Update: Trying AMR, extending the profile, exploring fluctuations, including velocity-damping
I) Invoking AMR (with smooth
Motivation:
Ultimately runs will be done with AMR so it is worth trying AMR at this stage and comparing with uniform grid runs.
Setup:
I introduced AMR with 5 levels of refinement for a run with dyne/cm , with refinement 'level 0' resolution equal to grid cells, in a box of physical dimension cm. This implies the equivalent of for level 5, which translates to a physical resolution equal (in principle) to that of Run (3) from the last blog post, which used grid cells for a box of dimension cm. So in a sense, this run combines the higher resolution of Run (3) with the larger box of Run (4). This run should in principle take about 10 days to complete on bluehive, but it did not actually complete (see below).
Results:
2d density slice full box
2d pressure slice full box
2d density slice central
2d pressure slice central
1d density central
1d pressure central
2d density fluctuations slice $6.6\times10^{-9}$ to $6.8\times10^{-9}$ g/cm$^3$
2d pressure fluctuations slice $9.98\times10^{6}$ to $1.04\times10^{7}$ dyne/cm$^2$
1d density fluctuations $10^{-8.3}$ to $10^{-8.0}$ g/cm$^3$
1d pressure fluctuations $10^{6.95}$ to $10^{7.1}$ dyne/cm$^2$
- The run got killed, initially because the time allotment was exceeded. But after restarting the run, it again got killed, possibly because of a problem in the calculation (unclear). But clearly the run has become unphysical by this time anyway. 80 of 101 frames completed up to a time of s.
- The pressure remains remarkably stable. The core pressure retains its initial value marginally better than Run (3), which had high resolution in a smaller box. Global oscillations are similar in amplitude and frequency to Run (4) (the larger box run), as would be expected, but seem to be less regular. This is best seen in the 1D pressure movie. This is not surprising given that the AMR grid is non-uniform, and refined patches are not spatially distributed in a regular manner.
- The density, on the other hand, becomes unstable on a time scale of about 1 dynamical time. The star first develops small indentations at the points on the surface closest to the boundary and then develops a boxy morphology with the orientation aligned with the grid. Finally, the instabilities develop at the corners of the boxy stellar profile in the 2d slice.
- The oscillations in the pressure stabilize somewhat after about half the simulation time (a few dynamical times). In the 1d plots that zoom in on the boundary, there seems to be at least two changes to the profiles of density and pressure: first at about 1 dynamical time ( s) and then again at a few dynamical times, when the density is seen to become unstable in the 2d density plot.
Discussion:
- The fact that the density becomes unstable but the pressure distribution remains stable is consistent with what we found happens last blog post when the resolution is reduced (comparison plots of Run C). So it seems likely that somewhere in the box, there is a lack of resolution that leads to the instabilities in density. The instability seems to be strongest at the "corners" of the boxy profile that develops, which suggests that the resolution is too low to resolve these sharp corners.
- It is worthwhile looking at the low-level fluctuations for the previous simulations as a comparison:
For the small box run we have (in 1d plots purple=smoothed pressure, green=unsmoothed pressure):
2d density slice boundary $6.6\times10^{-9}$ to $6.8\times10^{-9}$ g/cm$^3$
2d pressure slice boundary $9.98\times10^{6}$ to $1.04\times10^{7}$ dyne/cm$^2$
1d density boundary $10^{-8.3}$ to $10^{-8.0}$ g/cm$^3$
1d pressure boundary $10^{6.95}$ to $10^{7.1}$ dyne/cm$^2$
For the large box run we have (in 1d plots purple=smoothed pressure):
2d density slice boundary $6.6\times10^{-9}$ to $6.8\times10^{-9}$ g/cm$^3$
2d pressure slice boundary $9.98\times10^{6}$ to $1.04\times10^{7}$ dyne/cm$^2$
1d density boundary $10^{-8.3}$ to $10^{-8.0}$ g/cm$^3$
1d pressure boundary $10^{6.95}$ to $10^{7.1}$ dyne/cm$^2$
Conclusions:
The AMR fails to resolve the sharp density gradients that appear at certain points, causing the simulation to crash. Instead of developing 'clean' global pressure waves, like in the uniform grid simulations, the waves have a complex structure, with many small-scale fluctuations embedded within them. This probably helps to damp regular global osciallations that develop in the uniform grid models.
