Summary of discussion at Climate Etc.

Anastassia Makarieva | February 5, 2013 at 12:01 am | Reply
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I’d like to provide my summary of the discussion in that part that concerns our exchange with Nick Stokes and how it relates to paper content and the broader picture of our work.

I. The “0=1” argument.
It was repeated by Nick here, here, here and elsewhere.
This argument presumes that our system of equations (32)-(34) is mathematically inconsistent (contains a “mathematical fallacy”). This argument is false. The system is mathematically consistent. Those willing to consult an independent opinion please see the comment of Tomas Milanovic.

When challenged to derive the contradiction “1=0” explicitly from (32)-(34) Nick made two elementary algebraic errors (see here and here). At one place he seemingly implicitly conceded that there is no mathematical contradiction (but still made a later effort to derive one).

My take on the “mathematical fallacy” issue is here, here and here.

The argument “1=0” profoundly misinforms the reader about the actual issues that are worthy of discussion.

II. The derivation of Eq. (34).

It should be made clear that Eq. (34) in the paper was not derived from any other known equation. It was formulated based on several key physical propositions. These propositions are summarized here. They deserve a thoughtful physical consideration by anyone willing to understand the underlying physics.

1. The scenario of discussing these propositions developed as I suggested in an earlier comment here:

So, I suggest that you should instead insist that our paper contains a physical fallacy. … You should say that the assumptions that we involve to justify Eq. (34) are not convincing or anything obvious to you. To this I will respond that to us those physical assumptions do seem plausible…

In line with this, Nick once called our propositions arm-waving, another one that they are of “no relevance”. His most specific claim was

The formula is the same in all cases. Therefore it does not depend on gravity.

When asked if any formula that contains something depending on gravity must explicitly contain g, Nick changed topic. To this I responded that “to see no relevance” and “find an error” is not the same.

2. Additionally, Nick produced an alternative derivation of S (the one also presented by Dr. Held in his review with a reference to Nick). This derivation was independently reproduced by Cees de Valk. See here for a relevant thread.

Note that Eq. (34) is S = wN∂γ/∂z, where γ = pv/p is the relative pressure of water vapor and N is total air molar density. The alternative derivation of Nick et al. produces what we can call SNick = wNd∂γd/∂z, where Ndis dry air molar density and γd = pv/pd is the ratio of water vapor to dry air pressure. Since pv << p, pd and p are very close, so S (34) and SNick (Sd in our paper) differ by a small magnitude.

Now important: we do not have any objections as to from what principles SNick was derived. I accept a priori that it can be correct (which would mean that our S (34) is wrong). That would be fine. But, as everybody agrees, when put into the continuity equations (32)-(33), SNick produces a non-sensical result:
u∂N/∂x = 0. (1C)
This result is mathematically valid (no contradiction in the equations), but it implies that generally, under any possible circumstances on an isothermal surface, winds cannot have a velocity component parallel to the pressure gradient ∂p/∂x = RT∂N/∂x.
So this result is invalidated by observational evidence (e.g. radial convergence in hurricanes). By inference, whatever were the physical grounds from which SNick was derived, they were incorrect.

In contrast, our S (34), despite different from SNick by a small relative magnitude, when fed into the system (32)-(33) produces instead of the above equation
-u∂N/∂x = S. (2C)
It is a classical case where a small detail matters (“devil in details” etc.) To understand why it matters requires a deeper insight into the underlying physics, not merely manipulating with the continuity equations.

3. There is an alternative physical derivation of S (34) as presented in the blog and here based on energy consideration. This complementary view, although explicitly present in the paper, might not have been sufficiently clearly articulated. This alternative derivation does not involve any small factors in deriving (2C) above. These considerations provide independent support for the physical arguments used in deriving (34). None of the commentators here, Nick included, has ever commented on that.

In the meantime, we now think that this second view on how (2C) (the main result) is obtained is more easy to understand from the physical viewpoint. We recommend people who just have their first look on the subject to evaluate it first.

4. The argument about S = CNv (see here, here, and here) is confusing.

Otherwise this argument is a distraction. When challenged during the ACPD discussion to better explain the physical foundations of Eq. (34), we showed that Eq. (34) can be interpreted as S = wkvNv, where kv is the degree to which water vapor partial pressure deviates from hydrostatic equilibrium. This proportionality is physically consistent with S being a first-order reaction in Nv, but certainly S = wkvNv cannot be formally derived from chemical kinetics.

From the empirical viewpoint, since C is not a constant (it does not depend on Nv, i.e. it is a constant with respect to Nv only), the formula does not presume rain from dry air. Since to appreciate this linearity argument requires an understanding of what kv is, this argument is of little help for those who want to get a first idea of the physics behind (34). Again, I recommend this account.

III. Issues of interest

1. Since S (34) is not formally derived from any pre-existing equations but formulated based on several plausible physical propositions (two independent sets of them), it cannot (and has not been) refuted mathematically. It has not been shown to be in conflict with any physical law either. The only way to falsify Eq. (34) consists in checking the result it yields
-u∂N/∂x = S. (2C)
against empirical evidence, as I outlined here. Nick made a few attempts, but they have so far been inconclusive.

