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For almost any form of the function a(T) between values that are constant when the planet is completely ice covered or completely ice free, this expression has at least three solutions for T.

flG. 11.1. A 'single-slab atmosphere' model of the greenhouse effect, in which the atmosphere is treated as a homogeneous layer of temperature Ta that is perfectly transparent to solar radiation and perfectly opaque in the thermal infra-red. The surface receives the equivalent of two solar constants, raising its mean temperature from 255 to 303 K.

flG. 11.1. A 'single-slab atmosphere' model of the greenhouse effect, in which the atmosphere is treated as a homogeneous layer of temperature Ta that is perfectly transparent to solar radiation and perfectly opaque in the thermal infra-red. The surface receives the equivalent of two solar constants, raising its mean temperature from 255 to 303 K.

While such a simple model has no great significance for the real Earth, it shows how multiple equilibria, some stable and some unstable, can arise in a system that has some attributes of the climate. In a more realistic, i.e. complicated, representation, there could be quasistable equilibria that are stable against small or short-lived perturbations but respond catastrophically to larger or more sustained forcing. There is some evidence from the geological record that Earth's climate has behaved in this way in the not-too-distant past, for instance, during the 'Younger Dryas' event around 12000 years ago.

11.2.2 Simple greenhouse models

The energy-balance model can be extended so that the surface temperature and the effective emitting temperature are calculated separately. The actual behaviour of the cloud-free atmosphere, which is largely transparent at wavelengths corresponding to most of the incoming solar energy, and largely opaque at those wavelengths at which the Earth emits most of its thermal energy, is approximated by a bimodal transmission function with t = 1 at wavelengths shorter than about 4 ¡m, and t = 0 at longer wavelengths. The flux of energy F from the Sun that is absorbed by the planet is the same as before, and all of it heats the surface. The outgoing energy to space, which in this model must be entirely from the atmosphere, since its opacity prevents the surface from radiating to space, must also be equal to F in order to achieve energy balance. However, the flux from the atmosphere occurs in both the upward and downward directions (Fig. 11.1), and so in equilibrium the surface receives a second contribution equal to that from the Sun, raising its temperature to 21/4x 255 or 303 K. This calculated greenhouse enhancement of 48 K is larger than the observed 33 K, not surprisingly in view of the simplicity of the model.

If the atmosphere were homogeneous in reality, this enhancement would be the

Table 11.1 Constants used to calculate model temperature profiles, and the radiative equilibrium, stratospheric and surface temperatures that result for present-day CO2 abundances.

Distance, Earth to Sun

r0

150000000

km

Radius of Sun

Rq

696000

km

Radius of Earth

Re

6380

km

Effective temperature of Sun

Tq

5777

K

Solar constant

Sq

1366

W m"2

Albedo of Earth

a

0.30

Effective temperature of Earth

Te

255

K

Temperature of stratosphere

Ts

215

K

Temperature of surface

T o

288

K

Pressure at tropopause

p*

227

mb

Tropospheric lapse rate

r

6.6

K km"1

Scale height

H

8.45 (surface)

km

6.35 (tropopause)

upper limit on what could be achieved. In fact, it is clear (from studying Venus, for example) that the greenhouse enhancement can be not one but many times the direct solar input. The effective emitting temperature of Venus is about 240 K, while the surface temperature is 730 K. The downward flux at the surface must therefore be of the order of (730/240)4 or about 85 times that at the top of the atmosphere. This becomes possible in more realistic models in which the lower atmosphere, still opaque in the infra-red, is warmer in its lower regions (which warm the surface) than its upper (which radiate to space). This means representing the atmosphere by many layers, rather than just one. If a large number of isothermal layers is specified, the temperature profile becomes quasi-continuous and a realistic representation can be achieved.

Alternatively, a model with a more accurate representation of the vertical structure of the atmosphere can be constructed very simply by assuming an adia-batic (convective) lapse rate in the troposphere (§11.3.2), and a radiative equilibrium temperature in the stratosphere (§11.3.3). Such a model is completely characterized by a fixed tropospheric lapse rate r = g/cp, a stratospheric temperature, related to the effective radiative temperature of the Earth by Ts = Te/21/4 = 255/21/4 « 215 K, and either a tropopause height z* or a surface temperature To (Table 11.1). The independent variables are the amount of infra-red absorber present in the atmosphere, and the mean albedo A of the Earth.

If the absorber amount in the atmosphere increases, the temperature at every level in the troposphere must move to a lower pressure to keep the same overlying optical depth and hence the same net emission to space. For the case where the amount of absorber is doubled, assuming the strong limit of absorption, see

Table 11.2 Model results for surface temperature (K) corresponding to different values of Earth's albedo a, and different atmospheric CO2 mixing ratios (pre-industrial 290 ppmv, present 367 ppmv, possible future 580 ppmv), assuming that the abundances of all greenhouse gases (including water vapour) vary in the same proportion.

