in collaboration with the CCSE/LBL group.
We've developed a new algorithm for low speed flows in white dwarfs, which includes the compressiblity effects due to the background stratification. This is a generalization of the pseudo-incompressible model of Durran (1989), accounting for the thermodynamics of the stellar equation of state. The advantage is that the low Mach number solver filters out soundwaves, allowing for much larger timesteps to be taken compared to a fully compressible algorithm.
Some comparisons with compressible and anelastic solvers are available. In all cases, a 500 km portion of a one-dimensional white dwarf initial model was mapped into the domain and given a small perturbation near the base. The initial density profile is:
The red line is the original model, direct from the 1-d stellar evolution code. The blue line is the adjusted model, interpolated onto a uniform mesh and adjusted into hydrostatic equilibrium with a constant gravitational acceleration (for simplicity).
The model file is here: model_hse.out
The gravitational acceleration, g, is taken as -1.9e10 -- this is the value one would get from the midpoint of the radius range we are considering.
The domain size is:
xmin = 0.e0 xmax = 5.e7 ymin = 5.e7 ymax = 1.e8
The left and right boundaries are periodic. The upper and lower boundaries provide hydrostatic support -- this greatly damps out the noise from soundwaves as the model adjusts to HSE.
The initial temperature perturbation is done in pressure equilibrium. The temperature is specified as:
pert_ellipse = ((xcc - xctr)/r_pert)**2 + &
((ycc - yctr)/r_pert)**2
tempZone = temp_ambient + &
(temp_perturb - temp_ambient)* &
(0.5d0 + 0.5d0*tanh((2.0 - sqrt(pert_ellipse))/0.9d0))
tempZone = max(min(temp_perturb,tempZone), temp_ambient)
with
xctr = 2.5e7 yctr = 6.25e7 r_pert = 1.25e6 temp_pert = 6.e9
Here, temp_ambient is the temperature in the zone before any perturbation -- this just ensures that the perturbation does not decrease the temperature anywhere. The pressure in the perturbation is unchanged, so a new density is found with the new temperature through the EOS.