Some basic solutions

When the scattering and absorption properties of the atmosphere are independent of t*, the two stream equations have simple exponential solutions. We'll begin with an elementary solution for conservative scattering, which provide quite useful estimates of the effect of clear-sky and cloudy atmospheres on the solar-spectrum albedo of planets. We shall specify an incoming direct-beam flux of solar radiation, and we seek the outgoing reflected flux at the same wavenumber. For conservative scattering, = 1. In that case 71 = y2. Then, Eq 5.28 becomes d-*(I+ — I_) = L- exp( — (t* — t*)/ cos Z) (5.32)

where C is a constant. This equation states that, for conservative scattering, the net of the diffuse flux and the vertical component of the surviving direct-beam flux is constant. As the upper boundary condition we require that the diffuse incoming radiation be zero; hence C = I+j00 — L- cos Z, and

I+ — I_ = I+, 00 + Lq cos Z(exp( — (t* — t*)/cos Z) — 1) (5.34)

Deep in the atmosphere, I+ — I_ becomes constant, and is equal to the difference between the top-of-atmosphere incoming minus outgoing flux, i.e. I+j00 — Lq cos Z. When the atmosphere is optically thick, or when cos Z is small (i.e. when the sun is close to the horizon) the exponential term is significant only near the top of the atmosphere, It represents the conversion of the direct beam into diffuse radiation by scattering, which occurs within a conversion layer of depth 1/ cos Z in optical depth units.

I+ 00 is the reflected flux we wish to determine, and we must close the problem by applying a boundary condition at the ground. For this we need I+(t*), which we obtain by using the equation for I+ + I_. Let's restrict attention to the symmetric scattering case, g = 0, for which 7+ = y_ and 71 = y2 = 7. Then

(I+ + I_) = — 2y(I+ — I_) = — 2y(I+,oo — Lq cos Z + Lq cos Zexp(—(t* — t*)/cos Z)) (5.35)

The solution which satisfies I_ = 0 at the top of the atmosphere, is

I+ +1_ = I+, 0 + 27(I+,oo — Lq cos Z)(t* — t*) + 27Lq cos2 Z(1 — exp(—(t* — t*)/ cos Z)) (5.36)

Let's suppose that the ground is perfectly absorbing. This calculation characterizes the albedo of the atmosphere alone. Later, we'll compute how much the surface albedo enhances the planetary albedo. If the ground is perfectly absorbing, then we require I+ = 0 at t* = 0. To apply the boundary condition, we add Eqns. 5.34 and 5.36 and evaluate the result at the ground. Applying the boundary condition and solving for I+j00 we find

(2 — Y cos Z )/#q + Yt* I+, ro = —-—---Lq cos Z = a„i0 cos Z (5.37)

where = 1 — exp(—t*/ cos Z); this quantity is the proportion of the direct solar beam that has been lost to scattering by the time the beam reaches the ground. The fraction multiplying L- cos Z in Eq. 5.37 is the planetary albedo. Since any flux reaching the ground is absorbed completely, this albedo is in fact the albedo of the atmosphere alone, which we will call aa. In the optically thin limit, « t*/ cos Z, so the albedo approaches zero like 2t*/ cos Z. Half of the small amount of flux scattered by the atmosphere exits the top of the atmosphere, but the other half is scattered into the ground, where it is absorbed.

As the atmosphere is made more optically thick, the albedo increases in two stages. The first stage is an exponential adjustment, as the direct beam is converted to diffuse radiation. Some simple algebra shows that the numerator of Eq. 5.37 always increases with t*, regardless of the value of cos Z. However, when the incident beam is relatively near the horizon, so that 7 cos Z < 2, the conversion term leads to an exponential increase of albedo with t*. The effect becomes more pronounced when the Sun is more nearly on the horizon. This effect comes from the direct scatter of the incident beam to space. When t* becomes appreciably larger than 1/cos Z, however, the direct beam has been completely converted, 3o « 1, and the albedo no longer varies exponentially. For large t*, the albedo approaches unity like 1 — (27-1 + cos Z)/t*0 . The rather slow approach to a state of complete reflection is due to multiple scattering. In contrast to the exponential decay of the direct beam, the diffuse radiation surviving to be absorbed at the surface decays only like 1/t* because much of the radiation scattered upward is latter scattered back downward. The exponential decay of the direct beam just represents a conversion to diffuse radiation,and therefore does not materially alter the conclusion that scattering is a relatively ineffective way of preventing radiation from reaching the surface. That is why one cannot rely on Rayleigh scattering alone to shield life at the surface from harmful ultraviolet radiation, despite the fact that the Rayleigh optical thickness of an Earthlike atmosphere is quite high in the ultraviolet.

