifugal effects are small,
remembering that inward accretion produces a uniform profile of specific
angular momentum. With a rotation rate of $\Omega\sim 1.3$ rad/s at $1000$ km,
core-collapse simulations showed an earlier onset of neutrino-driven
convection that produced a stronger explosion in the equatorial plane, earlier
than the non-rotating case (Nakamura et al. 2014).

Even a modest amount of rotational kinetic energy $T$ compared to the
potential energy $|W|$ can trigger a spiral instability known as “low-$T/|W|$
instability” in the interior of isolated neutron stars (Shibata et al. 2002;
Watts et al. 2005; Passamonti & Andersson 2015). The mechanism of this
instability relies on the extraction of energy and angular momentum from
internal regions rotating faster than the spiral pattern towards external
region rotating slower (Cairns 1979; Saijo & Yoshida 2006). This instability
has been observed in 3D simulations of stellar core-collapse, where it
enhances the energy transport from the PNS to the shock and can lead to
stronger explosions (Ott et al. 2005; Cerdá-Durán et al. 2007; Takiwaki et al.
2016, 2021).

The interplay of centrifugal effects, SASI, convection and the low-$T/|W|$
instability can be difficult to disentangle in numerical simulations
considering the diversity of progenitors and numerical approximations (Ott et
al. 2008). Our current understanding relies on a very sparse sampling of this
diversity. During the collapse of a $27M_{\odot}$ progenitor, the dynamic of
the shock is driven by SASI and the convection for small rotation rates, and
dominated by the low-$T/|W|$ instability for high enough rotation rates
($\Omega=2$ rad/s) (Takiwaki et al. 2021). The enhancement of SASI by
differential rotation can compensate for the loss of neutrino energy due to
the centrifugal force during the collapse of a $15M_{\odot}$ progenitor (Summa
et al. 2018). The gravitational wave (GW) analysis of these simulations can
help identify the physical processes, such as the enhancement of SASI for high
rotation rates (Andresen et al. 2019). The 3D simulations of stationary
accretion by Iwakami et al. (2014) considered a range of rotation rates, mass
accretion rates, and neutrino luminosities. The structure of the dominant
instability and the observed patterns were classified according to three main
categories: spiral, buoyant bubbles, or spiral and buoyant bubbles. However,
the simultaneous variation of both rotation and neutrino luminosity made it
difficult to disentangle their respective effects.

To have a better understanding of the effect of each parameter on the growth
of instabilities, we propose to use a linear analysis, varying the rotation
and the heating rate separately. Doing this, we will be able to disentangle
the effect of these parameters in the linear regime. By focusing on the
accretion region above the surface of the PNS, we do not include the
interaction with the low-$T/|W|$ instability developing inside the PNS. The
aim of this paper is thus to study the impact of rotation on the onset of the
convective instability and its interplay with SASI, when both the neutrino
heating and the rotation rate are varied. In Sect. 2, we detail the numerical
set-up and define the stationary and perturbed flows. We study in Sect. 3 the
effect of rotation on SASI, convection and their interplay. Finally, we focus
in Sect. 4 on the features that might be observable in a gravitational wave
signal coming from an exploding supernova.

##  2 Methods

###  2.1 Numerical setup

To study the growth of the convective and SASI instabilities, we solve
numerically the system of perturbed equations corresponding to an idealised
model of a perfect gas, in spherical geometry restrained to the equatorial
plane, using the coordinates $(r,\phi)$. The parameters we vary are the
reference shock radius $r_{{\rm sh}0}$ obtained without neutrino heating,
dissociation and rotation, the rate $\varepsilon$ of nuclear dissociation
across the shock, the neutrino luminosity and the specific angular momentum
$J$ reaching the surface of the PNS.

####  2.1.1 Stationary flow

In spherical geometry, the system of stationary equations describing the
conservation of mass, the entropy profile $S(r)$ and the profile of the
Bernoulli parameter in the equatorial plane are:

$\displaystyle\diffp{}{r}\left(\rho vr^{2}\right)$ $\displaystyle=$
$\displaystyle 0,$ (1) $\displaystyle\diffp{S}{r}$ $\displaystyle=$
$\displaystyle\dfrac{\mathcal{L}}{Pv},$ (2)
$\displaystyle\diffp{}{r}\left(\dfrac{v^{2}}{2}+\dfrac{J}{2r^{2}}+\dfrac{c^{2}}{\gamma-1}-\dfrac{GM}{r}\right)$
$\displaystyle=$ $\displaystyle\dfrac{\mathcal{L}}{\rho v},$ (3)

where $G$ is the universal gravity constant and $M$ is the mass of the PNS,
$v$ the radial velocity and $c$ the sound speed in the gas with pressure $P$
and density $\rho$. The self-gravity of the infalling matter is neglected
compared to that of the PNS. Non-adiabatic heating and cooling processes are
described by a local function $\mathcal{L}\equiv\mathcal{L}_{\rm
h}+\mathcal{L}_{\rm c}$, as in Fernández et al. (2014). The cooling function
$\mathcal{L}_{\rm c}$ used in Houck & Chevalier (1992) is a parametric
function of $P$ and $\rho$:

$\mathcal{L}_{\rm c}=-A_{\rm c}\ \rho^{\beta-\alpha}P^{\alpha}.$ (4)

We use $\alpha={3/2}$ and $\beta={5/2}$ as in Blondin & Mezzacappa (2006),
Foglizzo et al. (2007), Yamasaki & Foglizzo (2008), Fernández & Thompson
(2009), Fernández et al. (2014), Guilet & Foglizzo (2012) and Blondin et al.
(2017). The normalisation constant $A_{\rm c}$ describes the micro-physical
processes responsible for the cooling by neutrino emission. Its value is
determined so that the radial velocity vanishes at the surface $r_{\rm PNS}$
of the PNS. Using dimensional analysis, we express $A_{\rm c}$ as a function
of the PNS radius $r_{\rm PNS}$, the surface gravity $GM/r_{\rm PNS}^{2}$ and
the mass accretion rate $\dot{M}$:

$\displaystyle A_{\rm c}=\tilde{A}_{\rm
c}\times\dot{M}^{1-\beta}\left({GM\over r_{\rm
PNS}^{2}}\right)^{1-\alpha+{\beta\over 2}}r_{\rm PNS}^{{5\beta\over
2}-2-\alpha},$ (5)

where $\tilde{A}_{\rm c}$ is a dimensionless quantity. The value of
$\tilde{A}_{\rm c}$ sets the value of the shock radius without dissociation,
rotation, or heating. $\tilde{A}_{\rm c}$ does not vary when these parameters
are changed. The heating function $\mathcal{L}_{\rm h}$ is expressed as

$\mathcal{L}_{\rm h}=A_{\rm h}\dfrac{\rho}{r^{2}}.$ (6)

The normalisation constant $A_{\rm h}$, which is proportional to the neutrino
luminosities and the neutrino opacities per unit mass, is varied as a free
parameter in our model. The heating and cooling functions are effective in the
post-shock region and are turned off above the shock, as in Fernández et al.
(2014).

In order to study the influence of stellar rotation on the growth of the
instabilities, we assume a constant specific angular momentum $J$. Its
dimensionless measure $j$ is defined using the Keplerian specific angular
momentum at the PNS surface

$j\equiv\frac{J}{\left(GMr_{\rm PNS}\right)^{1/2}}.$ (7)

We note that $j^{2}$ measures the ratio of the centrifugal and gravitational
forces at the PNS radius:

$\frac{J^{2}}{r_{\rm PNS}^{3}}\frac{r_{\rm PNS}^{2}}{GM}=j^{2}.$ (8)

In our analysis, we vary $j$ from $0$ to $0.5$ which corresponds to a maximum
centrifugal force equal to $25\%$ of the gravitational force at the PNS
boundary. The angular momentum at $50\%$ of the Keplerian rotation can be
expressed as

$\displaystyle\frac{J}{1.5\times 10^{16}{\rm cm^{2}/s}}$ $\displaystyle=$
$\displaystyle\frac{j}{0.5}\left(\frac{r_{\rm PNS}}{50{\rm
km}}\frac{M}{1.4{\rm M_{\odot}}}\right)^{1/2},$ (9) $\displaystyle=$
$\displaystyle\frac{\Omega}{1.5{\rm rad/s}}\left(\frac{r}{1000{\rm
km}}\right)^{2},$ (10)

where $\Omega$ in the second equation is the angular frequency in the
progenitor at a reference radius $r$. For reference, the strong magnetic field
mentioned in Raynaud et al. (2020) could be generated for $j\gtrsim 0.13$. Our
hydrodynamical study neglecting magnetic fields assumes that dynamo processes
are too slow to interfere.

