The temperature range of the 20th century spans about 0.74 C. Of this, about 40% or 0.3 C
has excellent correlation with the sunspot time-integral. An equation has been derived
that calculates average global temperature based on the physical phenomena involved.
With inputs of accepted measurements (source web links are given) from government
agencies, it calculates the average global temperatures (agt) since 1895 with 88.4%
accuracy (87.9% if CO2 is assumed to have no influence). This research is presented at
Reference 1 and links given there.
Clouds radiate energy from the planet. Cloud elevation determines cloud temperature and
thus the energy rate. The analysis presented here determines that an increase in average
cloud altitude of only about 72 meters results in an increase of steady-state average global
temperature of 0.3 Celsius degrees. Svensmark2 has shown that more sunspots correlate
with fewer low-level (below 3 km) clouds. If there are fewer low-level clouds then
average cloud altitude must be higher, average cloud temperature lower, less energy
radiated from the planet and thus the planet warms.
Steady-State Energy Balance
The overall approach is to determine base values that balance energy flows and then to
determine how each of these values must change in response to a change in average cloud
As a starting point in the analysis, the flow of energy is determined partly using estimates
from other sources, some rational refinements, and conservation of energy
considerations. A graphic that was copied from a NASA site3 is useful as a qualitative
description of the various energy flows. It is shown below with some revisions. Some
energy flows are more explicitly defined than at the NASA site and the values presented
there are replaced by symbols (alpha characters) with values determined herein.
Svensmark2 determined that more sunspots resulted in fewer low level (<3 km) clouds. If
there are fewer low-level clouds then the average cloud altitude would be higher. This
gives rise to the question that is addressed in this paper of how the steady-state surface
temperature would be influenced by change to average cloud altitude.
The general approach used here to determine this is to establish a baseline of values that
balance energy flows on the above graphic and then to determine how the surface
temperature changes as the cloud temperature changes.
The following demonstrates that higher average cloud altitude would result in increasing
average global temperature as occurred during the solar grand maximum of the last half
of the 20th century. All energy rates are in watts per square meter.
Calculation of the Magnitudes of Energy Flow
Measured values of energy from the sun vary only slightly depending on the solar cycle
and slight other long-term variations in the ‘solar constant’. The value used here is 1365
W/m2. The energy from the sun is intercepted by the planet according to the cross section
area but is distributed according to the surface area so this is divided by 4 to account for
the ratio of surface area to cross-section area of a sphere.
Q = 1365/4 = 341.25 W/m2
Part of this incident energy is reflected. The fraction of the incident energy that is
reflected is often assumed to be determined by the earth’s albedo which, from what
appears to be a credible source4, is about 0.297. However, albedo is primarily diffuse
reflection and does not include Fresnel-type low-incident-angle specular reflection8 from
the 71% of the planet that is covered by water. The fraction of the total energy reflected
from the planet is slightly higher than that determined by albedo alone and is about 0.308
R = Q * 0.308 = 105.105 W/m2
The amount of energy reflected from the clouds and atmosphere was determined as
required to maintain the same ratio of cloud+atmosphere reflection to surface reflection
as is reported by Kiehl and Trenberth5.
A = R/(1+23/79) = 81.4 W/m2
Part of the remaining energy is absorbed by clouds and atmosphere. I selected a value of
48% for this to be consistent with the absorbed energy rate assessment by Kiel &
Trenberth5. A comparison of the 1997 and 2008 K & T values along with corrections are
shown on page 3 of Ref 6. An additional factor of 0.6 accounts for the observation that
clouds cover about 60% of the planet.
B = (Q-A) * 0.48 * 0.6 = 74.84 W/m2
The energy that gets through the atmosphere is simply that which is not reflected or
absorbed by the clouds and atmosphere.
C = Q – A – B = 185.01 W/m2.
The part of this that is reflected by the surface is simply that part of the total reflection
that was not reflected by the clouds and atmosphere.
D = R – A = 23.71 W/m2.
The energy that is absorbed by the surface is the part of C (the entering energy that got
through the atmosphere) that is not reflected.
E = C – D = 161.31 W/m2.
