# 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

(11)\begin{split}\begin{aligned} y_{1} &= x^{2 - \ell}\, \frac{\txir}{r}, \\ y_{2} &= x^{2-\ell}\, \frac{\tP'}{\rho g r}, \\ y_{3} &= x^{2-\ell}\, \frac{\tPhi'}{gr}, \\ y_{4} &= x^{2-\ell}\, \frac{1}{g} \deriv{\tPhi'}{r}, \\ y_{5} &= x^{2-\ell}\, \frac{\delta \tS}{c_{p}}, \\ y_{6} &= x^{-1-\ell}\, \frac{\delta \tLrad}{L}. \end{aligned}\end{split}

## Oscillation Equations

The dimensionless oscillation equations are

(12)\begin{split}\begin{aligned} x \deriv{y_{1}}{x} &= \left( \frac{V}{\Gammi} - 1 - \ell \right) y_{1} + \left( \frac{\ell(\ell+1)}{c_{1} \omega^{2}} - \alphagam \frac{V}{\Gammi} \right) y_{2} + \alphagrv \frac{\ell(\ell+1)}{c_{1} \omega^{2}} y_{3} + \upsT \, y_{5}, \\ % x \deriv{y_{2}}{x} &= \left( c_{1} \omega^{2} - \fpigam \As \right) y_{1} + \left( 3 - U + \As - \ell \right) y_{2} - \alphagrv y_{4} + \upsT \, y_{5}, \\ % x \deriv{y_{3}}{x} &= \alphagrv \left( 3 - U - \ell \right) y_{3} + \alphagrv y_{4} \\ % x \deriv{y_{4}}{x} &= \alphagrv \As U y_{1} + \alphagrv \frac{V}{\Gammi} U y_{2} + \alphagrv \ell(\ell+1) y_{3} - \alphagrv (U + \ell - 2) y_{4} - \alphagrv \upsT \, U y_{5}, \\ % x \deriv{y_{5}}{x} &= \frac{V}{\frht} \left[ \nabad (U - c_{1}\omega^{2}) - 4 (\nabad - \nabla) + \ckapad V \nabla + \cdif \right] y_{1} + \mbox{} \\ & \frac{V}{\frht} \left[ \frac{\ell(\ell+1)}{c_{1} \omega^{2}} (\nabad - \nabla) - \ckapad V \nabla - \cdif \right] y_{2} + \mbox{} \\ & \alphagrv \frac{V}{\frht} \left[ \frac{\ell(\ell+1)}{c_{1} \omega^{2}} (\nabad - \nabla) \right] y_{3} + \alphagrv \frac{V \nabad}{\frht} y_{4} + \mbox{} \\ & \left[ \frac{V \nabla}{\frht} (4 \frht - \ckapS) + \dfrht + 2 - \ell \right] y_{5} - \frac{V \nabla}{\frht \crad} y_{6} \\ % x \deriv{y_{6}}{x} &= \left[ \alphahfl \ell(\ell+1) \left( \frac{\nabad}{\nabla} - 1 \right) \crad - V \cepsad - \alphaegv \cegv \nabad V \right] y_{1} + \mbox{} \\ & \left[ V \cepsad - \ell(\ell+1) \crad \left( \alphahfl \frac{\nabad}{\nabla} - \frac{3 + \dcrad}{c_{1}\omega^{2}} \right) + \alphaegv \cegv \nabad V \right] y_{2} + \mbox{} \\ & \alphagrv \left[ \ell(\ell+1) \crad \frac{3 + \dcrad}{c_{1}\omega^{2}} \right] y_{3} + \mbox{} \\ & \left[ \cepsS - \alphahfl \frac{\ell(\ell+1)\crad}{\nabla V} + \ii \alphathm \omega \cthk + \alphaegv \cegv \right] y_{5} - \left[ 1 + \ell \right] y_{6}, \end{aligned}\end{split}

where the dimensionless oscillation frequency is introduced as

(13)$\omega \equiv \sqrt{\frac{R^{3}}{GM}},$

(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 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

\begin{split}\begin{aligned} 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{aligned}\end{split}

## Boundary Conditions

### Inner Boundary

When inner_bound='REGULAR', GYRE applies regularity-enforcing conditions at the inner boundary:

\begin{split}\begin{aligned} c_{1} \omega^{2} y_{1} - \ell y_{2} - \alphagrv \ell y_{3} &= 0, \\ \alphagrv \ell y_{3} - (2\alphagrv - 1) y_{4} &= 0, \\ y_{5} &= 0. \end{aligned}\end{split}