Our input parameters and normalisation are similar to Fernández et al. (2014),
with the addition of stellar rotation. The dissociation parameter
$\varepsilon$ corresponds to the fraction of kinetic energy of the incoming
matter that is used to photo-dissociate the iron nuclei. $\varepsilon$ is
expressed in units of the specific kinetic energy $v_{{\rm ff}0}^{2}/2$
associated to free-fall at the reference radius $r_{{\rm sh}0}$. The Mach
number is imposed so that $\mathcal{M}_{1}=5$ above the shock, without
heating, rotation, or dissociation. The Bernoulli parameter is set to zero
above the shock, and the adiabatic index is set to $\gamma=4/3$. The default
values of the shock radius and the dissociation rate are $r_{{\rm
sh}0}=5r_{\rm PNS}$ and $\varepsilon=0$. For each figure, the value $r_{{\rm
sh}0}=5r_{\rm PNS}$ is used and the only exception is the use of $r_{{\rm
sh}0}=3.2r_{\rm PNS}$ for comparison in Figs. 1 and 5. The value of $r_{\rm
sh}$ specified in each figure corresponds to the shock radius taking into
account the parameters $(A_{\rm h},\varepsilon,J)$. The shock radius decreases
when dissociation increases. As the rotation or the heating rate $A_{\rm h}$
is increased, the shock radius increases (blue curve in Fig. 1) and the
corresponding Mach number immediately above the shock decreases.

Distances are normalised with either the PNS radius $r_{{\rm PNS}}$ or the
reference shock radius $r_{{\rm sh}0}$, densities with the density immediately
above the shock $\rho_{10}$, and velocities with the free-fall velocity
$v_{{\rm ff}0}$ at $r_{{\rm ff}0}$. These units are chosen for $(A_{\rm
h},\varepsilon,J)=(0,0,0)$ and they do not vary when $(A_{\rm
h},\varepsilon,J)$ are varied. With these units

$A_{\rm h}=\tilde{A_{\rm h}}v_{{\rm ff}0}^{3}r_{{\rm sh}0},$ (11)

where $\tilde{A_{\rm h}}$ is a dimensionless parameter characterising the
heating rate. Another way to measure the heating rate is to use the $\chi$
parameter defined in Foglizzo et al. (2006) as

$\chi\equiv\int_{r_{\rm g}}^{r_{\rm sh}}N(r)\dfrac{dr}{|v|},$ (12)

where $r_{\rm g}$ is the gain radius above which absorption of neutrino energy
exceeds the losses by neutrino emission, and $N(r)$ is the local Brunt-Väisälä
frequency:

$N\equiv\left(\dfrac{\gamma-1}{\gamma}g\nabla S\right)^{1\over 2}.$ (13)

In this equation, the gravity term $g$ was not corrected by the centrifugal
force for the sake of simplicity, which is acceptable because of the low value
of the ratio $j^{2}$ of those two forces. Indeed, Eq. (8) shows that even for
our rapid rotation rate, the ratio of those forces $\sim 7\%$ in the middle of
the post-shock region.

Figure 1: Evolution of the shock (blue) and gain (red) radii as a function of
$\chi$. Unless stated otherwise, the values of the parameters are $r_{{\rm
sh}0}=5r_{\rm PNS}$, $\varepsilon=0$ and $j=0$. For the moderate value of
angular momentum $j=0.08$, the stationary flow is almost unaffected by
rotation. The case $(j,\varepsilon)=(0,0)$ for $r_{{\rm sh}0}=3.2r_{\rm PNS}$
is plotted in dashed lines for comparison with the case including
dissociation. As expected, the shock radius increases with $\chi$ and the
introduction of rotation increases the shock radius when the cooling
normalisation $A_{\rm c}$ is conserved.

Figure 1 indicates that the gain radius varies little with the strength of
neutrino heating or the dissociation rate when the heating rate is high
enough. The gain radius is located near the shock for small values of $\chi$,
approaches twice the PNS radius for $\chi\sim 2$ and remains constant for
$\chi\gg 2$ while the radius of the stationary shock is pushed further out by
the strong heating of the matter. For $\chi\gtrsim 2$, the gain radius stalls
at $\sim 2r_{\rm PNS}$ and the size of the gain region increases
proportionally to the shock radius.

The stronger the dissociation, the lower the energy after the shock. This
leads to lower velocities of sound and matter than in the case without
dissociation. As a result, the integrated intensity of the cooling function
needed to decelerate across the shocked region is reduced. As $\tilde{A}_{\rm
c}$ is determined with $\varepsilon=0$, the size of the shocked region
decreases when the dissociation rate increases, as observed in Fig. 1. The
impact of rotation is barely visible in this figure with $j=0.08$ because of
the quadratic dependence of the centrifugal force with the rotation frequency
($=j^{2}<1\%$, see Eq. (8)).

However, we will see in the next section that this modest rotation is large
enough to have significant effects on SASI and the convective instability.

####  2.1.2 Perturbed flow

For the perturbed system, we use the same set of perturbed variables as in
Yamasaki & Foglizzo (2008). $\delta f$, $\delta h$, $\delta S$, and $\delta q$
are defined as follows:

$\displaystyle\delta f$ $\displaystyle\equiv$ $\displaystyle v\delta
v+\dfrac{J}{r}\delta v_{\phi}+\dfrac{2c}{\gamma-1}\delta c-\delta q,$ (14)
$\displaystyle\delta h$ $\displaystyle\equiv$ $\displaystyle\dfrac{\delta
v}{v}+\dfrac{\delta\rho}{\rho},$ (15) $\displaystyle\delta S$
$\displaystyle\equiv$ $\displaystyle\dfrac{2}{\gamma-1}\dfrac{\delta
c}{c}-\dfrac{\delta\rho}{\rho},$ (16) $\displaystyle\delta q$
$\displaystyle\equiv$ $\displaystyle\delta\left(\int\dfrac{\mathcal{L}}{\rho
v}dr\right).$ (17)

The time and angular dependence of the perturbations are assumed to follow the
form ${\rm exp}\left(-i\omega t+im\phi\right)$. Their radial dependence is
then governed by the same differential equations as Eqs. (6-11) in Yamasaki &
Foglizzo (2008) with $k_{z}=0$, despite the different geometry and the
inclusion of neutrino heating in the local function ${\cal L}$:

$\displaystyle{{\rm d}\over{\rm d}r}{\delta f\over\omega}$ $\displaystyle=$
$\displaystyle\dfrac{ic^{2}}{v\left(1-\mathcal{M}^{2}\right)}\left\\{\mathcal{M}^{2}\left(\delta
h-\dfrac{\omega^{\prime}}{c^{2}}\dfrac{\delta f}{\omega}\right)\right.$
$\displaystyle\left.+\left[1+\left(\gamma-1\right)\mathcal{M}^{2}\right]\dfrac{\delta
S}{\gamma}-\dfrac{\delta q}{c^{2}}\right\\},$ $\displaystyle\diff{\delta
h}{r}$ $\displaystyle=$
$\displaystyle\dfrac{i\omega}{v\left(1-\mathcal{M}^{2}\right)}\left(\mu^{2}\dfrac{\omega^{\prime}}{c^{2}}\dfrac{\delta
f}{\omega}-\mathcal{M}^{2}\delta h-\delta S+\dfrac{\delta q}{c^{2}}\right),$
(19) $\displaystyle\diff{\delta S}{r}$ $\displaystyle=$
$\displaystyle\dfrac{i\omega^{\prime}}{v}\delta
S+\delta\left(\dfrac{\mathcal{L}}{Pv}\right),$ (20) $\displaystyle\diff{\delta
q}{r}$ $\displaystyle=$ $\displaystyle\dfrac{i\omega^{\prime}}{v}\delta
q+\delta\left(\dfrac{\mathcal{L}}{\rho v}\right),$ (21)

where

$\displaystyle\omega^{\prime}$ $\displaystyle\equiv$
$\displaystyle\omega-\dfrac{mJ}{r^{2}},$ (22) $\displaystyle\mu^{2}$
$\displaystyle\equiv$ $\displaystyle 1-{m^{2}\over
r^{2}}\dfrac{c^{2}}{\omega^{\prime 2}}\left(1-\mathcal{M}^{2}\right).$ (23)

In these equations $\omega^{\prime}$ corresponds to the Doppler shifted value
of $\omega$ and is thus a function of radius. Note that this effect of
rotation is linear with respect to $\Omega$, whereas the centrifugal force is
proportional to $\Omega^{2}$ as discussed above. A linear dependence allows
for strong effects of rotation for slow rotation, as will be discussed later.

