Non-Adiabatic Oscillations

This section discusses how to undertake non-adiabatic oscillation calculations using the gyre frontend. Asteroseismic studies typically rely on adiabatic calculations, because the frequencies of oscillation modes are the primary focus. However, for heat-driven modes the linear growth or damping rates can also be of interest — and evaluating these requires that non-adiabatic effects are included in the oscillation equations.

Note

Not all types of stellar mode include the necessary data (e.g., thermodynamic coefficients, opacity partial derivatives) to undertake non-adiabatic calculations. The Model Capabilities section summarizes this information.

Overview

To include non-adiabatic effects gyre augments the linearized mass, momentum and Poisson equations with the linearized heat and radiative diffusion equations (see the Linearized Equations section for full details). With these additions, the equations and their solutions become complex quantities. The assumed time dependence for perturbations is \(\propto \exp (-\ii \sigma t)\); therefore, the real part \(\sigmar\) and imaginary part \(\sigmai\) of the eigenfrequency are related to the mode period \(\Pi\) and growth e-folding time \(\tau\), respectively, via

\[\Pi = \frac{2\pi}{\sigmar}, \qquad \tau = \frac{1}{\sigmai}.\]

Solving the non-adiabatic equations proceeds using the same general approach as in the adiabatic case, by searching for the roots of a discriminant function \(\Dfunc(\omega)\) (see the Numerical Methods chapter for more details). However, a challenge is that there is no simple way to bracket roots in the complex plane. Instead, gyre must generate initial trial roots that are close to the true roots, and then refine them iteratively. Currently, gyre offers three methods for establishing the trial roots.

Adiabatic Method

The adiabatic method involves adopting the (real) roots found from adiabatic calculations as the initial trial roots for the non-adiabatic problem. This works well as long as the adiabatic and non-adiabatic roots lie close together in the complex plane — typically, when the oscillation modes are only weakly non-adiabatic, with \(|\sigmai/\sigmar| \ll 1\).

To perform non-adiabatic calculations with the adiabatic method, set the following options in the &osc namelist group:

and the following options in the &num namelist group:

You may also wish to use the following setting in the &num namelist group:

This tells gyre to evaluate the finite-difference equations using the 2nd order Magnus scheme; experience suggests that this gives the most reliable convergence for the root refinement.

An example of the adiabatic method in action can be found in the $GYRE_DIR/test/nad/mesa/bcep/gyre.in namelist input file, which is set up to find \(\ell=0,\ldots,3\) modes of a \(20\,\Msun\) \(\beta\) Cephei model using the adiabatic method. The important parts are as follows:

&osc
  nonadiabatic = .TRUE.
/

&num
  diff_scheme = 'MAGNUS_GL2'
  restrict_roots = .FALSE.
/

&scan
  grid_type = 'LINEAR'
  freq_min = 3.0
  freq_max = 10.0
  n_freq = 50
/

Note the nonadiabatic option in the &osc namelist group, and the diff_scheme option in the &num namelist group. The restrict_roots = .FALSE. option tells gyre not to reject any modes that have \(\sigmar\) outside the frequency range specified by the &scan namelist group; this ensures that modes whose non-adiabatic frequencies fall just outside the frequency grid are still found.

Minmod Method

The minmod method involves evaluating the discriminant function along the real-\(\omega\) axis, and then adopting local minima in its modulus \(|\Dfunc|\) as the initial trial roots for the non-adiabatic problem. The method is described in full in Goldstein & Townsend (2020); as shown there, it does not perform significantly better than the adiabatic method, and is included in gyre for the sake of completeness.

To perform non-adiabatic calculations with the minmod method, set the following options in the &osc namelist group:

and the following options in the &num namelist group:

As with the adiabatic method, you may also wish to use the following setting in the &num namelist group:

An example of the minmod method in action can be found in the $GYRE_DIR/test/nad/mesa/bcep-minmod/gyre.in namelist input file, which is equivalent to $GYRE_DIR/test/nad/mesa/bcep/gyre.in but using the minmod method. The important parts are as follows:

&osc
  adiabatic = .FALSE.
  nonadiabatic = .TRUE.
/

&num
  diff_scheme = 'MAGNUS_GL2'
  nad_search = 'MINMOD'
  restrict_roots = .FALSE.
/

&scan
  grid_type = 'LINEAR'
  freq_min = 3.0
  freq_max = 10.0
  n_freq = 250
/

Note the additional nad_search = 'MINMOD' option in the &num namelist group, stipulating that the minmod method should be used.

Contour Method

The contour method involves evaluating the discriminant function on a grid in the complex-\(\omega\) plane, and then adopting intersections between the real zero-contours \(\Dfuncr=0\), and the corresponding imaginary ones \(\Dfunci=0\), as the initial trial roots for the non-adiabatic problem. The method is described in full in Goldstein & Townsend (2020); it is very effective even for strongly non-adiabatic modes with \(|\sigmai/\sigmar| \sim 1\), although there is an increased computational cost.

To perform non-adiabatic calculations with the contour method, set the following options in the &osc namelist group:

and the following options in the &num namelist group:

You must also ensure that at least one &scan namelist group with axis = 'REAL' is present, and likewise at least one with axis = 'IMAG'. Together, these groups define the real and imaginary axes of the discriminant grid in the complex-\(\omega\) plane. As a rule of thumb, the resolution along the imaginary axis should be comparable to that along the real axis; this ensures that the contour-tracing algorithm behaves well.

Finally, as with the adiabatic method, you may also wish to use the following setting in the &num namelist group:

Note

Because g modes are spaced uniformly in period (in the asymptotic limit of large radial order), it would seem sensible to set grid_type = 'INVERSE' in the &scan namelist group(s) that correspond to the real axis (i.e., with axis = 'REAL'). However, this typically results in a mismatch between the resolution of the real and imaginary axes, and the contour method doesn’t perform well. A fix for this issue will be forthcoming in a future release of GYRE, but in the meantime it’s probably best to avoid the contour method for g modes.

An example of the minmod method in action can be found in the $GYRE_DIR/test/nad/mesa/bcep-contour/gyre.in namelist input file, which is equivalent to $GYRE_DIR/test/nad/mesa/bcep/gyre.in but using the minmod method. The important parts are as follows:

&osc
  adiabatic = .FALSE.
  nonadiabatic = .TRUE.
/

&num
  diff_scheme = 'MAGNUS_GL2'
  restrict_roots = .FALSE.
  nad_search = 'CONTOUR'
/

&scan
  axis = 'REAL'
  grid_type = 'LINEAR'
  freq_min = 3.0
  freq_max = 10.0
  n_freq = 50
/

&scan
  axis = 'IMAG'
  grid_type = 'LINEAR'
  freq_min = -0.28
  freq_max = 0.28
  n_freq = 5
/

Note the additional nad_search = 'CONTOUR' option in the &num namelist group, stipulating that the contour method should be used; and, the fact that there are now two &scan namelist groups, one with axis = 'REAL' and the other with axis = 'IMAG'.

Footnotes