Global Warming Made Simple (rev 12/7/15)

See also average global temperature assessment at http://globalclimatedrivers2.blogspot.com

Energy balance

The only way that significant energy can arrive at or leave the planet is by electromagnetic radiation. Energy which is received by the planet is the part of the radiation from the sun that is not reflected. The total energy being radiated from the planet is the sum of the energy radiated between the clouds plus the energy radiated through the clouds and from the clouds themselves.

Simple calculations in radiation heat transfer demonstrate that much of the change in average global temperature that has been called Global Warming can be accounted for by a small reduction of low altitude clouds.

Generally accepted knowledge:

1.      Clouds cover about 60% of the planet all the time¹.

2.      Average radiation from the sun at earth’s orbital distance is fairly constant at about 1367 W/m².

3.      About thirty percent of the energy from the sun is reflected² by the planet (albedo plus specular reflection) and therefore does not need to be radiated from the planet.

4.      The planet radiates energy over its spherical surface which is four times its cross section area, which is the effective area that receives (intersects) radiation energy from the sun.

5.      On average over a century or so, the average global temperature does not change very much so the energy received by the planet from the sun must average out to about the same as that radiated from the planet.

6.      Clouds, because they consist of bits of liquid or solid water, radiate energy according to their emissivity and the fourth power of their absolute temperature.

7.      Average emissivity, ε, of the surface is about 0.98.

8.      The average emissivity of clouds³, ε_c, is about 0.5.

9.      At average cloud altitude, air temperature, and thus cloud temperature, declines with increasing altitude at a rate of 0.0065 °K/m. (1962 US Standard Atmosphere)

10.  Average global temperature before global warming was about 287.4 °K.

11.  Average global temperature increased about 0.74 °K during the 20^th century (the trend has been flat since before 2001).

Change in cloud reflectance

Part of the radiation that comes from the sun is reflected; most as albedo from clouds and a bit from land and some as specular reflection from water. A small change in the area of cloud cover causes reflectance change which produces an easily calculable change in average global absolute temperature, T. T can be determined from:

T = ((1-a)*S/(4*ε1*σ))^0.25  °K

Where:

a = planet average reflectance ≈ 0.3

S = Solar constant = 1367 W m^-2

ε1 = over-all planet average emissivity including effects of clouds and ghg = 0.612

σ = Stephan-Boltzmann constant = 5.6697E-8 W m^-2 K^-4

From this we calculate that a decrease of planet average reflectivity (as a result of fewer clouds) from 0.3 to the very slightly reduced value of 0.2964 would result in an average global temperature increase of 0.37 °K.

Change in cloud area (added 12/7/17)

The basic equation relating solar heating to average earth temperature is

S*(1-a)=ε*σ*A*T⁴                                                                               (1)

Where:

a = albedo =  fc*rc+(1-fc)*rnc. ≈ 0.3

Credible values:

fc = 0.62 = fraction of area covered by clouds

rc = 0.47 = average reflectivity of clouds

rnc = 0.03 = average reflectivity of area not covered by clouds

T=average global temperature ≈ 288 K

albedo at T:

a = 0.62*0.47 + (1-0.62)*0.03 = 0.3028                      (OK)                            (2)

Combine this in (1) with T = 288

S*(1-(0.62*0.47 + (1-0.62)*0.03)) = ε*σ*A*288        ⁴                                      (3)

Albedo at T-0.5 =287.5

a1 = (0.62*0.47)*(1+x) + (1-0.62)*0.03*(1-x)                                                (4)

Combine this with (1) with T = 287.5

S*(1-((0.62*0.47)*(1+x) + (1-0.62)*0.03*(1-x) = ε*σ*A*287.5⁴                      (5)

Divide (5) by (3) and solve for x

x = 0.017247 ≈ 1.72%

Thus, all else equal, an increase in cloud cover of 1.72% would result in an average global temperature decrease of 0.5 K

Change in cloud altitude.

If the decline caused by greenhouse gases (mostly water vapor) in effective surface emissivity is ignored and it is assumed that none of the radiation from the surface under the clouds leaves the planet, the energy balance in equation form is:

H = H_nc + H_c

Where:

H = total energy received from the sun and thus radiated from the planet.

H_nc = Energy radiated from the planet surface (below the atmosphere) between clouds.

H_c = Energy radiated from clouds at average cloud altitude.

The only unknown in this energy balance is average cloud temperature so it can be solved for. The Equation can be solved for average cloud temperature before global warming and then for average cloud temperature after the surface temperature increased 0.74 °K.

