.. _osc-dimless-form: Dimensionless Formulation ========================= To improve numerical stability, GYRE solves the :ref:`separated equations ` and :ref:`boundary conditions ` by recasting them into a dimensionless form that traces its roots back to :ads_citet:`dziembowski:1971`. .. _osc-dimless-vars: Variables --------- The independent variable is the fractional radius :math:`x \equiv r/\Rstar` (with :math:`\Rstar` the stellar radius), and the dependent variables :math:`\{y_{1},y_{2},\ldots,y_{6}\}` are .. math:: :label: e:dimless-vars \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}{\cP}, \\ y_{6} &= x^{-1-\ell}\, \frac{\delta \tLrad}{\Lstar} \end{aligned} (with :math:`\Lstar` the stellar luminosity). .. _osc-dimless-eqns: Oscillation Equations --------------------- The dimensionless oscillation equations are .. math:: :label: e:dimless-eqns \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} where the dimensionless oscillation frequency is introduced as .. math:: :label: e:omega \omega \equiv \sqrt{\frac{\Rstar^{3}}{G\Mstar}}\sigma (with :math:`\Mstar` the stellar mass). These differential equations are derived from the separated equations, with the insertion of 'switch' terms (denoted :math:`\alpha`) that allow certain pieces of physics to be altered. See the :ref:`osc-physics-switches` section for more details. For non-radial adiabatic calculations, the last two equations above are set aside and the :math:`y_{5}` terms dropped from the first four equations. For radial adiabatic calculations with :nml_n:`reduce_order`\ =\ :nml_v:`.TRUE.` (see the :ref:`osc-params` section), the last four equations are set aside and the first two replaced by .. math:: \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} .. _osc-dimless-bc: Boundary Conditions ------------------- Inner Boundary ^^^^^^^^^^^^^^ When :nml_n:`inner_bound`\ =\ :nml_v:`'REGULAR'`, GYRE applies regularity-enforcing conditions at the inner boundary: .. math:: \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} (these are the dimensionless equivalents to the expressions appearing in the :ref:`osc-bound-conds` section). When :nml_n:`inner_bound`\ =\ :nml_v:`'ZERO_R'`, the first and second conditions above are replaced with zero radial displacement conditions, .. math:: \begin{aligned} y_{1} &= 0, \\ y_{4} &= 0. \end{aligned} Likewise, when :nml_n:`inner_bound`\ =\ :nml_v:`'ZERO_H'`, the first and second conditions are replaced with zero horizontal displacement conditions, .. math:: \begin{aligned} y_{2} - y_{3} &= 0, \\ y_{4} &= 0. \end{aligned} Outer Boundary ^^^^^^^^^^^^^^ When :nml_n:`outer_bound`\ =\ :nml_v:`'VACUUM'`, GYRE applies the outer boundary conditions .. math:: \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} (these are the dimensionless equivalents to the expressions appearing in the :ref:`osc-bound-conds` section). When :nml_n:`outer_bound`\ =\ :nml_v:`'DZIEM'`, the first condition above is replaced by the :ads_citet:`dziembowski:1971` outer boundary condition, .. math:: \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 :nml_n:`outer_bound`\ =\ :nml_v:`'UNNO'` or :nml_v:`'JCD'`, the first condition is replaced by the (possibly-leaky) outer 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 boundary condition derived from a local dispersion analysis of waves in an isothermal atmosphere. Finally, when :nml_n:`outer_bound`\ =\ :nml_v:`'GAMMA'`, the first condition is replaced by the outer momentum boundary condition described by :ads_citet:`ong:2020`. Internal Boundaries ^^^^^^^^^^^^^^^^^^^ Across density discontinuities, GYRE applies the boundary conditions .. math:: \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} (these are the dimensionless equivalents to the expressions appearing in the :ref:`osc-bound-conds` section). Here, + (-) superscripts indicate quantities evaluated on the inner (outer) side of the discontinuity. :math:`y_{1}`, :math:`y_{3}` and :math:`y_{6}` remain continuous across discontinuities, and therefore don't need these superscripts. .. _osc-struct-coeffs: Structure Coefficients ---------------------- The various stellar structure coefficients appearing in the dimensionless oscillation equations and boundary conditions are defined as follows: .. math:: \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}}{\Rstar^{3}} \frac{\Mstar}{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}{\Lstar} \qquad \crad = x^{-3} \frac{\Lrad}{\Lstar} \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}{\Lstar} \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{G\Mstar}{\Rstar^{3}}} \qquad \dcthn = \deriv{\ln \cthn}{\ln r} \\ % \cthk = x^{-3} \frac{4\pi r^{3} \cP T \rho}{\Lstar} \sqrt{\frac{G\Mstar}{\Rstar^{3}}} \qquad \cegv = x^{-3} \frac{4\pi r^{3} \rho \epsgrav}{\Lstar} \end{gathered} .. _osc-physics-switches: Physics Switches ---------------- 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 (see the :ref:`osc-params` section). The table below summarizes the mapping between the switches appearing in the expressions above, and the corresponding namelist parameters. .. list-table:: :widths: 20 20 60 :header-rows: 1 * - Symbol - Parameter - Description * - :math:`\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 * - :math:`\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 * - :math:`\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`) * - :math:`\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` * - :math:`\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` * - :math:`\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 :math:`\kappa` mechanism * - :math:`\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 :math:`\kappa` mechanism * - :math:`\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 * - :math:`\alphatrb` - :nml_n:`alpha_trb` - Scaling factor for the turbulent mixing length. Set to the convective mixing length to include the turbulent damping term (see the :ref:`osc-conv` section), and to 0 to ignore the term