I emphasize (2C) is an extraordinarily strong statement (see Eq. (4) in the post which is the same). It predicts that where condensation is absent, winds cannot blow along the pressure gradient or that the pressure gradient must be absent. It also predicts the reverse (lack of condensation where u∇p = 0), but the latter prediction is less informative as it does not specify the scale at which this lack of condensation should be manifested (it can be very narrow).

The meaning of the differential form of 2C (see Eq. 4 in the post) is that all potential energy released from condensation is locally converted into the power of the large-scale horizontal pressure gradient force u∇p. This is very strong. At what scale it is actually true remains to be seen. It will also help to discriminate between condensation-induced dynamics and other mechanisms at work in the atmosphere (e.g. forced convection can be different). As a bottom line, the integral form of (2C) has already produced meaningful results, so it can serve as an integral limitation on the dynamic power of circulation.

For us the main point is that our theory (unlike the existing models) yields empirically falsifiable predictions. It is a working theoretical concept for a moist atmosphere.

2. The last but one section in our blog is very important for future theoretical analyses.

3. How the theory can be empirically tested is outlined here and here in response to manacker (Max). It is said quite enough for anyone who got a basic physical idea to publish original papers based on observational analysis. There seem to be lots of relevant data around.

IV. Miscellaneous

1. My personal view on why the paper was accepted.

2. I would like to express our gratitude to Nick Stokes for his persistent attention to our work. My personal view: For people who like us have a clear picture of underlying physics Nick’s comments can provide additional details and angles. For those people who do not have a clear physical picture and make their first acquaintance with the idea, Nick’s comments are paralyzing and preventing any further understanding. Cannot be recommended for students.

Thank you very much for this exciting discussion.

Anastassia

3 responses to “Summary of discussion at Climate Etc.”

1. Pekka Pirilä

Anastassia,

This is my reply at Climate etc.

1) You have not presented any real independent derivation of equation (34). You have only some arm waving to support that.

2) The only credible basis for ending up with that equation is that it’s picked from some derivation of an approximate continuity equation. It’s exactly such an equation and you have not presented any alternative explanation for its form.

3) Whatever the origin of the equation of (34), combining it with equations (32) and (33) leads to series of results. Many of these results are so totally against real world situation that the set of equations (32), (33), and (34) is totally refuted by observation. The most obvious case is perhaps that it leads to the result that condensation is fully controlled by the horizontal pressure gradient. Without such a gradient there could not be any rain. That’s just absolutely false result at the extreme. (This statement about the prediction on rain is confirmed by equations given by you in the above comment, it’s not my invention. Just notice that S is the source of rain.)

Thus you have no derivation for (34) and (34) leads to totally false results.

The only possible conclusion is that the equation (34) is totally wrong when it is presented as a third equation with equations (32) and (33).

2. 2s3c

This is my response to
Pat Cassen | February 12, 2013 at 1:36 pm | Reply

Pat, thank you for your clarifications.

The point is that your expression for ∂p/∂x is incompatible with that derived from the exact thermodynamic constraints, i.e., the first equation in this comment. That this is the case is readily apparent in the limit γ → 0, for which your pressure gradient vanishes, but that given in the first equation goes to a ‘dry’ limit.
From these considerations, I conclude that your prescription for S, whether derived from an independent hypothesis or an approximation, is invalid.

I still think I fully addressed your concern in my first reply. Your expression for u∂p/∂x is derived from the first law of thermodynamics, Clausius-Clapeyron law and the ideal gas law, plus assuming horizontal isothermy ∂T/∂x = 0. As I said before, the first law of thermodynamics, being an equation of equlibrium thermodynamics, is inexact when applied to real-time dynamic systems. It errs precisely about the effect that we are trying to estimate — the rate of generation of kinetic energy. This rate is neglected in the first law — any work performed on the gas goes to increase its internal energy, not kinetic energy. Therefore, any conclusion about dynamics inferred from the 1st law must be different from the correct one. So the fact that you derive something from the 1st law that differs from our result does not in any way undermine our conclusions.

That is what I said before. But I can go further and be more specific. Your equation for u∂p/∂x is not merely derived from the 1st law, it is derived from the 1st law for an adiabatic case, i.e. for dQ = 0. Thus you somehow presume that the air changes pressure adiabatically in both x and z directions, but temperature changes in only one direction. This is unphysical, but I leave it to you to analyze why it is so. The main point is that horizontal isothermy in the real world (e.g. in Hadley cell, where it is a very good approximation as admitted even by Dr. Held) is certainly not adiabatic. It is ensured by efficient horizontal mixing that ensures a poleward flux of heat. I.e. for every air parcel dQ is not zero.

Therefore, whatever one derives from the 1st law assuming that the motion is adiabatic is incorrect for two reasons (a) the motion is known not to be adiabatic and (b) we are not considering a system in equilibrium. Therefore, the difference between your u. ∂p/∂x and ours does not in any way invalidate the latter.

I hope this might be is clear now, but I’d welcome further comments from you. Regarding the last word, as you might have noticed I am not keen about having the last word. E.g. until Pekka explicitly complained that I was boycotting him, I had not been interfering with his numerous last words about our work. So if you are interested in my feedback, it’d be a pleasure for me to provide one. If not, it is my pleasure to leave the last word to you whatever it is. In any case, thanks again for your interest.

3. solvingtornadoes

Why Water is Weird
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