II) Adding an isothermal atmosphere (with smooth
Motivation:
The profiles used in previous runs are characterized by a constant ambient density and pressure.
The pressure profile was smoothed at the star-ambient boundary, but not the density profile.
Adding a thick hydrostatic atmosphere for which the density and pressure decline with radius may be more realistic.
In its simplest form, such an atmosphere would be isothermal, continuous in pressure, density and their respective gradients across the star-ambient boundary, and in hydrostatic equilibrium if self-gravity is neglected. The derivation of the profile, with plots, is available in the file profile_atmosphere.pdf.
Setup:
The atmosphere is added by simply modifying the density and pressure profiles inputted. It is ensured that these new profiles are defined over the entire box, so there is no need for any ambient values (either defined explicitly or imposed implicitly by astrobear).
Results:
Below are movies from two runs with the same physical resolution but different box size:
a) small box (with smooth
2d density slice extended color bar
2d pressure slice extended color bar
2d density slice old color bar
2d pressure slice old color bar
1d density
1d pressure
b) large box (with smooth
2d density slice extended color bar
2d pressure slice extended color bar
2d density slice old color bar
2d pressure slice old color bar
1d density
1d pressure
- Clearly, boundary effects lead to sharp gradients in density, which probably is what causes the code to crash in (b).
- Morphologies appear unphysical, if one looks at small densities and pressures, but at the density/pressure range we were looking at before, the density slice remains quite circular and stable until a few dynamical times, when the density becomes suddenly unstable.
Conclusions:
These runs are somewhat less stable than the constant ambient density/pressure runs above, as sharp gradients develop more easily, probably as a result of reflection from the boundary. If this problem could be overcome, then this idea of an atmosphere may be more useful, because it seems to succeed (at least transiently) in transfering the disturbances to very low density and pressure where they are negligible.
New results (post-March23 meeting):
c) small box (with smooth , Multipole expansion BCs for Poisson solver, and run time seconds)
2d density slice extended color bar
2d pressure slice extended color bar
2d density slice old color bar
2d pressure slice old color bar
2d density slice extended color bar with velocity (scaled)
1d density
1d pressure
d) small box (with smooth
2d density slice extended color bar with velocity (scaled)
2d density slice extended color bar with velocity (unscaled)
2d density slice old color bar
1d density
III) Implementing damping (with smooth
Motivation:
We wish for the star to be stable. As we have seen this is rather difficult to achieve. We must then resort to damping the motions by, in effect, adding a damping source term to the momentum equation. We follow the idea discussed by Ohlmann+17. That is to add a term of the form
,
where
is a parameter.
Setup:
The most direct way to implement this form of damping would be to add the extra term directly to the momentum equation. However this would involve changing the code at the lowest level. For the time being we try an alternative prescription, whereby we set
,
with
,
since
,
which leads to
.
However, it was unclear to me exactly how to implement this in the code, so I tried two different methods below.
Preliminary Results:
a) small box, with resolution
2d density 2d pressure 1d density 1d pressure 2d hydrostatic equilibrium 2d sound speed 2d Mach number
b) small box, with resolution
2d density
2d pressure
1d density
1d pressure
2d hydrostatic equilibrium
2d sound speed
2d Mach number
c) small box, with resolution
2d density
2d pressure
1d density
1d pressure
2d hydrostatic equilibrium
2d sound speed
2d Mach number
…and the fiducial run for comparison (small box,
2d density
2d pressure
1d density
1d pressure
2d hydrostatic equilibrium
2d sound speed
2d Mach number
Note that the fractional hydrostatic equilibrium
does not include the gravity of the point source (I must correct this).
d) small box, with resolution
2d density
2d density and velocity
2d pressure
1d density
1d pressure
e) small box, with resolution
2d density
2d density and velocity
2d pressure
1d density
1d pressure
f) small box, with resolution
2d density
2d density and velocity
2d pressure
1d density
1d pressure
g) small box, with resolution
2d density
2d density and velocity
2d pressure
1d density
1d pressure
Comments
No comments.