The current magnitude of the greenhouse effect is measured by the
difference between the emission from the surface and the emission from the top of the
atmosphere,
€
G =σTs
4 −σTe
4 ≈ 150 W m-2. What would be the required magnitude of
the greenhouse effect to maintain the surface temperature at 288 K if the solar constant
were reduced by 30%? By what distance would the effective emission level of the
atmosphere need to rise if the lapse rate is approximately 6 K km-1?
To determine the required magnitude of the greenhouse effect to maintain the surface temperature at 288 K if the solar constant were reduced by 30%, we can use the Stefan-Boltzmann law.
The Stefan-Boltzmann law states that the power radiated by a blackbody is proportional to the fourth power of its temperature. Mathematically, it can be written as:
P = σT^4
Where P is the power radiated, σ is the Stefan-Boltzmann constant (5.67 x 10^-8 W m^-2 K^-4), and T is the temperature.
In this case, we want to find the required magnitude of the greenhouse effect (G) to maintain the surface temperature at 288 K. Let's assume the solar constant is S.
Given that G = σTs^4 - σTe^4 (where Ts is the surface temperature and Te is the emission from the top of the atmosphere), we can rearrange the equation:
G = σ(Ts^4 - Te^4)
To find G, we need to calculate Ts and Te.
First, let's calculate Ts. Given that Ts is 288 K, we can substitute it into the equation:
Ts = 288 K
Next, we need to calculate Te. Since the greenhouse effect is due to the difference between the emission from the surface and the emission from the top of the atmosphere, we can determine Te by subtracting the greenhouse effect G from σTs^4:
Te = (σTs^4 - G)^(1/4)
Now, we can calculate Te using the given magnitude of the greenhouse effect G:
Te = (σTs^4 - 150)^(1/4)
With Te calculated, we can now find the required magnitude of the greenhouse effect G to maintain the surface temperature at 288 K if the solar constant were reduced by 30%. The solar constant after reducing it by 30% becomes S' = 0.7S.
The new greenhouse effect G' can be calculated using the same formula:
G' = σTs^4 - σTe'^4
Where Te' is the new emission from the top of the atmosphere.
Since the reduction in the solar constant affects the emission from the surface and the emission from the top of the atmosphere, we can assume that the new values of Ts and Te remain the same. Therefore, G' = G.
Next, let's calculate the required magnitude of the greenhouse effect G' to maintain the surface temperature:
G' = σ(Ts^4 - Te^4)
Now, to calculate the required magnitude of the greenhouse effect, substitute the values of Ts and Te into the equation:
G' = σ(288^4 - Te^4)
Finally, to find the required magnitude of the greenhouse effect G', calculate G':
G' = σ(288^4 - Te^4)
Now considering the second question, to find the distance by which the effective emission level of the atmosphere would need to rise if the lapse rate is approximately 6 K km^-1, we need to calculate the change in temperature (ΔT) corresponding to that distance.
The lapse rate is the rate at which the temperature decreases with an increase in altitude. In this case, the given lapse rate is approximately 6 K km^-1.
The change in temperature (ΔT) for a given change in altitude (Δh) can be calculated using the lapse rate:
ΔT = Lapse Rate × Δh
Where ΔT is the change in temperature, Lapse Rate is the given lapse rate, and Δh is the change in altitude.
In this case, we want to find the change in altitude (Δh) corresponding to a given change in temperature (ΔT) of 6 K.
Rearranging the equation, we have:
Δh = ΔT / Lapse Rate
Substituting the given values, we get:
Δh = 6 K / 6 K km^-1 = 1 km
Therefore, the effective emission level of the atmosphere would need to rise by 1 km if the lapse rate is approximately 6 K km^-1.