The simple albedo formula given above has many physically important ramifications. Before computing the albedo for various conservatively scattering atmospheres, though, it is necessary to bring in the effect of asymmetric scattering if we are to deal with clouds, as scattering from cloud particles is strongly forward-peaked. A nonzero asymmetry factor simply adds a direct beam source term to the equation for d(I+ + U)/dr*, since 7+ — no longer vanishes. Some straightforward algebra shows that, allowing for a nonzero asymmetry factor, the albedo formula becomes

This differs from the symmetric scattering form only in that the optical thickness is multiplied by 1 — g, which reduces the effective optical thickness when g > 0. Thus, in the context of the two stream equations, the effect of asymmetric scattering is not very surprising or subtle. Since forward scattering just adds back into the forward radiation as if scattering had not occured at all, forward-dominated scattering simply has the effect of reducing the optical thickness of the atmosphere. The only reason one can't get by with simply redefining the optical depth is the presence of the direct beam; the direct beam transmission factor 3o is computed using the unmodified optical depth, rather than the rescaled optical depth, because even forward-scattered radiation is transferred out of the direct beam and into the diffuse component. The behavior of the albedo formula is explored in Problems ?? and ??.

In Table 5.4 we use Eq.5.38 to compute the albedos of a number of clear-sky and cloudy planetary atmosphere, based on optical depths computed from the Rayleigh scattering or Mie-scattering cross-sections in the visible spectrum. Results are shown for both the hemispherically isotropic and the Eddington approximations; the results differ little between the approximations for the symmetric Rayleigh scattering cases. In clear sky conditions, Earth's atmosphere reflects about 8% of the incoming solar energy back to space. This represents nearly a third of Earth's observed albedo, and is a significant player in the energy budget. The thick CO2 atmosphere postulated for Early Mars has an even more significant effect on albedo, reflecting fully 43% of the solar energy. Further increases in CO2 lead to even greater reflection, making it hard to warm Early Mars with the gaseous CO2 greenhouse effect alone. The case of Venus is particularly interesting. We see from the table that if the thick clouds of Venus were removed, Rayleigh scattering alone would be sufficient to keep the albedo high. The value in the table is an overestimate of the albedo Venus would have in the no-cloud case, since it ignores the solar absorption of CO2, but


Early Mars



Water cloud

Sulfate aerosol

T * ' 00














albedo, hemi-isotropic







albedo, quadrature







albedo, Eddington







Table 5.4: Albedos for purely scattering atmospheres. Values given for Earth, Early Mars, Venus and Titan are for hypothetical clear-sky atmospheres consisting of 1 bar of air for Earth, 2bar of CO2 for Early Mars, 90bar of CO2 for Venus and 1.5bar of N2 for Titan. The water cloud case assumes a path of 10 grams of water per square meter in droplets of radius 10 ^m, having a scattering efficiency of 2. The sulfate aerosol case assumes a path of 2 gram per square meter in sulfuric acid droplets of radius 1^m, having a scattering efficiency of 3. Albedos are computed at a wavelength of .5^m, with a zenith angle of 45o for the direct beam.

Table 5.4: Albedos for purely scattering atmospheres. Values given for Earth, Early Mars, Venus and Titan are for hypothetical clear-sky atmospheres consisting of 1 bar of air for Earth, 2bar of CO2 for Early Mars, 90bar of CO2 for Venus and 1.5bar of N2 for Titan. The water cloud case assumes a path of 10 grams of water per square meter in droplets of radius 10 ^m, having a scattering efficiency of 2. The sulfate aerosol case assumes a path of 2 gram per square meter in sulfuric acid droplets of radius 1^m, having a scattering efficiency of 3. Albedos are computed at a wavelength of .5^m, with a zenith angle of 45o for the direct beam.

it suffices to show that virtually all of what escapes absorption would be scattered back to space by Rayleigh scattering. This is important to the evolution of Venus-like planets, which might not have atmospheric chemistry that supports sulfuric acid clouds like those of Venus at present; for that matter, it is not completely certain that thick clouds are a perennial feature of our own Venus. When one factors in that rather little solar radiation penetrates the present thick clouds of Venus, it is evident that the strong Rayleigh scattering of the surviving flux implies that only a trickle of solar radiation reaches the surface of a planet. It is only a trickle, but as we have seen in Chapter 4, it is a very important trickle, since the surface could not be so hot if most of the solar energy were absorbed aloft.

