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 and a top boundary, and four side boundaries.

All boundary conditions of the hydrodynamic case are available in the MHD module.

• 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. Like the velocity field, the magnetic field is mirrored at the boundary. This boundary condition is unphysical, because the magnetic field is an axial vector and it violates the divergence free property of the magnetic field. Therefore, this boundary condition should not be used in MHD simulations. Use closed boundary conditions instead.
  • constant: open boundary with constant extrapolation of all values, no gravity, no radiation
  • closed, closedtop: closed wall, can handle gravity, open for outward radiation Magnetic field lines are orthogonal to the boundary, i.e. the tangential component of the magnetic field vanishes at the boundary.
  • closedbottom: closed wall, handles gravity, radiation in diffusion approximation. Magnetic field lines are orthogonal to the boundary, i.e. the tangential component of the magnetic field vanishes at the boundary.
  • periodic: periodic boundaries for hydrodynamics, radiation, and magnetic fields
  • transmitting: transmitting boundary for hydro and outward radiation. Magnetic field lines are orthogonal to the boundary, i.e. the tangential component of the magnetic field vanishes at the boundary.
  Any of these values can be specified. But in fact, not all of them are recognized by all modules. Therefore some parameters are for test purposes (e.g. shock calculations) only. In simulations of a solar-like star with the 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 instance
  character top_bound    f=A80 b=80 n='top boundary conditions'
  transmitting
  
  Possible values are:
  • reflective: closed wall, no gravity, no radiation. Like the velocity field, the magnetic field is mirrored at the boundary. This boundary condition is unphysical, because the magnetic field is an axial vector and it violates the divergence free property of the magnetic field. Therefore, this boundary condition should not be used in MHD simulations. Use closed boundary conditions instead.
  • constant: open boundary with constant extrapolation of all values, no gravity, no radiation
  • closed, closedtop: closed wall, can handle gravity, open for outward radiation. Magnetic field lines are orthogonal to the boundary, i.e. the tangential component of the magnetic field vanishes at the boundary.
  • periodic: periodic boundaries for hydrodynamics, radiation, and magnetic fields
  • transmitting: transmitting boundary for hydro and outward radiation. Magnetic field lines are orthogonal to the boundary, i.e. the tangential component of the magnetic field vanishes at the boundary.
  In almost every simulation of stellar convection a 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 by
  character bottom_bound f=A80 b=80 n='bottom boundary conditions' &
    c0=closedbottom
  transmitting
  
  Possible values are:
  • reflective: closed wall, no gravity, no radiation. Like the velocity field, the magnetic field is mirrored at the boundary. This boundary condition is unphysical, because the magnetic field is an axial vector and it violates the divergence free property of the magnetic field. Therefore, this boundary condition should not be used in MHD simulations. Use closed boundary conditions instead.
  • constant: open boundary with constant extrapolation of all values, no gravity, no radiation
  • closed, closedtop: closed wall, can handle gravity, open for outward radiation. Magnetic field lines are orthogonal to the boundary, i.e. the tangential component of the magnetic field vanishes at the boundary.
  • closedbottom: closed wall, handles gravity, radiation in diffusion approximation. Magnetic field lines are orthogonal to the boundary, i.e. the tangential component of the magnetic field vanishes at the boundary.
  • periodic: periodic boundaries for hydrodynamics, radiation, and magnetic fields
  • transmitting: transmitting boundary for hydro and outward radiation. The parameters real c_tchange, real c_tsurf, and real c_hptopfactor have to be specified. Magnetic field lines are orthogonal to the boundary, i.e. the tangential component of the magnetic field vanishes at the boundary.
  • 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. Magnetic field lines are orthogonal to the boundary, i.e. the tangential component of the magnetic field vanishes at the boundary.
  • inoutflow2: variant of the open lower boundary condition. The parameters real s_inflow, real c_schange, real c_pchange, and real B1_inflow have to be specified.
  In simulations of a solar-like star with the MSrad radiation transport module the bottom boundary is typically of type ``inoutflow''. A supergiant simulation will have a transmitting lower boundary.

• character heat_mode:
  The mode in which energy is supplied can be adjusted with this parameter. The classical choice is to leave it empty, in which case the mode is chosen from s_inflow (see Sect. 5.3.4) and luminositypervolume (see Sect. 5.3.4). Example:
  character heat_mode  f=A80 b=80  n='Heating mode' &
    c0='-/bottom_entropy1/bottom_energy1'
  bottom_entropy1
  
  Possible values, so far:
  • : (empty). The classical value. For local models the energy comes through the lower boundary, either by radiation (for a closed bottom boundary closedbottom) or by convection + radiation (for an open bottom boundary inoutflow).
  • bottom_entropy1: The entropy in the bottom layers (defined as being less than r0_grav above the bottom of the model) is adjusted towards s_inflow on a rate given by c_schange.
  • bottom_energy1: Energy in the bottom layers is added according to luminositypervolume.

• real luminositypervolume:
  The luminosity of a ``Star-in-a-Box'' or a local model with the appropriate heat_mode can be set with this parameter. To avoid numbers that do not fit into a 4 Byte real the luminosity per volume has to be specified as e.g. in
  real luminositypervolume f=E15.8 b=4 n='Luminosity per core volume' &
    u='erg/cm^3/s'
  4.5E-02
  
  Reference volume is $4/3 \pi r0_{\rm grav}^3$. If this parameter is set to a value of 0.0 or below the entropy of the material within the core (defined by as all cells within radius r0_grav) is adjusted instead.

• real s_inflow:
  The entropy of the material streaming through an open boundary of type ``inoutflow'' into the model can be specified e.g. with
  real 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. A value of 0.0 (default) or below disables this energy input.

• 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. with
  real 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. by
  real 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. with
  real 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 $H_{\rho}$ scales with the pressure scale height $H_p$ as $H_{\rho}$=$H_p$/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 $\leq$ 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 $\geq$ 1.0: Density scale height is smaller than pressure scale height. Not really useful.

• 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. with
  real c_radhtautop f=E15.8 b=4 n='Scale height of optical depth at top' u=cm
  60.0E+05
  

• real rho_min:
  During long periods of matter infall the density at an open outer boundary can become very low. To limit the decrease of the density a lower limit in the extrapolated ghost cells can be set e.g. with
  real rho_min f=E15.8 b=4 n='Minimum boundary density' u=g/cm^3
  1.0E-25
  
  The density within the model will typically not fall much below this value. A value of 0.0 (default) or below deactivates this feature.

• real c_coredrag:
  To damp the flow in the core of models with central potential a drag force restricted to the inner part of the model ($r$$<$r0_grav) can be applied. It is controlled e.g. with
  real c_coredrag f=E15.8 b=4 n='Core drag force parameter' u=1
  1.0
  
  A value of 0.0 (default) or below deactivates this feature.