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To improve numerical stability, gyre solves the separated equations <osc-sep-eqns> by recasting them into a dimensionless form that traces its roots back to :ads_citet:dziembowski:1971.
Variables
The independent variable is the fractional radius x ≡ r/R and the dependent variables {y1, y2, …, y6} are
These equations are derived from the separated equations, but with the insertion of 'switch' terms (denoted α) that allow certain pieces of physics to be altered. See the osc-physics-switches section for more details
For non-radial adiabatic calculations, the last two equations above are set aside and the y5 terms dropped from the first four equations. For radial adiabatic calculations with :nml_n:reduce_order=:nml_v:.TRUE. (see the osc-params section), the last four equations are set aside and the first two replaced by
$$\begin{aligned}
\begin{align}
x \deriv{y_{1}}{x} &=
\left( \frac{V}{\Gammi} - 1 \right) y_{1} - \frac{V}{\Gamma_{1}} y_{2}, \\\
%
x \deriv{y_{2}}{x} &=
\left( c_{1} \omega^{2} + U - \As \right) y_{1} + \left( 3 - U + \As \right) y_{2}.
\end{align}
\end{aligned}$$
Boundary Conditions
Inner Boundary
When :nml_n:inner_bound=:nml_v:'REGULAR', GYRE applies regularity-enforcing conditions at the inner boundary:
When :nml_n:outer_bound=:nml_v:'UNNO'|:nml_v:'JCD', the first condition is replaced by the (possibly-leaky) outer mechanical boundary conditions described by :ads_citet:unno:1989 and :ads_citet:christensen-dalsgaard:2008, respectively. When :nml_n:outer_bound=:nml_v:'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 :nml_n:outer_bound=:nml_v:'GAMMA', the first condition is replaced by the outer mechanical boundary condition described by :ads_citet:ong: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. y1, y3 and y6 remain continuous across discontinuities, and therefore don't need these superscripts.
Structure Coefficients
The various stellar structure coefficients appearing in the dimensionless oscillation equations are defined as follows:
GYRE offers the capability to adjust the oscillation equations through a number of physics switches, controlled by parameters in the :nml_g: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
$\alphagrv$
:nml_n:alpha_grv
Scaling factor for gravitational potential perturbations. Set to 1 for normal behavior, and to 0 for the :ads_citet:cowling:1941 approximation
$\alphathm$
:nml_n:alpha_thm
Scaling factor for local thermal timescale. Set to 1 for normal behavior, to 0 for the non-adiabatic reversible (NAR) approximation (see :ads_citealp:glatzel:1990), and to a large value to approach the adiabatic limit
$\alphahfl$
:nml_n:alpha_hfl
Scaling factor for horizontal flux perturbations. Set to 1 for normal behavior, and to 0 for the non-adiabatic radial flux (NARF) approximation (see :ads_citealp:townsend:2003b)
$\alphagam$
:nml_n:alpha_gam
Scaling factor for g-mode isolation. Set to 1 for normal behavior, and to 0 to isolate g modes as described by :ads_citet:ong:2020
$\alphapi$
:nml_n:alpha_pi
Scaling factor for p-mode isolation. Set to 1 for normal behavior, and to 0 to isolate p modes as described by :ads_citet:ong:2020
$\alphakar$
:nml_n:alpha_kar
Scaling factor for opacity density partial derivative. Set to 1 for normal behavior, and to 0 to suppress the density part of the κ mechanism
$\alphakat$
:nml_n:alpha_kat
Scaling factor for opacity temperature partial derivative. Set to 1 for normal behavior, and to 0 to suppress the temperature part of the κ mechanism
$\alpharht$
:nml_n:alpha_rht
Scaling factor for time-dependent term in the radiative heat equation (see :ads_citealp:unno:1966). Set to 1 to include this term (Unno calls this the Eddington approximation), and to 0 to ignore the term