The boundary conditions at the shock are expressed using conservation laws
across a perturbed shock:

$\displaystyle{\delta f_{\rm sh}\over\omega}$ $\displaystyle=$ $\displaystyle
iv_{1}\Delta\zeta\left(1-\dfrac{v_{\rm sh}}{v_{1}}\right),$ (24)
$\displaystyle\delta h_{\rm sh}$ $\displaystyle=$
$\displaystyle-i\dfrac{\omega^{\prime}}{v_{\rm
sh}}\Delta\zeta\left(1-\dfrac{v_{\rm sh}}{v_{1}}\right),$ (25)
$\displaystyle\delta S_{\rm sh}$ $\displaystyle=$ $\displaystyle
i\dfrac{\omega^{\prime}v_{1}}{c_{\rm sh}^{2}}\Delta\zeta\left(1-\dfrac{v_{\rm
sh}}{v_{1}}\right)^{2}-\dfrac{\mathcal{L}_{\rm sh}-\mathcal{L}_{1}}{\rho_{\rm
sh}v_{\rm sh}}\dfrac{\Delta\zeta}{c^{2}_{\rm sh}}$
$\displaystyle+\left(1-\dfrac{v_{\rm
sh}}{v_{1}}\right)\dfrac{\Delta\zeta}{c^{2}_{\rm sh}}\left(\dfrac{2v_{1}v_{\rm
sh}}{r_{\rm sh}}+\dfrac{J^{2}}{r_{\rm sh}^{3}}+\dfrac{GM}{r_{\rm
sh}^{2}}\right),$ $\displaystyle\delta q_{\rm sh}$ $\displaystyle=$
$\displaystyle-\dfrac{\mathcal{L}_{\rm sh}-\mathcal{L}_{1}}{\rho_{\rm
sh}v_{\rm sh}}\Delta\zeta,$ (27)

where $\Delta\zeta$ is the shock displacement, the indices “$1$” and “sh”
refer to the values above and below the shock, respectively. The only
difference with the boundary conditions (12-15) in Yamasaki & Foglizzo (2008),
who assumed a cylindrical symmetry, is the factor $2$ implied by our choice of
spherical-equatorial geometry, in the term $2v_{1}v_{\rm sh}/r_{\rm sh}$ in
Eq. (2.1.2).

We use the same lower boundary condition at the PNS surface, where the radial
velocity vanishes. We use the Newton-Raphson method to find the eigenvalues of
this system when $\varepsilon$, $\mathcal{M}_{1}$, $\chi$, and $r_{\rm sh}$
(or $\tilde{A}_{\rm c}$) are given. This method determines the perturbation
growth rate ($\omega_{i}$) and frequency ($\omega_{r}$) for each azimuthal
number $m$.

In order to identify the effect of the four parameters $(\varepsilon,r_{{\rm
sh}0},J,\chi)$ on the growth of instabilities, we select a mode and follow its
evolution when the parameters are varied. For a given value of $m$, we study
the evolution of each harmonic without rotation and select the overtone with
the highest growth rate, which happens to be the fundamental mode as obtained
by Yamasaki & Yamada (2007). We follow the properties of this fundamental mode
when parameters $(\varepsilon,r_{{\rm sh}0},J,\chi)$ are varied. When rotation
exceeds $j\sim 0.15$, we noticed that some higher harmonics can be slightly
more unstable than the fundamental mode, but this effect is marginal.

##  3 Results

###  3.1 SASI and the convective instability without rotation

Figure 2: Evolution of the modes as a function of $\chi$ parameter. Two
instability domains can be identified: SASI ($\omega_{r}\neq$0) for modest
neutrino heating and the convective instability ($\omega_{r}=$0) for stronger
heating.

Figure 2 shows the effect of the heating parameter $\chi$ on the growth rate
and the frequency of the fundamental mode corresponding to several azimuthal
numbers $m=1$ to $5$. Similarly to Yamasaki & Yamada (2007) and Fernández et
al. (2014), for each value of $m$, the instability is oscillatory (like SASI)
for modest heating and purely growing (like the convective instability) for
stronger heating. The most unstable perturbations in the oscillatory regime
have a large angular scale corresponding to a small azimuthal number $m=1$, as
expected for SASI. For each mode, the SASI frequency decreases abruptly to
zero when the heating rate is increased. The transition to a mode with zero
frequency corresponds to a steep increase of the growth rate. The azimuthal
scale of the dominant convective modes is smaller (i.e. larger $m$, here
$m=4$) than SASI modes. The heating rate corresponding to the transition
between SASI and convection is $\sim$7% smaller than expected by Fernández et
al. (2014). This may be due to their use of an entropy cut-off in the
heating/cooling function, ${\rm exp}-(S/S_{\rm min})^{2}$, where $S_{\rm min}$
is the entropy at the PNS surface. Except for this small quantitative
difference, the general behaviour is the same as in Fernández et al. (2014).

To study the influence of the parameters $\varepsilon$ and $J$ on the growth
of the convective instability (Fig. 3), we fixed the value of $\chi=4$ such
that the dominant mode is convective for every dissociation rates. We compare
the value $m_{\rm num}$ corresponding to the most unstable mode obtained
numerically, to the expected value estimated analytically by Foglizzo et al.
(2006):

$m_{\rm ana}=\dfrac{\pi}{2}\dfrac{r_{\rm sh}+r_{\rm g}}{r_{\rm sh}-r_{\rm
g}}.$ (28)

This geometric formula is based on the assumption that convective cells have a
circular shape, radially centred in the gain region. It neglects the geometry
of the star and the fact that the local value of the Brunt-Väisälä frequency
is highest near the gain radius and the width of its radial profile is
inferior to the gain region size. As a result, the limiting scale for the
convective cells is smaller than the whole gain region and comparable to the
size of the most buoyant region, as can be seen in Fig. 10. This may explain
the offset between the blue triangles and the blue curve in Fig. 3, indicating
that $m_{\rm ana}$ over-estimates the angular size of the dominant convective
scale.

Figure 3: Influence of the dissociation on the azimuthal number of the most
unstable fundamental mode $m_{\rm num}$ (triangles) at $\chi=4$ for three
different values of the rotation rate. The numerical values of the most
unstable modes (triangle) are compared to the analytical predictions of Eq.
(28) (dashed lines) as a function of the dissociation rate. We allow for
quarter-integer values of $m_{\rm num}$ for an easier comparison with Eq.
(28).

The magnitude of the increase of $m_{\rm num}$ with $\varepsilon$ in Fig. 3 is
consistent with the increase predicted by Eq. (28) based on the increase of
the size of the gain region visible in Fig. 1.

We also varied $r_{{\rm sh}0}$ between $2$ and $4.5$ times the PNS radius by
changing the cooling intensity $\tilde{A}_{\rm c}$ and adapting the pre-shock
Mach number $\mathcal{M}_{1}$ such that $\mathcal{M}_{1}=5$ for $r_{\rm
sh}=5r_{\rm PNS}$, $\varepsilon=0$. For this variable, the agreement between
$m_{\rm num}$ and $m_{\rm ana}$ is similar to the one obtained in Fig. 3
without dissociation.

