RMPh 2014, Теория — Задача 2. Theoretical Problem no. 2 (10 points) — Terrestrial climate modeling

Task no. 1 (1 т.)

In the task no. 1 uses simple modeling. Suppose that the energy is taken from the Sun and that the Earth loses energy as a black body.

1.a. Determine the expression of average temperature $T_P$ that the Earth's surface should have , in accordance with simple modeling used. (0,70p) [0,7 т.]

1.b. Calculate the numerical value $T_P$ of the average temperature that the Earth's surface should have, according to this model. (0,30p) [0,3 т.]

Task no. 2 (1 т.)

Simple modeling of the task no. 1 one assumes that the Earth is a black body. Assumption is unrealistic, because all images taken from space show the Earth as a luminous body.

Earth's atmosphere (especially clouds) reflects approximately 24% of the energy coming from the Sun and the Earth's surface (especially ice areas) still reflects 6% of the incident energy. Feature called albedo measures the ratio of reflected radiant flux and incident radiant flux. Considers that terrestrial albedo is $A = 30\%$.

2.a. Determine the expression of average temperature $T_P'$ that the Earth's surface should have, in accordance with simple modeling used in these workload. (0,70p) [0,7 т.]

2.b. Calculate the numerical value $T_P'$ of the average temperature that the Earth's surface should have, according to the model in workload 2a. (0,30p) [0,3 т.]

Task no. 3 (4 т.)

While the Sun emits energy mostly in the visible, relatively low temperature of the Earth's land surface causes its emission to be localized in the infrared spectrum. A careful modeling of climate takes account of the atmosphere. The atmospheric gas is a mixture having specific absorbent properties. The Spectrum illustrated in Figure 1 shows that the different atmospheric gases absorb radiation in different spectral domains. Characterization of the spectral absorption can be achieved by introduction of transmission coefficients in the visible $\alpha_{vis}$ and infrared $\alpha_{ir}$, respectively, representing the ratio of the energy passing through the air (in the visible or infrared), and the energy that enters the atmosphere. If radiation in a specified spectral range is completely absorbed, then the corresponding transmission coefficient is zero, and if radiation is not absorbed at all, then the transmission coefficient is one.

Task no. 3 proposes to determine the Earth's temperature $T_P''$ using a modeling that takes into account the partial reflection of light coming from the sun (albedo $A$) and absorptive properties of the atmosphere (transmission coefficients $\alpha_{vis}$ and $\alpha_{ir}$).

3.a. Determine the expression of average temperature $T_P''$ that the Earth's surface should have, in accordance with modeling used in these workload. (2,50p) [2,5 т.]

3.b. Calculate the numerical value $T_P''$ of the average temperature that the Earth's surface should have when $A = 0{,}3$, $\alpha_{vis} = 0{,}8$ and $\alpha_{ir} = 0{,}1$. (0,30p) [0,3 т.]

3.c. Using modeling proposed in this workload, calculate the average temperature on the Earth surface, where albedo and transmission coefficients have the values given in Table 1 (1,20p)

Table 1

[1,2 т.]

Task no. 4 (1 т.)

Use modeling proposed in the task no. 3 and assume that the distance between Earth and the Sun would rise by $f = 1\%$.

4.a. Determine the expression of average temperature $T_P'''$ that the Earth's surface should have, if $A = 0{,}3$, $\alpha_{ir} = 0{,}3$ and $\alpha_{vis} = 0{,}6$. (1,00p) [1 т.]

Task no. 5 (3 т.)

Assume that some of the sand found in the Sahara desert would be "transformed" into a "mirror glass".

5.a. Using the modeling proposed in task no. 3 estimates the surface that "mirror glass" should have so that the average temperature at the Earth's surface to fall $1^{\circ}C$ against the value determined in the task 3.b. Consider that the radius of the Earth is $R_P = 6400\ \mathrm{km}$. (3,00p) [3 т.]