In discussions of climate, the use of a statistic “temperature anomaly” has become ubiquitous. Further, the statistic has taken on the role of temperature in the minds of many people and in many NOAA press releases and web sites. But “temperature anomaly” is not a temperature in any technical sense of the word “temperature”.

“Temperature” is only defined rigorously for a system at equilibrium, so the action of adding different temperatures by definition denies the validity of all but, at most, one.

If an effect of CO2 is to somehow change the water content of air or to change wind speeds, averaging temperatures will not show the correct change in atmospheric heat content.

The measure of heat energy for a fluid is enthalpy. In joules per kilogram, the expression for total specific energy, enthalpy + potential + kinetic is

h = (C_{p} * T – .026) + q * (L(T) + 1.84 * T) + g * Z + V^{2}/2

Cp is heat capacity, T is temperature in Celsius, q is specific humidity in kg H20/kg dry air, g is gravity, L(T) is latent heat of water ~2501 kJ/kg , Z is altitude, V is wind speed.

This can be converted back to an equivalent temperature with or without the potential and kinetic energy terms. Adding the wind energy seems to make a “wet stagnation” temperature.

T equivalent = h /C_{p}

An interesting study is https://pielkeclimatesci.files.wordpress.com/2009/10/r-290.pdf

Pielke shows that the real world difference between equivalent temperature h/Cp and thermometer temperature can be tens of degrees Celsius.

Classical climate data does not include what is necessary to calculate energy to an accuracy better than several percent. This inaccuracy is greater than effects attributable to CO2. Hurricane velocity winds add single degrees of effective temperature, but modest winds can add tenths of degrees. Evaporation or condensation of water can change temperature by tens of degrees.

I speculate that part of the reason for the never ending adjustment of historical temperatures is an attempt to compensate for the inaccuracies inherent in temperature only estimates of energy.

For a more formal discussion of wet enthalpy,

http://www.inscc.utah.edu/~tgarrett/5130/Topics_files/Thermodynamics%20Notes.pdf

**Regional Equivalent Temperatures**

Since enthalpy is an extensive property, enthalpies of different systems can be added. To construct a regional or global equivalent temperature from a variety of enthalpies, we need two sums: the enthalpies times their mass weight factor, and the weight factors.

T equivalent = (∑ h_{i} ρ / ∑ ρ )/ Cp

where ρ is the mass density and h_{i} is the enthalpy at a point i in the atmosphere.

While we do not have mass densities, we do have the numbers to calculate it:

ρ = P / (R_{M} T) where Rm is R/ effective molecular weight.

Effective molecular weight is Q Mair + (1-Q) Mwater where Mair and Mwater are molecular weights, Q is the ratio of water density to air density, and P is pressure at the point of measure of temperature T. Note that meteorological pressures are often corrected to sea level and need to be corrected back to the relevant altitude.

T equivalent = ∑ h_{i} P / (R_{M} T) / ( ∑ P / (R_{M} T) ) / Cp

P, T, and Rm will all vary from point to point as well as h.

The following URLs cover lots of issues.

https://pielkeclimatesci.wordpress.com/2010/07/22/guest-post-calculating-moist-enthalpy-from-usual-meteorological-measurements-by-francis-massen/

http://meteo.lcd.lu/papers/Enthalpy/calculating_moist_enthalpy_revisited.pdf

http://www.uoguelph.ca/~rmckitri/research/globaltemp/GlobTemp.JNET.pdf

Parts of this post were presented at https://wattsupwiththat.com/2017/02/22/through-the-looking-glass-with-nasa-giss/#comment-2435373 and https://wattsupwiththat.com/2017/04/23/the-meaning-and-utility-of-averages-as-it-applies-to-climate/#comment-2484011