###  3.2 Influence of stellar rotation

In this part, we characterise the dependence of the growth rate $\omega_{i}$
and the oscillation frequency $\omega_{r}$ on the rotation rate for both SASI
and the convective instability.

####  3.2.1 Effect of rotation and heating on SASI

The growth rate of SASI is expected to increase approximately linearly with
the rotation rate, and the angular scale of the dominant mode is expected to
diminish when the rotation rate increases. The corotation radius, defined as

$r_{\rm corot}\equiv\left(\dfrac{mJ}{\omega_{r}}\right)^{1/2},$ (29)

is also expected to increase with the rotation rate. These results were
obtained without neutrino heating using a perturbative analysis and numerical
simulations both in cylindrical geometry (Yamasaki & Foglizzo 2008; Kazeroni
et al. 2017) and in spherical geometry (Blondin et al. 2017; Walk et al.
2022).

Figure 4: Evolution of the growth rate $\omega_{i}$ in units of $v_{\rm
sh}/r_{\rm sh}$ (top), and the corotation radius in units of $r_{\rm PNS}$
(bottom) as a function of the rotation rate, measured as in Yamasaki &
Foglizzo (2008) with $f_{\rm p}$, the pulsar frequency at $10$ km. The
azimuthal numbers $m=1$ (blue) and $m=2$ (red) are displayed for $\chi=0$
(solid line), and $\chi=2$ (dash dotted line). To allow for a comparison with
Fig. 1 of Yamasaki & Foglizzo (2008), we considered the same parameter $r_{\rm
sh}=5r_{\rm PNS}$ with a strong adiabatic shock.

Our perturbative analysis in spherical equatorial geometry, shown in Fig. 4,
confirms these results for $\chi=0$ and measures the impact of neutrino
heating for $\chi=2$. The lower plot in Fig. 4 shows the same dependence of
the corotation radius on the rotation rate as in Yamasaki & Foglizzo (2008).
We note that increasing the rotation rate introduces a corotation radius
$r_{\rm corot}>r_{\rm PNS}$ associated to the mode $m=2$ before the mode
$m=1$. This hierarchy is conserved when neutrino heating is taken into
account. In addition, in Yamasaki & Foglizzo (2008), the frequencies for which
$r_{\rm corot}=r_{\rm PNS}$ are $f_{p2}\sim 0.15$kHz and $f_{p1}\sim 0.25$kHz
for $m=2$ and $m=1$, respectively. These values are very close to our results,
where $f_{p2}\sim 0.17$kHz and $f_{p1}\sim 0.28$kHz.

The growth rate of the mode $m=1$, dominant without rotation, becomes inferior
to the growth rate of the mode $m=2$ with higher rotation rates. The slope of
the $m=2$ mode in the upper plot in Fig. 4 is indeed steeper than that of the
$m=1$ mode. We find that this behaviour is robust when neutrino heating is
taken into account, as demonstrated in Fig. 4 for $\chi=2$.

In the absence of dissociation and heating (solid lines in Fig. 4), the
observed transition with the mode $m=2$ dominating the instability for
$j\in[0.3,0.5]$ is consistent with the results of Blondin et al. (2017) where
the mode $m=2$ dominates for $j\sim 0.36$ (corresponding to $h=0.115$ in their
units). When neutrino heating is taken into account, Fig. 4 shows that the
transition from $m=1$ to $m=2$ requires a slightly larger amount of rotation.
On the other hand, this transition requires a smaller amount of rotation when
the shock radius is smaller or when dissociation is taken into account (Fig.
5). The influence of dissociation can be explained by its effect on the
advection timescale. For a given shock radius (kept constant in Fig. 5), the
matter velocity decreases with $\varepsilon$ because of the larger compression
through the shock. As a consequence, the advection timescale $r_{\rm
sh}/v_{\rm sh}$ increases with $\varepsilon$ and the frequency of the SASI
modes decreases. A slower rotation is therefore sufficient to be comparable to
the SASI frequency, which may explain the steeper effect of rotation. This
interpretation is confirmed by the bottom panel of Fig. 5 where the rotation
rate has been normalised using the advection timescale. This renormalisation
indeed collapsed the curves with and without dissociation, at least for the
small rotation rates. Note, however, that this argument cannot explain the
dependence on the shock radius, which remains mysterious.

Figure 5: Influence of rotation on the growth rate of SASI with dissociation
(solid lines) and without dissociation (dashed lines). The bottom panel
presents the rotation rate normalised with a proxy for the advection timescale
from the shock to the PNS calculated for $(\chi,j)=(0,0)$. Other parameters
are $\chi=0$ and $r_{\rm sh}\left(j=0\right)=3.2r_{\rm PNS}$.

####  3.2.2 Effect of rotation on the convective instability

Figure 6: Evolution as a function of $\chi$ of the growth rate $\omega_{i}$
(top) and the frequency $\omega_{r}$ (bottom) of the fundamental mode, with an
azimuthal number $m=4$ for different values of the rotation rate.

We consider in Fig. 6 the effect of the heating rate on the eigenfrequency of
the mode $m=4$ for different rotation rates. The mode $m=4$ was chosen because
it is the first mode to exhibit a convective instability when $j=0$. The most
striking feature regarding the oscillation frequency (lower plot in Fig. 6),
compared to Fig. 2, is the loss of a sharp transition between an oscillatory
instability for modest neutrino-heating and a purely growing instability for
strong enough neutrino-heating. This effect blurs the distinction between SASI
and convection, which becomes difficult for $j\gtrsim 0.1$ as will be
discussed in the next section. In this section, we focus on the regime of slow
rotation ($j\lesssim 0.1$), in which the two instabilities can be still be
distinguished.

Even with modest angular momentum, the differential nature of rotation implies
that convective motions are going to be sheared in addition to being entrained
in rotation with a frequency intermediate between the shock and PNS rotational
frequency. The region most unstable to buoyant motions, i.e. the radius
$r_{\rm BV}$ corresponding to the largest Brunt-Väisälä frequency $N(r)$ (Eq.
13) is expected to contribute most to defining the phase frequency of the
convective mode, i.e. the rotation frequency of the frame in which the
convective pattern is stationary. We compare in Fig. 7 $r_{\rm BV}$ with the
corotation radius $r_{\rm corot}$ defined by Eq. (29). For small rotation
rates ($j\lesssim 0.1$), we remark in Fig. 7 that $r_{\rm corot}$ is close to
$r_{\rm BV}$ as anticipated for a purely convective mode.

Figure 7: Evolution of the corotation radius $r_{\rm corot}$ as a function of
the rotation rate for $\varepsilon=0$ (top) and $\varepsilon=0.3$ (bottom).
The corotation radius of the most unstable mode is represented with a blue
line (numbers above the line specify the corresponding azimuthal number $m$).
For comparison, grey lines show the PNS radius $r_{\rm PNS}$, the gain radius
$r_{\rm gain}$, the radius of the Brunt-Väisälä frequency maximum $r_{\rm BV}$
and the shock radius $r_{\rm sh}$. With both values of the dissociation, the
corotation radius of the most unstable mode is close to $r_{\rm BV}$ for slow
rotation and moves away for higher rotation rates ($j>30\%$).

Figure 8: Growth rate evolution depending on the rotation rate for several
modes at $\chi=6.5$ with $\varepsilon=0$ (top panel) and $\varepsilon=0.3$
(bottom panel). For $j=0$, $r_{\rm sh}(\varepsilon=0)=7.8r_{\rm PNS}$ and
$r_{\rm sh}(\varepsilon=0.3)=4.1r_{\rm PNS}$. For this heating rate, non
axisymmetric convective modes are quenched by rotation. In the absence of
rotation, all modes are above the transition from SASI to convection, except
for the mode $m=1$ in the bottom panel.

