A Review of the Mixing Length Theory of Convection in 1D Stellar Modeling

Round 1

Reviewer 1 Report

This is an excellent review of the MLT as used in 1D stellar models.  It is written at a time when interest on the uses of MLT and limitations on 1D stellar models is increasing.

The description and discussion of the "classical" MLT as used in the literature of stellar structure is very clear.  It is also clearly written from the point of view of the Dartmouth researchers.

It is suggested that the authors clarify and sharpen the discussions of the impact of asteroseismology and of the 3D simulations of the outer layers of 1D stellar models on the future improvements in the art of stellar modeling.

 

Author Response

Please see the attachment.

Author Response File: Author Response.pdf

Reviewer 2 Report

Results such as those shown in fig. 4 and 5, prove to anyone and without any doubts, the failure of the MLT.

MLT was invented in 1925 by Prandlt who had no intention of applying it to stellar interiors. It was E. Bohm-Vitense in 1953 who did that to the permanent detriment of scientific progress.

The number of eddies in a highly turbulent flow such as stellar interiors, is very large, specifically, the ratio of the largest to the smallest eddy scales like Re^3/4 and thus in a 3D case, the number of grid points or degrees of freedom, scales like Re^9/4.

The more viscous the flow, the smaller is the range of eddies and since Prandtl dealt with engineering flows with large viscosity, he dealt with relatively small Re and relatively narrow number of eddies which he extrapolated to one eddy, and thus the MLT.

 

Stellar interiors have exactly, a diametrically opposite behavior, large Re and large number of eddies, millions of eddies. Reducing them to one eddy is an intrinsic contradiction to the physics of the problem. There is one and only one solution: forget about MLT and consider all the eddies. That constituted the only true progress after 38 years of misguided MLT and yet, the authors refer to it only after 25 references. Such a paper has been cited 951 times! Playing games with the parameter alpha is a terrible kind of science, it is not even science. This review propagates a physically misguided MLT model perpetrating an error instead of correcting it which I view as a disservice to the stellar structure community.  

 

 

Author Response

The authors were invited to review MLT, not the full spectrum turbulence model. This report is unprofessional and does not provide any specific criticism or suggestions for improvement. We will not be engaging further.

Reviewer 3 Report

This paper contains a welcome review of the mixing length theory of
convection, as employed in 1D stellar models.

There are many things to like about this review: for example, it has a good
mix of references to both classic and modern papers, and manages to
synthesize these into a coherent story. It is well-written.  I found parts
of it very interesting: for example, there is a very useful and
well-illustrated discussion of the effects (on evolution, structure, etc.)
induced by changing the mixing length. This bit alone might have been worth a whole review!  For these reasons, it ought to be published in due course, and I believe will serve as a useful reference.

That said, there are a couple of places in which I thought the discussion
in the paper was incorrect, or at least quite misleading, and I believe these
ought to be addressed prior to acceptance. These are detailed below.

My major concern has to do with the discussion of where MLT is
"applicable."  Figure 1, describing the "applicability of MLT on the main
sequence," has a giant X drawn through stars with convective cores and
radiative envelopes; the same point is made at various places in the text,
but particularly in section 4 (e.g. page 5, lines 154 to 177).

This is certainly an unusual way of putting things, and I believe it gives
readers the wrong impression. I fully agree that in practice, the convective cores of massive stars are very close to adiabatic, so in that sense -- if all you cared about was the entropy gradient -- you could get away with just setting nabla = nabla_ad, rather than going through the usual MLT solve.  But this is not, in practice, what is done in the 1D evolution codes with which I have any acquaintance. MLT is still used in these regions -- i.e., there's no flag in the code to say "assume it's just exactly adiabatic instead."  Of course, the values of superadiabaticity predicted by MLT in the core of a massive star are very small (i.e., nabla is very close to nabla_ad), just as they are in (e.g.) the deeper parts of the solar convection zone --but the transition between these "efficient" layers and relatively less-efficient ones near the surface is a smooth one. More conceptually: if anything, I would have said that it's in these deep, efficient regions where MLT is arguably fine. It's almost certainly a much poorer approximation in regions with high superadiabaticity, even if -- annoyingly -- that's precisely where we would really like a good theory!

Also, as a practical matter, even in regions where the superadiabaticity is
very small, MLT is still used for (e.g.) convective velocity estimates by
some authors, so again I think it is misleading to say the theory is inapplicable in these regions. The bottom line is that I believe this part of the discussion, and the associated figure, really needs to be substantially reworked.

Apart from that, I have very few other criticisms. I have listed these
below (including very minor things, like typos) in the order they appear in
the text.

Page 1, line 4 "review of attempts" -> "review attempts" ?

line 34-40ish: I found this description a little confusing; please consider
rewording.  What exactly does "denature" mean in this context, for example?
You might also want to note that hot parcels are (for typical equations of
state) under-dense, which is why they feel a buoyancy force that makes them rise in the first place. 

lines 102-107.  Momentum conservation isn't really a separate principle or
equation from hydrostatic equilibrium; HEQ is just the "v=0" solution to
the momentum equation.  Consider rephrasing this.

line 128ish: Might consider adding a few references here on the
Schwarzschild vs Ledoux criteria, where they are applicable, etc.

equation 4: I think there is a typo here -- please check.  (Specifically I
think you are missing a factor of T.) Also, the definition of Hp later in
the paragraph is a bit odd (this one would be dimensionless, and not equivalent to the definition later in the paper).

lines 152-177: see my more substantive criticisms above

line 190ish: Although I agree with the broad spirit of this paragraph, I'd
consider reworking slightly to avoid calling MLT a "diffusive
approximation."  I guess it is just semantics, but to me this implies that
it would obey a diffusion equation (dT/dt ~ second order spatial
derivatives), which isn't really what MLT does.  (As a practical matter, models of MLT in which the heat really does diffuse out of the parcel during its rise or descent are essentially indistinguishable from those in which the parcel simply gives up its heat at a specified height.)

line 223: not sure what you mean by "disrupt the density gradient"?

caption to figure 2: again, I find the use of "denaturing" odd here

line 255: another place where the "MLT is not applicable" idea creeps
in.

More general comment on section 6: This section doesn't quite deliver on
what it seemed to promise at the beginning ("a standard derivation for the amount of flux carried by convection").  I would either reword slightly, or else
finish off your derivation by getting to an actual formula for the flux.

Figure 4: perhaps I'm being dense, but what the fainter tracks in the background showing me?

line 604: "age as several" -> "age of several"?

In summary, there are a number of useful elements in this (well-written and comprehensive) paper and I believe it should be published.  But I think the discussion of where MLT is applicable needs to be altered before acceptance.

Author Response

Please see the attachment

Author Response File: Author Response.pdf