5.3.4 Boundary Conditions

The boundary conditions at the six sides of the computational box cannot be specified independently. For the naming convention of the boundaries a gravitational acceleration in -x3 direction is assumed. Accordingly, there is a bottom, a top, and four side boundaries.

`character side_bound`

:

The boundary condition at all four sides is given by e.g.character side_bound f=A80 b=80 n='side boundary conditions' & c0='closed, transmitting, periodic' transmitting

Possible values are:`reflective`

: closed wall, no gravity, no radiation`constant`

: open boundary with constant extrapolation of all values, no gravity, no radiation`closed`

,`closedtop`

: closed wall, can handle gravity, open for outward radiation`closedbottom`

: closed wall, handles gravity, radiation in diffusion approximation`periodic`

: periodic boundaries for hydrodynamics and radiation`transmitting`

: transmitting boundary for hydro and outward radiation

`MSrad`

radiation transport module the side boundaries*have*to be`periodic`

. In simulations of a red supergiant all boundaries (including the sides) will typically be`transmitting`

. As an alternative,`closed`

boundaries can be chosen in this case.`character top_bound`

:

The boundary condition at the top of the model is given by for instancecharacter top_bound f=A80 b=80 n='top boundary conditions' transmitting

Possible values are:`reflective`

: closed wall, no gravity, no radiation`constant`

: open boundary with constant extrapolation of all values, no gravity, no radiation`closed`

,`closedtop`

: closed wall, can handle gravity, open for outward radiation`periodic`

: periodic boundaries for hydrodynamics and radiation`transmitting`

: transmitting boundary for hydro and outward radiation

`transmitting`

top boundary will be selected, the`closed`

one is an alternative. The`periodic`

condition is only recognized by the hydrodynamics routines and not by any radiation transport routine.`character bottom_bound`

:

The boundary condition at the bottom of the model is given for instance bycharacter bottom_bound f=A80 b=80 n='bottom boundary conditions' & c0=closedbottom transmitting

Possible values are:`reflective`

: closed wall, no gravity, no radiation`constant`

: open boundary with constant extrapolation of all values, no gravity, no radiation`closed`

,`closedtop`

: closed wall, can handle gravity, open for outward radiation`closedbottom`

: closed wall, handles gravity, radiation in diffusion approximation`periodic`

: periodic boundaries for hydrodynamics and radiation`transmitting`

: transmitting boundary for hydro and outward radiation. The parameters`real c_tchange`

,`real c_tsurf`

, and`real c_hptopfactor`

have to be specified.`inoutflow`

: "classical" open lower boundary for deep convection, gravity and radiation possible. The parameters`real s_inflow`

,`real c_schange`

, and`real c_pchange`

have to be specified.

`MSrad`

radiation transport module the bottom boundary is typically of type ```inoutflow`

''. A supergiant simulation will have a`transmitting`

lower boundary.`real s_inflow`

:

The entropy of the material streaming through an open boundary of type ```inoutflow`

'' into the model can be specified e.g. withreal s_inflow f=E15.8 b=4 n='Entropy of core material' & u=erg/K/g 3.25E+09

In the case of a`central`

potential the entropy in a sphere with radius`r0_grav`

is adjusted towards this entropy value. In both geometry (supergiant as well as solar) this value is very important as it finally (but indirectly) determines the luminosity and effective temperature of the star.`real c_schange`

:

The entropy`s_inflow`

of the material in the bottom layer (solar case,`inoutflow`

boundary condition) or the central region of the model (supergiant case) is not just set to the specified but adjusted towards it. The adjustment rate can be controlled with e.g.real c_schange f=E15.8 b=4 & n='Rate of entropy change for open lower boundary' u=1 0.3

Guide values are`1.0`

: fast adjustment`0.3`

: typical value`0.1`

: slow adjustment`<=0.0`

: not allowed

`real c_pchange`

:

The`inoutflow`

boundary condition not only controls entropy and velocity but also the pressure in the bottom layers: It is locally adjusted towards the global average to damp out possible instabilities. The adjustment rate can be specified e.g. withreal c_pchange f=E15.8 b=4 & n='Rate of pressure change for open lower boundary' u=1 1.0

`real c_tchange`

:

In the case of a`transmitting`

upper or outer boundary the temperature of the material streaming into the model is adjusted with a rate given e.g. byreal c_tchange f=E15.8 b=4 & n='Rate of temperature change for open upper boundary' u=1 0.3

`real c_tsurf`

:

In the case of a`transmitting`

upper or outer boundary the temperature of the material streaming into the model is adjusted towards a temperature`teff`

*`c_tsurf`

. This temperature can be specified as fraction of the effective temperature e.g. withreal c_tsurf f=E15.8 b=4 n='Temperature factor for open upper boundary' u=1 0.62

The value depends on where the outer boundary is located relative to the photosphere: If the boundary lies at a point where the solar photospheric minimum temperature is located, it can be fairly small. If the boundary is far away from the photosphere of a red supergiant, the value can be even smaller. On the other hand, if the boundary lies somewhere within the solar chromosphere even values above 1.0 might be reasonable.`real c_hptopfactor`

:

In the case of a`transmitting`

upper or outer boundary the density stratification outside the model has to be extrapolated properly. Assumptions about this density affects the amount of mass flowing into the model. For the extrapolation it is assumed that the density scale scales with the pressure scale height as =/`c_hptopfactor`

.real c_hptopfactor f=E15.8 b=4 & n='Correction factor for surface pressure scale height' u=1 0.8

Possible values are- C
`0.0`

: No effect (actually, a value of`1.0`

is chosen). `0.0`

C`1.0`

: The density scale height is enlarged to account for possible effects of turbulent pressure on the scale height: The density decays less rapidly with height than in an (isothermal) hydrostatic stratification.- C
`1.0`

: Density scale height is pressure scale height. - C
`1.0`

: Density scale height is smaller than pressure scale height. Not really useful.

- C
`real c_radhtautop`

:

The`MSrad`

radiation transport module needs the specification of the scale height of the optical depth at the upper boundary, e.g. withreal c_radhtautop f=E15.8 b=4 n='Scale height of optical depth at top' u=cm 60.0E+05