Figure 8 illustrates the effect of rotation on the growth rate of convective
modes at $\chi=6.5$, which is significantly larger than the marginal stability
threshold of the most unstable mode ($\chi=3.5-4$). For both dissociation
rates ($\varepsilon=0$ and $\varepsilon=0.3$), the differential rotation
reduces the growth rate of the convective instability, in particular for the
smaller scale modes. The adverse effect on the convective instability is less
abrupt for larger scale modes, which might be interpreted by a smaller effect
of the shearing due to differential rotation between the shock and the PNS. As
a result, larger angular scales become dominant for high rotation rates. At
the largest scales ($m=1$ for $\epsilon=0$, $m=1-2$ for $\epsilon=0.3$),
rotation can on the contrary have a beneficial effect on the growth rate,
which will be discussed in the next paragraph and section.

Figure 9: Impact of rotation on the parameter $\chi_{\rm marg}$ characterising
the advective stabilisation of the convective mode. We consider several
azimuthal numbers $m$ and $\varepsilon=0.3$. The modes $m=1$ and $m=2$ are not
convective in this region and are not shown. The curves are cut when the mode
is unstable for all heating rates and $\chi_{\rm marg}$ does not exist
anymore.

Fig. 9 shows the effect of rotation on the heating rate at marginal stability,
expressed in terms of $\chi_{\rm marg}$. This threshold for unstable
convective modes decreases with rotation for all $m$ considered, thus showing
a _destabilising_ effect on convection near its marginal stability. Such a
destabilising effect is surprising and stands in striking contrast to the
adverse effect of rotation at $\chi=6.5$ discussed in the last paragraph. We
propose that this effect can be interpreted by the mixed nature of the modes
near the transition between SASI and convection. In this interpretation, the
destabilising effect of rotation on convection is a remnant of the rotational
destabilisation of SASI in the regime where the convective mode retains some
characteristics of SASI near the transition between the two instabilities. The
behaviour of a mode with respect to rotation would then be expected to depend
on how close the heating rate is to the SASI/convection transition. Indeed,
Fig. 9 shows that the heating rate at marginal stability $\chi_{\rm marg}$
decreases faster for smaller $m$ when rotation is increased. In the absence of
rotation, when $m$ increases, the transitional $\chi_{\rm trans}$ from SASI to
convection decreases (see Fig. 2) while the marginal stability depends little
on the azimuthal number. As a result, the difference $\chi_{\rm
trans}-\chi_{\rm marg}$ increases with $m$, and the residual effect of SASI
decreases. This highlight the mixed SASI/convection state of the modes near
$\chi_{\rm trans}$ for small rotation rates.

####  3.2.3 Mixed SASI-convective mode induced by rotation

For intermediate rotation rates ($0.1<j<0.3$), the behaviour of the modes is
modified by rotation to such an extent that it becomes impossible to
distinguish clearly between convective and SASI modes. As will be described in
this section, the modes in this regime retain characteristics of both
instabilities and should be understood as mixed SASI/convection/rotation
modes.

Figure 10: Spatial structure of the eigenmode entropy perturbation in the
equatorial plane, for $\varepsilon=0.3$ and $\chi\sim 6$. Each row corresponds
to a value of $m=(1,2,4,5)$ (from top to bottom) and each column to a rotation
rate with $j=(0,0.1,0.4)$ (from left to right). In magenta and cyan, we
represented the gain and corotation radii, respectively. The diagonal sequence
of framed plots corresponds to the most unstable modes. Coordinates $x$ and
$y$ are expressed in $r_{\rm PNS}$ unit. The deformation induced by rotation
is stronger for smaller scale modes.

In order to gain a deeper understanding of the transition between the SASI
dominated and the buoyancy dominated mode including rotation, we show in Fig.
10 the entropy structure of the eigenfunction for the modes $m=1,2,4,5$ for
three values of the angular momentum $j=0$, $0.1$ and $0.4$ and $\chi\sim 6$.
When there is no rotation (first column in Fig. 10), the entropy structure of
SASI (first line) is uniformly spread in the radial direction as a spiral
pattern. It is clearly distinct from the structure of convection, whose
perturbations are more localised in the vicinity of the gain radius and do not
display any spiral pattern. In the second column of this figure, we note that
the $m=1$ spiral SASI pattern is little affected by a moderate rotation with
$j=0.1$ (compare left and middle columns). This can be interpreted by the fact
that for such moderate rotation, the corotation radius is close to the PNS
surface. For the same rotation rate, the differential rotation shears the
convective cells with $m=2-5$. This leads to spiral structures that are
smaller but similar to a SASI pattern. However, the convective nature of the
mode is still visible by the localisation of the perturbations near the gain
radius. These similar structures can be explained if we consider that the mode
is not purely convective, but becomes a mixed mode SASI/convection/rotation.

Figure 11: From left to right, maps of the growth rate $\omega_{i}$, frequency
$\omega_{r}$, azimuthal number $m$ and corotation radius of the most unstable
fundamental mode with three dissociation rates $\varepsilon=0$ (top row),
$0.3$ (middle row) and $0.5$ (bottom row). Above the magenta dashed line, the
growth rate of equatorial perturbations decreases with rotation. In the two
left columns, the white lines show where the corotation radius equals $r_{\rm
PNS}$ and $r_{\rm gain}$. In the two right columns, the black lines highlight
the variation of the dominant azimuthal number $m$ (numbers).

A global panorama of the parameter space is shown in Fig. 11 where we vary
$\chi$ and $j$ for three values of the dissociation parameter $\varepsilon=0$,
$0.3$ and $0.5$ (upper to lower rows). The four columns represent the growth
rate and oscillation frequency of the most unstable mode, its azimuthal number
$m$ and the corotation radius $r_{\rm corot}$. The main effect of dissociation
is to diminish the radius of the stationary shock and thus the size of the
gain region and the timescale of advection.

For small values of $j$, we can see a sharp jump in frequency and $m$ when
neutrino heating exceeds the threshold $\chi\sim 4$ corresponding to the
transition from SASI to the convective instability as the most unstable mode.
However, as rotation increases, the frequency of the convective instability
increases until its frequency is similar to the SASI frequency. At angular
momentum $j_{\rm crit}\sim 0.08-0.12$, the jump disappears and it becomes
impossible to distinguish clearly convective from SASI modes: mixed modes
appear.

The appearance of mixed modes can also be recognised in the evolution of the
corotation radius shown in Fig. 7. Above a critical rotation rate, the
corotation radius indeed starts to move away from the Brunt-Väisälä radius,
where it is located for a convective mode. We remark that $r_{\rm corot}$
starts to shift outwards for rotation rates exceeding $j\sim 10\%$, indicating
a lower frequency than expected for a purely convective mode. This slower
oscillation frequency compared to the rotation rate of the most buoyant layer
suggests that some physical process associated to rotation and/or SASI
contributes significantly to the instability mechanism. However, the change of
the azimuthal number of the dominant mode (see below) keeps the corotation
close to the Brunt-Väisälä radius, indicating that convection still plays a
role in the instability mechanism.

We remark in the 3rd column of Fig. 11 that the azimuthal number $m$ of the
most unstable mode decreases with the rotation rate for $\chi>4$, while it
increases for $\chi<4$. The change of dominant convective scale induced by
rotation is not predicted by Eq. (28): comparing the magenta and blue
triangles in Fig. 3 indicates a shift to larger angular scales induced by a
moderate angular momentum $j=0.08$, despite a negligible change in the size of
the gain region. For this small rotation rate, the shortcoming of Eq. (28) is
not surprising when considering the rotation-induced distortion of the
convective cells in Fig. 10 and the residual influence of SASI. The rotational
stabilisation of perturbations with a large azimuthal wave number $m$ can be
understood as a consequence of their strong shear (Eq. 22) which induces a
short radial wavelength $2\pi v/\omega^{\prime}$ for advected perturbations
near the PNS, and thus a phase mixing particularly visible in the lower right
corner of Fig. 10.

Figure 12: Evolution of the critical rotation rate $j_{\rm crit}$ depending on
the dissociation using two methods. This critical rotation corresponds to the
apparition of the mixed-modes. The higher the dissociation, the later the
transition. For the blue points, we used the discontinuity end of the dominant
m at $\chi\sim 4$. For the magenta points, we used the advection timescale
$\tau_{\rm adv}$ from the shock to the gain radius for $m=4$ and $\chi\sim 4$.

