B.2.1 Conservation of energy in Earth’s atmosphere Notes

Energy Balance: Emissivity and Albedo

The Earth’s energy balance is a delicate equilibrium between the energy it receives from the Sun and the energy it radiates back into space.

Two key concepts, emissivity and albedo, play critical roles in this balance.

Emissivity: How Efficiently a Surface Radiates Energy

1. A black body is a perfect emitter with an emissivity of 1, meaning it radiates the maximum possible energy at a given temperature.
2. Real surfaces have emissivity values between 0 and 1.
3. The formula for emissivity is: $$
   e = \frac{\text{Power radiated per unit area}}{\sigma T^4}
   $$ where:
   1. $e$ is the emissivity.
   2. $\sigma$ is the Stefan-Boltzmann constant ($5.67 \times 10^{-8} \, \text{W m}^{-2} \text{K}^{-4}$).
   3. $T$ is the absolute temperature of the surface in Kelvin.

How Emissivity Affects Radiation

1. High emissivity: Surfaces with emissivity close to 1, like black bodies, radiate energy efficiently.
2. Low emissivity: Surfaces with emissivity near 0, like polished metals, radiate much less energy.

Emissivity in the Real World

Different surfaces have varying emissivity values:

1. Oceans: High emissivity (~0.8), radiating energy efficiently.
2. Ice: Low emissivity (~0.1), radiating less energy.

Albedo: How Much Energy is Reflected

1. It is the fraction of incoming radiation that is reflected back into space.
2. The formula for albedo is: $$
   \alpha = \frac{\text{Total scattered power}}{\text{Total incident power}}
   $$

Albedo values range from 0 (no reflection) to 1 (all radiation is reflected).

How Albedo Affects Energy Balance

1. High albedo: Surfaces like snow and ice reflect most of the incoming radiation, contributing to cooler temperatures.
2. Low albedo: Surfaces like oceans and forests absorb more radiation, contributing to warmer temperatures.

Factors Influencing Albedo

1. Surface Type: Snow and ice have high albedo, while forests and oceans have low albedo.
2. Cloud Cover: Clouds increase albedo by reflecting sunlight.
3. Angle of Incidence: Radiation striking a surface at a shallow angle is more likely to be reflected.

Relationship Between Emissivity and Albedo

For a surface that only absorbs or reflects radiation (without transmitting it), the sum of emissivity and albedo is 1: $$
e + \alpha = 1
$$

This relationship highlights the balance between absorption, emission, and reflection of energy.

Solar Constant ($S$): The Sun’s Energy Reaching Earth

Its value is approximately 1,400 W m⁻².

Calculating the Solar Constant

1. The Sun emits a total power $P = 3.9 \times 10^{26} \, \text{W}$.
2. To find the intensity of solar radiation at Earth, consider a sphere with radius $d$, the average distance from the Sun to Earth ($1.5 \times 10^{11} \, \text{m}$).
3. The intensity $I$ at this distance is: $$
   I = \frac{P}{4\pi d^2}
   $$
4. Substituting the values gives: $$
   I = \frac{3.9 \times 10^{26}}{4\pi (1.5 \times 10^{11})^2} \approx 1,400 \, \text{W m}^{-2}
   $$
5. This is the solar constant.

Average Intensity on Earth’s Surface

1. Since Earth is a rotating sphere, the solar energy is distributed over its entire surface.
2. To find the average intensity received by Earth, divide the power passing through the cross-sectional area ($\pi R^2$) by the total surface area ($4\pi R^2$): $$
   I_{\text{avg}} = \frac{S}{4}
   $$

This accounts for day-night cycles and varying angles of sunlight.

Energy Balance and Equilibrium Temperature

Energy Balance Equation

1. For Earth to maintain a stable average temperature, the energy it absorbs must equal the energy it radiates back into space.
2. The absorbed energy per unit area is: $$
   I_{\text{in}} = (1 - \alpha) \frac{S}{4}
   $$
3. The radiated energy per unit area, assuming Earth behaves as a black body, is: $$
   I_{\text{out}} = \sigma T^4
   $$
4. At equilibrium, these two intensities are equal: $$
   (1 - \alpha) \frac{S}{4} = \sigma T^4
   $$

Calculating Earth’s Equilibrium Temperature

Solving for $T$ gives:

$$
T = \left(\frac{(1 - \alpha) \frac{S}{4}}{\sigma}\right)^{1/4}
$$