Water Vapor Multiplier of Carbon Dioxide

Transcription

1 Water Vapor Multiplier of Carbon Dioxide Harvey S. H. Lam October 10, 007 Abstract The temperature rise of the earth due to the direct greenhouse effects of atmospheric carbon dioxide can be amplified by some indirect consequences such as water vapor feedback. The IPCC reports provide climate sensitivity numbers to quantify the total temperature rise driven by atmospheric carbon dioxide. How big a water vapor multiplier does the IPCC numbers imply? This question is quantitatively examined in some detail, and the relevance of some observational data is discussed. 1 Introduction The earth receives radiant energy from the sun, and its long term energy balance is maintained by its own radiation outward into space. After appropriately averaging over the horizontal spatial scales and the short time scales, the earth s long term vertical energy balance equation per unit area is: F in solar = F out total(t ) = σ ɛ total (T ;...) T 4, (1) where Fsolar in is the incoming solar flux (after properly accounting for the earth s albedo), Ftotal out is the outgoing earth s flux, σ = is the Stefan Boltzmann constant, and T (K) is the earth s surface temperature. The factor ɛ total (T ;...) represents all the complications of the earth s natural climate dynamics including the direct radiation absorptions by atmospheric Professor emeritus, Department of Mechanical and Aerospace Engineering, Princeton University. lam@princeton.edu 1

2 October 10, 007 carbon dioxide which concentration has been steadily increasing since the industrial revolution. It is convenient to explicitly single out the the direct effects of anthropogenic carbon dioxide absorption and rewrite Eq.(1) as follows: F in solar = F out total(t ) = σ ɛ total (T o + T )(T o + T ) 4 F out CO, () where T is the total temperature rise needed to balance FCO out the amount of radiant flux which fails to escape the earth because of direct absorption by atmospheric carbon dioxide. Both numbers are offsets from their preindustrial revolution values. Note that T here includes both the direct and the indirect consequences of FCO out such as water vapor feedback, clouds, ice and snow, impacts on ocean circulations, etc. It is assumed that ɛ total (T ) does not explicitly depend on atmospheric carbon dioxide concentration. Thus ɛ total (T ) is now a catch-all term which carries the full responsibility for all the indirect consequences which could be induced by a change in temperature T. The impact of the earth s albedo can also be formally absorbed by ɛ total (T ) but no attention is given to this issue here. The science of global warming is essentially the study of FCO out and ɛ total (T ). The IPCC Numbers In recent years, the Intergovernmental Panel on Climate Change (IPCC) has issued several well-publicized reports [1,, 3]..1 The IPCC F out CO In its 001 and 007 reports, IPCC recommends the following simplified expression for FCO out which is directly attributed to anthropogenic carbon dioxide absorption [, see its Table 6.]: F out CO 5.35 ln X X o watts/meter, (3) where X is the atmospheric carbon dioxide concentration (and X o is its preindustrial revolution value, 85 ppm). Eq.(3) was first reported by Myhre et. al. in 1998 [4]. It is also possible to derive it analytically [5].

3 October 10, The IPCC Climate Sensitivity The IPCC recommendation for T is [1]: T = β ln ln X X o, (4) where β is the earth s equilibrium climate sensitivity the total surface temperature rise T (above the pre-industrial revolution value) when the atmospheric carbon dioxide concentration X reaches twice its pre-industrial revolution value. The IPCC s likely range for β is between C and 4.5C; its very unlikely range for β is below 1.5C; its best estimate is β 3C. 3 Multiplier of Indirect Effects Using IPCC s Eq.(3) for FCO out in Eq.(), assuming Fsolar in constant and doing some linearization algebra, one obtains : fractional FCO out change {}}{ F out CO ɛ total (T o )σt 4 o T o T = ln X, (5) 4 + E(T o ) 4 + E(T o ) ɛ total (T o )σto 3 X o where E(T o ) is dimensionless and is shorthand for: ( ) d ln ɛtotal E(T o ). (6) d ln T o Comparing Eq.(4) and Eq.(5), one obtains a formula for β(t ): ln 5.35 β(t o ) = C. (7) 4 + E(T o ) ɛ total (T o )σto 3 Using T o 0C = 93K and plugging in numbers, one has: β(t o ) = multiplier {}}{ µ(e) direct {}}{ 0.65 ɛ total (T o ) C. (8) where the dimensionless µ(e) is the multiplier of indirect effects: µ(e) E(T o ). (9) The direct term depends only on IPCC s Eq.(3) and ɛ total (T o ). See [5]. The value of β(t o ) depends on both the values of E(T o ) and ɛ total (T o ).