The third column of Fig. 11 can be used to better constrain the domain of
existence of these mixed modes. For small rotation rates, the azimuthal number
contours converge to a line at $\chi\sim 4$. It corresponds to the transition
from purely SASI modes to purely convective modes, across which the azimuthal
number and frequency of the dominant mode is discontinuous. Above this
boundary, the high $m$ modes dominate, and their frequency depends linearly on
the rotation rate. However, this frontier disappears for higher rotation
rates, and the transition from small $m$ to higher $m$ is continuous. The
value of the rotation rate $j_{\rm crit}$ when the frontier disappears
slightly increases when the dissociation rate increases (blue points in Fig.
12). This behaviour can be quantitatively reproduced if we assume that mixed
modes appear when the frequency of the convective mode $\simeq m\Omega(r_{\rm
BV})$ matches the frequency of the corresponding SASI mode in the absence of
rotation or heating $\simeq 2\pi/\tau_{\rm adv}$. This criterion defines a
critical specific angular momentum

$J_{\rm mix}=\frac{2\pi r_{\rm BV}^{2}}{m\tau_{\rm adv}},$ (30)

which reproduces remarkably well the variation of $j_{\rm crit}$ when
considering the mode $m=4$ (magenta points of Fig. 12). This is the signature
of the growth of mixed modes SASI/convection/rotation.

The first column in Fig. 11 shows the adverse effect of rotation on the growth
rate of the convective instability for $\chi>5$, as discussed in Sect. 3.2.2.
For each rotation rate, the magenta line highlights the value of $\chi$ above
which the rotation has an adverse effect on the growth of convection in the
equatorial plane. In this region, the dynamic of the flow will be dominated by
axisymmetric convective modes in the plane $(r,z)$, that are expected to be
little affected by the rotation for $j\in[0,0.3]$. This stands in contrast
with the beneficial effect of rotation on the growth rate of SASI for
$\chi<3-4$, as discussed in Sect. 3.2.1, where spiral modes are therefore
expected to dominate the dynamics. Interestingly, rotation has a beneficial
effect on the growth rate of the dominant non-axisymmetric convective modes
for moderate convection parameters $\chi\simeq 4-5$ (Fig. 11). The same
behaviour can be observed for the $m=4$ mode in the upper plot of Fig. 6,
where the growth rate for $\chi\sim 3-4$ is significantly enhanced by
rotation. It is also linked to the easier onset of convection with rotation
shown by the decrease of $\chi_{\rm marg}$ in Fig. 9, as discussed in Sect.
3.2.2. All these observations suggest that the perturbation of the shock
induced by the convective instability produces an advective-acoustic cycle
favourable to enhance the convective instability, as already suggested by the
entropy structure in Fig. 10. The mode $m=1$ in Fig. 8 calls for a particular
attention, with a significant enhancement of the growth rate by rotation (this
is also true for the $m=2$ mode in the lower panel of the same figure) in a
regime $\chi=6.5$ a priori dominated by convection with an adverse effect of
rotation. This peculiar behaviour of the $m=1-2$ modes can also be explained
by a mixed state of this mode with the advective-acoustic cycle interacting
constructively with the convective mode. Indeed, for these low values of $m$
the transition from SASI to convection takes place at a rather large value of
$\chi$ (e.g. $\chi\simeq 6$ for $m=1$ in Fig. 2). As a consequence, mixed
modes with a beneficial effect of rotation may be expected for larger values
of $\chi$ close to this transition.

Figure 13: Growth rate (top) and frequency (bottom) evolution of several modes
depending on the rotation rate $j$, for $\chi=4.5$, $\varepsilon=0$ and
$r_{\rm sh}(j=0)=6.8r_{\rm PNS}$. Figure 14: Same as Fig. 13 but for
$\varepsilon=0.3$ and $r_{\rm sh}(j=0)=3.9r_{\rm PNS}$.

Figs. 13 and 14 illustrate the effect of rotation on both instabilities in two
configurations mainly differing by their shock radius, induced by the
dissociation parameter. The value $\chi=4.5$ is chosen in the region where the
most unstable mode is convective and the growth rate increases with rotation
(below the magenta line of the first column in Fig. 11). In the bottom panel
of these figures, the oscillation frequency without rotation ($j=0$) reveals
the convective ($\omega_{r}=0$) or SASI ($\omega_{r}\neq 0$) nature of the
modes. We can see that the convective frequency, enhanced by rotation, is
similar to the SASI frequency for $j\in[0.02,0.08]$. The top panels of this
figure show that the effect of rotation varies depending on the nature of the
instability. The growth rate of SASI modes is enhanced by the rotation, as in
Yamasaki & Foglizzo (2008) and Blondin et al. (2017). For convective modes,
the effect of rotation depends on the proximity of $\chi$ to the transition
value $\chi_{\rm trans}$ defining the transition of a mode between SASI and
convection. Defining $\Delta\chi\equiv\chi-\chi_{\rm trans}$, the rotation can
induce an increase of the growth rate, similar to the effect of rotation on
SASI, if $\Delta\chi<1$. Otherwise, the convective modes are quenched by
rotation, as observed in Fig. 8. Comparing Figs. 13 and 14, the increase of
the frequency and growth rate induced by rotation is stronger for
$\varepsilon=0.3$.

Figure 15: Evolution of the growth rate depending on the azimuthal number $m$
for several rotation rates and $\chi=4.5$.

Figure 15 illustrates the complexity of the impact of rotation depending on
the azimuthal number. The growth rate of SASI modes ($m=1,2$) is increased.
However, the effect of rotation on convective modes depends on both the
rotation and the azimuthal number. For $m\sim 3$, the destabilising effect of
rotation on the convective instability can be clearly seen by continuity with
the increase of the SASI growth rate. For $m\in[3,5]$, the effect of rotation
on the convection is similar to its effect on SASI, but with a smaller
magnitude. For higher $m\gtrsim 6$ and low rotation rates, the rotation has an
adverse effect and leads to a decrease of the growth rate. The effect changes
for high rotation rates and will be discussed in Sect. 3.3.

As a summary, the rotation smoothens the transition from SASI to convection,
leading to a mixed mode phase. These mixed modes appear when the corotation
radius moves away from the Brunt-Väisälä radius and exist up to $j\sim 30\%$
where the properties of the modes starts to change.

###  3.3 Strong rotation-induced instability

The mixed state of the modes seems to disappear when the rotation becomes too
strong. When the rotation rate exceeds $j\sim 20\%$, Fig. 6 shows that both
the frequency and the growth rate of $m=4$ become strikingly insensitive with
respect to the heating rate when expressed in units of $v_{\rm sh}/r_{\rm
sh}$. Rotation also changes the behaviour of the dominating $m$ as a function
of heating. While the dominant $m$ increases with $\chi$ for small and
intermediate rotations, it decreases slightly with heating for higher rotation
rates. Also, when considering the most unstable mode, the rotation seems to
lessen the influence of the parameter $\chi$ on the frequency and growth rate
of the dominant mode for $j\gtrsim 0.3$ (Fig. 11). This suggests that the mode
loses its convective nature and that the buoyancy driven by the entropy
gradient no longer plays an important role in the strong rotation regime.
Another indication that the role of buoyancy in the instability mechanism has
been erased by rotation comes from the fact that the corotation radius moves
definitively away from the Brunt-Väisälä radius for high rotation rate
$j\gtrsim 30\%$ (Fig. 7).