The energy leaving the surface as convection (thermal), evaporation (latent) and radiation
from the surface that directly leaves the planet are all as presented on the K & T chart.
F = 17 W/m2
G = 80 W/m2
J = 40 W/m2
The radiation from the surface is gray-body radiation as calculated by the Stephan-
Boltzman equation. Most (71%) of the surface of the planet is covered by sea water with
an emissivity of about 0.995 at water temperature. Part is from snow with emissivity of
about 0.99 and the rest is from land that is mostly about 0.99 but with a few small local
areas that are substantially lower. The over-all average emissivity from the surface is
taken to be 0.99. The average global temperature is taken to be 288 K as typically
reported. The S-B constant, σ, is 5.6697E-08 W/m2/T4. (Temperature in degrees Kelvin)
U = 0.99 * 5.6697E-8 * 288^4 = 386.16 W/m2
Clouds are fine particles of liquid water or ice and thus also radiate according to the S-B
equation. Clouds cover about 60% of the planet and have an average emissivity7 of about
0.5. Their average temperature was determined to be 258 K in an entirely different
analysis that produced an average global temperature of 288K. The thin air and low
temperature at high altitude means that there is very little water vapor so radiation up
from the clouds nearly all gets directly to space. The small part that doesn’t is ignored
P = 0.6 * 0.5 * 5.6697E-8 * 258^4 = 75.36 W/m2.
The same energy flux as from the top of the clouds exists also from the bottom of the
clouds. The down flux encounters substantial absorption prior to reaching the ground.
K & T report 40 W/m2 or 40/396 = 10.1 % of the radiation making it from the surface
through the ghg to space. A calculation using data from Barrett9 for 60% cloud cover
produces 10.8 %. Since about half of the source radiation is stopped by clouds in going
from the surface to space and not stopped by clouds when going from clouds to the
surface, the fraction getting to the surface is about 21% of that going from the surface
directly to space.
K = 0.21 * P = 15.83 W/m2.
The total energy being radiated from the planet must be the amount from the sun minus
the amount reflected.
M = Q – R = 236.10 W/m2.
The energy radiated from the atmosphere must be the total for the planet minus that just
from the clouds to space minus that which goes directly from the surface to space.
N = M – P – J = 120.78 W/m2.
The amount of energy that is thermalized can now be calculated from the other values
entering and leaving the atmosphere system.
H = M – J + K – F – G – B = 40.13 W/m2.
The fraction of radiation from the surface (that does not go directly to space) that is
TH = H/(U-J) = 0.1159 or 11.59%.
The energy that is absorbed by ghg and returned to the surface is that leaving the surface
minus that going directly to space minus that which is thermalized.
I = U – J – H = 306.02 W/m2.
Effect of cloud altitude change
Cloud temperature varies with altitude and, on average, varies as presented in the
Standard Atmosphere Tables that are widely available. Using the graphic above and the
values calculated above as a baseline, required cloud temperature change to produce a
given surface temperature change was determined.
The new value for the amount of energy that is thermalized starts with the baseline value.
Adjustments are made to it to account for changes to the other values entering and
leaving the atmosphere system. Subscript 0 refers to baseline values, s refers to surface
and c refers to cloud
H = H0 + (N – N0) – (F – F0) – (G – G0) + (K – K0)
Thermalization does not change so it is used to determine the new value for the energy
radiated from the surface.
U = H/TH0 + J
Each of the baseline values of energy rate was varied according to a function of a new
temperature to the baseline temperature as follows:
The above numerical data for the steady-state average global (surface) temperature that
results from change in average cloud altitude is shown in the following graph:
This graph is actually slightly curved. It appears to be straight because it covers such a
small range, although the range is large compared to the magnitude of change in average
global temperature that has actually been experienced since accurate temperature
measurements have been made.
As previously determined1 the part of agt change that is not accounted for by effective sea
surface temperature change and the tiny bit that might be attributed to the change in the
level of atmospheric carbon dioxide is about 40% of the total or about 0.3 C. A cloud
average altitude change of only about 72 meters results in this amount of steady-state
average global temperature change.
2. Phys. Rev. Lett. 85, 5004–5007 (2000) (a very brief summary is at