Table 11.2 Model results for surface temperature (K) corresponding to different values of Earth's albedo a, and different atmospheric CO2 mixing ratios (pre-industrial 290 ppmv, present 367 ppmv, possible future 580 ppmv), assuming that the abundances of all greenhouse gases (including water vapour) vary in the same proportion.

Albedo

290 ppmv

367 ppmv

580 ppmv

0.25

288

294

306

0.30

284

290

302

0.35

280

287

299

§4.2.4, eqn (4.52), the temperature T(p) will now be found at pressure p', where p' = p/y/2. Since the tropospheric lapse rate g/cp must remain the same, there has to be an increase AT in the surface temperature in the model. Physically what is happening is that the depth of the convective layer increases to offset the greater opacity of the atmosphere, so that heat is still brought up to the level where it can radiate to space. The stratospheric temperature remains unchanged because, in the optically thin approximation, this depends only on the equilibrium temperature of the planet, T®.

If, however, the composition of the atmosphere is constant and the albedo of the planet changes, the temperatures at every pressure level must adjust to maintain an overall energy balance between the planet and the Sun. If the albedo increases, T® will decrease, according to eqn (11.3), and this requires the same increase in temperature at every level (including the surface) since the lapse rate must remain the same, at g/cp in the troposphere and zero in the stratosphere.

Table 11.2 and Fig. 11.2 show some calculated values for this scheme, which show that changes in the planetary albedo from 0.25 to 0.35 alter the surface temperature by about 7 K; while doubling the greenhouse-gas abundance has an effect that is more than twice as large, at 18 K. The implicit assumption has been made that doubling of CO2 is accompanied by doubling of all other greenhouse gases, including water vapour, since the simple model cannot separate the individual contributions (and indeed we have no way of knowing what change has taken place in mean concentration of the most important greenhouse gas, tropospheric water vapour, between pre-industrial times (left-hand column in Table 11.2) and the present (central column). Nor do we have much idea how the albedo of the Earth has changed during that period.

Suppose, however, purely as an exercise, that the water-vapour and other greenhouse-gas abundances do roughly track that of CO2 as the atmosphere and surface warm up. Suppose further that the increased temperature and humidity give rise to extra cloud cover that has increased the mean albedo of the planet, from 0.25 in 1760 to 0.3 today, and that the increase will continue until a value of 0.35 is reached when CO2 and the other gases reach twice their pre-industrial value later in the present century. Finally, account needs to be taken of the lag that the surface temperature experienced following greenhouse-induced changes, due to the thermal inertia of the climate system, primarily in the oceans. On a timescale of the order of a century, only about half of any change in forcing is reflected in the surface air temperature.

With all of these assumptions, the change in mean surface temperature of the Earth has increased by about 1 K since 1760, and will increase by a further 4.5 K in the twenty-first century. These figures are close to observations and IPCC 2001 predictions, respectively. Of course, we must recognize that this model is too crude to tell us with certainty what is really happening in the extremely complicated system that it represents. It can only give us a very general idea of how the climate might respond to known changes, and a feeling for how those responses might reinforce each other, or alternatively tend to cancel each other out.

The next step is to recognize that possible changes are linked to each other, not only in the sense that an increase in aerosol is expected along with an increase in CO2, because of the nature of the industrial processes involved, but because certain changes will force other quantities to change, with further consequences for the climate. An example already considered is where increased air pollution induces higher surface temperatures, resulting in increased evaporation of water from land and sea, which in turn results in higher atmospheric humidities and more rapid cloud formation. Assuming that more cloud means a higher planetary albedo, this is an example of negative feedback in the climate system.

Of course, water vapour is itself a powerful greenhouse gas. So, in the above scenario, the increased humidity of the atmosphere is tending to amplify any warming through adding water vapour to the atmosphere, at the same time as it might be reducing it through increased albedo due to clouds. The water-vapour enhancement is an example of positive feedback.

Another example of positive feedback would be the reduction in the solubility of carbon dioxide in water with temperature. It is known that a large fraction (about one third) of the CO2 released into the atmosphere is taken up by the ocean and does not contribute directly to global warming. As the ocean gets warmer, however, and as it gets saturated with CO2, it will tend to respond by adding, rather than removing, CO2 from the atmospheric greenhouse, thus amplifying the warming. To be useful as predictors of future climate change, models have to include these and other feedback processes in their computational schemes. What is difficult is to understand the physics of the feedback process in enough detail to produce a formulation that can be incorporated into the model. In the simple model just described, a parameterization was implicitly assumed where the formulae

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