4 October 10, Radiation Physics Planck s black body radiation intensity is denoted by B λ (T ): πhc B λ (T ) = λ 5 (exp( hc ) 1) watts/meter per wave length, (10) λkt where λ is wave length (meters), h is Planck s constant, c is the speed of light, and k is the Boltzmann constant. The Wien Displacement Law is: λ peak T = meter K (11) where λ peak is the wave length at the peak of the black body curve. Fig. 1 shows B λ (T ) versus λ for T = 15C = 88K and 0C = 93K. It is seen that both peaks are near λ peak 10 5 meters, or approximately 10 microns. Carbon dioxide has three absorption bands with wave length λ below 5 microns. For the earth, these bands can be ignored because Fig. 1 says their values are quite small. There is another absorption band between 13 and 17 microns which is optically thick at sea level. When water vapor, methane and nitrous oxide are included, the earth s atmosphere is essentially transparent to radiations between 7.5 microns and 13 microns except for a narrow spike of ozone absorption just below 10 microns [6, see its Fig. 1(b)]. The earth s B λ (T o ) peak is inside this transparent window. For wave lengths longer than 17 microns, there are many water vapor absorption bands (and thus some minor semi-transparent windows). See Fig Estimating ɛ total (T o ) ɛ total (T o ) is the earth s pre-industrial revolution effective emissivity the fraction of the total black body radiation flux (at T o ) that had managed to escape the earth. This number can be computed from actual historical measurements. The usual quoted values range from 0.6 to 0.8. Since the direct factor in Eq.(8) is roughly unity, the value of earth s climate sensitivity β(93k) is thus numerically close to the value of her multiplier of indirect effects µ(e). 4. Estimating E(T o ) E(T o ) would be zero if the atmosphere were completely transparent to all radiations i.e. ɛ λ = 1 for all wave lengths. Assume, for the sake of simplicity, that the atmosphere has only one single dominant transparent window

5 October 10, (i.e. ɛ λ 1) located between λ microns and λ 13 microns (ignoring the narrow ozone spike). Outside this window, ɛ λ 0. 1 Then ɛ total (T ; λ 1, λ ) is simply: ɛ total (T ; λ 1, λ ) 1 λ B σt 4 λ (T )dλ. (1) λ 1 Introducing ξ = λt as a convenient variable in place of λ, one obtains: ɛ total (T ; λ 1, λ ) 1 σ ξ =λ T ξ 1 =λ 1 T B(ξ)dξ, (13) where B(ξ) as given below now has no explicit temperature dependence: B(ξ) πhc ξ 5 (exp( hc (14) ) 1). kξ It is duly noted that 0 B(ξ)dξ = σ so that ɛ total (T ; 0, ) = 1. Thus the temperature dependence of ɛ total (T ; λ 1, λ ) with fixed λ 1 and λ comes only from the integration limits in Eq.(13). The fraction of outward total radiation flux absorbed (outside this single transparent window) by the atmosphere is 1 ɛ total (T ; λ 1, λ ) > 0. Given any fixed width transparent window, E(T ; λ 1, λ ) can be computed by taking the logarithmic differential of Eq.(13) with respect to T while λ 1, λ are held fixed. One obtains (using straightforward calculus): ( ) ln ɛtotal E(T ; λ 1, λ ) = (ξb(ξ)) ξ=λt (ξb(ξ)) ξ1=λ1t ln T λ T λ 1,λ λ 1 T B(ξ)dξ. (15) Since ξ > ξ 1 always, it is seen that B(ξ ) B(ξ 1 ) is a sufficient condition for E(T ; λ 1, λ ) > 0. This is in fact the case for the earth at T o 93K in the 7.5 microns to 13 microns transparent window. The numerator of Eq.(15) can be interpreted graphically as the difference of the areas of two rectangles which upper right corners are pinned to the B(ξ) curve. An ad hoc estimate of E(93K; λ 1, λ ) for this major transparent window is around +1. Fig. 1 shows clearly that B λ (T ) responds more vigorously to a T perturbation (when T o 90K) in this transparent window than elsewhere. 1 Even when the atmosphere is optically thick at sea level, ɛ λ inside the absorption band is finite because at sufficiently high altitudes outward radiations always manage to escape the earth. Since the temperature there is much lower, the amount of energy flux involved is small.

