On partial derivatives

People have been arguing about partial derivatives (ATTP, Stoat, Lucia). It arises from a series of posts by David Evans. He is someone who has a tendency to find that climate science is all wrong, and he has discovered the right way. See Stoat for the usual unvarnished unbiased account. Anyway, DE has been saying that there is some fundamental issue with partial derivatives. This can resonate, because a lot of people, like DE, do not understand them.

I don't want to spend much time on DE's whole series. The reason is that, as noted by many, he creates hopeless confusion about the actual models he is talking about. He expounds the "basic model" of climate science, with no reference to a location where the reader can find out who advances such a model or what they say about it. It is a straw man. It may well be that it is a reasonable model. That seems to be his defence. But there is no use setting up a model, justifying it as reasonable, then criticising it for flaws, unless you do relate it to what someone else is saying. And of course, his sympathetic readers think he's talking about GCMs. When challenged on this, he just says that GCM's inherit the same faulty structure, or some such. With no justification. He actually writes nothing on how a real GCM works, and I don't think he knows.

So I'll focus on the partial derivatives issue, which has attracted discussion. Episode 4, is headlined Error 1: partial derivatives. His wife says, in the intro:

"The big problem here is that a model built on the misuse of a basic maths technique that cannot be tested, should not ever, as in never, be described as 95% certain. Resting a theory on unverifiable and hypothetical quantities is asking for trouble. "

Sounds bad, and was duly written up in ominous fashion by WUWT and Bishop Hill, and even echoed in the Murdoch press. The main text says:

The partial derivatives of dependent variables are strictly hypothetical and not empirically verifiable

He expands:

When a quantity depends on dependent variables (variables that depend on or affect one another), a partial derivative of the quantity “has no definite meaning” (from Auroux 2010, who gives a worked example), because of ambiguity over which variables are truly held constant and which change because they depend on the variable allowed to change.

So even if a mathematical expression for the net TOA downward flux G as a function of surface temperature and the other climate variables somehow existed, and a technical application of the partial differentiation rules produced something, we would not be sure what that something was — so it would be of little use in a model, let alone for determining something as vital as climate sensitivity.<

So I looked up Auroux. The story is here. DE has just taken an elementary introduction, which pointed out the ambiguity of the initial notation and explained what more was required (a suffix) to specify properly, and assumed, because he did not read to the bottom of the page, that it was describing an inadequacy of the PD concept.

Multivariate Calculus

Partial derivatives can seem confusing because they mix the calculus treatment of non-linearity with dependent and independent variables. But there is an essential simplification:

• The calculus part simply says that locally, non-linear functions can be approxiumated as linear. The considerations are basically the same with one variable or many.
• So all the issues of dependence, chain rule etc are present equally in the approximating linear systems, and you can sort them out there

That is a great relief, because partial derivatives are messy to write down on a blog, so I won't try.

Dependence

I'll use a simplified version of his radiative balance example, with G as nett TOA flux, here taken to depend on T, CO2 and H2O. The gas quantities are short for partial pressure, and T is (at least for DE) surface temperature. So, linearized for small perturbations,

G = a1*T + a2*CO2 + a3*H2O

Now there may be dependencies, but that is a stand-alone equation. It expresses how G depends on those measurable quantities. It is true that the measured H2O may depend on T, but you don't need to know that. In fact, maybe sometimes the two are linked, sometimes not. If you put a pool cover over the oceans, the dependence might change, but the equation which expresses radiative balance would not.

If you do want to add a dependence relation

H2O = a4*T

then this is simply an extra equation in your system, and you can use it to reduce the number of variables:

G = (a1+a3*a4)*T + a2*CO2

And since at equilibrium you may want to say G=0, then

T =- a2*CO2/(a1+a3*a4)

expresses the algebra of feedback. But this is just standard linear systems. It doesn't say anything about the validity or otherwise of partial derivatives.