Radiation

Objectives:

To mathematically demonstrate radiation relationships for shortwave and longwave radiation. This will be done through calculations demonstrating incident shortwave radiation and how it is affected by solar angle; and by using a simple model to understand the role played by the atmosphere in radiation exchanges and the thermal behaviour of the Earth-Atmosphere system. Use the information provided for each part, as well as other support materials to help answer the questions.

Part A: Solar Radiation Inputs

Background (also review radiation concepts covered in your text and notes):

1. Useful formulae and constants:
   Area of a sphere = 4$\pi$r²
   Stefan Boltzmann constant: $\sigma$ = 5.67 x 10^-8 W m^-2 K^-4
   Cosine Law of Illumination: I = I_ocosZ
   Mean distance between Earth and Sun: r = 1.50 x 10¹¹ m

2. Remember these key concepts:
   
   Flux is a rate, in our case the rate of flow of ${\frac{{\mbox{energy}}}{{\mbox{time}}}}$ or J s^-1 or W (remember 1 J s^-1 = 1 W)
   
   Flux density is a rate per area, in our case ${\frac{{\mbox{energy}}}{{\mbox{(time $\times$ area)}}}}$ or J s^-1 m^-2 or W m^-2

3. Solar Radiation:
   I_o , the ``Solar Constant'', is the flux density (in W m<sup>-2</sup> ) of solar radiation the Earth receives from the Sun, as it enters the atmosphere. It is the maximum flux density, and it occurs when the Sun's rays (i.e. solar beam) is perpendicular to the upper boundary of the atmosphere (i.e. directly overhead). I<sub>o</sub> can be approximated by the amount of radiation reaching the inner surface of a spherical shell which is concentric about the Sun, whose radius is the mean distance between the Earth and the Sun. The approximation works because shortwave radiation is not diminished by travelling through space. [It is ironic that I<sub>o</sub> is known as the ``Solar Constant'' as fluctuations in the Sun's energy, and variations in the Earth's orbit (Milankovitch Cycles) that are responsible for long-term climate change occur.]

   Two factors diminish I_o to produce the flux density that reaches the Earth's surface. These factors are attenuation (i.e. scattering, reflection, absorption) due to the atmosphere, and further intensity reduction due to the solar angle, determined by the Cosine Law of Illumination.

4. Cosine Law of Illumination:
   Not all solar energy entering the atmosphere strikes the Earth at 90^o to the surface (i.e. comes from directly overhead). The Cosine Law of Illumination allows you to calculate the radiation intensity (i.e. flux density) when the sun is not directly overhead. It describes the relationship between the sun's rays (i.e. solar beam) and the Earth's surface, and states:
   
   I = I_ocosZ
   
   where:

   I =

   the radiation flux density on the surface. In Figure 3.1 it is the solar beam intensity or flux density, on Plane AB which represents the surface.

   I_o =

   the radiation flux density on a surface that is perpendicular to the solar beam (i.e. incoming flux density). This is the maximum radiation flux density (i.e. intensity). It occurs when the solar beam is directly overhead. In Figure it is shown by the solar beam striking the Plane CD.

   Z =

   the zenith angle which measures how far away the solar beam is from being directly overhead. It is shown in Fig. as the angle between the solar beam and a line that is perpendicular, or 90 ^o , to the surface.

   Figure 3.1: Radiation relationships.

   

   
   

   Determining the solar radiation intensity (I ) is important for a number of applications. Some examples are: calculating the energy that can be obtained from solar panels, determining the amount of energy available for plants, and determining the heating and cooling needs for buildings. To determin solar flux density, the Cosine Law of Illumination can be used in a number of ways. One could directly measure Z , and then calculate I for every time, day, and place needed. This is very labour intensive and would require a year of measurements for each place. So we need to be able to determine the solar intensity for a particular time, date, and place without having to take measurements. The Sun's seasonal and daily change in position in relation to the Earth, allows cosZ to be defined using other easily available information. Mathematically, this is expressed as:

   cosZ = sin$$\phi$$sin$$\delta$$ + cos$$\phi$$cos$$\delta$$cosh
   Where:

   $\phi$

   = the latitude of your location in degrees

   h

   = the hour angle, which is 15^o for every hour from solar noon. At solar noon, the hour angle is zero.

   $\delta$

   = the Solar Declination (i.e. the latitude where the sun is directly overhead). It is measured in degrees North or degrees South. A positive value represents a latitude of degrees North, while a negative value represents a latitude of degrees South. A reasonable estimate of $\delta$ can be calculated by:

   $$\delta$$ = - 23.4 cos$${\frac{{360(T_{J} + 10)}}{{365}}}$$

   where: T_J = the Julian day (an alternate way of counting days in a year with January 1 having a Julian day of 1, Feb 1 having a Julian day of 32, and December 31 having a Julian day of 365, or 366 if it is a leap year.)

Questions:

1. The flux of radiation emitted by the Sun is about 3.865 x 10²⁶ W.
   1. Calculate I_o the flux density of solar radiation reaching the Earth in W m^-2 .
   2. What is another name for this energy input to the earth?
   