6 October 10, The next semi-transparent window for the earth is located on the longer wave length side of the B λ (93K) peak, and thus its E(T ; λ 1, λ ) is negative. Using the Rayleigh-Jeans Law for long wave lengths, one can analytically show that E(T ; λ 1 >> λ peak, ) 3. The effective E(T o ) for multiple transparent windows is simply the weighted sum of the individual E(T ; λ 1, λ ) s the weighting factor being the relative contribution of each window to the total integral. Since the 7.5 to 13 microns window is the major transparent window for the earth, the earth s effective multi-transparent windows E(T o ) is expected to be nearer +1 than 3. The value (and sign) of the multiple transparent window s effective E(T o ) is critical to the value of β(t o ). 4.3 Water Vapor Feedback The above estimate of E(93K) when used in Eq.(8) would not give earth s β in the IPCC s likely range. The bottom line is that the IPCC-endorsed climate sensitivity needs earth s multiplier µ to be between to 4, or her E(93K) to be between and 3. If E(93K) were positive, the earth s multiplier µ would be less than unity. Water vapor feedback which was not included in the above fixed width transparent windows estimates would modify the ɛ λ s and push E(93K) in the negative direction: more absorption in unsaturated water vapor bands, widening of already saturated water vapor bands, albedo affected by clouds and ice, etc. Could these and other indirect effects [7] overwhelm the firmly positive (radiation physics) contribution of the 7.5 to 13 microns dominant transparent window to get E(93K) into the and 3 range? 5 Observations The earth s actual responses must include all direct and indirect effects. 5.1 Earth s Historical Experience The current carbon dioxide atmospheric concentration is roughly one-third the way to doubling of the pre-industrial revolution value. The earth s average surface temperature has risen roughly 0.7C since the industrial revolution. If all of the observed historical temperature rise is attributed to the

7 October 10, increase of carbon dioxide in the atmosphere (33% increase), then β 1.85C, which is slightly below the likely range and slightly above the very unlikely range of the IPCC recommended numbers. The IPCC s likely range of climate sensitivity is consistent with the observed warming and emissions data in the last few decades of the Twentieth Century. However, roughly the same warming rate was also observed in the first few decades of the same century when the annual amount of carbon dioxide then being emitted into the atmosphere was much smaller. 5. A Grand Experiment What would happens if we move the earth further away or closer to the sun as an experiment? The water vapor multiplier must come into play via the natural climate dynamics of the earth! The earth s orbit is known to be elliptical, and the ellipticity is approximately 3%. So, near the equator, the amount of incoming solar radiation in January (perihelion) is approximately 6% more than that in July (aphelion). Averaging over days and nights, Fsolar/F in solar in is roughly It is duly noted that the incremental radiative forcing by the doubling of atmospheric carbon dioxide is estimated to be approximately 5.35/(σ ɛ total To 4 ) 0.013/ ɛ total, a smaller number than Under the very drastic assumption that the energy transport in such short time scales (and very limited horizontal spatial domains) is still dominated by vertical radiant energy balance, the theoretical temperature at Singapore (which is located very near the equator; mean T 30C) in January should be about.c hotter than July using E total = 0. This disagrees with the actual recorded Singapore data which says that its July is 1C warmer than its January. The disagreement can of course be attributed to the stronger role played by horizontal energy transport for such small horizontal domains and short time scales ( probably the obliquity of the earth s rotation also plays a role). Nevertheless, the Singapore data is not consistent with the proposition that the indirect water vapor feedback could amplify the direct radiative forcing by a factor of to 4. Such multipliers would raise the theoretical January/July Singapore temperature difference to 4 to 8 C in the wrong direction. The data suggests the multiplier is probably close to unity in Singapore. (there is no dispute that water vapor feedback would amplify globally).

