## Solar Radiation and the Earth's Energy Balance.

### The Physics of Radiative Heat Transfer.

#### 1. Forms of thermal energy transfer in the climate system:

Energy moves in the climate system from one form to another. Particularly important in the climate system is the transfer of thermal energy within and between the different components of the climate system. Thermal energy or heat can move from one place to another in three different forms:

• Conduction: The spreading of heat through molecular vibrations. This form of heat transfer requires the presence of matter and can occur in solids, liquids, and gases. In the climate system conduction occurs mainly over small distances. In the atmosphere and oceans it is taken over by advection and convection.
• Advection/convection: The spreading of heat in fluids and gases through the flow of matter from one place to another. Advection occurs in the horizontal plane and convection in the vertical. In the climate system these forms of heat transfer are manifested in the motion of the atmosphere and ocean, on both local and global scales.
• Radiation: The transfer of energy through electromagnetic waves. This form of energy transfer does not require the presence of matter to occur. In this form energy can travel through empty space from the Sun to the Earth and other planets in the solar system. Radiation also occurs within the climate system between the earth's surface and the atmosphere, and within the atmosphere and ocean.

All forms of heat transfer move heat from one body to another of from one part of a system (such as the climate system) to another. Heat always moves from where the temperature is relatively high to where it is relatively low. This is true for any form of heat transfer.

When electromagnetic radiation spreads from a localized source, such as the Sun, it usually does so in a directionally uniform way (if there are local non-unifromities in the amount of energy they become less important the further away one gets from the source). Far enough from the source, the radiated energy will thus be equally distributed on a surface of a sphere centered on the emitting object. Assuming that the total radiative energy emitted from the source is fixed, then as the distance from the emitting object increases the same total amount of radiation is distributed over a larger sphere. Thus the radiative energy flux (energy per unit time per unit area) decreases as the energy moves further away from the source. The rate of energy flux decrease with increasing distance from the emitting object, is directly related to the rate of increase in the area of the sphere, centered on that object with distance. Because the surface area of a sphere increases in proportional to the square of the radius we have:

I (at distance r from source) = I (at the source) / r2

Where I is the energy flux and r is the distance from the source. In the same way:

I (r2) / I(r1) = r12 / r22

Where r1 and r2 are two distances along the path of the radiation from the source. This dependence of the radiative energy flux on distance from the source is demonstrated in Figure 3.

#### 3. Interaction of radiation with matter:

As long as electromagnetic radiation travels through empty space it remains intact. When electromagnetic radiation encounters a parcel of matter, be it a solid, liquid or gas, it goes through one or all of three processes:

• transmission
• reflection
• absorption

(see Figure 4). To represent the degree to which each process occurs we associate a dimensionless coefficient with each. These are the coefficients of tansmissivity, reflectivity, and absorptivity of matter. All these coefficient determine the fraction of incident radiation that is either transmitted, reflected or absorbed.

When put into a field of radiation, a body with a capability to absorb electromagnetic waves will warm up until it emits as much heat as it absorbs and then stop warming, reaching a state of thermal equilibrium. The heat loss by the body may occur through any form of heat transfer, but if the process takes place in empty space, the only way in which the body can loose heat is through radiation. In that case the body is said to be in a radiative balance and its radiated energy flux will be equal to the absorbed flux.

#### 5. How does radiation depend on temperature - the concept of Blackbody:

Origin of radiation - the discrete spectrum:

The radiation of electromagnetic energy has its origin in changes in the subatomic or submolecular state of matter. For example, when an electron changes its orbit around the nucleus and goes from a high energy state to one with lower energy, the excess energy is released in the form of a finite (quantum) amount of radiation in a particular wavelength. If the change of orbit is from low to high state, the energy has to be supplied from the outside the atom, either through radiating the atom with the necessary amount of energy or through collisions between neighboring atoms (remember that under normal conditions atoms are continuously moving around - a process reflected in temperature). Similarly, quantum amounts of electromagnetic energy are released when atoms bound together in a molecular structure change their binding electron orbits or their intermolecular distance. Here too a transition from high to low energy states will result in the emission of radiation at a particular wavelength.

We could cause matter to radiate by heating it up, inducing rapid molecular motion. That increase motion will lead to increased collisions that will tend to raise the energy state of the molecules and atoms. As the molecules and atom fall back into their original energy state, they will emit that excess energy as electromagnetic radiation. We expect suchemission occurs in a discrete set of frequencies depending on the composition of the radiating matter creating a so-called line spectrum. The intensity of radiation emitted from a parcel of matter is directly related to the number of energetically excited atoms/molecules within it. That number is related to the to the kinetic energy of the atoms/molecules, which is in turn related to the absolute temperature of the parcel.

The continuous spectrum:

But how do we go from a discrete spectrum to a continuous one such as found in most circumstances (see Figure 1 for the spectrum of Sun and Earth, for example)?

The broadening of discrete spectral lines occur through several processes that are quite common in natural systems and are more likely to occur when many atoms and molecules interact strongly with each other. Such spectral line broadening finally gives way to the continuous spectrum. The shape of the spectrum (energy as a function of wavelength) radiated from any object depends only on one variable: its absolute temperature.

The dependence of the intensity and wavelength (color!) of radiation on temperature can be demonstrated by a simple experiment: Consider an iron bar placed in a hot fire. At first its color does not change, but if taken out from the fire, it will warm its surroundings because it radiates in the infrared range (invisible radiation in the wavelength range of 0.7-100 μm). When we continue to heat the iron bar it will begin glowing read and then, as it continues to warm, turn brighter to orange, yellow, white, and finally blue-white in color (the wavelength becoming shorter and shorter within the visible light range, 0.4-0.7 μm).

Blackbody:

A body that absorbs all the radiative energy it received, regardless of wavelength, is called "blackbody". This is because in the absence of self-emitted radiation the color of various objects depends on the wavelength of visible radiation that they do not absorb and are, hence reflected. In radiative balance a blackbody emits a continuous spectrum of radiation with a shape determined by its absolute temperature.

It was the physicist Max Planck who determined the relationship between the radiative energy flux emitted from a blackbody and its absolute temperature. This expression is known as the Planck blackbody radiation law. It is by using this law that the spectra of Sun and Earth emitted radiation were calculated in Figure 1. In that figure we substituted in Planks law values of 5780 K and 288 K for the Sun's and Earth's temperatures, respectively.

Planck's law states a complex (and non-linear) relationship between the energy flux per unit wavelength, the wavelength and the temperature. Two derivatives of this law, useful for our discussion, are the Wien law, stating the relationship between the wavelength corresponding to the maximum energy flux output by a blackbody (λmax) and its absolute temperature (T), and the Stefan-Boltzman law stating the relationship between absolute temperature and the total energy flux emitted by a blackbody, over the entire wavelength range (I).

Wien's law states that:

λmax= a / T

where λmax is given in μm, T is in units of K, and a is a constant equal 2897 μm K.

The Stefan-Boltzman law states that:

I = σT4

where I is in units of W/m2, T is in units of K, and σ (the Greek letter sigma) is a constant equal to 5.67 x 10-8 with units of W m-2 K-4.

Text by Yochanan Kushnir, 2000. Revised 2004.