Atmospheric Sciences 321 Science of Climate Lecture 9: Radiation Transfer Community Business Check the assignments HW #3 due next Wednesday
Questions? Radiative Transfer Equation For absorption and emission only
dIv cosθ = ρakv (Bv (T ) − Iv ) dz Now define optical depth a little differently, from the surface up.
z τ (z) = ρ k dz v ∫0 a v What is Optical Depth? It is a coordinate transformation. Instead of using physical distance (vertical or horizontal), we rescale to a dimensionless coordinate, where optical depth = 1 means that only e-1 = 0.368 of the energy is passed without being extincted (absorbed or scattered) in passing through one unit of optical depth. 1.2
1 z Exp(-tau) τ (z) = ρ k dz 0.8 v ∫0 a v 0.6 Exp(-tau) −τ 0.4 F(τ ) = F e 0.2 ∞ 0 0 0.5 1 1.5 2 2.5 3 3.5 4 Tau = Optical Depth Example
Suppose you look out the window of ATG 610 at the Space Needle on a clear day. The optical depth distance to the Space Needle is almost zero at visible wavelengths, you see it clearly even though it is almost 3 miles away SW. On a very foggy day the optical depth distance to the Space Needle might be effectively infinite. You have no chance of seeing the Space Needle, but the physical distance is the same. (the optical depth is the more relevant distance for seeing, or transmitting radiation.) In the infrared the optical distance in the middle of the 15 micron CO2 band might be quite large, since there is a lot of CO2 between here and the Space Needle. Two Photons
Two Photons, 9.6Micron and 12Micron, arrive at 30km altitude on their way to space from Los Angeles. 12Micron has lots of friends with him, basically the whole group who left LA. 9.6Micron is alone, all his friends from LA were absorbed and became heat, but he picked up a few strangers who were emitted along the way.
12Micron says to 9.6Micron, “You 9.6Micron types are weak, my whole group made it from LA.”
9.6Micron says, “We actually have more energy than you 12Micron types, but we had to travel hundreds of optical depths to get here and you all had to travel less than one optical depth to get here.” Another Example: You are flying over the ocean. There is a small sand island in the sea below you, which has an albedo of 0.5 compared to the surrounding ocean albedo of 0.1. A cloud with an optical depth of 1 is between you and the surface. Can you see the island?
The cloud transmits exp(-1) of the incident light and reflects back 1-exp(-1), assuming that which is not transmitted is reflected back by the cloud (no absorption) Cloud with unit optical depth I(1-e-1) I
-2 0.1Ie 0.5Ie-2
-1 Ie -1 0.1Ie-1 0.5Ie
Blue Water Island
Island = I*(1-e-1+0.5*e-2)=0.6997*I Ocean = I*(1-e-1+0.5*e-2)=0.6455*I Difference is about 8%, probably detectable with human eye, but clearly might not be for optical depth = 2 Without cloud contrast is a factor of 5; 0.5 vs 0.1 Some formulas for Clouds and Radiation Combine our formula for liquid water content 4 LWC = πr 3ρ hN 3 L with our formula for optical depth τ = 2π h r 2 N To get a new formula for optical depth
3 LWC τ = 2 ρL r So the optical depth is proportional to the liquid water content and inversely proportional to the average particle radius. What Clouds do Cloud particle radius matters for solar reflection Twomey Effect Clouds are water collected on cloud condensation nuclei. So if I double the number of cloud condensation nuclei and keep the liquid water content the same, then the optical depth will do what? τ = 2π h r 2 N
