5.4.4 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. The velocity vector is mirrored at the boundary.
  • 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, radiation.
  • transmitting: transmitting boundary for hydro and outward radiation.
  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. The velocity vector is mirrored at the boundary.
  • 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, radiation.
  • transmitting: transmitting boundary for hydro and outward radiation: exponential decrease of density (standard open boundary condition).
  • transmitting2: transmitting boundary for hydro and outward radiation: exponential decrease of density and extra velocity treatment.
  • transmitting3: transmitting boundary for hydro and outward radiation: constant density extrapolation.
  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. The velocity vector is mirrored at the boundary.
  • 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.
  • closedbottom2: closed wall, handles gravity, radiation in diffusion approximation. In this version, the extrapolation of quantities should be smoother than for closedbottom.
  • periodic: periodic boundaries for hydrodynamics, 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.
  • inoutflow2: variant of the open lower boundary condition. The parameters real s_inflow, real c_schange, real c_pchange have to be specified. In this version, the extrapolation of quantities should be smoother than for inoutflow.
  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. 5.4.4) and luminositypervolume (see Sect. 5.4.4). 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.
  • core_entropy1: The entropy in the core is adjusted towards s_inflow on a rate given by c_schange.
  • core_energy1: Energy in the core is added according to luminositypervolume.
  • core_energy2: Energy in the core is added according to luminositypervolume with a Gaussian distribution of the energy source.

• 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_v3changelinbottom:
  For the open lower boundary condition (inoutflow and inoutflow2), a damping of the vertical velocity at the open boundary can be specified, e.g., with
  real c_v3changelinbottom f=E15.8 b=4 &
    n='Linear velocity reduction rate at bottom' u=1
  0.0025
  
  The open lower boundary condition tends to produce a slight reduction of the horizontal velocities at the very bottom. As the HD solver does not apply any extra viscosity at the bottom layers (the MHD solver has this option), there is a tendency to produce a slight predominance of vertical velocities in the regions where matter enters the computational box through the lower boundary. By introducing a little bit of damping of the vertical velocity component, this tendency can be significantly reduced. Guide values are
  • 0.0: off: no linear damping
  • 0.002: small reasonable value
  • 0.005: large, possible useful value

• real c_v3changesqrbottom:
  For the open lower boundary condition (inoutflow and inoutflow2), an additional damping of the vertical velocity at the open boundary can be specified, e.g., with
  real c_v3changesqrbottom f=E15.8 b=4 &
    n='Quadratic velocity reduction rate at bottom' u=1
  0.002
  
  This damping is stronger for velocities that exceed the rms value of the velocities averaged over the entire bottom layers.

• real c_rhochangetop:
  The transmitting upper boundary condition can smooth density fluctuations with this parameter. It is locally adjusted towards the global average to damp out possible instabilities. It appears to be useful for the HLLMHD solver. For simulations without magnetic fields, there is no need to set this parameter, so far. The adjustment rate can be specified e.g. with
  real c_rhochangetop f=E15.8 b=4 &
    n='Rate of density change for open upper boundary' u=1
   0.2
  

• 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='1'
  60.0E+05
  
  Possible values are
  • C $>$ 0.0: Older version: $\tau_0(x1,x2) = C * \kappa(x1,x2)$.
  • C $\le$ 0.0: New version: $\tau_0(x1,x2) = \mathrm{abs}(C) * H_p(x1,x2) * \kappa(x1,x2)$. In this case, a value of C_radHtautop=-1.0 might be a good choice.

• 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.

The following parameters are specific to MHD simulations. For MHD calculations they must be set additional to the hydrodynamic boundary parameters.

