vmt-award-2024_47--2-_purple

using equation (8)

Equation (28) means that a decrease in emissivity ( δε < 0) leads to an increase in average surface temperature ( δ T > 0). This is true because such a decrease in emissivity makes it harder for the sur face to emit the infrared radiation, which leads to warming. This is important because an increase in CO₂ leads to a decrease in emissivity. In fact, to slightly paraphrase from the book by Randall (2012), It is known from measured optical proper ties of CO₂ that, for the current climate, a doubling of CO₂ relative to its preindustrial concentration would reduce the outgoing long wave radiation by 4 W/m², so that σ T 0 ⁴ δε ≈ - 4 W/m². We also know, from satellite observations, that the outgoing long wave radiation ε 0 σ T 0 ⁴ = 240 W/m². Forming the ratio, we find This means, that doubling CO₂ creates an approxi mate 1.7% change to the outgoing long - range radi ation. From this we see that using the current glob ally averaged surface temperature of 288 K, equa tion (28) implies that doubling CO₂ in the atmos phere would lead to a change in temperature of ap proximately, - δε/ε 0 = (4/(240) ) ≈ 0.017. (29)

R = ( Ω /4)(1 - a ) - εσ T ⁴ ≡ R in - R out , (24)

If the Earth were in perfect energy equilibrium, then R = 0. This is essentially another form of the “ energy in = energy out ” model we used earlier. In this case, we use the zero subscript to denote the equilibrium values, at equilibrium,

0 = ( Ω 0 /4)(1 - a 0 ) – ε 0 σ T 0 ⁴ . (25)

We can now investigate how small changes in the various terms in equation (24) change the value of R, by the amount δ R . Using differentials again, we find

δ R ≈ (1/4)(1 – a 0 ) δΩ - (1/4) Ω 0 δ a - 4 ε 0 σ T 0 ³ δ T – σ T 0 ⁴ δε. (26)

It is important to note that we are examining a change from one equilibrium state to another. This means, the initial net radiation balance, R = 0, is perturbed by an amount δ R . This causes the system to evolve, and a new radiation balance is achieved. We can set δ R to zero in equation (26). Then we can rearrange the resulting expression, using equa tion (25), to relate fractional changes in the Earth's surface temperature to the corresponding fractional changes in, respectively, solar output Ω , planetary albedo a , and emissivity ε , namely,

δ T ≈ - (1/4( δε/ε 0 ) T 0 = (1/4)(0.017)(288) ≈ 1.2 K. (30)

We leave it to the reader to find this temperature change in Celsius and Fahrenheit degrees.

δ T / T 0 ≈ (1/4)( δΩ / Ω 0 - δ a /(1 - a 0 ) - δε / ε 0 ). (27)

Conclusion

We examine a special case of this result below by ignoring any changes in the solar flux and the aver age Earth albedo.

A recent paper by Loeb et al. (2021) is very timely, and I draw on this author for my concluding re marks: Climate is determined by how much of the sun's energy the Earth absorbs and how much energy Earth sheds through emission of thermal infrared radiation. Their sum de termines whether Earth heats up or cools down. Continued increases in concentra tions of well - mixed greenhouse gasses in the atmosphere and the long time - scales time required for the ocean, cryosphere, and

Special Case

In this section we examine the changes in the tem perature of the Earth that is induced by an increase in CO 2 . The increase in CO 2 uses a crude measure. We begin by setting δΩ = 0 and δ a = 0 in equation (27). This means there are no changes in either so lar output or albedo. Then, from (27), we find

δ T / T 0 ≈ - δε /4 ε 0 . (28)

Virginia Mathematics Teacher vol. 47, no. 2

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