Earth’s Energy Balance Overall:

*light from the sun striking the earth = heat radiated from earth’s surface to space.*

This relation is not instantaneous, but represents a quasi-equilibrium condition, and is virtually true when averaged over surface of the earth and over one year.

( The largest exception is heat absorbed and retained by the oceans, which will be shown to have an effect lowering earth’s equilibrium temperature by 0.16C.)

From the figure above, notice that sunlight strikes the earth outer atmosphere as a parallel (collimated) beam, therefore the amount of sunlight striking the earth at any time is equivalent to light striking a disk perpendicular to the radiation, at earth’s distance from the sun, the radius “r” of which is earth’s:

S_{o} (watts/m^{2}) * π r^{2}

S_{o} varies very slightly around 1367 watts/m^{2}

Much light is reflected off the earth back to space. This reflective property is known as “Albedo”, symbolized herein by “A”. For earth, A is typically accepted as 0.30. Therefore earth’s surface receives 1.00 – 0.30 = 70% of radiation incident at the top of atmosphere.

**Compute Heat In = Heat Out**

So net heat (as mostly visible light) striking the earth’s surface is

(0.7)*S_{o} (watts/m^{2}) * πr^{2}

but per unit area of earth’s spherical surface (= 4π r^{2}) the average absorbed sunlight is

(0.7)*S_{o} /4 (watts/m^{2}) = 0.7 * 1367 / 4 watts/m^{2} = **239 watts/m ^{2}**

**Compute Earth Surface Temperature with no Greenhouse Effect**

Please notice that earth radiates its heat (as infrared) to space in all directions. The equation for radiation would follow the Stefan-Boltzmann Law if there were no Greenhouse Effect:

Q = heat loss to space (watts/m^{2}) = εσT_{a}^{4 } = 239 w/m^{2} ^{ }

Where surface area of a sphere = 4πr^{2
}ε = emissivity of earth, approximately 0.97

σ = Stefan-Boltzmann constant = 5.67E-8 watts/m^{2}/deg-K^{4
}T_{a}= equivalent earth surface temperature, deg K.

Heat (as infrared light) is radiated in different amounts from the earth’s surface, clouds, and atmosphere. Therefore, the energy balance for earth, averaged over all its surface for a year, if there were no Greenhouse Effect is approximately described by:

0.70*S_{o} * πr^{2} = 4πr^{2} εσT_{a}^{4
}solving for temperature:

**T _{a} = 257 K = -16C = 3F**

**Greenhouse Effect, as Temperature Differential**

The resistance imposed by the atmosphere to loss of earth’s heat to space is called the Greenhouse Effect. It can be expressed as increase in temperature referenced to no resistance or as in radiation, as w/m^{2} ^{ .}

Expressed as temperature, the Greenhouse Effect would be

288K (present Ts)- 257K (Greenhouse effect = 0, calculated above ) =** 31K**

Mathematically, heat balance including the Greenhouse is:

239 w/m^{2} =^{ }0.70*S_{o } /4 = εσTs^{4} – Greenhouse Effect

**Greenhouse Effect as watts/m ^{2}**

Since Earth Surface Temperature is known to be 288 K, the above equation is used to compute a Greenhouse Effect of **139 watts/m ^{2}.**

Detailed Energy Flow is depicted below, from International Panel on Climate Control (IPCC). Please note that regardless of all the processes beneath the top of the atmosphere, including convection, evaporation, condensation, downward re-irradiation, all the heat **must leave earth as infrared radiation**.

Effect of Heat Retained by Oceans

The chart below shows a recent history of net heat gained by the oceans. This reduces the amount of heat earth must radiate to space to maintain equilibrium:

The heat added to the ocean in the 26 year period from 1986 to 2012 is 16E22 Joules (J). The average rate of addition, per second is:

16E22 J / [26 yrs * 8760 hr/yr * 3600 sec/hr] = 1.95E14 J/sec = 1.95E14 watts

The surface area of the oceans is 3.8E08 km^2, x 1E6 m^2/km^2 = 3.8E14 m^2

The average heat flux diverted, ΔH, to the oceans is then

1.95E14 w / 3.8E14 m^2 = 0.54 w/m^2.

Using the formula to compute temperature change from heat flux:

ΔT = 0.301 w/m^2/C * ΔH

Results in** ΔT = 0.16C**

The formula ΔT = 0.301 w/m^2/C * ΔH is derived here.