ESE/Ge 148a Problem Set 21 Spectral Lines and Optical DepthIn this problem, you will solve for the spectral irradiance for one spectral line of CO2using an idealizedprofile of temperature and density.1.1 Number density of absorberAssume the atmosphere has a constant scale height, H, of 7.5 km. Plot ρa, the absorber density, as afunction of height assuming that it is well-mixed with a mixing ratio of 300 ppm. Compare this to thecorresponding figure in lecture 8.1.2 Spectral line shapeThe absorption coeffecient for a single spectral line has the formκ(ν, p, T ) =Sγfν − νcγ(p), (1)where νcis the frequency of the center of the linear, S is the linear intensity, and γ(p) = γ(p0) × p/p0is the line width. Assume a Lorentzian line shape for f with a width of γ(p0)/p0= 0.1 cm−1atm−1(convert from wavenumber to frequency), an integrated intensity, S/γ(p0), of 5×10−20cm2molecule−1,and a line center at 4.3 µm. A Lorentzian line shape is defined by the formulaf(x) =1π(1 + x2). (2)Plot the spectral line.1.3 Optical depthForm a wavenumber-dependent optical depth using the absorption coefficient and the absorber density.Plot the optical depth from the top of the atmosphere to z = (0, 1, 2) × H.1.4 Schwarzschild’s equationUse the Matlab code on the course website to solve Schwarzschild’s equation (or an appropriate alterna-tive such as Mathematica’s NInt) to determine the irradiance at the top of the atmosphere (e.g., 100 km)as a function of wavenumber using the optical depth and the idealized temperature profile given by asurface temperature of 290 K and the following lapse rate:Γ = −∂T∂z=(6.5 K km−1if z < 10 km−1.0 K km−1otherwise(3)It is fine to ignore that the density and temperature profiles are inconsistent.11.5 Brightness temperatureDetermine the brightness temperature as a function of wavelength using the Planck function (i.e., ablackbody at what temperature gives the same spectral irradiance).Compare this to the the temperature where the optical depth is unity. What accounts for the differ-ence?1.6 Changing CO2Double the CO2concentration and redo the calculation. What is the sign of the integrated change in theatmosphere’s emission? What spectral regions have the largest changes (e.g., close to the line center)?1.7 Changing atmopsheric pressureConsider the case of an atmosphere that has 1% of the total mass as Earth’s atmosphere, but has thesame mass of CO2. What is the surface pressure and CO2mixing ratio of such an atmosphere? Redo theabove calculation for the radiance. What does this suggest about how efficient the greenhouse effect ison Mars?1.8 Changing atmospheric temperatureConsider the effect of the stratosphere on the brightness temperature by removing the change in the lapserate at the tropopause.Γ = −∂T∂z=(6.5 K km−1if z < 10 km6.5 K km−1otherwise(4)How is the brightness temperature different in this case compared to the original case? Explain how youmight have anticipated the structure of the changes.2 Response of climate to volcanic eruption (from Hartmann 1994)The transient temperature response, T0, to an imposed climate forcing, Q, can be modelled with thefirst-order differential equation:cp∂T0∂t= −λ−1rT0+ Q. (5)Suppose that a volcanic eruption gives rise to a stratospheric aerosol cloud that changes the Earth en-ergy balance by 4 W m−2initially but then decreases exponentially with an e-folding time of 2 years:Q = 4 W m−2exp(−t/(2 years)). Assume the heat capacity of the climate system on this time scale isequivalent to a layer of water 50 m deep and the sensitivity of the climate is characterized by a sensitivityparameter λr= 1 K (W m−2)−1. What is the temperature response after 2 years? How does this com-pare to a case where the volcanic forcing was constant over 2 years (i.e., Q = 4 W m−2for all times)?How does this compare to the equilibrium response to constant volcanic forcing (i.e., T0(t = ∞) withQ = 4 W m−2for all