(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,

\begin{split}\begin{aligned} y_{1} &= 0, \\ y_{4} &= 0. \end{aligned}\end{split}

Likewise, when inner_bound='ZERO_H', the first and second conditions are replaced with zero horizontal displacement conditions,

\begin{split}\begin{aligned} y_{2} - y_{3} &= 0, \\ y_{4} &= 0. \end{aligned}\end{split}

### Outer Boundary

When outer_bound='VACUUM', GYRE applies the outer boundary conditions

\begin{split}\begin{aligned} y_{1} - y_{2} &= 0 \\ \alphagrv U y_{1} + (\alphagrv \ell + 1) y_{3} + \alphagrv y_{4} &= 0 \\ (2 - 4\nabad V) y_{1} + 4 \nabad V y_{2} + 4 \frht y_{5} - y_{6} &= 0 \end{aligned}\end{split}

(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,

$\left\{ 1 + V^{-1} \left[ \frac{\ell(\ell+1)}{c_{1} \omega^{2}} - 4 - c_{1} \omega^{2} \right] \right\} y_{1} - y_{2} = 0.$

When outer_bound='UNNO' or 'JCD', the first condition is replaced by the (possibly-leaky) outer 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 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

\begin{split}\begin{aligned} U^{+} y_{2}^{+} - U^{-} y_{2}^{-} &= y_{1} (U^{+} - U^{-}) \\ y_{4}^{+} - y_{4}^{-} &= -y_{1} (U^{+} - U^{-}) \\ y_{5}^{+} - y_{5}^{-} &= - V^{+} \nabad^{+} (y_{2}^{+} - y_{1}) + V^{-} \nabad^{-} (y_{2}^{-} - y_{1}) \end{aligned}\end{split}

(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:

$\begin{split}\begin{gathered} V = -\deriv{\ln P}{\ln r} \qquad V_{2} = x^{-2} V \qquad \As = \frac{1}{\Gamma_{1}} \deriv{\ln P}{\ln r} - \deriv{\ln \rho}{\ln r} \qquad U = \deriv{\ln M_{r}}{\ln r} \\ % c_1 = \frac{r^{3}}{R^{3}} \frac{M}{M_{r}} \qquad \fpigam = \begin{cases} \alphapi & \As > 0, x < x_{\rm atm} \\ \alphagam & \As > 0, x > x_{\rm atm} \\ 1 & \text{otherwise} \end{cases}\\ % \nabla = \deriv{\ln T}{\ln P} \qquad \clum = x^{-3} \frac{\Lrad+\Lcon}{L} \qquad \crad = x^{-3} \frac{\Lrad}{L} \qquad \dcrad = \deriv{\ln \crad}{\ln r} \\ % \frht = 1 - \alpharht \frac{\ii \omega \cthn}{4} \qquad \dfrht = - \alpharht \frac{\ii \omega \cthn \dcthn}{4 \frht} \\ % \ckapad = \frac{\alphakar \kaprho}{\Gamma_{1}} + \nabad \alphakat \kapT \qquad \ckapS = - \upsT \alphakar \kaprho + \alphakat \kapT \\ % \ceps = x^{-3} \frac{4\pi r^{3} \rho \epsnuc}{L} \qquad \cepsad = \ceps \epsnucad \qquad \cepsS = \ceps \epsnucS \\ % \cdif = - 4 \nabad V \nabla + \nabad \left(V + \deriv{\ln \nabad}{\ln x} \right) \\ % \cthn = \frac{\cP}{a c \kappa T^{3}} \sqrt{\frac{GM}{R^{3}}} \qquad \dcthn = \deriv{\ln \cthn}{\ln r} \\ % \cthk = x^{-3} \frac{4\pi r^{3} \cP T \rho}{L} \sqrt{\frac{GM}{R^{3}}} \qquad \cegv = x^{-3} \frac{4\pi r^{3} \rho \epsgrav}{L} \end{gathered}\end{split}$

## 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

$$\alphagrv$$

alpha_grv

Scaling factor for gravitational potential perturbations. Set to 1 for normal behavior, and to 0 for the Cowling (1941) approximation

$$\alphathm$$

alpha_thm

Scaling factor for local thermal timescale. Set to 1 for normal behavior, to 0 for the non-adiabatic reversible (NAR) approximation (see Gautschy et al., 1990), and to a large value to approach the adiabatic limit

$$\alphahfl$$

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 Townsend, 2003b)

$$\alphagam$$

alpha_gam

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)

$$\alphapi$$

alpha_pi

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)

$$\alphakar$$

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 $$\kappa$$ mechanism

$$\alphakat$$

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 $$\kappa$$ mechanism

$$\alpharht$$

alpha_rht

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

$$\alphatrb$$

alpha_trb

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