• 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