Post on 21-Feb-2016
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• ENERGY BALANCE MODELS
• Balancing Earth’s radiation budget offers a first approximation on modeling its climate
• Main processes in Energy Balance Models (EBMs) are:
1) Radiation fluxes2) 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 1) Earth’s temperature2) Transparency of the atmosphere to this
outgoing thermal radiation
• There are two forms of EBM:
1) Zero-dimensional modelThe Earth is considered as a single point with a
mean effective temperature
1) First-order modelThe temperature is latitudinally resolved
Zero-dimensional EBM
• Solar radiation input:Si = R2S• Reflected solar radiation:Sr = * Si• Emitted infrared radiation:E = 4R2Te4
R = distance between Earth and Sun, Te = effective temperature, Stefan-Botlzman constant, S = solar constant = 1370 W/m2
• Therefore,
(1-)*(S/4)=Te4
Example:T = 33 K, = 0.3 Ts = 288 K
Note is the albedo. When describing models we will use a terminology according to McGuffie and Henderson-Sellers
• Note that Ts = Te + Twith 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 = 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 (T/t)=(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 ?1) Let us calculate the warming for each day (t = 1)2) T is our unknown3) Ae = 30 m x 10 m 4) Depth = 2 m5) 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
6) (R↓-R↑) = 20 W/m2 in 24 hours7) 2.52*109 = 20 x 30 x 10 x 24 x 60 x 60 T (1 day) =
0.2K8) T (1 month) = 0.2 x 30 = 6 K
What about the Earth ?Remember : mc (T/t)=(R↓-R↑)Ae
R↑ Stefan-Boltzman R↑ T4a
With a accounting for the infrared atmospheric transmissivity
R↓ = (1-)*S/4
T/t =((1-)*S/4 - T4a) /C
C = fw*ro*c*d*Ae = 1.05*1023 J/Kfw = fraction water 0.7, d = 70 m (depth of mixed layer)
One-dimensional EBM
(1- (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
(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
(Ti) =b(phi)-0.009Ti Ti < 283K(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) = i4 [ 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 T changes in response to the: 1) change in surface thermal forcing Q, 2) atmospheric feedback, expressed in terms of a climate feedback parameter , 3) the leakage of energy permitted to the underlying water
• The equations describing the rates of heating in the two layers are therefore:
1) Mixed layer (total capacity Cm)Cm d(T)/dt = Q- T-M2) Deeper watersT0/ t2T0/ z2
With K being the turbulent diffusion coefficient and assumed constant
M acts as a surface boundary condition to the eq. 2 of the previous slide
• If we assume that T0(0,t)=T(t)then M can be computed as:M = -wcwK(T0/ z)z=0
And can be used in the previous Eq. 1. is a parameter used to average over land and ocean and ranges between 0.72 and 0.75. w 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 Q is assumed to increase exponentially Q=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: 1) Includes polar sinking ocean water into deep ocean2) Seasonally varying mixed layer depth3) Seasonal forcing• Cons1) Hemispherically averaged cloud fraction2) No opportunity to incorporate temperature-surface
albedo feedback mechanism (as land is hemispherically averaged)
• Readings:McGuffie and Henderson-Sellers Chapter 3, pp 81 - 116