• character side_bound_mag_x1 and character side_bound_mag_x2:
  
  The boundary condition at the sides perpendicular to the x1 direction and perpendicular to the x2 direction, respectively. They are given by e.g.
  character side_bound_mag_x1   f=A80 b=80 n='side boundary conditions magnetic x1'
  fixed
  character side_bound_mag_x2   f=A80 b=80 n='side boundary conditions magnetic x2'
  periodic
  
  Possible values are:
  • constant: constant extrapolation of all magnetic field components into the ghost cells.
  • periodic: periodic continuation of all magnetic field components into the ghost cells.
  • fixed: the component normal to the boundary is kept fixed at its inital value. Constant extrapolation applies for the transversal components.
  • vertical: constant extrapolation for the vertical component. The transversal cell-centered field is mirrored, with the opposite sign. That should result in transversal components of the boundary-centered field being zero at the boundary.
  • vertical2: constant extrapolation for the vertical component. The transversal cell-centered field is set to zero.
  • reflective: The magnetic field is mirrored at the boundary. This boundary condition is unphysical, because the magnetic field is an axial vector and because it violates the divergence free property of the magnetic field. Therefore, this boundary condition should be used with caution.
  The fixed conditions are realized by setting the electric field at those cell edges that coincide with the physical boundary zero. This is done in the constrained transport module.

• character top_bound_mag:
  The boundary condition at the top of the model is given by for instance
  character top_bound_mag f=A80 b=80 n='bottom boundary conditions magnetic'
  vertical
  
  Possible values are:
  • constant: constant extrapolation of all magnetic field components into the ghost cells.
  • periodic: periodic continuation of all magnetic field components into the ghost cells.
  • fixed: the component normal to the boundary is kept fixed at its inital value. Constant extrapolation applies for the transversal components.
  • vertical: constant extrapolation for the vertical component. The transversal cell-centered field is mirrored, with the opposite sign. That should result in transversal components of the boundary-centered field being zero at the boundary.
  • vertical2: constant extrapolation for the vertical component. The transversal cell-centered field is set to zero.
  • oblique: magnetic fields with a given inclination at the boundary. The inclination is specified through parameters C_magthetaB and C_magphiB.

• character bottom_bound_mag:
  The boundary condition at the bottom of the model is given by for instance
  character bottom_bound_mag f=A80 b=80 n='bottom boundary conditions magnetic'
  inoutflow
  
  Possible values are:
  • constant: constant extrapolation of all magnetic field components into the ghost cells.
  • periodic: periodic continuation of all magnetic field components into the ghost cells.
  • fixed: the component normal to the boundary is kept fixed at its inital value. Constant extrapolation applies for the transversal components.
  • vertical: constant extrapolation for the vertical component. The transversal cell-centered field is mirrored, with the opposite sign. That should result in transversal components of the boundary-centered field being zero at the boundary.
  • vertical2: constant extrapolation for the vertical component. The transversal cell-centered field is set to zero.
  • oblique: magnetic fields with a given inclination at the boundary. The inclination is specified through parameters C_magthetaB and C_magphiB.
  • inoutflow: magnetic field can be advected into the computational domain by ascending material flow. Its strength can be specified with the parameter b1_inflow. The boundary condition for the hydrodynamic variables must be set to inoutflow too, otherwise this boundary condition is the same like constant.
  In the case of inoutflow the magnetic field, which is advected into the computational domain has a unique component which is in the x1 direction and it is only present where a velocity in the positive x3 direction exists. In all other places, the magnetic field components are constantly extrapolated into the ghost cells.

• real b1_inflow:
  This parameter controls the strength of the inflowing horizontal magnetic field at the lower boundary in connection with bottom_bound_mag=inoutflow. The default value is 0.0. Example:
  real b1_inflow f=E15.8 b=4 &
      n='Strength of inflowing horizontal magnetic field'           u=G
  10.0
  
  The unit of this parameter is gauss (no factor $\sqrt{4\pi}$!).

• real C_magthetaB:
  This parameter specifies the angle between the magnetic field vector and the x3-axis (in radians) in the case of oblique boundary conditions. The default value is 0.0. Example:
  real c_magthetab f=E15.8 b=4 &
    n='angle magnetic field w.r. to the vertical direction' &
    u=rad c0='used in combination with oblique conditions'
  0.523598775598299
  

• real C_magphiB:
  This parameter specifies the angle between the horizontal component of the magnetic field vector and the x1-axis (in radians) in the case of oblique boundary conditions. The default value is 0.0. Example:
  real c_magphib f=E15.8 b=4 n='angle magnetic field w.r. to the x-axis' &
    u=rad c0='used in combination with oblique conditions'
  0.0