Lesson 2Radiative equilibriumSlide 3Greenhouse effect - one atmospheric layer modelGreenhouse effectSlide 6Slide 7Perturbation to the radiative forcingClimate sensitivityCO2 doublingChange in the solar constantChange in the solar constant 2General expression for αSlide 14Radiative forcing of greenhouse gasesRadiative forcingNegative feedbackPositive feedbackPolar ice coverage - positive feedbackCLIMATE FEEDBACK MECHANISMSSlide 22Using Total Ozone measurementsJet Streams on March 11, 1990 Sub-tropical = Blue, Polar = RedMean latitude of the Sub-tropical frontMean Latitude of the Polar FrontSummary of values of the tropical belt expansion21st Century Predicted Widening Ensemble mean ~2 deg. latitude Lu et al. (2007)Observed vs Predicted WideningTemperature deviation (GISS) versus latitudeNumber of tropical pixels versus time for longitude -80 to -60Number of tropical pixels for 58 to 62 latitude vs longitudeLesson 2AOSC 621Radiative equilibrium€ The total amount of solar radiation intercepted by the Earth is πRe2, where Re is the radius of the earth. This energy will be spread over the entire area of the Earth - 4πRe2. Hence the average solar fluxthat reaches the top of the atmosphere (TOA) is given by : FS= πRe2S /4πRe2= S /4Where S is the solar constant. The earth reflects some of this radiation . Let the albedo be ρ, then the energy absorbed by the earth is (1- ρ)Fs.The thermal energy emitted by the earth is σBTe4Hence we get 4/1)1(⎥⎦⎤⎢⎣⎡−=BseFTσρRadiative equilibrium•The spherical albedo of the earth is about 30%. This gives an effective temperature of 255 K. This is considerably lower than the measured average temperature at the surface Ts of about 288 K.•The fact that Ts>Te is seen in all of the terrestrial planets with substantial atmospheres. It is due to the fact that whereas the planets are relatively transparent in the visible part of the spectrum, they are relatively opaque in the infra-red.•A portion of the IR emitted by the surface and atmosphere is absorbed by polyatomic gases and re-emitted in all directions – including back to the earth.Greenhouse effect - one atmospheric layer model•  is the fraction of the IR radiation absorbed by the atmosphereGreenhouse effect€ Consider the radiative equlibrium in the atmosphere.The amount of energy absorbed is εσbTE4 and the amountemitted is 2σbTA4 . Hence in equilibrium: TA4= TE4/2Now consider the energy balance at the surface : (1− ρ)S4+ εσbTA4= σbTE4substituting for TA we get (1− ρ)S4+εσbTE42= σbTE4which gives TE=(1− ρ)S4σb(1−ε/2) ⎡ ⎣ ⎢ ⎤ ⎦ ⎥1/ 4Greenhouse effect•An examination of the previous equation shows that as e increases the ground temperature increases.•This is what is referred to as greenhouse warming.•As we shall show later, not all of the additional energy trapped by the additional absorption goes inot heating the surface. Some , for example goes into evaporation of water - which stores energy in the atmosphere as additional latent heat.•So one should view the inference of the equation with some caution.•This issue will be discussed later when climate feedback is addressed.Radiative equilibrium•First Law of Thermodynamics - the time rate of change of the column energy isTOAsFFNtE−−==∂∂)1( ρ• and4SBeffTOATF σΤ=• Where N is called the radiative forcingWhere Teff is the effective transmission of the atmosphere to thermal radiation ( (1-e) in the simple model)Perturbation to the radiative forcing•What happens if N(TS) is changed by Δ N(TS) •Assume that the atmosphere moves to a new equilibrium temperature, then we can write0=Δ∂∂+ΔssTTNNWe define the climate sensitivity α asNTdsΔ=Δ αClimate sensitivity⎥⎦⎤⎢⎣⎡∂−∂−∂∂=∂∂−=−sssTOAsTFTFTNv)1()/(1ραCO2 doubling•The calculated radiative forcing from doubling CO2 is ~4 Wm-2•Assuming that the solar flux is constant we get { }TOAseffsBseffsBFTTTTTT44)(1314==⎪⎭⎪⎬⎫⎪⎩⎪⎨⎧∂∂=−−σσα• FTOA= 240 Wm-2 and Ts = 288 K, hence α=0.3 Wm-2K-1• Hence the increase in surface temperature is 1.2 KChange in the solar constant•Suppose the solar constant were to decrease by 1%•Assume no feedbacks other than the reduced thermal emission. If there is no albedo change due to the decreased solar flux then the second term in the climate sensitivity is zero. Hence we get:{ }SSSsTOAFTFNTFΔ−=ΔΔ−=Δ∂∂=)1()1(/ραραChange in the solar constant 2•Substituting for FTOA SSsdsTOAsssBeffFFTTFTTTTΔ⎭⎬⎫⎩⎨⎧=Δ⎭⎬⎫⎩⎨⎧=⎪⎭⎪⎬⎫⎪⎩⎪⎨⎧∂∂=−−44(114σα• Change in temperature is -0.72 KGeneral expression for α•Consider a parameter Q ( such as albedo) that depends on the surface temperature•The direct forcing ΔN is augmented by a term€ ∂N∂Q.∂Q∂Ts.ΔTs• Solving for the climate sensitivity we find1)}/)(/(/4{−∂∂∂∂−=ssTOATQQNTFαGeneral expression for α•If we have an arbitrary number of forcings Q then one can derive a general expression for the climate sensitivity[ ]1)//()/(/4−∑∂∂∂∂−=ssTAOTQQNTFα It should be noted that the above equation assumes that the individual forcings are independent of one anotherRadiative forcing of greenhouse gasesRadiative forcingNegative feedbackPositive feedbackPolar ice coverage - positive feedbackCLIMATE FEEDBACK MECHANISMS•·POSITIVE AND NEGATIVE FEEDBACKS •·WATER VAPOR - POSITIVE •·ICE COVER - POSITIVE •.CLOUDS - POSITIVE AND NEGATIVE - MAINLY NEGATIVEFig. 7-27, p. 191Positions of the jet streamsFig. 8-30, p. 231Relation between jet stream and high and low pressure systemsUsing Total Ozone measurements• Contributions to the column ozone amount (total ozone) come mainly from the lower stratosphere. In this altitude region ozone has a long chemical lifetime compared to its dynamic lifetime. Hence one can treat Total Ozone acts as a dynamic tracer. •The subtropical and polar fronts correspond to a rapid change in the tropopause height, and total ozone shows abrupt changes.•Hudson et. al, 2003, 2006 developed a method to determine the position of the fronts (jet streams) from satellite total ozone data.Jet Streams on March 11, 1990Sub-tropical = Blue, Polar = RedMean latitude of the Sub-tropical frontShift from 1980 to 2004 is 1.7 degreesMean Latitude of the Polar FrontShift from 1980 to 2004 is 3.1 degreesSummary of values of the tropical belt expansion•Zonal mean