4 3 LWC = πr ρ hN 3 L for fixed LWC 1 3 3 LWC ⎛ N1 ⎞ τ = r2 = r1 ⎜ ⎟ 2 ρL r ⎝ N2 ⎠ Twomey Effect Clouds are water collected on cloud condensation nuclei. So if I double the number of cloud condensation nuclei and keep the liquid water content the same, then the optical depth will do what? 4 3 3 LWC LWC = πr ρLhN τ = 3 2 ρL r Double 1 1 number, 3 3 ⎛ N1 ⎞ ⎛ N1 ⎞ radius r2 = r1 ⎜ ⎟ = r1 ⎜ ⎟ = 0.79 r1 decreases ⎝ N2 ⎠ ⎝ 2N1 ⎠ a little τ τ = 1 = 1.26τ Optical depth increases 26% 2 0.79 1 Aerosols and Cloud Reflection Ship Tracks and other pollution brighten clouds Remotely Sensed Cloud Droplet Radius Cloud droplet radius Aerosols and Cloud Reflection Ship Tracks and other pollution brighten clouds First indirect effect of aerosols: more aerosols reduce cloud particle radius, so for fixed LWC clouds reflect more solar radiation
Second indirect effect of aerosols: More aerosols reduce cloud particle radius, so precipitation size particles less easily formed, LWC increases due to reduced precipitation. Aerosols and Cloud Reflection Ship Tracks and other pollution brighten clouds My FavoriteAEROSOL EFFECTS Image of
smoke bright over land Aerosol Effectsdirect reflection by aerosols
ship tracks first and second indirect effects
smoke dark over clouds semi-direct effect Zoo of aerosol effects
Tables from IPCC
IN = Ice Nuclei - the small particles on which cloud ice can form Thermal Equilibrium in Realistic Model We have a one-dimensional (z-only) model of the global atmosphere that solves for the vertical distribution of temperature in radiative or thermal equilibrium. In radiative equilibrium all the radiative energy flux convergences add to zero.
∂T 1 ∂F | = − = 0.0 ∂t rad c ρ ∂z ρ In thermal equilibrium we add an adjustment process that moves energy vertically to conserve energy while keep the lapse rate at less than some value. ∂T 1 ∂F |rad = − + Convection = 0.0 ∂t cρ ρ ∂z Radiative and Thermal Equilibria - Global Here are four solutions, pure radiative, Dry adiabatic lapse rate, moist adiabatic lapse rate and global mean observed lapse rate
We specified the water vapor, carbon dioxide and ozone profiles – no clouds Role of H20, CO2 and O3 You can see that water vapor only gets close
Carbon dioxide warms about 10˚C and Ozone produces the stratosphere Heating Rates in Thermal Equilibrium
Radiation cools the troposphere and the stratosphere is in radiative equilibrium The stratosphere is a balance by shortwave heating by ozone and longwave cooling from carbon dioxide 1-D Model: Total Solar Irradiance sensitivity Sensitivity depends on what we assume about humidity, Fixed absolute or fixed relative humidity
Sensitivity to TSI more than doubles for fixed relative humidity compared to fixed absolute humidity
0.5 K/% for FAH
1.36 K/% for FRH
These are clear-sky calculations, but surface albedo has been adjusted to make surface temperature close to
observed for TSI/TSI0=1.0 1-D Model Cloudless CO2 Sensitivity
Change CO2 concentration, Fixed Absolute and Fixed Relative Humidity.
Temperature is not linear in
CO2.
Opacity of CO2 is more like log of concentration, not linear in concentration.
