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