Mapping a modified RGB profile to the grid: modifying the surface and ambient pressure

One of the next steps identified in the last post was to try increasing the ambient pressure to help to prevent the near-surface layers of the star from expanding outward. There are two problems with the (modified) MESA profile near the surface :
1) Due to a sudden drop in pressure as , the pressure scale height drops drastically, quickly becoming unresolvable.
2) The pressure profile is not smooth at ; i.e. is discontinuous across the surface. This could potentially lead to numerical problems.

We comment on (2) at the end of this post; here we focus on (1). Problem (1) can possibly be addressed by increasing the ambient pressure. This would have the affect of moving the effective surface of the star () inward. It is important to be aware how the minimum pressure scale height would be affected. Here I have plotted the pressure vs radius (top) and pressure scale height vs radius (bottom) for both modified and unmodified profiles. Three horizontal lines in the bottom plot show the chosen scale height cutoff, while the horizontal lines in the top plot show the corresponding pressure cutoffs. For example, if we want the minimum scale height to be , then we should set the ambient pressure to dyne/cm. This would cause the pressure to equal the ambient pressure at rather than at .

Given that plot, it seems reasonable to try imposing ambient pressures of , and dyne/cm. (Recall that our resolution for is about .) This was implemented by simply setting the pressure equal to the maximum value of the profile and the specified ambient pressure at each radius. The movies are shown here, with initial and final frames compared below each movie. All units are CGS:
A) set implicitly by AstroBear to a low value: 3d density 1d density 3d pressure 1d pressure 1d
3d density start and end comparison 1d density start and end comparison 3d pressure start and end comparison 1d pressure start and end comparison

B) dyne/cm: 3d density 1d density 3d pressure 1d pressure 1d
3d density start and end comparison 1d density start and end comparison 3d pressure start and end comparison 1d pressure start and end comparison

C) dyne/cm: 3d density 1d density 3d pressure 1d pressure 1d
3d density start and end comparison 1d density start and end comparison 3d pressure start and end comparison 1d pressure start and end comparison

D) dyne/cm: 3d density 1d density 3d pressure 1d pressure 1d
3d density start and end comparison 1d density start and end comparison 3d pressure start and end comparison 1d pressure start and end comparison

Discussion
Density: Comparing run (B) with the fiducial run (A), we see that the density profile ends up deviating more from that at than in run (A). The situation seems to improve as we increase in run ©. Low amplitude density waves are seen outside the star, but the density profile remains reasonably constant. In run (D), density waves outside the star are of lower amplitude than in ©, but the final density profile has been squeezed slightly in comparison with the initial profile (the star appears slightly smaller). The 1D density comparison shows the presence of an unphysical kink in the profile. Here the density wave is propagating inward, and the profile has not yet stabilized. In fact, none of the density profiles have stabilized by the end of the simulation.

Pressure: Increasing can be seen to reduce the expansion of the star's outer layers. On the other hand, the profile in the inner part of the star gets perturbed as pressure disturbances propagate inward.

We see then that imposing an ambient pressure helps to prevent the outer layers from expanding, but also causes larger inward-propagating perturbations.

The sudden transitions in and at the stellar surface (Problem (2) above) should also be addressed. Probably some combination of increasing and smoothing the profile to satisfy mass continuity and hydrostatic equilibrium at all radii (even outside the current stellar surface) is needed. The next step is therefore to think about how to smooth the profile near the surface in a reasonable way.

Update 1
It occurred to me that another possible way to get around the problem of the scale height dropping to low values near the surface is to simply truncate the stellar profile at some value. I did this by truncating the profile where the pressure drops below dyne/cm. This happens at about (so the star loses 8 of its radial extent). The movies are shown here, with initial and final frames compared below each movie. All units are CGS:
E) set implicitly by AstroBear to a low value and cut off profile at dyne/cm: 3d density 1d density 3d pressure 1d pressure
3d density start and end comparison 1d density start and end comparison 3d pressure start and end comparison 1d pressure start and end comparison

Outgoing pressure and density waves are visible. Note also that the ambient values of pressure and density set implicitly by AstroBear are larger than in Model (A), approximately by the factor . The results are not very encouraging… and we conclude that truncating the star just below the surface does not seem to offer any signficant benefit.

Update 2
After discussing with Eric, one idea that came up was to try a model like (D) but with the density also held constant where the pressure is held constant. This would avoid having a small (unresolvable density scale height). This makes the situation worse, probably because it is the pressure gradient that determines the force, so might as well keep the density profile as realistic as possible.
F) set to dyne/cm and set to a constant in the same region: 1d

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