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

[Arlette Grotsch]

Sorry Bill, I forgot to attach the file.

Let me add this comment: Finding the zero of a discontinuous function by locating a change of sign (or checking the sign layer by layer) leads to a infinity of solutions. Whatever the location of the discontinuity (i.e. the location of the boundary), it can satisfy the condition on the change of sign. The only solution acceptable in the frame of MLT is the one for which grad_rad = grad_ad on the convective side. This indeed means that the radial convective velocity is zero and that the convective flux is accordingly zero at the boundary. This is the condition that I write L_rad = L(total).
Arlette

Début du message réexpédié :

> De: Arlette Grotsch <Arlette.Noels at ulg.ac.be>
> Objet: Rép : Misuse of Ledoux criterion in mesa MLT
> Date: 19 juin 2014 10:17:35 UTC+2
> À: Bill Paxton <paxton at kitp.ucsb.edu>
> Cc: MESA users group <mesa-users at lists.sourceforge.net>
> 
> 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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