By Charles T Blaisdell PhD ChE

Abstract

The mathematical world of climate change is dominated by radiation, W/m^2, shortwave, SW, and longwave, LW.  Many diagrams show radiation arrows going up and down all entering or exiting the top of the atmosphere, TOA, see Figure 1 for an example.  The NOAA diagram (2) show SW in is equal to LW out.  If SW in does not equal LW out we have climate change.  The 20 year separate studies by Loeb (5) and Dubal (4) both show that the SW in and LW out are not equal over the 20 years.  Leaving the questions: when did the departure from equality start, how long has it been going, what caused it, when will it stop, and is it real?

Figure 1 NOAA The Earth – Atmosphere Energy Balance, (2) (all # are in % of sun’s irradiation)

Irradiance is radiation over time, W/m^2-yr. That is: the average radiation over one year.  Radiation is not energy.  Irradiance is energy.  This essay will show that the earth’s Enthalpy (energy) can be related to LW irradiance out.  The Loeb and Dubal data only covers 20 years.   The Physical Science Laboratory, PSL, cover data since 1948 from which enthalpy can be calculated.

This essay will calculate the total energy of all the annual global sources of enthalpy since 1948 from NOAA’s PSL (1) data.  The enthalpy, kJ/m^2 – yr, results were converted to irradiance, W/m^2-yr.  The 75 year plot of the calculated irradiance shows a good correlation to Loeb and Dubal results and suggests that the departure in SW and LW radiation started about 1975 (+/- 10).  No suggestion of CO2 radiative forcing was found (no decreasing LW irradiance).  The results show that atmospheric water has a small contribution to total irradiance but a significant contribution to the change in irradiance.

Methods and calculations

The diagram above shows that the total outgoing LW (TOA) irradiance is about 70% ( 240 W/m^2) of the sun’s total SW irradiance and is equal to the non-reflected incoming SW irradiance.   This outgoing LW (TOA) irradiance can be divided into 4 sources that account for the 240 W/m^2:

En(total) = En(1) + En(2) + En(3) + En(4) = H * En(total) = 240 W/m^2 = Irr(total)  Eq(1)

Where En(num) is the annual enthalpy of one of the 4 TOA sources of enthalpy below.

Conversion factor, H,  for kJ/m^2 – yr to W/m^2 – yr         Eq(2)

D = 0.278  W-hr/kJ  conversion factor

E = 8760 hr/year   

G = 5.15/E+14 m^2  surface area of earth

H = D * E / G  W/m^2/kJ for one year                                      Eq(3)

H = 6.16E-20 W/m^2/kJ – yr

• The atmosphere’s temperature (no water) TOA Enthalpy.  The earth’s atmosphere is a complex zone where on an annual basis many energy forces come to equilibrium and the surviving irradiation makes it to TOA.  The enthalpy can be calculated from the temperature, mass, and absolute heat capacity of an atmospheric profile and converted to irradiation.  NOAA’s Physical Science Laboratory (1), PSL, provides annual temperature data for 6 atmospheric pressures (zones).  (The temperatures were converted from anomalies to actual if needed.)

En(1) = (T(z1) + T(z2))/2 *) * Cs(air) +  Cp(air) ) * M(z1) = H * En(1) = Irr      Eq(4)

(repeated for each pressure zone and summed)

Where En is annual enthalpy, Irr is annual Irradiance , T is temperature in kelvin, M is mass, Cs is the specific heat, and Cp is absolute heat capacity, and the number in () is one of the 6 atmospheric pressure zones.  The enthalpy for all 6 zones was added and converted to annual irradiance for atmospheric temperature. Note that the Cp is the absolute heat capacity of air from 0’K not the relative heat capacity at 0’C.

These 6 zones are far from ideal, but it is all we have.  Let’s see how it works out.  This analysis covers about 99% of the atmosphere’s mass.  (Ending in the middle of the stratosphere at a temperature of -58.5’C , assumed to be constant with time.)  There is no accounting for the remaining mass of the stratosphere and mesosphere were ozone in absorbing UV radiation.

