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## ENERGY BALANCE MODELS

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**Balancing Earth’s radiation budget offers a first**approximation on modeling its climate • Main processes in Energy Balance Models (EBMs) are: • Radiation fluxes • Equator-to-pole transport of energy**The simplest way is looking at the Earth’s climate in**terms of its global energy balance • Over 70 % of the incoming energy is absorbed at the surface surface albedo plays a key role , being the ratio between outgoing and incoming radiation • The output of energy is controlled by • Earth’s temperature • Transparency of the atmosphere to this outgoing thermal radiation**There are two forms of EBM:**• Zero-dimensional model The Earth is considered as a single point with a mean effective temperature • First-order model The temperature is latitudinally resolved**Solar radiation input:**Si = pR2S • Reflected solar radiation: Sr = a* Si • Emitted infrared radiation: E = 4pR2sTe4 R = distance between Earth and Sun, Te = effective temperature, s Stefan-Botlzman constant, S = solar constant = 1370 W/m2**Therefore,**(1-a)*(S/4)=sTe4 Example: DT = 33 K, a = 0.3 Ts = 288 K Note a is the albedo. When describing models we will use a terminology according to McGuffie and Henderson-Sellers**Note that**Ts = Te + DT with Te being the effective temperature and DT the greenhouse increment. In other words, the effective temperature (e.g., in a simplistic way the ‘body planet’ temperature) is lower than Ts (the Earth+greenhouse temperature)**Trip to Venus**• S = 2619 W/m2 • a = 0.7 • Te = ?**Te = 242 K**• Though Venus is closer to the Sun, it has a lower Te than Earth because of the high albedo as it is completely covered by clouds • Besides, Venus atmosphere is very dense and made mostly of carbon dioxide (CO2) • Ts was found to be 730 K ! • The difference between Te and Ts is partially due to greenhouse and partially to adiabatic warming of descending air**Rate of change of temperature**mc (DT/Dt)=(R↓-R↑)Ae Where Ae = area of the Earth, c = specific heat capacity of the system, m = mass of the system, R↓ and R↑ are the net incoming and net outgoing radiative fluxes (per unit area)**Swimming pool warming**• How long would it take for your swimming pool to warm by 6 K ? • Let us calculate the warming for each day (Dt = 1) • DT is our unknown • Ae = 30 m x 10 m • Depth = 2 m • c = 4200 J/(Kg*K) total heat capacity C = ro*c*V = ro*c*d*Ae=1000*4200*2*30*10=2.52*109J/K with ro = water density • (R↓-R↑) = 20 W/m2 in 24 hours • 2.52*109 = 20 x 30 x 10 x 24 x 60 x 60 DT (1 day) = 0.2K • DT (1 month) = 0.2 x 30 = 6 K**What about the Earth ?**Remember : mc (DT/Dt)=(R↓-R↑)Ae R↑ Stefan-Boltzman R↑ esT4ta With taaccounting for the infrared atmospheric transmissivity R↓ = (1-a)*S/4 DT/Dt =((1-a)*S/4 - esT4ta) /C C = fw*ro*c*d*Ae = 1.05*1023 J/K fw = fraction water 0.7, d = 70 m (depth of mixed layer)**One-dimensional EBM**(1-a (Ti))*S(Ti)/4= R↑(Ti)+F(Ti)**The term F(Ti) refers to the loss of energy by a latitude**zone to its colder neighbor or neighbors • Plus, any ‘storage’ system have been ignored so far since we have been considering time-scale where the net loss or gain of stored energy is small. • Any stored energy would appear as an additional term Q(Ti) on the right side of the previous equation**Parametrization of the climate system**• Albedo a(Ti) = 0.6 if Ti < Tc or 0.3 if T > Tc Tc = critical temperature, ranges between -10ºC and 0ºC**Albedo II**Another way for parametrizing albedo is a(Ti) =b(phi)-0.009Ti Ti < 283K a(Ti) =b(phi)-0.009x283 Ti ≥ 283K b(phi) is a function of latitude phi**Outgoing radiation**R ↑(Ti) = A+BTi with A and B being empirically determined constants designed to account for the greenhouse effect of clouds, water vapour and CO2**Outgoing radiation II**• R(Ti) = sTi4 [ 1-mi*tanh(19*Ti6x10-16)] With mi representing atmospheric opacity**Rate of transport of energy**F(Ti) = Kt(Ti-Tav) where T is the global average temperature and Kt is an empirical constant**Box Models: another from of EBM**• Ocean – atmosphere system with 4 boxes • 1) Atm over Ocean, 2) Atm. Over land, 3) Ocean mixed layer, 4) Deep ocean**The heating rate of the mixed layer is computed assuming a**constant depth of the mixed layer in which the temperature difference DT changes in response to the: 1) change in surface thermal forcing DQ, 2) atmospheric feedback, expressed in terms of a climate feedback parameter l, 3) the leakage of energy permitted to the underlying water**The equations describing the rates of heating in the two**layers are therefore: • Mixed layer (total capacity Cm) Cm d(DT)/dt = DQ- lDT-DM 2) Deeper waters DT0/ t = K 2 DT0/ z2 With K being the turbulent diffusion coefficient and assumed constant**DM acts as a surface boundary condition to the eq. 2 of the**previous slide • If we assume that DT0(0,t)=DT(t) then DM can be computed as: DM = -grwcwK(DT0/ z)z=0 And can be used in the previous Eq. 1. g is a parameter used to average over land and ocean and ranges between 0.72 and 0.75. rw and cw are the density and specific heat capacity of water**Using this approach it is possible to estimate the impacts**of increasing atmopsheric CO2. • If DQ is assumed to increase exponentially DQ=b*t*exp(wt) b and w are coefficients to be determined.**The level of complexity can be increased by including, for**example, separate systems for the Northern and Southern hemisphere land, ocean mixed layer, ocean intermediate layer and deep oceans.**Pros:**• Includes polar sinking ocean water into deep ocean • Seasonally varying mixed layer depth • Seasonal forcing • Cons • Hemispherically averaged cloud fraction • No opportunity to incorporate temperature-surface albedo feedback mechanism (as land is hemispherically averaged)**Readings:**McGuffie and Henderson-Sellers Chapter 3, pp 81 - 116