   

2. Using the Cosine Law of Illumination, calculate the intensity of solar radiation arriving at the top of the atmosphere in Prince George on the afternoon of September 21 at 3:00 p.m. PST (15:00 PST). For simplicity assume the year is not a leap year, and that local standard time is the same as solar time. Prince George is located at 122 ^o 41' W longitude (about 123 ^o W) and 53 ^o 53' N latitude (about 54 ^o N).
3. Identify the latitude and longitude of the location where the Sun is directly overhead at the start of your lab period. Use information given in the question above, the background information, and knowledge that the Earth's daily revolution is 15 degrees of longitude per hour.
   (HINT: If you want to know the longitude where the sun is directly overhead you need to realize that solar noon is defined as the time when the sun is at it's highest point in the sky. In the northern hemisphere (north of 23.5 degrees N), solar noon occurs when the sun is directly to the south. At the same time at a lower latitude, the sun would be directly overhead. As the earth revolves, solar noon moves along a line of latitude by 15 degrees of longitude per hour.)

Part B: Radiation Balances

Background

The Radiation Balance of the Earth-Atmosphere System consists of shortwave and longwave radiation fluxes which can be visualized as transfers through various atmospheric layers. The Greenhouse Effect (also called the Atmosphere Effect) is caused by radiatively active gasses such as CO₂ , water vapour, methane, (and others) that absorb outgoing longwave radiation from the Earth and effectively prevent most of it from reaching space. This absorbed outgoing longwave radiation warms the atmosphere resulting in increased atmospheric temperatures and subsequent longwave radiation emissions from the atmosphere back to the Earth. This increases the earth's average temperature by about 33 K.

Radiation models, such as the simple model below, quantify radiation exchanges, and develop an understanding of radiative balance concepts and temperature in the Earth-Atmosphere system. A more elaborate version of this model could be used to study global warming. This part of the Lab demonstrates the Greenhouse Effect by presenting a simple model of radiation transfers through layers of the atmosphere. The model assumes:

• the temperature in each layer is uniform
• all layers of the atmosphere are totally transparent to solar radiation
• all infra-red radiation entering a layer is absorbed by that layer (i.e. layers are ``black bodies'')
• all energy transfer is accomplished by radiation; there is no convective energy transfer
• there is radiative equilibrium in each layer so that radiative gains equal radiative losses
• there is no infra-red radiation entering the atmosphere from space

Figure 3.2 illustrates the model. It shows an atmosphere consisting of two ``black body'' layers, each absorbing all the infra-red radiation entering from above or below. No emissivities are included because a ``black body'' has complete absorption, (i.e. a = $\epsilon$ = 1 ). From the Stefan-Boltzmann Law we know that Layer 2 radiates $\sigma$T₂⁴ upward and $\sigma$T₂⁴ downward, and receives $\sigma$T₁⁴ from Layer 1 below but nothing from above as there is no IR from space.

So, for Layer 2 (under the conditions of radiative equilibrium - what goes in = what goes out) you get:

$$\sigma$$T₁⁴ = $$\sigma$$T₂⁴ + $$\sigma$$T₂⁴

simplified to:

T₁⁴ = 2T₂⁴

Since each layer absorbs all the radiation entering that layer, the only radiation leaving the model comes from Layer 2. Therefore, T₂ must equal T_E-A , the radiation (or equilibrium) temperature of the Earth-Atmosphere system. T_E-A is approximately 255 K.

Figure 3.2: Schematic of a two layer atmosphere.

Questions

1. Using the Model:
   1. Write an equation for Layer 1 that relates T₀ (the surface temperature), T₁ , and T₂ .
      

   2. Determine model (i.e. calculated) values for T₁ and T₀ , and complete the following table to compare observed and modelled atmospheric temperatures. The observed values in the table below are obtained as follows: T₀ is the global average surface temperature from observations, T₁ is the estimated average temperature in the Troposphere and T₂ is the maximum temperature in the Stratosphere from text Fig. 1-9).
      

      	T0 (K)	T1 (K)	T2 (K)
      Calculated from Model
      Observed E-A system values	288	$\approx$ 253	$\approx$ 271

      
      

   3. Explain why there are discrepancies between each of the model (calculated) and observed temperature values above. (HINT: Models don't always give the correct answer. Think about the model assumptions, whether they are valid, and what the consequences would be if they are not valid.)

2. To quantify your understanding of the Earth-Atmosphere Radiation Balance, complete the following table. The rows in the table represent the components of the Earth-Atmosphere (E-A) system. The columns of the table represent the components of the net radiation balance. All units are in GJ m^-2 year^-1 (G = giga = 10⁹ ). The following information is also needed:
   • the annual average solar radiation flux density entering the E-A system, = 342 W m^-2
   • the global (E-A) albedo ( $\alpha_{{E-A}}^{}$ ) = 0.30
   • K^* + L^* = Q^*
   • Radiation values for the Earth System and the Atmosphere System add to total values for the Earth-Atmosphere System

   Start by determining the value of Q^*_E-A (the net radiation for the entire E-A system). Remember that on an annual basis the E-A system is neither warming nor cooling, and that radiation is the only way energy enters or leaves the E-A system. Use this information to find the value of Q^*_E-A .

   
   

   	Net solar	Net infra-red	Net all-wave
   System	(K* )[GJ m-2 y-1 ]	(L* )[GJ m-2 y-1 ]	(Q* )[GJ m-2 y-1 ]
   Atmosphere	K*A =	L*A =	Q*A = - 3.02
   Earth	K*E =	L*E = - 2.12	Q*E =
   Earth-Atmosphere	K*E-A =	L*E-A =	Q*E-A =

   
   (NOTE: A `+' sign indicates an energy gain, and a `-' sign indicates an energy loss.)