måndag 5 november 2012

A stepwise axiomatic approach

This, I guess, is in part a response to Postma's latest article which I find well written and addresses several key questions concerning the GE. In particular I like the approach which starts off with simple reasoning and simple equations, adding details when necessary, rather than a heavy first principles approach from the very start. I will not go into direct polemic, instead I will try to summarize my own thoughts in what I call a "stepwise axiomatic approach". The present version of it is probably not the most succinct, but I think that this or similar approaches could be helpful in disentangling disagreements, to see where the viewpoints really depart.

Consider the following set of assumptions (axioms), each of which may be subject to amendments or cancellation. (Some are inconsistent with each other and are therefore numbered.)

A.  For the atmosphere the ideal gas law T = PV/nR holds true with an accuracy sufficient for our considerations. 

B. Thermometers measure temperature with an accuracy sufficient for our considerations.

C. There is always a (net) heat flow from higher to lower temperature

D1. The incoming sunlight is a heat flow

D2. The incoming sunlight is not a heat flow but should instead be considered as an energy source (thereby functionally equivalent to a radiator, burning of fuel. radioactive decay and so on)

Axiom C is one formulation of the second law of thermodynamics. Obviously we have to choose between D1 and D2 so lets start with D1.

Consider the set of assumptions (A, B, C, D1). Here we encounter a problem. Since the upper atmosphere is colder than the surface (B) assumption C says that there must be a net heat flow between the surface and the upper atmosphere. However, D says that there is no such net heat flow. (We have neglected the fact that the sunlight is asymmetrically distributed, but adding the asymmetry doesn't help, think about it.) So how do we resolve this? A significant number of people resolve this by an amendment to the second law of thermodynamics, stating that in a gravity field there need not be a net heat flow from higher to lower temperature. In the end they usually arrive at the following formula for the lapse rate:

The details of the reasoning behind this, and its problems, can not be covered here, instead I refer to other articles (for example on this blog). An alternative is to abandon D1 in favour of D2.

Consider the set of assumptions (A,B,C,D2).  In this case the lapse rate simply pops out in the form of the heat equation:

where F is the solar forcing. If this simple observation has evaded you until now don't be ashamed. It took me months. The heat equation I think provides a powerful tool to analyze certain topics infected with misunderstandings. I believe for example that back-radiation (in the most general sense) is neatly incorporated in the left hand side of the above equation, see for example my post "The heat equation revisited". It is not necessary to incorporate back-radiation for example in the following way:

where B is a fictitious extra forcing term supposedly arising from back-radiation. Moreover, if we test one of the predictions of the GE theory, namely that an atmosphere without greenhouse gases becomes isothermal. Given the set of assumptions (A,B,C,D2) that would correspond to an atmosphere with infinite thermal conductivity. Now ask yourself the question: Does dry air have infinite thermal conductivity?

It should also be noted that the formula for the adiabatic lapse rate may become useful also under the set of assumptions (A,B,C,D2) but then in the context of the limit it puts on convection, see previous posts. 

In summary, I think that an approach like this can resolve many misunderstandings. Try it! 

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