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
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 parameters in the &osc
namelist group:
nonadiabatic
=.TRUE.
adiabatic
=.TRUE.
[1]
and the following parameters in the &num
namelist group:
You may also wish to use the following setting in the &num
namelist group:
diff_scheme
='MAGNUS_GL2'
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
parameter in the &osc
namelist
group, and the diff_scheme
parameter in the &num
namelist group. The restrict_roots
=.FALSE.
setting, also in the &num
namelist group, 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 parameters in the &osc
namelist group:
nonadiabatic
=.TRUE.
adiabatic
=.FALSE.
[2]
and the following parameters in the &num
namelist group:
nad_search
='MINMOD'
As with the adiabatic method, you may also wish to use the following
setting in the &num
namelist group:
diff_scheme
='MAGNUS_GL2'
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'
parameter
in the &num
namelist group, which stipulates 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 (see here for one strategy for mitigating this cost).
To perform non-adiabatic calculations with the contour method, set
the following parameters in the &osc
namelist group:
nonadiabatic
=.TRUE.
adiabatic
=.FALSE.
[2]
and the following parameters in the &num
namelist group:
nad_search
='CONTOUR'
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:
diff_scheme
='MAGNUS_GL2'
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.,
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'
parameter in the &num
namelist group, which stipulates 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