8 October 10, Planets in the Solar System Let R denote the distance of a planet from the sun. Setting Fsolar in 1/R and E 0, ɛ total constant for all the planets in the solar system, one can easily recover from Eq.() the textbook result that the surface temperatures of the planets in the solar system are proportional 1/R 1/. This well-known R-scaling has been amply confirmed (on a log-log plot) observationally. The glaring discrepancies are the earth and Venus both observed surface temperatures are significantly higher than the theoretical temperature. Both discrepancies can be explained by the greenhouse effect. The atmosphere of Venus is 95% carbon dioxide, has very little water vapor, and its surface pressure is about 90 times that of the earth. At Venus observed temperature 740K, λ peak (740K) is just below 5 microns. The transparent carbon dioxide window (between 5 and 13 microns) is now on the longer wave length side of the B λ (740K) peak. Thus, unlike the earth, Venus E(740K; λ 1, λ ) for this transparent window is expected to be negative. Thus Venus needs a much higher surface temperature rise than the earth to radiate out the same amount of radiation energy through her transparent window without help from indirect effects. The amount of carbon dioxide in the Venus atmosphere is irrelevant to the amount of warming so long as her atmosphere is already optically thick (at ground level) in the relevant absorption bands. 6 Summary The value of E(93K) is all important. The factor 4+E(93K) is the earth s effective Stefan Boltzmann temperature exponent, and 1/( E(93K)) is her multiplier of indirect effects. Unlike carbon dioxide, water vapor in the earth s atmosphere is not well mixed either spatially or in time. Instead of direct estimates which would involve a lot of statistics the present analysis points out that water vapor feedback would need to overwhelm a significant positive contribution to E(93K) coming from the specifics of the radiation physics of the main transparent window of the earth s atmosphere. The multiplier of indirect effects, being the response of the natural climate dynamics of the earth, must also operate on perturbations from any other external radiative forcings. Whenever one contemplates the possibility of the earth s µ(e) being in the range of + to +4, one should also consider its prospects of quantitative consistency with available observational data.

9 October 10, References [1] Climate Change 007, The Physical Science Basis by WG I, and Mitigation of Climate Change by WG III. Cambridge University Press, 007. [] Climate Change 001, The Scientific Basis by WG I. Cambridge University Press, 001. [3] Houghton, J.T., Jenkins, G. J., and Ephraums, J. J. (Ed), Climate Change 1990: The IPCC Scientific Assessment, Cambridge University Press, [4] Myhre, G., Highwood, E. J., Shine, K. P. and Stordal, F., New Estimates of Radiative Forcing due to Well Mixed Greenhouse Gases, Geophysical Research Letters, 5, No. 14, pp , [5] Lam, S. H., The Logarithmic Response and the Equilibrium Climate Sensitivity of CO, Submitted for publication [6] Goody, R. M. and Robinson, G. D., Radiation in the Troposphere and Lower Stratosphere, Quart. J. Meteorol. Soc. 77, issue 33, pp [7] Philipona, R., Anthropogenic Greenhouse Forcing and Strong Water Vapor Feedback Increase Temperature in Europe, Geophysical Research Letters, 3, 005.

10 October 10, Figure 1: Planck s Blackbody Radiation versus Wave Length for the Earth

11 October 10, Figure : From Goody and Robinson [6]. The 93K curve in (a) is shifted slightly to the left of the 50K curve, so that it peaks at λ 10 microns. In (b), the fully saturated absorption to the right of the bump near 0 microns is due to water vapor. The atmospheric absorption properties below 5 microns are not important for T 93K