Dimensionless Formulation
To improve numerical stability, GYRE solves the separated equations and boundary conditions by recasting them into a dimensionless form that traces its roots back to Dziembowski (1971).
Variables
The independent variable is the fractional radius \(x \equiv r/R\) (with \(R\) the stellar radius), and the dependent variables \(\{y_{1},y_{2},\ldots,y_{6}\}\) are
Oscillation Equations
The dimensionless oscillation equations are
where the dimensionless oscillation frequency is introduced as
(with \(M\) the stellar mass). These differential equations are derived from the separated equations, 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 nonradial 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
regularityenforcing conditions at the inner boundary:
(these are the dimensionless equivalents to the expressions appearing in the Boundary Conditions section).
When inner_bound
='ZERO_R'
, the first and second
conditions above 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 the
outer boundary conditions
(these are the dimensionless equivalents to the expressions appearing in the Boundary Conditions section).
When outer_bound
='DZIEM'
, the first condition
above is replaced by the Dziembowski (1971) outer boundary condition,
When outer_bound
='UNNO'
or 'JCD'
, the
first condition is replaced by the (possiblyleaky) outer boundary
conditions described by Unno et al. (1989) and
ChristensenDalsgaard (2008), respectively. When
outer_bound
='ISOTHERMAL'
, the first condition is
replaced by a (possiblyleaky) outer boundary condition derived from a
local dispersion analysis of waves in an isothermal atmosphere.
Finally, when outer_bound
='GAMMA'
, the first
condition is replaced by the outer momentum boundary condition
described by Ong & Basu (2020).
Internal Boundaries
Across density discontinuities, GYRE applies the boundary conditions
(these are the dimensionless equivalents to the expressions appearing in the Boundary Conditions section). Here, + () superscripts indicate quantities evaluated on the inner (outer) side of the discontinuity. \(y_{1}\), \(y_{3}\) and \(y_{6}\) remain continuous across discontinuities, and therefore don’t need these superscripts.
Structure Coefficients
The various stellar structure coefficients appearing in the dimensionless oscillation equations and boundary conditions 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 (see the Oscillation Parameters section). 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 nonadiabatic reversible (NAR) approximation (see Gautschy et al., 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 nonadiabatic radial flux (NARF) approximation (see Townsend, 2003b) 
\(\alphagam\) 

Scaling factor for gmode isolation. Set to 1 for normal behavior, and to 0 to isolate g modes as described by Ong & Basu (2020) 
\(\alphapi\) 

Scaling factor for pmode isolation. Set to 1 for normal behavior, and to 0 to isolate p modes as described by Ong & Basu (2020) 
\(\alphakar\) 

Scaling factor for opacity density partial derivative. Set to 1 for normal behavior, and to 0 to suppress the density part of the \(\kappa\) mechanism 
\(\alphakat\) 

Scaling factor for opacity temperature partial derivative. Set to 1 for normal behavior, and to 0 to suppress the temperature part of the \(\kappa\) mechanism 
\(\alpharht\) 

Scaling factor for timedependent 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 
\(\alphatrb\) 

Scaling factor for the turbulent mixing length. Set to the convective mixing length to include the turbulent damping term (see the Convection Effects section), and to 0 to ignore the term 