Going from zero to 300 has a much bigger effect than going from 300 to 600 ppmv. Lines saturate. Clouds and Earth’s Temperature All the previous results were for clear skies, but clouds have a substantial effect on the TOA energy balance. Clouds in 1-D model
Here are some clouds Clouds in 1-D model
Here are some clouds. and their effects Clouds in 1-D model
Here are some clouds. and their effects Clouds in 1-D model Here are some clouds. and their effects Observed Cloud Fractions High Clouds (p<440mb) Max High Cloud in tropical rain areas Observed Cloud Fractions
Low Clouds (p > 680mb)
Max low cloud subtropical stratus Observed Cloud Fractions Total Clouds
Max total cloud Midlatitude Oceans Cloud High Fractional Coverage
Low Cloud Fractions High Low
All Total Heuristic Model of Cloud Radiative Effect (CRE) a.k.a. Cloud Forcing
TOA Energy Balance
S0 ↑ RTOA = (1−α p ) − F (∞) 4 ΔR = R − R = ΔQ − ΔF ↑ (∞) TOA cloudy clear abs Cloud Radiative Effect – Add Clouds, what changes? S S ΔQ = 0 (1−α ) − 0 (1−α ) abs 4 cloudy 4 clear
S0 S0 = (α cloudy −α clear ) = − Δα p 4 4 Heuristic Model of Cloud Radiative Effect (CRE) a.k.a. Cloud Forcing
• Cloud Radiative Effect – Add Clouds, what changes? ΔR = R − R = ΔQ − ΔF ↑ (∞) TOA cloudy clear abs • Shortwave bit S S ΔQ = 0 (1−α ) − 0 (1−α ) abs 4 cloudy 4 clear
S0 S0 = (α cloudy −α clear ) = − Δα p 4 4 • Longwave bit
ΔF ↑ (∞) = F ↑ (∞) − F ↑ (∞) cloudy clear Heuristic Model of Cloud Radiative Effect (CRE) a.k.a. Cloud Forcing
• Longwave bit ΔF ↑ (∞) = F ↑ (∞) − F ↑ (∞) cloudy clear • Expand using grey absorption integral equations
T {z ,∞} ↑ 4 4 ct 4 ΔF (∞) = σTz T {zct ,∞}−σTs T {zs ,∞}− σT(z′) dT {z′,∞} ct ∫T {z ,∞} s • Assume cloud top is above most of water vapor, then OLR is emission from top of cloud T {z ,∞}≈ 1.0 ct 1 ↑ 4 4 4 ΔF (∞) = σTz −σTs T {zs ,∞}− σT(z′) dT {z′,∞} ct ∫T {z ,∞} s ↑ 4 ↑ ΔF (∞) = σTz − Fclear (∞) ct Heuristic Model of Cloud Radiative Effect (CRE) a.k.a. Cloud Forcing
• Putting the pieces together, ↑ ΔR = R − R = ΔQ − ΔF (∞) TOA cloudy clear abs • becomes
S0 ↑ 4 ΔRTOA = − Δα p + Fclear (∞) −σTz 4 ct • The solar and longwave parts tend to be of opposite sign and we can calculate the cloud top temperature at which they will exactly cancel.
1/4 ⎧ ↓ ⎫ ⎪−(S0 / 4)Δα p + Fclear (∞) ⎪ Tz = ⎨ ⎬ ct σ ⎩⎪ ⎭⎪ Net CRE = zero
We had from our heuristic model
1/4 ⎧ ↓ ⎫ ⎪−(S0 / 4)Δα p + Fclear (∞) ⎪ Tz = ⎨ ⎬ ct σ ⎩⎪ ⎭⎪ Assuming a linear lapse rate, we can solve for the altitude of the cloud top that will make the net CRE zero.
Tz = Ts − Γ zct ct Net CRE = zero
From our heuristic model 1/4 ⎧ ↓ ⎫ ⎪−(S0 / 4)Δα p + Fclear (∞) ⎪ T = T − Γ z T zct s ct z = ⎨ ⎬ ct σ ⎩⎪ ⎭⎪
16 5 75 2 0 175 125 50 150 100 14
On the right is a graph of 12 75 25 0 25 125 50 the net radiative effect at 100 − 50 − the top of the atmosphere 10 − of putting in a cloud at 75 25 0 8 50 −25 25 50 −75 −1 altitude z, that creates an − −100 6 albedo contrast as specified Cloud Altitude (km) 25 0 −25 75 125 175 in the abscissa. 4 −50 − − −150 − What is the take-away Approximate Altitude Limit of Validity message from this graph? à 2 0 0 0.1 0.2 0.3 0.4 0.5 0.6 Albedo Change Thanks!
16 5 75 2 0 175 125 50 150 100 14
12 75 25 0 25 125 50 100 − 50 − 10 −
75 25 0 8 50 −25 25 50 −75 −1 − −100 6
Cloud Altitude (km) 25 0 −25 75 125 175 4 −50 − − −150 − Approximate Altitude Limit of Validity 2
0 0 0.1 0.2 0.3 0.4 0.5 0.6 Albedo Change