# Earth’s Energy Balance, Land Surface Temperatures and Outgoing Longwave Radiation

A E Banner, January, 2021

Introduction
The energy balance of the Earth and its importance in global warming considerations has been studied extensively for many years.  Unfortunately, this has been based upon values of energy averaged over the whole surface of the Earth, and so cannot give accurate results for specific locations on the land surface. This must be of significance in many applications, such as meteorology.

The present paper deals instead with areas of one square metre at specific latitudes.  Whereas NASA takes the energy incoming from the Sun, suitably modified by transmission through the atmosphere, and simply divides by 2 to obtain an average for the illuminated part of the Earth’s surface, and then divides by 2 again for the diurnal effect of day and night, the present paper concentrates on an individual square metre, but at each principal latitude.

The calculated temperatures apply to the land surface itself, NOT to the atmosphere close to the surface, and are mid-day values.

It is required to find the temperature of the surface for which the power it emits is equal to the total power which is absorbed by the surface.  This is the condition of energy balance and the “energy balance temperature”.

Theory
In line with NASA, the proportion of the energy from the Sun which is transmitted through the atmosphere and reaches the Earth’s surface is taken to be 0.48.  This is the atmosphere transmission factor, atf.

With the usual value of 1366 Watts per square metre for the Solar Constant, this gives a gross incoming power of 656 Wats per square metre.  NASA maintains that 30% of the power absorbed by the surface is given up to the atmosphere in non-radiative form by convection and evaporation.  Therefore, only 70% is absorbed by the surface and re-emitted as electromagnetic radiation, which is 458.98 Wm^-2.

At higher latitudes, however, the effective power is reduced by the cosine of the latitude.

A further modification must be made because of the effect of the changing tilt of the Earth’s axis as the Earth revolves in its orbit around the Sun.

The net incoming power takes into account these three items.

Power radiated from the surface
At the longer wavelengths, the radiated energy is absorbed by the greenhouse gases in the atmosphere, in line with the Greenhouse Gas Theory, but for the shorter wavelengths only some is absorbed, and the remainder travels freely out to space.  This is known as the Atmospheric Window, and operates at wavelengths below 14 microns.

The energy absorbed by the greenhouse gases is subsequently re-emitted, and the overall effect is that 50% of the emission travels upwards to space, and 50% travels down to the Earth’s surface.

Let  P be the total power emitted from the surface.
w =  the total emitted power at short wavelengths, 14 microns and less
f  = the proportion of the shortwave radiation w which escapes directly to space

Hence, the atmospheric window is fw, and so
this power escapes directly to space………… fw……….Space (1)
This leaves a power (P – fw) in the atmosphere.
By the Greenhouse Gas Theory, this is absorbed by the atmosphere and is re-emitted, half going upwards to space, and half going downwards to be absorbed by the Earth.

That is    power to space = 0.5( P – fw)…………………..Space(2)

And        power to Earth = 0.5( P – fw)…………………..Earth(3)

So, from (1) and (2),
the total power to space =  fw + 0.5(P – fw)
=  0.5(P + fw)
Also,
the power to Earth’s surface  =  0.5( P – fw) ….Earth Return
This Earth Return is of vital importance, because it is needed to augment the power from the Sun absorbed into the surface, and thus provide sufficient power for Planck’s Law to be viable.

The total input power to the surface is equal to the net incoming power from the Sun, PLUS the Earth Return, and thus depends upon the term fw.

The total input power to the surface
is compared with power P radiated by the land surface, in accordance with Planck’s Law, which varies with the fourth power of the Absolute Temperature.  This also provides the value of  fw, and so the energy balance temperature can be established.

Results
The net incoming power is taken to be 459 Wm^-2; the atmosphere transmission factor is 0.48, and the window transmission factor  f  is 0.50, throughout.

A computer model based on this theory and Planck’s Law has provided data for monthly, mid-day land surface temperatures, for the range of latitudes from 0 to 90 degN.

This is for the land surface itself, NOT the atmosphere close
to the surface

The data is shown in Fig 1.

In the winter months, at latitudes of 60 degN and more, very little of the Sun’s energy is absorbed into the surface, and so energy balance can only be attained at very low temperatures, tending to Absolute Zero.  In such cases, the surface temperature for energy balance is indeterminate, and so no data is shown.  (Also applies later in other figures.)

Such surface temperatures will, of course, be moderated by the usual atmospheric effects of circulating air currents.

It is seen by interpolation, that at a latitude of 35 degN, the mid-day surface temperature for June, month 6, is found to be between 62 and 59 deg C.  This is almost, but not quite, hot enough to fry an egg on the sidewalk, as is tried in a competition each year in Oatman, Arizona.  Success requires a temperature of 62 degC

These results are shown graphically in Fig 2.

Fig 3 shows the temperatures displayed against latitude for the three chosen months.

The outgoing longwave radiation (OLR) is shown graphically against latitude in Fig 4, and in table form in Fig 5.

Fig 6 shows both input power, Inp, and the associated outgoing radiation, OLR, for the range of months, and latitudes from 0 to 60 degN.  The paired values are virtually identical.

Conclusion

The theory outlined above can readily be validated by comparison of the calculated temperatures with measured values of the radiative land surface temperatures at various latitudes.  (Similarly with outgoing longwave radiation.)

The theory has assumed that all of the power radiated from the Earth’s surface is absorbed by the atmosphere, apart from the power of the atmospheric window, and is then re-emitted in line with the Greenhouse Gas theory.

Therefore, if the calculated temperatures are equal to the measured temperatures, this means that all of the power is indeed being absorbed, and so the temperatures will not increase with further addition of carbon dioxide.

If the measured temperatures are less than the calculated values, then more CO2 would produce an increase in temperature, but this would be limited to the calculated values.