Climate Sensitivity Exercise: Assume that the atmosphere can be regarded as a thin layer We consider sensitivity in a very simple context. with an absorbtivity of aS = 0.1 for shortwave (solar) radia- • We consider a single-layer isothermal atmosphere. tion and aL = 0.8 for longwave (terrestrial) radiation. • We assume the system is in radiative balance. Assume the Earth’s is A = 0.3 and the solar constant −2 • We assume the atmosphere is almost transparent is Fsolar = 1370 W m . to shortwave radiation. Assume that the earth’s surface radiates as a blackbody at • We assume the atmosphere is relatively opaque to all wavelengths. longwave radiation. ???

• We assume the Earth radiates like a blackbody. Calculate the radiative equilibrium temperature TE of the Problems: surface and the sensitivity of TE to changes in the following parameters: • Compute the equilibrium temperature of the surface • Absorbtivity of the atmosphere to shortwave radiation and of the atmosphere. • Absorbtivity of the atmosphere to longwave radiation • Investigate the effects of changing parameters on the temperature. • Planetary albedo • Solar constant 2

Solution: Solution: The net FS absorbed by the earth-atmosphere system is equal to the solar constant reduced by the albedo and by the areal factor of four: 1 − A 0.7 F = F = × 1370 = 240 W m−2 S 4 solar 4 Therefore, the incoming flux of solar radiation at the top of the atmosphere, averaged over the whole Earth, is −2 FS = 240 W m .

The absorbtivity for solar radiation is aS = 0.1. We define the transmissivity as τS = 1 − aS. The downward flux of short wave radiation at the surface is the incoming flux multiplied by the transmissivity, τS FS.

Let FE be the longwave flux emitted upwards by the surface.

Since the absorbtivity for terrestrial radiation is aL = 0.8, the longwave transmissivity is τL = 1 − aL = 0.2. 3 4 Thus, there results an upward flux at the top of the atmo- Repeat: to find FE and FL, we must solve sphere of τL FE. FE − FL = τS FS Let FL be the long wave flux emitted upwards by the atmo- τL FE + FL = FS sphere; this is also the long wave flux emitted downwards. This gives the values Thus, the total downward flux at the surface is τ F + F S S L     1 + τS 1 − τSτL Radiative balance at the surface (upward flux equal to down- F = F F = F E 1 + τ S L 1 + τ S ward flux) gives: L L FE = τS FS + FL ??? The upward and downward fluxes at the top of the atmo- Assuming that the Earth radiates like a blackbody, the sphere must also be in balance, which gives us the relation Stefan-Boltzman Law gives σT 4 = F FS = τL FE + FL surface E using the expressions derived for F and F , this is To find FE and FL, we solve the simultaneous equations S E 1 + τ 1 − A F − F = τ F 4 S E L S S σTsurface = Fsolar 1 + τL 4 τL FE + FL = FS

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Again,   Sensitivity to Shortwave Absorbtion 4 1 + τS 1 − A σTsurface = Fsolar 1 + τL 4 Suppose that some change causes an increase in the absorb- Taking logarithms, we have tion of solar radiation by the atmosphere. For example, an increase in ozone concentration in the strato- log σ+4 log Tsurface = log(1+τS)−log(1+τL)+log(1−A)−log 4+log Fsolar sphere would result in greater absorbtion of incoming solar Now differentiate: radiation. dT dτ dτ dA dF So, 4 surface = S − L − + solar Tsurface 1 + τS 1 + τL 1 − A Fsolar aS ⇒ aS + d aS τS ⇒ τS + d τS This equation enables us to investigate the sensitivity of the Clearly, if d aS > 0 then d τS = −d aS < 0. surface temperature to changes in various parameters. The Blue Equation reduces to For reference, let’s call this the Blue Equation. dTsurface dτS ??? 4 = Tsurface 1 + τS

7 8 Again, dT dτ Sensitivity to Longwave Absorbtion 4 surface = S Tsurface 1 + τS Suppose next that some change causes an increase in the Suppose the transmissivity decreases from 0.9 to 0.8. absorbtion of terrestrial radiation by the atmosphere. Then τS = 0.9 and dτS = −0.1. For example, a pall of ash in the stratosphere, following a Suppose also that the equilibrium temperature of the Earth major volcanic eruption, could absorb or scatter a signifi- with τS = 0.9 is 288 K (as we found above). cant proportion of outgoing longwave radiation. Then So, dT −0.1 −0.1 288 surface aL ⇒ aL + d aL 4 = or dTsurface = = −3.8 K 288 1.9 1.9 4 τL ⇒ τL + d τL

Thus, the assumed increase in shortwave absorbtivity has Clearly, if d aL > 0 then d τL = −d aL < 0. ◦ resulted in a decrease in surface temperature of about 4 C. The Blue Equation reduces to dT dτ 4 surface = − L Tsurface 1 + τL

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Again, dT dτ Sensitivity to Planetary Albedo 4 surface = − L Tsurface 1 + τL Suppose next that some change causes an increase in the Suppose the longwave transmissivity decreases from 0.2 to albedo or reflectivity of the atmosphere. 0.1. Then τL = 0.2 and dτS = −0.1. For example, an increase in condensation nuclei might result Suppose once more that the equilibrium temperature of the in a greater coverage of high-level cirrus , which could Earth with τL = 0.2 is 288 K. reflect a higher proportion of incoming solar radiation. Then So,     A ⇒ A + dA dTsurface −0.1 0.1 288 4 = − or dTsurface = = 6.0 K 288 1.2 1.2 4 The Blue Equation reduces to Thus, the assumed increase in longwave absorbtivity has dTsurface dA ◦ 4 = − resulted in an increase in surface temperature of about 6 C. Tsurface 1 − A

11 12 Again, dT dA Sensitivity to Solar Constant 4 surface = − Tsurface 1 − A Suppose next that the solar energy flux varies. Suppose the albedo increases from 0.3 to 0.4. Then A = 0.3 This could be the result of a major solar anomaly, or due and dA = 0.1. to a secular or cyclic variation associated, for example, with Suppose once more that the equilibrium temperature of the the sun-spot cycle. Earth with A = 0.3 is 288 K. So, Then Fsolar ⇒ Fsolar + d Fsolar dT 0.1 0.1 288 4 surface = − or dT = − = −10.3 K The Blue Equation reduces to 288 0.7 surface 0.7 4 dT dF 4 surface = solar Thus, the assumed increase in albedo has resulted in an Tsurface Fsolar decrease in surface temperature of about 10◦C.

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Again, dT dF Review of Sensitivities 4 surface = solar Tsurface Fsolar • An increase in shortwave absorbtivity of 0.1 resulted in a Suppose the solar output increases by 1%. Then dFsolar/Fsolar = decrease in surface temperature of about 4◦C. 0.01. • An increase in longwave absorbtivity of 0.1 resulted in an Suppose once more that the equilibrium temperature of the increase in surface temperature of about 6◦C. Earth is 288 K. • An increase in albedo of 0.1 resulted in an decrease in Then surface temperature of about 10◦C.   dTsurface 0.01 Fsolar 288 4 = or dTsurface = 0.01× = 0.7 K • A 1% increase in solar flux has resulted in an increase in 288 Fsolar 4 surface temperature of less than 1◦C. Thus, the assumed 1% increase in solar flux has resulted in In general, all parameters undergo small changes. More- an increase in surface temperature of less than 1◦C. over, we have completely neglected the effects of water vapour and liquid water. The results give an indication of how difficult it is to gauge the consequences for climate of any changes which may oc- cur. 15 16