The right column of Fig. 10 illustrates the structure of the modes in the
strong rotation regime. The localisation around the Brunt-Väisälä radius, a
characteristic of the convective instability that was still visible at
intermediate rotation, is absent in this regime. The radial profile of entropy
perturbations suggests that it is driven by the deformations of the shock
rather than by the region of maximum buoyancy. The spiral mode structure of
SASI is also clearly perturbed by rotation due to the corotation radius, which
is well inside the domain. The pattern is similar to the one obtained by
Blondin et al. (2017) at high rotation rates without heating. In an adiabatic
flow where entropy perturbations are simply advected, the entropy pattern
would trace the flow lines and display a change of direction at the corotation
radius since the pattern rotates faster than the outer flow and slower than
the inner flow. In Fig. 10, a change of direction of the spiral pattern is
clearly visible in the strong rotation case (right column). The radial
location of the pattern extrema is, however, slightly above the corotation
radius, which can be interpreted as due to the non-adiabatic heating/cooling
functions.

For these high rotation rates ($j>0.3$), it seems that the modes are not mixed
SASI/convection/rotation any more, but result from a rotation-induced
instability. Some characteristics of this instability can be inferred from the
evolution of the growth rate. Comparing Figs. 13 and 14, we remark that when
the modes properties do not depend on the heating rate, the dependence of the
modes on the rotation rate seems to be stronger for increasing dissociation
rates. In addition, the slope of this increase seems to be independent of the
value of the azimuthal number. This behaviour of the mode for high rotation
rates can also be seen in Fig. 15. We notice that the rotation enhances the
growth rate of every mode. In particular, for initially convective modes, this
effect is opposite to the rotation effect for low rotation rates.

##  4 Expected consequences on the gravitational wave signature

The mode frequencies computed with a linear analysis have been shown to
reproduce well the frequencies present in the gravitational waves spectra in
non-linear numerical simulations. We first focus on the properties of the
fundamental mode $m=2$ which has a direct impact on the production of
gravitational waves according to the quadrupole formula. We include the effect
of dissociation with $\varepsilon\sim 0.3$ (Huete et al. 2018) and consider
the typical range of shock radius $r_{\rm sh}\sim(3-7)r_{\rm PNS}$.

Figure 7 (bottom) shows that the mode $m=2$ is dominant in the high rotation
regime $j\in\left[0.2,0.5\right]$ for $\chi=6.5$. The mode $m=2$ is actually
dominant over a large domain of the parameter space ($j,\chi$) according to
Fig. 11 (third column with $\epsilon=0.3$).

Figure 16: Frequency of the $m=2$ fundamental mode as a function of the
specific angular momentum $j$ for several values of $\chi$. The grey-shaded
region illustrates the frequency range corresponding to a corotation radius
located between the gain radius and the maximum of the Brunt-Väisälä
frequency.

Fig. 16 shows the impact of rotation on the $m=2$ mode frequency for different
values of $\chi$. In the case of SASI modes ($\chi=0$ and $\chi=2$), the
frequency increases almost linearly with the rotation rate for $j$ varying
from $0$ to $0.5$. The convective mode corresponding to $\chi=6$ has a more
complex behaviour. For small rotation ($j\lesssim 0.1$), the frequency is
approximately proportional to the rotation rate similarly to the convective
modes in Fig. 13 and 14. This behaviour can be interpreted with an
approximately constant corotation radius located between the gain radius and
the radius where the Brunt-Väisälä frequency is maximum (Fig. 7), as expected
for the convective instability and visualised by the grey-shaded region in
Fig. 16. At higher rotation rates $j>0.2$, the frequency separates from this
linear trend, corresponding to the corotation radius moving to larger radii
(Fig. 7) when the mode takes a mixed nature. At still higher rotation
frequencies ($j\gtrsim 0.3$), the frequencies expressed in units of $v_{\rm
sh}/r_{\rm sh}$ converge toward a single linear trend that does not depend on
the heating rate. The effect of heating has been erased by rotation. This
behaviour is consistent with the behaviour of the dominant modes seen earlier.

Figure 17: Maps, for varying heating and rotation rates, of the growth rate,
the frequency and the corotation radius of the most unstable fundamental mode
between modes $m=1$ and $m=2$. In the second panel, we plotted $2\omega_{r}$
when $m=1$ is dominant to account for the doubled frequency expected in the
gravitational wave signal. In the third panel, the black line illustrates the
frontier between the modes.

To account for the possible signature of a dominant mode $m=1$ on the
gravitational wave signal, we include in Fig. 17 the doubled frequency of the
mode $m=1$ when it is more unstable than the $m=2$ mode. Note that this
frequency is higher than the frequency of the $m=2$ mode. This remarkable
property can be deduced from Figs. 5 and 6 in Foglizzo et al. (2007) where the
frequency of the SASI mode $m=2$ is always smaller than twice the frequency of
the mode $m=1$. We observe that the region of the parameter space dominated by
the mode $m=1$ is limited to moderate heating rates $\chi<5.2$ and modest
rotation rates $j<15\%$.

In the regime $j\geq 30\%$, an analytical form of the linear dependence
illustrated by Fig. 16 would be useful to directly translate the frequency
observed in the gravitational wave signal into a relation between $j$ and the
advection frequency at the shock $v_{\rm sh}/r_{\rm sh}$.

##  5 Discussion and conclusion

In this paper, we studied a spherical toy model where we considered the
dynamics of the equatorial plane between the shock and the surface of the PNS.
Thanks to a linear analysis, we computed unstable non-axisymmetric modes for
varying rotation and heating rates and different dissociation energies through
the shock. This led to several discoveries for the behaviour of the convective
instability with rotation and its interplay with SASI and rotation-induced
instabilities.

Relatively slow rotation rates (less than $10\%$ of the Keplerian rotation at
the PNS radius), for which centrifugal effects are very small, can have a
strong and complex influence on both SASI and the convection instability:

  * •

In the presence of rotation, convective modes have an oscillation frequency
that corresponds to a corotation radius located close to the maximum of the
Brunt-Väisälä frequency, where the convective engine is expected to be most
powerful. The appearance of this non-vanishing oscillation frequency blurs
somewhat the abrupt transition between SASI and convection.

  * •

For large values of the convection parameter $\chi\gtrsim 5$, the rotation
hampers the growth of non-axisymmetric convective modes. The decrease in
growth rate is more pronounced for larger $m$ modes, which can be interpreted
by the larger stabilising effect of the shear due to differential rotation. In
this regime, axisymmetric modes would be expected to dominate the dynamics, as
they are expected to be only slightly affected by rotation.

  * •

By contrast, for $\chi\lesssim 5$, where the advection has a strong influence
on the growth of convection, rotation increases the growth rate of both SASI
and the convective instability. The increase in the SASI growth rate with
rotation was well known in the absence of heating (Yamasaki & Foglizzo 2008;
Kazeroni et al. 2017; Blondin et al. 2017) and is here shown to hold in the
presence of neutrino heating. On the other hand, this behaviour was not
expected for the convective instability, which was generally thought to be
hampered by rotation. The destabilising effect of rotation on convection in
this regime leads to a decrease with rotation of the value of $\chi$ at
marginal stability. We interpret this effect as a residual influence of an
advective-acoustic cycle which acts to reinforce convective motions close to
the transition between convection and SASI.

  * •

We observe two different effects of rotation on the dominant scale, depending
on the instability mechanism that dominates without rotation. For SASI, the
dominant $m$ increases with rotation (as in Yamasaki & Foglizzo 2008 and
Blondin et al. 2017), while it decreases for the convective instability. The
diminution of the dominant azimuthal number $m$ as rotation is increased was
not expected to occur as it does. Indeed, in our case, not only are large $m$
disadvantaged by rotation, but there is a heating domain for which small $m$
are advantaged by rotation.

At moderate rotation rates ($10$ to $30\%$ of the Keplerian rotation at the
PNS surface), the modes are modified by rotation to such an extent that it
becomes impossible to distinguish clearly between convective and SASI modes.
When the heating rate is increased, the frequency and azimuthal number of the
most unstable mode change smoothly, with no clear transition from one
instability to another. Modes retain characteristics of both instabilities and
should therefore be understood as mixed SASI/convection/rotation modes. The
dominant mixed modes have a relatively large angular scale with $m=1-3$ and
their growth rate increases with faster rotation. This regime takes place
above a critical angular momentum such that the frequency of a convective mode
(corotating with the maximum of Brunt-Väisälä frequency) is comparable to the
frequency of SASI. The non-oscillatory nature of convection and the low
frequency associated to SASI both allow for a significant effect of moderate
differential rotation, with the introduction of a corotation radius in the
structure of non-axisymmetric perturbations which may participate in the
extraction of energy and angular momentum from the interior region rotating
faster than the region exterior to the corotation (Cairns 1979; Yoshida &
Saijo 2017; Saijo & Yoshida 2006). For a PNS radius of $r_{\rm PNS}=50$km, the
mixed mode regime corresponds to an interval of specific angular momentum
$3-9\times 10^{15}$ cm2/s. Extrapolated to the radius of a cold neutron star
$12\,{\rm km}$, this corresponds to relatively fast rotation periods of $\sim
1-3$ms.

