Dimensionless Formulation¶
To improve numerical stability, GYRE solves the separated equations by recasting them into a dimensionless form that traces its roots back to Dziembowski (1971).
Variables¶
The independent variable is the fractional radius \(x \equiv r/R\) and the dependent variables \(\{y_{1},y_{2},\ldots,y_{6}\}\) are
Oscillation Equations¶
The dimensionless oscillation equations are
These equations are derived from the separated equations, but with the insertion of ‘switch’ terms (denoted \(\alpha\)) that allow certain pieces of physics to be altered. See the Physics Switches section for more details
For non-radial adiabatic calculations, the last two equations above
are set aside and the \(y_{5}\) terms dropped from the first four
equations. For radial adiabatic calculations with
reduce_order
=.TRUE.
(see the Oscillation Parameters
section), the last four equations are set aside and the first two
replaced by
Boundary Conditions¶
Inner Boundary¶
When inner_bound
='REGULAR'
, GYRE applies
regularity-enforcing conditions at the inner boundary:
When inner_bound
='ZERO_R'
, the first and second
conditions are replaced with zero radial displacement conditions,
Likewise, when inner_bound
='ZERO_H'
, the first and
second conditions are replaced with zero horizontal displacement
conditions,
Outer Boundary¶
When outer_bound
='VACUUM'
, GYRE applies vacuum surface
pressure conditions at the outer boundary:
When outer_bound
='DZIEM'
, the first condition is
replaced by the Dziembowski (1971) outer mechanical
boundary condition,
When outer_bound
='UNNO'
|'JCD'
, the
first condition is replaced by the (possibly-leaky) outer mechanical
boundary conditions described by Unno et al. (1989) and
Christensen-Dalsgaard (2008), respectively. When
outer_bound
='ISOTHERMAL'
, the first condition is
replaced by a (possibly-leaky) outer mechanical boundary condition
derived from a local dispersion analysis of an isothermal atmosphere.
Finally, when outer_bound
='GAMMA'
, the first
condition is replaced by the outer mechanical boundary condition
described by Ong & Basu (2020).
Jump Conditions¶
Across density discontinuities, GYRE enforces conservation of mass, momentum and energy by applying the jump conditions
Here, + (-) superscripts indicate quantities evaluated on the inner (outer) side of the discontinuity. \(y_{1}\), \(y_{3}\) and \(y_{6}\) remain continuous across discontinuites, and therefore don’t need these superscripts.
Structure Coefficients¶
The various stellar structure coefficients appearing in the dimensionless oscillation equations are defined as follows:
Physics Switches¶
GYRE offers the capability to adjust the oscillation equations through
a number of physics switches, controlled by parameters in the
&osc
namelist group. The table below summarizes the mapping
between the switches appearing in the expressions above, and the
corresponding namelist parameters.
Symbol |
Parameter |
Description |
---|---|---|
\(\alphagrv\) |
|
Scaling factor for gravitational potential perturbations. Set to 1 for normal behavior, and to 0 for the Cowling (1941) approximation |
\(\alphathm\) |
|
Scaling factor for local thermal timescale. Set to 1 for normal behavior, to 0 for the non-adiabatic reversible (NAR) approximation (see Gautschy & Glatzel, 1990), and to a large value to approach the adiabatic limit |
\(\alphahfl\) |
|
Scaling factor for horizontal flux perturbations. Set to 1 for normal behavior, and to 0 for the non-adiabatic radial flux (NARF) approximation (see Townsend, 2003b) |
\(\alphagam\) |
|
Scaling factor for g-mode isolation. Set to 1 for normal behavior, and to 0 to isolate g modes as described by Ong & Basu (2020) |
\(\alphapi\) |
|
Scaling factor for p-mode isolation. Set to 1 for normal behavior, and to 0 to isolate p modes as described by Ong & Basu (2020) |
\(\alphakap\) |
|
Scaling factor for opacity partial derivatives. Set to 1 for normal behavior, and to 0 to suppress the \(\kappa\) mechanism |
\(\alpharht\) |
|
Scaling factor for time-dependent term in the radiative heat equation (see Unno & Spiegel, 1966). Set to 1 to include this term (Unno calls this the Eddington approximation), and to 0 to ignore the term |