The cloud cases in Table 5.4 could really apply to any planet with condensible water or sulfer compounds. The cloud parameters chosen are quite typical of Earth conditions. Scattering of solar radiation from cloud particles is quite different from Rayleigh scattering because of the strong asymmetry, which makes the albedo considerably lower than one would expect on the basis of optical thickness. Nonetheless, a small mass of cloud water, or a still smaller mass of sulfate aerosol in the form of micrometer-sized droplets, leads to a very significant albedo. To put the mass path of sulfate aerosol into perspective, we note that if we assume a 10 day lifetime for aerosol in the atmosphere, the assumed mass path is equivalent to a world wide sulfur emission of about 8 megatonnes/da, allowing for the proportion of S in H2SO4. Actual worldwide sulfur emissions for 1990 are estimated to have been more like 3 megatonne/da, which is why the albedo of Earth's sulfate aerosol haze is lower than the estimate in the table, though still a significant player in the radiation budget.

Note also that when the asymmetry is as pronounced as it is for cloud particles, the albedo predicted by the Eddington approximation is significantly greater than that for the hemispherically isotropic case. Although the asymmetry factor reduces the attenuation of diffuse radiation by the cloud, the decay of the direct beam is exponential in the optical depth itself, leading to near total attenuation of the direct beam by the water cloud. This is why a thick cloud looks bright, though you cannot easily discern the disk of the Sun. It is also why it is possible to get quite thoroughly sunburned on a cloudy day.

In the preceding calculation we assumed that the ground was perfectly absorbing. If the ground is instead partially reflecting, having albedo ag, it will reflect some of the light reaching the surface back upward. Some of this light reflected from the surface will make it through the atmosphere and escape out the top, increasing the planetary albedo. How does the albedo of the atmosphere combine with the albedo of the ground to make up the planetary albedo? Since we are assuming that there is no atmospheric absorption, if a proportion aa of incoming light is reflected by the atmosphere, then a proportion (1 - aa) reaches the ground. The proportion of this reflected back upward is (1-aa)ag, and if we were to simply add this upward proportion to the part reflected directly by the atmosphere we'd get a planetary albedo of aa + (1 - aa)ag. This simple estimate already illustrates the important result that putting a reflective (e.g. cloudy) atmosphere over an already reflective surface changes the planetary albedo much less than putting such an atmosphere over a dark surface - you can't make the planet whiter than white, as it were. The simple estimate overestimates the planetary albedo, though, since some of the upward radiation from the ground bounces back from the atmosphere whereafter some of the remainder is absorbed at the ground. The rest is reflected back upwards, and ever diminishing proportions remain to multiply scatter back and forth between the atmosphere and the ground. One doesn't need to actually sum an infinite series to solve the problem; all we need to do is to correctly specify the boundary condition on the upward radiation at the ground, which requires in turn a specification of the proportion of upward radiation which is reflected back to the surface by the atmosphere. The upward radiation reflected from the ground is, by definition, purely diffuse, so as a preliminary to this calculation we need the albedo of the atmosphere for upward-directed diffuse radiation. Since the radiation is purely diffuse, the form of this atmospheric albedo is somewhat simpler than the formula for incoming solar radiation. It is derived in Problem ?? , and is

Note that this has the same form as the expression for a, except that the direct beam term in the numerator, proportional to pr, has been dropped. In terms of a'a, the boundary condition on upward radiation at the ground is

I+(0) = ag • (I_(0) + L0 cos Z exp(-T^/ cos Z)) = a.g • ((1 - aa)Lr cos Z + a aI+(0)) (5.40)

which can be solved for I+(0). When this boundary condition is applied to the conservative two-stream equations, the resulting expression for I+, oo yields the following expression for the planetary albedo:

The details are carried out in Problem ??. When all three albedos, a, 'a and g are small, the expression reduces to the sum of the albedos a + g. Eq. 5.41 has very important consequences for the net effect of clouds on the planetary radiation budget. Clouds have both a warming and a cooling effect on climate. High-altitude clouds have a warming effect, since they strongly reduce OLR either by absorption and emission, or by scattering, of infrared radiation. Clouds at any altitude have a cooling influence, through increasing the planetary albedo in the solar spectrum. The net effect of clouds depends on how the competition between these two factors plays out. Eq. 5.41 shows that clouds increase the planetary albedo rather little, if they are put over a highly reflective surface (such as ice), or if they are put into an atmosphere which is already quite reflective (such as the dense atmosphere of Early Mars). In either case, introduction of clouds will tend to have a strong net warming effect, because the cloud greenhouse effect is relatively uncompensated by the cloud albedo effect. It is for this reason that clouds can greatly facilitate the deglaciation of a Snowball Earth, and that clouds of either water or CO2 can very significantly warm Early Mars.

