GEO 302C Climate: Past, Present and Future

Radiative Equilibrium and Forcing – The Ins and Outs (see page 44 of your textbook)

 

 Name:_________________                                                 

Definitions:

 

Blackbody: A blackbody is a hypothetical body (a coherent mass of any material) comprising a sufficient number of molecules absorbing and emitting electromagnetic radiation in all parts of the electromagnetic spectrum so that

 

·        all incident radiation is completely absorbed (hence the term black), and

·        in all wavelength bands and all directions the maximum possible emission is realized.

 

Solar radiation: Solar radiation is comprised of electromagnetic waves having magnetic and electrical properties.  They do not need molecules to propagate them.  In a vacuum, they travel at a constant speed of nearly 300,000 km (186,000 mi) per second – the speed of light.  Solar radiation releases "heat" energy when it is absorbed by an object; the energy emitted from earth into space is in the form of infrared energy.  Over the Earth as a whole, outgoing infrared energy equals incoming solar energy according to the Law of Conservation of Energy.

 

Albedo: The albedo of a planet is defined as the ratio of reflected solar radiation to incoming solar radiation. Snow has a high albedo, hence, a greater capacity to reflect light than say, a woodland forest, or a rocky desert.

 

 

 

                                               

 

Case 1: Blackbody Earth

 

Consider a spherical planet “Earth” with no atmosphere. Energy from the Sun (solar radiation) pours down onto the Earth, a portion of the solar radiation being reflected back to space, and a remainder absorbed by the Earth.

 

The amount of solar energy absorbed by the Earth is given by

 

Flux In = (1-a) pr²S

 

where:

a = albedo (reflectivity)

S =  intensity of sunshine (the solar constant), 1367 W/m²

r  =  radius of a circle (the Earth intercepting Sunlight)

pr² = the area of the circle onto which the sun’s radiation falls

 

Assuming the “Earth” is a blackbody, the Stefan-Boltzmann law describes that the rate of energy being lost by outgoing infrared (IR) energy as a function of surface temperature, by

 

Flux Out = 4pr² sT_e⁴

 

Where:

s = Stefan-Boltzmann constant, 5.67 x 10^-8 W/m²/K⁴

T_e = Temperature of the Earth’s surface

4pr² = the surface area of a sphere (the Earth emitting infrared radiation during the day and at night)

 

Since the radiation flux in must equal the flux out, we can write

 

(1-a) pr²S = 4pr² sT_e⁴

 

and simplify this to

 

S(1-a)/4 = sT_e⁴

 

Problems:

 

1. Express your answers in degrees Kelvin (K)

 

a)      Calculate Earth’s equilibrium temperature T_e for an average albedo of 0.33.  Why is the temperature lower/higher than the observed average temperature of about 15°C or 288 K?

 

b)      Calculate the Earth’s equilibrium temperature T_e during a glacial period, where albedo = 0.75.

 

2.  Refer to Table 2.3 (Typical Albedos of Various Surfaces) of your textbook or from other sources (e.g., websites), calculate reflected energy out (i.e., αS) for the following:

 

a)      average earth

b)      equatorial marine environment

c)      the Sahara Desert

d)      alpine forest

e)      Antarctica

 

Bonus:

f)        White Sands, New Mexico (gypsum sand known for its bright white color)

g)      What would you guess for the typical range of albedo values for Austin, TX and the surrounding area? You may use a range, i.e., more than one, of albedo values if you feel it is appropriate. Please include an explanation.

 

Please put your answers in the following table. Notice that the solar constant = 1367 W/m²

 

 

Surface	Albedo Value	Incident Light	Reflected Light
a) Average Earth		1367
b) Equatorial Marine		1367
c) The Sahara Desert		1367
d) Alpine Forest		1367
e) Antarctica		1367
f) White Sands, NM		1367
g) Austin, TX		1367

 

 

3. Thought Questions:

 

Having understood the Sun’s incoming radiant energy, its reflectance back to space, and the dependency of reflected light on the albedo of the planet,

 

a)      Would you expect the temperature of an ice-covered planet to increase or decrease over time? Why? What would be the result of this trend?

 

b)      Give some brief reasons other than the albedo as to why the Earth’s mean temperatures fluctuate through time. How is the Earth able to recover from extensive periods of glaciation or greenhouse conditions? A full treatise on the dynamics of Earth’s climate is not necessary, just a few brief observations taken from readings and lectures as to why the “Blackbody Earth” is not a complete picture.