3tx dHx 8tx dKx btx

On using the Eddington approximation we obtain

with thus and

4n dJx(rx)

3 drx

Substituting for the source function for coherent and isotropic scattering we have l^f1 = C1 " ^)(Mrx) ~ Bx(rx)), (3.104)

which has the form of a steady-state diffusion equation. The above equation can then be solved for the mean radiance J by any standard technique given the boundary conditions at the surface and deep in the atmosphere (see Chapter 6).

Another approach for removing the need to solve the transfer equation without explicitly taking into account the directional dependence of the radiation field is the Schuster-Schwarzschild or two-stream approximation. Here, the radiation field can be anisotropic but only two directions are considered, outgoing and incoming directions, corresponding to the radiances I + and Irespectively. For clarity we do not include the subscript for wavelength. If the scattering is symmetric, as in the case of Rayleigh scattering by atmospheric molecules, then the mean radiance J = \ (I+ +) and the net outgoing flux is given by f = n(I + — I-). For solar radiation absorbed and scattered by cloud particles, the Mie scattering is highly asymmetric. In both cases the source function is not isotropic (Chapter 6). However, in the two-stream approximation we consider two isotropic components of the source function for the outgoing and incoming directions, S+ and S-, respectively.

If we consider the scattering of visible solar radiation in the atmosphere, then the source function has no thermal emission component and the two components can be written as

where a, = ^(1 + g) is the fraction of the radiation that is scattered in the initial direction of the photons, i.e. scattered forwards (Sagan and Pollack 1967,

Irvine 1968). For Rayleigh scattering, g = 0 and the parameter a takes the value while for Mie scattering with g = 0.85, for example, a, = 0.925. In the original Schuster-Schwarzschild approximation g = 0 and w = 1, pure scattering, resulting in the two transfer equations

or and dl-

or with ¡j,e = ■j to allow for the effect of non-vertical radiation transfer on the optical depth, and we note that je is negative for incoming (downwelling) solar radiation. For asymmetric scattering Sagan and Pollack (1967) modified the two-stream approximation by introducing the above parameter a to preserve anisotropies and set ¡j,e = as suggested by Chandrasekhar (1960). Another approach is to set je = Jo where jo is the cosine of the angle at which the photons are incident on the scattering layer. The above equations for I + and I- are coupled linear first-order differential equations with constant coefficients and can be solved in closed form by well-known techniques. Sagan and Pollack give the solution of these equations for w =1 in terms of the fractions of the incident radiation reflected, R, transmitted, T, and absorbed, A, by such a scattering atmospheric layer. These fractions are

with the fraction absorbed, A =1 — R — T and where

and the effective optical depth is given in terms of the actual vertical optical depth t of the scattering layer by

For a purely scattering (w = 1) layer A = 0 and T = 1 — R, where R is given by

3.5.8 General solutions for upwelling radiation

We now consider the general solution of the plane-parallel equation of transfer

for upwelling radiation when the source function is a known function, S(tx), of the optical depth. If we multiply both sides of the equation by the integration factor, e-Tx/^, then the transfer equation can be integrated directly given a boundary condition. The equation of transfer then has the form

otx M

Let us consider an atmospheric layer whose base is defined by the optical depth t2X and top is defined by tix, as shown in Fig. 3.7. Integrating both sides of the above equation from tix to t2X with the boundary condition that the radiance at the base of the layer is IX (t2X, m), then the upwelling (m > 0) radiance at the top of the layer is given by

1 fT2X

h(rlx,p)=Ix(T2X^i)e-^-T^^ + - / S(tx)e-^-T^^dtx, (3.116)

where t2X > r\X. The first term on the right-hand side of the above equation represents the transmitted radiance incident at the base of the layer, while the second term represents the emission from the layer itself. If we now set tix = 0 and t2x ^ <x>, then the emergent or upwelling radiance at the top of the atmosphere (TOA) that arises from the emission of an optically thick atmosphere is given by

The upwelling radiance at TOA is thus the Laplace transform of the atmospheric source function. If, for simplicity, we take the source function to be a linear function of the optical depth, S(tx) = a + bTX, then the upwelling radiance Ix (0, )) = a+b) = S()). This is known as the Eddington-Barbier relation. Thus, the upwelling radiance at TOA is characteristic of the source function within the atmosphere where tx/) = 1. If the atmosphere is in local thermodynamic equilibrium (LTE) (valid at sufficiently high pressures) then the source function is given by the Planck function, Bx(T), and the above results apply to the thermal radiation emitted by the atmosphere. For a planetary atmosphere this corresponds to thermal infra-red radiation.

