# The Temperature of the Earth

## Conservation of Energy and Heat Flow

Firstly I will start with a simple energy balance diagram.

If there is no change in the internal or stored energy in a system then the energy in must equal the energy out. This concept is known as the First Law of Thermodynamics. We can apply this model to the Earth where the system is the Earth, the boundary is the outmost layer of the atmosphere and the surrounding are the vacuum of ‘space’. To understand the Energy in and out requires the introduction of some further concepts.

Heat transfer is the energy flow due to a temperature difference. Heat flows from a hot surface to a cold surface There are three modes of heat transfer: conduction, convection and radiation.

**Conduction** is a progressive exchange of energy between the molecules of a substance where the heat flow is from a hot region to a cold region.

**Convection** is heat transfer due to mass flow or movement. It is the mode of energy transfer between a solid surface and the adjacent liquid or gas that is in motion, and it involves the combined effects of conduction and fluid motion. For example when you stand in front of a fan on a hot day, energy is transferred by conduction to the air molecules in contact with you skin. Air flow induced by the fan removes these warm air molecules and replaces them with cooler molecules resulting in heat flow from your skin. Heat flow due phase change processes (evaporation and condensation) is essentially convective heat flow.

Radiation is heat transfer from the surface of one body to the surface of another due to electromagnetic radiation. The flow due to radiation can be described by a useful simplification of the Stefan-Boltzmann relation:

Where:

*J *= heat transfer per unit time per unit area (W/m^{2})

*σ* = Stefan-Boltzmann constant, = 5.67×10^{-8} W/m^{2}K^{4}

*e* = the ability of the surface to emit radiation. This is called its *emissivity*

*T _{H }*= absolute temperature of the radiating or hot surface (K)

*T _{C }*= absolute temperature of the receiving or cold surface (K)

It is also important to note that for a surface that is irradiated by another surface the above equation applies but the emissivity term is replaced by an absorptivity term *a*. The absorptivity is the ability of a surface to receive or absorb radiation. Generally *e * is equal to *a* but I will discuss this further at a later point.

These equations are important to understand the energy balance of the Earth which we will come to now.

## The Earth’s Effective Temperature

Going back to the previous diagram we can say that he energy in can be assumed to be just the radiant energy from the Sun. This has been measured and found to be about 1368 W/m^{2}. When this value is adjusted for the surface area of the Earth we see that there is an average of 342 W of solar radiation reaching each square meter of the Earth’s surface

According to our Energy balance diagram, energy in must equal energy out. The surrounding space prevents energy escaping by conduction or convection. There is no material or molecules to vibrate and exchange energy. There is no mass to flow moving heat energy with it. The only way energy can leave the Earth is by radiation. And this is where we need the Stefan-Boltzmann equation above.

The left hand side of the equation is the energy absorbed by the Earth, the right hand side is the energy radiated from the Earth. These must be equal for the conservation of energy.

If we re-arrange this equation to find T_{E} we get:

So to find the temperature of the Earth T_{E} we need to know the absorptivity and emissivity of the Earth. If these are equal as some would argue then *a*/*e* = 1 and :-

When these numbers are plugged into the calculator we get T_{E} = 279 K or 6 degrees C. The question is what does the temperature T_{E} indicate. From the Stefan-Boltzmann law it is the temperature of the radiating surface but what is that. We will see later that the Earth radiates energy from the surface and from the atmosphere. The average temperature of the surface is often said to be about 15 deg C, the atmosphere gets colder with altitude. Six degrees C sounds about right.

So how does the CSIRO arrive at the statement

Without heat-trapping greenhouse gases the surface would have an average temperature of –18°C rather than our current average of 15°C.

Simply this is done by measuring the absorptivity and putting this value into the Stefan-Boltzmann equation and then erroneously assuming the Earth to be a hypothetical “blackbody” or perfect emitter and set emissivity = 1.

So which is more correct, setting emissivity = absorptivity, or measuring absorptivity and setting emissivity to 1? (Note: absorptivity *a* is related to reflectivity *r* by *a*=1-r. Reflectivity has been measured and found to be about 0.3, which means absorptivity = 0.7)

It would seem that the best way to find the radiative temperature would be to use measured values of e and a. Absorptivity we know, emissivity is more difficult.

Emissivity has been measured for various surfaces. Water which is a major constituent of the Earths surface has a emissivity of about 0.95. Most solid surfaces have a high value e values? (i.e greater than 0.9) depending on colour, surface roughness etc. However gasses generally have a very low e value, and nitrogen and oxygen, the major gasses in the atmosphere, are poor emitters and have extremely low emissivity.

The problem is exacerbated by the fact that the Earth is not a solid homogenous surface, but has a fuzzy atmosphere.

If we consider the Stefan-Boltzmann equation again we can see that the Earth has an effective emissivity of about 0.7 if we say that the average radiative temperature of the Earth is 6 deg.C.

What does this mean? Is the emissivity of the Earth reduced by GH gasses leading to higher effective temperatures? We will come back to this after looking further at heat flows between the surface of the Earth and the atmosphere.