H = 1367/4*(1-0.3) = 239.22 W/m²

Then before global warming the energy radiated from between clouds is:

H_nc1 = (1-0.6)*ε*σ*T_s⁴  = 0.4*0.98*5.6697x10^-8 * (287.4)⁴ = 151.63 W/m²

And after global warming the energy radiated from between clouds is:

H_nc2 = 0.4*0.98*5.6697x10^-8 * (287.4+0.74)⁴ = 153.2 W/m²

Average cloud temperatures before and after the 20^th century warming can now be calculated:

Average cloud temperature before global warming:

239.22 = 151.63 + 0.6*ε_c*σ*T_c1⁴ = 151.63 + 0.6*0.5*5.6697x10^-8* T_c1⁴

T_c1 = 267.88 °K

This equates to an average cloud altitude of 3120 m which is reasonable.

 Average cloud temperature after global warming:

239.22 = 153.2 + 0.6*0.5*5.6697x10^-8* T_c2⁴

T_c2 = 266.67 °K

The change in average cloud temperature is then

ΔT_c = 266.67-267.88 = -1.21 °K

This equates to an increase in average cloud altitude, to account for the entire increase in average global temperature of 0.74 °K, of 1.21/0.0065 = 185.6 m or, for half the increase, 0.37 °K in 92.8 m.

The effective emissivity of the surface might be less than 0.98. Also, some of the radiation from the surface under the clouds makes it through the clouds. But the end result must always be the same; over a long period of time the sum of all the radiation leaving the planet must, on average, equal all of the radiation received by the planet. And also, in any calculation, average cloud altitude must be reasonable. 

If it is assumed that the effective emissivity of the surface between clouds (to space) is 0.65 and the surface beneath clouds (to space) is 0.22, the effect on average cloud altitude is exactly the same as determined above (185.6 m).

In fact, any combination of effective surface emissivities that passes the sanity test of energy balance and reasonable average cloud altitude will come to the same result: About half of the surface-temperature increase which has been called global warming can be accounted for by an increase of average cloud altitude of only about 100 meters.

Discussion and conclusions

Henrik Svensmark, a Danish physicist, discovered a natural mechanism that causes a change in the amount of low-level clouds: Sunspot number increase indicates solar magnetic field increase which increases shielding of the earth from galactic cosmic rays. Reduced cosmic rays results in fewer low altitude clouds. Fewer low-level clouds results in both lower reflectance and higher average cloud altitude.

An abstract of Svensmark’s paper, which was published in 2000, is at http://prl.aps.org/abstract/PRL/v85/i23/p5004_1

Because higher altitude clouds are colder and radiate less energy to space, and fewer clouds means lower reflectance, more sunspot activity causes global warming and less sunspot activity causes global cooling.

That is, both contribute to temperature change in the same direction so, when combined, they could easily account for a substantial part of the temperature change that has been called Global Warming.

Note that a low but wide solar cycle could have just as much influence as a tall but narrow one; so the time-integral of sunspot numbers, which accounts for both magnitude and time of a solar cycle, is the determining factor. Of course the time-integral of sunspot numbers must be reduced by the time-integral of radiation from the planet and a proxy factor must be applied.

The rest of average global temperature change in the 20^th century is accounted for by natural net global ocean surface temperature oscillation.

 A simple equation, using accepted values for sunspot numbers calculates the AGT trend since it has been accurately measured world wide (about 1895) with an R² of 0.9 (0.98 for 5-yr smoothed measured temperatures). CO₂ has no significant influence on climate. The equation is given at  http://globalclimatedrivers2.blogspot.com

Without human caused Global Warming there can be no human caused climate change.

References

1.      Average cloud cover http://www.nytimes.com/interactive/2012/05/01/science/earth/0501-clouds.html?_r=0

2.      Goode, P. R.; et al. (2001). "Earthshine Observations of the Earth's Reflectance". Geophysical Research Letters 28 (9): 1671–1674. Bibcode:2001GeoRL..28.1671G

3.      T.R. Shippert, S.A. Clough, P.D. Brown, W.L. Smith, R.O. Knuteson, and S.A. Ackerman. "Spectral Cloud Emissivities from LBLRTM/AERI QME". Proceedings of the Eighth Atmospheric Radiation Measurement (ARM) Science Team Meeting March 1998 Tucson, Arizona. http://www.arm.gov/publications/proceedings/conf08/extended_abs/shippert_tr.pdf.