[mesa-users] Convective temperature gradient from Canuto, Goldman & Mazzitelli (1996)

[Warrick Ball]

Hi all,

Following from a previous message (see below) where I describe my 
implementation of the Canuto & Mazzitelli (1991, CM1991) convective 
temperature gradient, I also went ahead and implemented the temperature 
gradient in Canuto, Goldman & Mazzitelli (1996, CGM1996) [1].  As with 
CM1991, this only involves modifying the traditional cubic equation with a 
non-linear term for which CGM1996 give an analytic fit (equation 73-75). 
This is basically a progress report of sorts for anyone interested.

I've attached my run_star_extras.f, with which the new convective 
temperature gradients can be selected with

   use_other_mlt = .true.
   MLT_option = 'CGM1996' ! or 'CM1991'

Don't forget to change alpha to a more appropriate value! (e.g. ~0.7 for 
solar-like values in CGM1996.)

I've also attached a plot of the pre-MS and MS evolution of a 1Msun star, 
modified from the default work folder, showing tracks for the default MLT 
with alpha=2.0 or 1.4, as well as alpha=0.7 with CM1991 or CGM1996 
convection.  I'm reasonably confident that the temperature gradients are 
being computed correctly, since these line up with my expectations. 
CM1991 originally compared alpha=0.7 in their model to alpha=1.4 in 
traditional MLT.  For CGM1996, I'm used to solar-calibrated values of 
alpha~=0.68, which is in line with the plot.  In addition, I'm running a 
(very slow and poorly-setup!) solar calibration, and it's currently 
settled on alpha~=0.66.

But, I caution against this being anything like production quality for two 
reasons.  First, it obviously needs much more thorough testing, which I 
will try given time.  Second, I haven't modified the convective velocities 
at all.  I'm not expert enough (but I'm trying to learn more) to know 
exactly what to do, but suffice it to say that the convective velocity is 
still being derived as in the existing MESA MLT, only with the temperature 
gradient given its new value.  (If you have the paper to hand, I believe 
that the mean squared velocity expression in equation (88) is the 
convective velocity I'm looking for.  It's of a similar form to the other 
analytic expressions, so it shouldn't be too hard to implement.)

Finally, as ever, thanks to the MESA team for writing such easily 
modifiable/extensible code!

Cheers,
Warrick

[1] http://adsabs.harvard.edu/abs/1996ApJ...473..550C

On Wed, 17 Dec 2014, Warrick Ball wrote:

> Hi MESAlians,
>
> What started off as a side interest following a conversation over a 
> conference lunch 12 days ago has ended up with me possibly having implemented 
> the temperature gradient from Canuto & Mazzitelli (1991) [1], or just CM1991.
>
> Much of the original paper is committed to determining the convective flux 
> according to a more detailed description of the spectrum of turbulent eddies 
> than in mixing length theory.  These details are still a bit beyond me, but 
> the results of those calculation are tabulated in the paper and an analytic 
> fit given in their equation (32).  This introduces a new factor (eqn 34), 
> involving the MLT convective flux (eqn 4), that must be incorporated in the 
> traditional cubic equation of Cox & Giuli (eqn 14.82 in the textbook; eqns 9 
> and 10 in CM1991).
>
> The new form of the "cubic" (eqn 36) isn't really a cubic, so I used the 
> safe_root routines from mesa/num to solve the equation, using the MLT value 
> of the parameter Gamma as an initial guess.  I solved for the gradient of 
> Gamma implicitly.  The parameter Zeta in the code becomes
>
> Zeta = Phi_CM/(1 + Phi_CM)
>
> (see eqn 63 in the paper) where Phi_CM is the new convective flux, given by 
> the analytic fit (eqn 32).  Again, the derivative can be computed implicitly.
>
> After some fumbling around, I seem to have succeeded in implementing the new 
> temperature gradient in MESA.  I have attached a run_star_extras.f that 
> contains a modified version of MESA's own MLT routine, in which the CM1991 
> temperature gradient is developed as a modification to the 'Cox' option. 
> Just set MLT_option = 'CM1991' and use_other_mlt = .true. in &controls. 
> (FYI, this was all developed in revision 7184.)
>
> Attached is an HR track from a default work folder, modified to a 1 solar 
> mass model.  The two tracks are for Cox MLT with alpha = 2 and CM1991 MLT 
> with alpha = 0.75, up to core hydrogen exhaustion.  They don't quite match, 
> which isn't totally surprising because the conversion between the two is 
> non-trivial.  I'm trying to calibrate a solar model now to see what value 
> comes out.
>
> Anyway, that's what I've done.  It started out as an exercise in teaching 
> myself a bit more about convection in the models, and ended up with something 
> I hope that someone, somewhere might find interesting or useful. I'd love to 
> know if I've actually done the right thing.  I've iterated on the "cubic" 
> equation (32), although the paper talks about iterating directly on the 
> convective flux (equation 63).  I assume that these are equivalent...
>
> The fact that I got somewhere with the code is testament to it's clarity: 
> thanks again Bill and other contributors!  Finally, note that I haven't done 
> anything regarding (a) the turbulent pressure, for which CM1991 also offers a 
> different expression (I think?), or (b) the suggestion of limiting the 
> mixing-length to the distance to the convective boundary.
>
> Cheers,
> Warrick
>
> [1] http://esoads.eso.org/abs/1991ApJ...370..295C
>
> ------------
> Warrick Ball
> Postdoc, Institut für Astrophysik Göttingen
> wball at astro.physik.uni-goettingen.de
> +49 (0) 551 39 5069


------------
Warrick Ball
Postdoc, Institut für Astrophysik Göttingen
wball at astro.physik.uni-goettingen.de
+49 (0) 551 39 5069
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