For high rotation rates (more than $30\%$ than the Keplerian frequency at the
PNS surface, i.e. $j\gtrsim 0.3$), the frequency and growth rate are
independent or weakly dependent on the heating rate, when they are expressed
in terms of the advection timescale $r_{\rm sh}/v_{\rm sh}$. This, together
with the significant deviation of the corotation radius from the most buoyant
region, suggests that the instability is dominated by rotational rather than
buoyancy effects. The study of Walk et al. (2022) without neutrino heating
pointed out the existence of a similar instability regime where the frequency
of the dominant mode depends too little on the advection time to be explained
by an advective-acoustic cycle. Our results suggest that these results
obtained without heating are still valid when heating is taken into account.
We note that the regime of rapid rotation appears for $j\sim 0.3$ which
corresponds to a small ratio $\sim 0.03$ of the centrifugal force to the
gravity at the corotation radius, even though the centrifugal displacement of
the stationary shock is not negligible. A precise understanding of this strong
rotation regime is still missing, but it seems probable that the corotation
radius plays an important role in the instability mechanism.

A future detection of gravitational waves is expected to give information on
the physical phenomena during stellar core-collapse. Our identification of
three different instability regimes depending on the rotation rate should help
clarify the still poorly known influence of rotation on the gravitational wave
signal. Non-axisymmetric convective modes become oscillatory in the presence
of rotation. It may therefore become possible to identify their frequency in
the low-frequency part of the gravitational wave spectrum. For low/moderate
rotations, the corotation radius of the $m=2$ convective/mixed mode is close
to the gain or Brunt-Väisälä radius. The identification of the mode frequency
would therefore give access to the rotation frequency at this radius. Non-
axisymmetric equatorial modes with a large angular scale $m=1,2$ are strongly
destabilised by rotation and dominate the dynamics in a wide region of the
parameter space in particular for moderate to strong rotation, which should be
favourable to a strong emission of gravitational waves. In the regime of
strong rotation, the frequency of the $m=2$ mode becomes independent of the
heating rate and depends only on the advection timescale $r_{\rm sh}/v_{\rm
sh}$ and the angular momentum $j$. This reduction of the parameter space
should help extract physical information from the measure of the mode
frequency. These modes should be incorporated in future asteroseismic studies
similar to Torres-Forné et al. (2018, 2019b) where non-axisymmetric
perturbations would be taken into account.

Our results can be compared to previous numerical simulations of core-collapse
supernovae, including rotation. Fryer & Heger (2000) and Fryer & Warren (2004)
found that convection was quenched by rotation, especially in the equatorial
plane. For models A and B of Fryer & Warren (2004), this may be explained by
the strong centrifugal effects for such strong rotation ($j^{2}>0.4$).
Convection in the slow rotating model C ($j\sim 0.03$) does not seem to be
confined to the poles, which is consistent with our study, where a purely
convective instability can be observed in the equatorial plane up to $j\sim
0.1$.

The setup of Iwakami et al. (2014) is similar to ours in that it did not
involve the interior of the PNS and considered a similar range of rotation
rates (corresponding to $j=0-0.33$ in our units). If we focus on the structure
in models D, E, and F as displayed in Fig. 5 of their paper, we see that for
higher rotation and smaller neutrino luminosity, the entropy structures are at
larger scales and that buoyant patterns become spiral ones. This structure
change is consistent with our result that rotation favours larger-scale
convection and transforms convective modes into mixed spiral modes for
moderate rotation rates. The precise interpretation of their results is,
however, complicated by the fact that both rotation and neutrino luminosity
are changed such that it is difficult to disentangle their respective
influence. In addition, we note that they identified a pattern referred to as
a spiral motion with buoyant-bubble (SPB), which may be related to the mixed
SASI/convection/rotation modes we have identified for moderate rotation rates.

To have a better understanding of the processes at work at the onset of the
shock revival, a non-linear analysis of the dynamics of the fluid would be
necessary. It would shed light on how rotation acts in the saturation
mechanism (Guilet et al. 2010) and show how the system evolves from the linear
phase to the explosion. This last part would give information on the
signatures of linear phenomena that may remain in the explosion signal.

One should keep in mind that our model is idealised in many respects. We
assume a very simple equation of state of ideal gas with $\gamma=4/3$,
dissociation is taken into account as a fixed energy sink, neutrinos are
parameterised through analytical cooling and heating terms. We also did not
describe the equatorial swelling of the PNS due to the centrifugal force. This
effect increases the neutrinosphere radius, leading to cooler neutrinos and a
smaller heating rate in the equatorial plane. As discussed above, such a
centrifugal effect is expected to be very small at low/intermediate rotation
rates ($j<0.3$) but can become significant for the fastest rotations
considered in this analysis. Part of this complexity is avoided by displaying
our results as a function of $\chi$ rather than the heating rate. Note that,
when $\chi$ is kept constant and the rotation increased, the heating constant
$\tilde{A}_{\rm h}$ decreases, in a qualitatively similar way to the expected
impact of the neutrinosphere swelling.

The centrifugal force induces a deformation with respect to the spherical
symmetry, which is challenging to take into account in our formalism because
it would couple different spherical harmonics. To avoid such a complexity, we
restrained our analysis to the equatorial plane while keeping the effect of
convergence in spherical geometry. As a result, our conclusions are limited to
modes with spherical harmonics indices $m=\pm l$ and cannot describe the
dynamics outside the equatorial plane, such as convective motions along the
polar axis. The equatorial non-axisymmetric modes described here are
nonetheless expected to dominate the dynamics in most of the parameter space
except at $\chi\gtrsim 5-5.5$ and slow to moderate rotation, where
axisymmetric convective modes are expected to be more unstable.

The highest specific angular momentum considered in this study would be
expected to be sufficient to allow for the development of the low-$T/|W|$
instability inside the PNS (Takiwaki et al. 2021; Bugli et al. 2022). Since it
is restricted to the post-shock region without including the PNS interior, our
linear analysis cannot describe the low-$T/|W|$ instability. It may therefore
miss the most unstable mode in the regime of strong rotation. A linear
stability analysis including both the post-shock region and the PNS interior
is therefore an important next step. Previous linear mode calculations focused
on the prediction of GW mode frequencies and included the post-shock region in
addition to the PNS interior, but they were restricted to axisymmetric modes,
and they assumed a hydrostatic equilibrium neglecting advection (Torres-Forné
et al. 2018, 2019b). To describe all unstable mode and their possible
interaction, a linear stability analysis will have to face the challenge of
combining the PNS in approximate hydrostatic equilibrium and the advection in
the post-shock region.

Finally, the magnetic field, which was neglected in this study, can be
expected to play an important role for strong rotation. A strong magnetic
field can quench the development of the low-$T/|W|$ instability (Bugli et al.
2022). In the absence of rotation, the magnetic field can have a complex
influence on the post-shock dynamics because of the propagation of vorticity
through Alfvén waves (Guilet & Foglizzo 2010; Guilet et al. 2011) and a small-
scale dynamo can amplify the magnetic field (Endeve et al. 2012; Müller &
Varma 2020). It would be interesting to investigate further how the magnetic
field would change the landscape of the post-shock instabilities in the
presence of rotation.

## Acknowledgements

JG acknowledges support from the European Research Council (MagBURST grant
715368). EA is supported by RK MES grant No. AP13067834 and NU Faculty
Development Grant No. 11022021FD2912.

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