• Atmospheric water TOA Enthalpy.  The water data in PSL was treated in a similar manner with PSL data for specific humidity

En(2) = ( (SH(z1)+SH(z2))/2 )*Cp(water)+Cs(water)*(T(z1)+T(z1)/2 ) )*M(z1) =

H*En(2)=Irr(1)         Eq(5)

(repeated for each pressure zone and summed)

Where SH is specific humidity, Cs(water) is the specific heat of water, and Cp(water) is the absolute heat capacity of water relative to 0’K not the relative heat capacity at 0’C.

•  Ocean’s TOA enthalpy.  The earth’s oceans where the majority sun’s SW irradiance (that is not reflected) is absorbed and readmitted to the atmosphere as water vapor, returned as rain, convection energy, and a small amount of LW irradiance makes to the TOA.  This small TOA LW irradiance is a function to the ocean’s surface thickness (assume 1 meter), effective area of SST measurement, heat capacity of water, and average surface sea temperature, SST, of the ocean’s surface.  The area of ocean involved in this source of irradiation will be changed to match Dubal data at year 2006.

En(3) = T(SST) * surface thickness * surface area * Cp(water)       Eq(6)

• Land’s TOA enthalpy.  The land where the sun’s SW irradiance (that is not reflected) is absorbed and readmitted to the atmosphere similar to the ocean’s where the bulk of irradiation goes to the atmosphere and a small amount makes it to TOA.  This small TOA LW irradiance is a function of thickness (assume 1 meter), effective area (same % as ocean’s), heat capacity of land, and lower atmosphere temperature.

En(4) = T(1) * surface thickness * surface area * Cp(land)         Eq(7)

Results

(attached is the excel work sheet, here)

(If this mathematical exercise has been done by someone else please let me know in the comments.)

              The assumptions of 1 meter surface thickness for both ocean and land and fit to the Dubal 2006  data point resulted in a reasonable 68% area where SST data average data for ocean was taken, land was set at the same 68%.  The 75 year plot of the calculated LW outgoing irradiance shows a detectable increase in irradiance after about 1975 and a flat trend from 1948 to 1975, see Figure 2.

When the Dubal data is overlaid over the calculated irradiance data’s slope the fit to Dubal data’s slope is reasonable, Figure 3. (The Dubal 2006 data was used to adjust the calculated irradiance to one point: therefore, only the slope of the data is significant.)   The calculated irradiance seems to have a lower standard deviation than the CERES data.

Temperature dominates the enthalpy and irradiance calculation thus Figure 2 is about the same shape as a temperature plot.  The calculation does come out very close to measured irradiance values which could be an indication of the accuracy of the PSL data.

The 2023 distribution of the earth’s irradiance shown in Figure 4 indicates that the atmosphere’s temperature is the main accounting of irradiance at TOA.  Water is a very small contributor to the total irradiance.

Looking at the change in irradiance from 1975 to 2022 the atmospheric temperature and water show a significant contribution to the change with water changing the most, Figure 5.  Since temperature and water are related, this observation suggests that the change is related to a change in the earth’s water cycle. 

If CO2 (or other greenhouse gases) where involved in the irradiance vs time graph there would be a decrease in irradiance with time, this was not observed

Of interest is the atmospheric profile of irradiance to TOA from each PSL zone.  Figure 6 shows the lower altitude zones with their higher mass have the lowest irradiation to TOA while the higher altitude zones with lower mass have higher irradiation to TOA.  This is the expected result, giving confidence to this analysis.

Discussion

This mathematical exercise was done to see how close calculated irradiance would come to measured irradiance and to prove that climate models using enthalpy instead of radiation or irradiation were valid.  (The CRGW (3) model is a enthalpy based model).  With the non-ideal PSL data for this task and the assumption this exercise came very close to the measured annual outgoing LW irradiation.  This gives some validity to the annual enthalpy correlations to cloud fraction and vapor pressure deficit, VPD.  used in the CRGW model. 

Annual enthalpy is pseudo annual LW irradiation.

              Bibliography

1. NOAA Physical Science Laboratory  web Monthly Mean Timeseries: NOAA Physical Sciences Laboratory