Sign relation between damping rate and Imaginary frequency

[pjuelsgaard]

When doing non-adiabatic frequencies, I am having a hard time figuring out the significance of the relation between the sign of the imaginary part of the eigenfrequency and the damping rate. Does anyone know the relation,or how to figure it out from the GYRE odcumentation?

Kind regards
- Peter Juelsgaard

[rhtownsend]

Hi Peter --

GYRE assumes a time dependence of exp(-i sigma t), where sigma is the mode angular frequency. Internally, GYRE works with the dimensionless frequency

omega = sqrt(R^{3}/GM) sigma

...but the signs of sigma and omega agree.

Modes with negative Im(sigma) are unstable and will grow with time, whereas modes with positive Im(sigma) are damped and will decay with time. The decay will follow the scaling

amplitude ~ exp(-Im(sigma) t)

Hope that helps!

cheers,

Rich

[pjuelsgaard]

Thanks for the detailed explanation. I was having a hard time wrapping my head around it.

Also, Günter Houdek sends his regards (he's my thesis advisor)

[rhtownsend]

Thanks for the detailed explanation. I was having a hard time wrapping my head around it.

Also, Günter Houdek sends his regards (he's my thesis advisor)

Glad I could help -- and do send regards back to Günter!

[pjuelsgaard]

Going through the final touches, I notice that part of this analysis doesn't quite add up for me.

If the mode angular frequency agrees with the dimensionless frequency, as in sign(sigma)=sign(omega), and the time dependence is
amplitude ~ exp(-i sigma t)
Then this would lead to
amplitude ~ exp(-i sigma t) = exp(-i (Re(sigma) + i Im(sigma)) t) = exp(i Re(sigma) t) * exp( Im(sigma) t)
Indicating that frequencies with Im(sigma)<0 should be stable and decay with time, while frequencies with Im(sigma)>0 grow exponentially in time.

Or am I missing something important here?

I can see that the frequencies I compute with GYRE agree with the convection you mentioned, but this leads me to conclude that the time dependence in the perturbation equations should be amplitude ~ exp(i sigma t), after all Re(exp(i Re(sigma) t) ) = Re(exp( -i Re(sigma) t) ).

Or am I misunderstanding something quite basic?

[rhtownsend]

Peter --

You're absolutely right -- I can't imagine what I was thinking when I wrote my reply above. Positive Im(omega) will grow with time, and vice versa.

cheers,

Rich

[taylor_m]

Hello,

I am having trouble with some inconsistency between the sign of Im(omega) and the growth rates gyre is reporting. I ran a suite of 13 WD models, and there is a spread in the sign of Im(omega) for a g-mode of radial order 2 (g2); some are positive and some are negative. However, the growth rates are reported to be positive for all 13 models. I confirmed the normalized growth rate values based on the Work Integrals (from the Stellingworth definition), and I believe them, but I am wondering why the Im(omega) sign values for g2 are not all negative if all the normalized growth rates are all positive? It seems that Im(omega) fairly small in magnitude... could this be a numerical issue, even with double precision?

Attached are 2 figures that may help explain my misunderstanding.

Thank you in advance for any help you can provide!

Morgan

[rhtownsend]

Hello,

I am having trouble with some inconsistency between the sign of Im(omega) and the growth rates gyre is reporting. I ran a suite of 13 WD models, and there is a spread in the sign of Im(omega) for a g-mode of radial order 2 (g2); some are positive and some are negative. However, the growth rates are reported to be positive for all 13 models. I confirmed the normalized growth rate values based on the Work Integrals (from the Stellingworth definition), and I believe them, but I am wondering why the Im(omega) sign values for g2 are not all negative if all the normalized growth rates are all positive? It seems that Im(omega) fairly small in magnitude... could this be a numerical issue, even with double precision?

Attached are 2 figures that may help explain my misunderstanding.

Thank you in advance for any help you can provide!

Morgan

Hi Morgan --

Quick question -- what are you using for diff_scheme? MAGNUS_GL2 is known to sometimes give unreliable growth rates and/or frequencies.

cheers,

Rich

[taylor_m]

Hi Rich,

Thank you for the reply! I was indeed using MAGNUS_GL2. I tried COLLOC_GL4 and saw similar results.
In trying the COLLOC_GL2 scheme, I got the same sign for the normalized growth rates and Im(omega) values for any given radial order, which appears more trustworthy.

In viewing a prior post, I saw that COLLOC_GL2 might be best for growth rates. Is this still the case for gyre 6.0.1?
If so, is COLLOC_GL2 best for both nonadiabatic and adiabatic calculations?
I found a small (~0.01%) difference in the adiabatic periods (radial orders <11, l=1) between the COLLOC_GL2 and COLLOC_GL4 schemes.
Not crazy concerning, just wondering what scheme is recommended for the adiabatic case.

Thanks in advance,

Morgan

[rhtownsend]

Hi Morgan --

Could you post the model and gyre inlist files, so I can have a dig into this myself?

Many thanks,

Rich