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

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

Oscillation Equations

The dimensionless oscillation equations are

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

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

\[\begin{split}\begin{align} x \deriv{y_{1}}{x} &= \left( \frac{V}{\Gammi} - 1 \right) y_{1} - \frac{V}{\Gamma_{1}} y_{2}, \\ % x \deriv{y_{2}}{x} &= \left( c_{1} \omega^{2} + U - \As \right) y_{1} + \left( 3 - U + \As \right) y_{2}. \end{align}\end{split}\]

Boundary Conditions

Inner Boundary

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

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

When inner_bound='ZERO_R', the first and second conditions are replaced with zero radial displacement conditions,

\[\begin{split}\begin{align} y_{1} &= 0, \\ y_{4} &= 0. \end{align}\end{split}\]

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

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

Outer Boundary

When outer_bound='VACUUM', GYRE applies vacuum surface pressure conditions at the outer boundary:

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

When outer_bound='DZIEM', the first condition is replaced by the Dziembowski (1971) outer mechanical 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'|'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

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

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:

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

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. 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 & Glatzel, 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