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”. A discussion is presented by Essex, McKitrick, and Andresen (EMA) below.

“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. All variables should be concurrent. Using average values does not produce an accurate “average enthalpy”, though a properly constructed average q might.

Enthalpy can be converted 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 often does not include the humidity data consistent with the temperature data needed to calculate atmospheric 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.

For a more formal discussion of wet enthalpy,

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

**Equivalent Temperature**

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

Largely because of the latent heat of water, the equivalent temperature can be much higher than thermometer temperature. The following graph is for a variety of cities and times of day using data from https://www.wunderground.com/history/, and uses the excellent formulas of Massen. The upper branch includes humid Key West FL. The lower branch includes the relatively dry Denver, CO.

**Errors in Global Temperature Anomalies**

The word “error” implies some standard from which the observed statistic differs.

It is common in statistics to use error as a measure of the extent to which a particular statistic varies from its expected value. In engineering, error means deviation from specification or design. So the extent of error depends on the intended use.

For purposes of estimating ground radiation, unmolested temperature is best but averaging of multiple sites should be weighted as T^4. For estimating global heat content enthalpy+wind energy+potential energy should be used.,

If the purpose of calculating global temperature anomalies is to observe some heating effect from alteration of radiative transport by CO2, and some heat is diverted to evaporation of water, then the observed temperatures do not accurately represent the heat effect. Using Massen’s equations (below) for enthalpy adjusted for relative humidity(Rh) and differentiating wrt measured temperature and humidity, we find that the delta in measured temperature from an delta in relative humidity evaluated at 50%Rh and 15 oC(59 oF) is:

delta observed Temp/ delta %Rh = (dh/d%Rh)/(dh/dTemp) = 0.83 degc/%Rh

This means that, if the actual Rh is 1% higher than some reference standard, the observed temperature will be lower than its energy equivalent by 0.83oC since heat is locked up in creating water vapor.

Since dTemp/d%Rh varies with both humidity and temperature, an estimate of the degree to which humidity changes at constant enthalpy requires consideration of all relevant variables. All relations are highly non-linear and there is no expectation that errors will “average out”.

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.

**References**

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

EMA: 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