Wednesday, October 7, 2015

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.


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.


  1. I'm (just) an electrical engineer from way back. It's been 40 years or so since I had to mess around with differential equations. But I'm still pretty handy with the ol' algebra since I use it extensively in my current schtick as a programmer. And so it was that I caught out David Evens a few years ago trying to put one over on his partner Jo Nova's largely uncritical audience (though, of course they like to imagine themselves as being critical). Now, of course, I may be completely wrong about this. But Nick will tell me if I am :-) H/T to Sou at Hot Whopper for first bringing this to our attention. Apologies in advance for the rather large cut n' paste. But it is my original work. Here goes...

    In the article, Sou refers to Gavin Schmidt as saying:

    Changing a unit to have a small sounding number doesn't actually change anything; neither the significance nor the accuracy. .... – gavin

    This reminds me so much of the time I caught 'rocket scientist' David Evans (Jo Nova's partner) doing something completely nonsensical in an effort to make a number look smaller. Towards the end of this article:

    Evans quotes a number from Levitus et. al. 2009:

    See NOAA's PDF, table T1 (on page 14): heat content change for the 0 – 700m layer of the world's ocean of 15.913 * 10^22 Joules corresponds to a change in mean temperature of 0.168 deg C, so an increase of 10^22 Joules in 0-700m of the world's oceans corresponds to a temperature rise of 0.168/15.913 = 0.0106 deg C.

    Why the heck would he take a number representing the volume mean ocean temperature increase from 1955 - 2008, and divide it by the number of joules (minus the 10^22 part)?! A volume mean temp increase in degrees C is what it is, an increase in temperature. Because Levitus refers to the .168 figure as: "TChange – total change in volume mean temperature [ºC]". Is there some difference between 'heat content' (in deg C) and 'temperature' (in deg C) that I am unaware of? Or is Evans just doing what tamino likes to refer to as 'mathturbation', in order to make a big number look less scary to the frightened AGW deniers?

    Anyway, for the lolz, I decided to try to replicate the Levitus et. al results for myself, just to make sure I understood it. I came pretty close on the first shot (and the numbers are scary. Look at the change in just 2 years!):

    Volume (1.37 * 10^9 km^3) and mean depth (4117m) of world oceans:

    So, percent of that in 0 - 700m layer is:

    700/4117 = 17%

    Then: .17 x 1.37 * 10^9 = 232,900,000 km^3

    1 km^3 of seawater weighs = 1,035,000,000,000 kilograms (seawater is denser than fresh water. Source:

    So weight of 0 - 700m layer is: 232,900,000 km^3 * 1,035,000,000,000 kg/km^3 ~= 2.41 x 10^20 kg

    Specific heat of seawater = 3900 j/kg (just for the lolz, specific heat of pure water = 4184 j/kg) is the amount of energy required to raise 1 kg of seawater 1 deg C (source:

    So finally... change in volume mean ocean temp for 0 - 700m layer, 1955 - 2008 is:

    15.913 x 10^22 j/(2.41 x 10^20 kg * 3900 j/kg) ~= 0.169 C (Levitus et. al. 2009 says: 0.168 C)

    Revised calculation for 1955 - 2010 from Levitus et. al. 2012:

    16.7 X 10^22 j/(2.41 x 10^20 kg * 3900 j/kg) ~= 0.177 C (Levitus et. al. 2012 says: 0.18 C)

  2. metzo,
    I think he's just working out a conversion factor, as in his table. Like saying, if I did 400 miles in 8 hours, I'm can do 50 miles per hour. If he'd wanted a really small number, he could have worked out ° per Joule :)

    I think you are right with the Levitus calc.

    The problem with all that is the non-uniformity of heating. The temperature gradient is steep at the surface and then diminishes. It doesn't suddenly change at 700m. I think 10^22 J would produce a much bigger rise in SST, and that would last a very long time. The heat will never be distributed uniformly down to 700 m. By the time (many years) 700 m has significantly warmed, al lot of heat will have gone on to lower depths. It may help with an order of magnitude calc, but no more.

    1. Thanks Nick. OK, but if he's just doing a conversion, his units don't make any sense:

      0.168C volume mean increase in ocean temp from 1955 - 2008 / 15.913 x 10 ^ 22j = 0.0106C per 10 ^ 22j. What kind of units are those?!

      So he's just doing it to make the number look less scary to the LOL WHUT?! innumerati :-)

  3. Models are certainly black boxes for me, but it's possible to test the outcome and see if it makes sense...
    Since you are discussing ocean heat content, models, etc. I have actually performed the ultimate test of models vs observations, and it looks like I have found Trenberths missing heat. Travesty no more... :-)

    To be precise, the heat accumulation during the ARGO-era is 13.4*10^22 J suggested by observations and 13,3 suggested by all models average. It is quite nice that Argo observations replicate the seasonal behaviour suggested by the models, at least during the recent years. Hence,I expect a downturn in heat content for the next quarter July-Sept. El Nino could also drain some of the ocean heat to the atmosphere and space. To be continued..

  4. Off topic...

    Nick Stokes's web page writes:
    "NCEP/NCAR reanalysis surface temp anomaly area weighted global average made 2015-10-07
    Recent days global anomaly
    Oct 4 0.697"


  5. I presume that is the record?

    1. I have put up a new post on this. I have only calculated daily averages back to start 2014 - I'll extend. I've posted that data. It's about 0.2°C higher than anything in 2014.

    2. Stuff that's in so far, AMO - major tick upward.

      ONI went from 1.2 to 1.5.

  6. Thanks! Typo: "calsulus"