At some level, tx, within a planetary atmosphere, we can obtain solutions for the mean radiance, Jx, and net upward radiation flux, fx, if the source function is a known function of the optical depth. The mean radiance for a plane-parallel atmosphere is given by

The upwelling radiance at tx, arising solely from atmospheric emission, can be written as

The upwelling radiance at tx, arising solely from atmospheric emission, can be written as

where ) > 0 and the downwelling radiance can be written as where ) < 0. The mean radiance is then given by

Performing the integration over all angles gives

The net upwelling flux is given by f 1

-i and can be calculated from fx(rx) = 2n j ^ Ix(tx, (3.123)

f ro r- rx fx(rx) = 2n Sx(tx)E2(\tx-Tx\)dtx-2W Sx(tx)E2(\tx-Tx\)dtx. (3.124)

For a planetary atmosphere with an emitting surface, the net upwelling flux at optical depth, tx, includes a component from the surface (here taken as a blackbody at temperature Tg) and so r ts*

fx(Tx) = 2nBx(Tg)E3(\tsx - tx\) + 2n Sx(tx)E2(\tx - Tx|)dtx

3.5.10 Grey atmospheres in radiative equilibrium

In radiative equilibrium the emission and extinction processes are balanced fdtij dA[^x - X-Tx]=0. (3.126)

Neglecting scattering, the source function will be isotropic and we can write fOO roo

and if we can replace the wavelength-dependent absorption coefficient, kx, by a grey or wavelength-independent value, k, then S = J, where

and the transfer equation in terms of the grey optical depth, t, has the form

which has the general solution

1 fro

known as the Milne equation and can be solved through iteration. In planetary atmospheres, where we consider thermal radiation, a common approach to obtaining a wavelength-independent absorption coefficient is to use the Planck mean absorption coefficient, k, defined by fOO fOO

which gives the correct thermal emission.

Under the Eddington approximation, the mean intensity, J, is related to the net upwelling flux, f, for a grey atmosphere by

3 dr where df and

For an atmosphere that is in radiative equilibrium J = S and hence f is constant and so

At the upper boundary, t = 0, it can be shown that under the Eddington approximation, f = 2nJ so that

If the atmosphere is also in local thermodynamic equilibrium (LTE), then S = B where b(t(t)) = / BxdX = -aT4, (3.137)

J o n and J = B for both LTE and RE. We can now solve for the temperature structure of the atmosphere under these conditions. At the upper boundary we have fo = 2nJo = 2nBo so that

The effective temperature of the planet-atmosphere system is defined by f = nB(TeS) = 2nB(To), (3.140)

where To is called the atmospheric skin temperature, related to the effective temperature through

For the Earth, Teff « 255 K so that To « 214 K, corresponding to the temperature of the tropopause. Thus, as t ^ 0 the atmospheric temperature becomes isothermal and approaches the skin-temperature value. We also note that the effective temperature of the planet-atmosphere system is located near where t = 2/3. Near the Earth's surface convective equilibrium determines the temperature structure and not radiative equilibrium. These concepts are discussed in more detail in Chapters 2 and 11.

3.5.10.1 Limb-darkening law Since J = S we can use eqn (3.136) for the source function in eqn (3.117) to obtain for the emergent radiance at zenith angle, z, where j = cos z, so that

so that the ratio of the emergent radiation in the direction j and that emerging normally (j = 1) to the atmospheric surface is

Thus, the ratio of the emergent radiance from near the limb (j = 0) of the atmosphere (i.e. from layers near the top of the atmosphere) and that emerging normally from the atmosphere (i.e. from deep in the atmosphere) is |. We remember that the optical depth in the direction of the radiation is t/j.

3.6 Bibliography

Atmospheric radiative-transfer theory had its origin in early work on stellar atmospheres, as described by Eddington and more recently by Mihalas. The book by Chandresekhar is the classical reference for the mathematics of radiation transfer, and gives analytic methods for the solution of the radiative-transfer equation that were particularly important in the days before digital computers. The Goody and Yung book, itself an update of R.M. Goody's classic text dating from 1964, is solidly based in the physics and chemistry of atmospheric radiation, and is a standard reference for approximate methods such as spectral-band models that took advantage of early computers of limited power. The book by Liou is recommended for its treatment of radiation in cloudy atmospheres.

3.6.2 References and further reading

Chamberlain, J. W. and Hunten, D. M. (1978). Theory of planetary atmospheres. Academic Press, New York.

Eddington, A. S. (1926). The internal constitution of the stars. Cambridge University Press, Cambridge.

Chandrasekhar, S. (1960). Radiative transfer. Dover Publications Inc., New York.

Goody, R. M. and Yung, Y. L. (1989). Atmospheric radiation. Oxford University Press, Oxford.

Irvine, W. M. (1968). Multiple scattering by large particles. Astrophys. J., 152, 823-834.

Liou, K. (1980). An introduction to atmospheric radiation. Academic Press, New York.

Mihalas, D. (1978). Stellar atmospheres. W. H. Freeman and Company, San Francisco.

Sagan, C. and Pollack, J. (1967). Anisotropic non-conservative scattering and the clouds of Venus. J. Geophys. Res., 72, 469-477.

Vardavas, I. M. (1976). Non-Isotropic redistribution effects on spectral line formation. J. Quant. Spectrosc. Radiat. Transfer, 16, 1-13.

Vardavas, I. M. (1976). Redistribution effects on line formation in moving stellar atmospheres II. J. Quant. Spectrosc. Radiat. Transfer, 16, 781-788.

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