[mesa-users] Misuse of Ledoux criterion in mesa MLT

[Bill Paxton]

Hi Arlette,

Let me restate the 2 questions from the end of my last email since I don't find direct answers to them in your response (the answers are probably implicit in your reply, but I'm trying to stick to answers directly from you).


Begin forwarded message:

> Can there be a convective-semiconvective boundary in the model? [Yes/No]
> 
> If the answer is yes, then we have grad_rad > grad_ad at the boundary by definition (Schw unstable on both sides) which goes against my understanding of your answer to question 2.  So I don't think this will be your answer.
> 
> If the answer is no, then please enlighten us at to why such a case is impossible.  I think this is equivalent to asking how it is that B > 0 can only happen in places that are Schw stable.  That will be educational -- for me at least.


My previous guess that you don't allow a convective-semiconvective boundary seems to be right since you state in your last email that grad_rad = grad_ad at the convective boundary, and that leaves no possibility for semiconvection between convective and radiative regions for reasons I covered in detail in my last email. 

That leads to my previous question as to why such a case is impossible.  Here your answer seems to be that it can't happen because MLT theory doesn't include it.    And interestingly, in older versions of mesa before we added an option for semiconvection, that is exactly how mesa worked.  And it still works that way if you turn off Ledoux --- in that case it uses the Schw criterion to decide the convective boundary and agrees with you that the boundary is defined by the zero point for grad_ad - grad_rad.  In that case, we can consider how well the cell boundary in mesa matches the location of the zero of that function -- there will undoubtedly be a subcell difference that we can look at to see if it is significant compared to the other approximations in the model.

But in current versions of mesa, when Ledoux is turned on the code uses the Ledoux criterion to determine the convection boundary (which is at the zero of grad_rad - grad_L) and allows there to be a semiconvective region that is Schw unstable but Ledoux stable.  You say that's wrong --- there cannot be such a semiconvective region since that would imply grad_rad > grad_ad at the convective boundary.  By definition, both the convective and semiconvective zones are Schw unstable, meaning grad_rad > grad_ad, so if grad_rad = grad_ad at the convective boundary, there is no semiconvection.

At risk of misstating your implicit argument, you seem to be suggesting that there cannot be a stabilizing composition gradient producing a semiconvective region because the semiconvective region doesn't exist; it is actually part of the convective region and therefore cannot have a composition gradient because we are assuming perfect convective mixing that wipes out composition gradients.  But that argument leaves me unsatisfied -- it seems there cannot be a semiconvection region because somehow the convection region has devoured any candidate for semiconvection --- but how did that happen?    So I hope that isn't in fact your argument, and you can clarify for me why semiconvection is impossible.

Also, I'm curious about the implications for the 77 papers on ADS that have "semiconvection" in their titles.  Here's a recent example:

H.C. Spruit, "Semiconvection: theory", A&A 552, A76 (2013)


And here is the "classic" by Langer et al that inspired the implementation of semiconvection in mesa:

Langer, El Eid, and Fricke, "Evolution of massive stars with semiconvective diffusion", A & A, 145,, 179-191 (1985)


If as you say, grad_ad = grad_rad at the convective boundary, then these papers seem to be about something that doesn't exist.  I'm sure the authors will be interested to learn this! 

For their benefit as well as mine, please explain -- where did the semiconvection go?

Thanks,
Bill






On Jun 19, 2014, at 1:17 AM, Arlette Grotsch wrote:

> Hi Bill,
> 
> I comment the first part of your mail, which is the most important in order to understand what is going on. My comments are based on MLT theory.
> 
> 
>> 
>> In your 1st answer, you say your convective region is "pure", no semiconvection.  According to the definitions I think we've agreed to use, that means it is unstable both for Schw and Ledoux (in contrast to semiconvection which is unstable for Schw but not for Ledoux).  So far just definitions, but i'll spell it out in excessive detail as is my usual style.  For cases involving semiconvection, the boundary of the convection zone is where it changes from Ledoux unstable (convective) to Ledoux stable (semiconvective).  Both convective and semiconvective are Schw unstable, so there is no change in that criterion at the boundary.  It looks like this (again using grad_L = grad_ad + B, where B is the composition gradient term).
>> 
>> 		Convective			<boundary>	Semiconvective		
>> 	Schw unstable (grad_rad > grad_ad)	   |		Schw unstable (grad_rad > grad_ad)
>> 	Ledoux unstable (grad_rad > grad_L)	   |		Ledoux stable (grad_rad < grad_L)
> 
> Important: At the boundary —>                             grad_rad = grad_ad  | grad_rad > grad_ad
>                                                                                     L_rad = L               | L_rad = L
> Since the mixing is assumed to be perfect in the convective region,        |
> grad_mu = 0 and grad_L = grad_ad. 
> 
> This comes directly out of the MLT theory, as we discuss in Gabriel et al. (attached file). See our Fig. 6 (left panel) where convective boundary is located at the jump.



>                                                      
>>         
>> Hurray!  Now we clear about the convective boundary in the presence of semiconvection.  We expect a change in sign of (grad_L - grad_rad), but we require (grad_rad > grad_ad) to be positive on both sides.
> 
>>  
> We require grad_rad - grad_ad to be positive on the semi convective side and zero on the convective side. So the function grad_rad - grad_ad jumps from zero to a positive value at the boundary if there is a discontinuity in the chemical composition (this happens when the convective core is growing). The function grad_rad - grad_L jumps from zero to a positive value at the boundary if there is a gradient of chemical composition (when the convective core shrinks).
> 
>> Of course if there is no semiconvection, then it would look like the following with both Schw and Ledoux switching together from unstable to stable at the boundary.
>> 
>> 		Convective			<boundary>	Radiative		
>> 	Schw unstable (grad_rad > grad_ad)	   |		Schw stable (grad_rad < grad_ad)
>> 	Ledoux unstable (grad_rad > grad_L)	   |		Ledoux stable (grad_rad < grad_L)
>> 
> Important: At the boundary —>                             grad_rad = grad_ad  | grad_rad < or = grad_ad
>                                                                                     L_rad = L               | L_rad = L
> Since the mixing is assumed to be perfect in the convective region,        |
> grad_mu = 0 and grad_L = grad_ad. 
> 
> If the check on (grad_rad - grad_ad (or grad_L)) is made layer by layer, it will be impossible in cases of discontinuities in mu or grad_mu to have zero on the convective side of the boundary.
> 
> Do we agree on this?
> 
> Cheers,
> Arlette
> 
> 
> 

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