Now let's do an infrared scattering problem, one that illustrates the scattering greenhouse effect in its simplest form. Consider an atmosphere made of a gas that is completely transparent

(hence also non-emitting) in the infrared. Suspended in the atmosphere is a cloud made of a substance such as CO2 ice or liquid methane that is almost non-absorbing in the infrared; we'll idealize it as being exactly non-absorbing, and assume the scattering to be symmetric. Since neither the gas nor the cloud emit infrared, the temperature profile of the atmosphere is immaterial. This atmosphere lies above a blackbody surface with temperature T. What is the OLR? This problem is also a case of conservative scattering (w° = 1), but with different boundary conditions. Since there is no incoming infrared, the upper boundary condition is I_ = 0. At the ground, the upward flux boundary condition is I+ = nB(v, T). Without any direct beam or blackbody source term, I+ — I_ is a constant, which is equal to the outgoing radiation I+j00. at the frequency under consideration. The equation for d(I+ +1_ )/-t* then tells us that I+ +1_ = 2nB — (1 + yt*)I+j00 . Finally, imposing the boundary condition that I_ = 0 at t* = t* , we conclude that

Hence, the infrared scattering reduces the outgoing infrared by a factor 1/(2 + yt*) relative to what it would be in the absence of an atmosphere. This increases the surface temperature of the planet in the same fashion as the OLR reduction from the more conventional absorption/emission greenhouse effect. However, the scattering greenhouse effect works quite differently, since it reduces the OLR regardless of whether the atmospheric temperature goes down with height.

Next we'll extend the preceding scattering greenhouse problem to allow < 1, so the atmosphere can absorb and emit. We'll assume the atmosphere to be isothermal at the same temperature T as the ground. In this case, I+ = I_ = nB(v, T) is a particular solution satisfying the boundary condition on I+ at the ground, though it does not satisfy the boundary condition I_ = 0 at the top of the atmosphere. We must add a homogenous solution to the particular solution, which cancels I_ at the top of the atmosphere for the particular solution, but leaves the bottom boundary condition intact. The homogeneous equation is obtained by taking the derivative of Eq. 5.28 and substituting the derivative of I+ + I_ using Eq. 5.29, dropping the source terms from both. Assuming and g to be independent of height, the homogeneous equation is then

The general solution to this is aexp(—K • (t* — t*)) + bexp(K • (t* — t*)), where

One term grows exponentially in optical depth, while the other decays exponentially. The solution for I+ +1_ is then obtained from the solution for I+ — I_ using Eq. 5.28, allowing us to obtain the two fluxes individually for use in applying the boundary conditions. First, the homogeneous solution we add in must not disturb I+ at the ground, since the particular solution already satisfies the boundary condition there. To keep the algebra simple, let's assume t* ^ 1. In that case, we approximately satisfy the boundary condition at the ground by taking the solution which decays toward the ground, i.e. b = 0. The boundary condition on I_ (0) then determines the value of the coefficient a. Carrying out the algebra and adding the homogeneous to the particular solution, we find the outgoing radiation to be i+,oo = 2-Y'(1 — -nB (5.45)

In this equation, nB is the emission the planet would have in the absence of an atmosphere, and the the coefficient multiplying it gives the reduction in emission due to the atmosphere. Note that for a non-scattering atmosphere, the isothermal atmosphere assumed would have no effect whatever on the outgoing radiation. In contrast to the conservative scattering case described by Eq 5.42, the emission in the partially absorbing case does not approach zero in the optically thick limit, but rather approaches the nonzero value given by Eq. 5.45. When there is no scattering, i.e.

= 0, the atmosphere should have no effect on emission, and this limit shows the shortcomings of the Eddington approximation. For wo = 0, the factor reducing the emission is 2y'/(y' + %/Yl'), which reduces to unity only when y = y' . Both the quadrature and the hemispherically isotropic assumptions satisfy this requirement, but as we have seen before, the Eddington approximation gives the wrong answer when scattering is weak, and is not suitable for such cases. For any of the approximations, as the scattering is made stronger relative to absorption, wo ^ 1 and the emission goes to zero in proportion to ^J(y/y')(1 — ^o)/(1 — g).

Finally, we'll use an elementary solution to show how scattering affects the vertical distribution of solar absorption. We'll suppose that thermal emission is negligible at the frequency under consideration, as is the case for solar radiation on planets at Earthlike (or even Venus-llke) temperatures. The basic idea is that scattering increases the net path traveled by radiation in going from one altitude to another, because the radiation bounces back and forth many times rather than proceeding in a straight line. This allows more radiation to be absorbed within a thinner layer, as compared to the no-scattering case. At the same time, however, scattering reflects some radiation back to space before it has any opportunity to be absorbed at all. As will be shown in the forthcoming derivation, the net result is to reduce the solar absorption while at the same time concentrating it more in the upper atmosphere, as compared to a no-scattering case with the same distribution of absorbers.

The full problem with arbitrary surface albedo, optical depth, and asymmetry factor is analytically tractable so long as wo is constant. The full solution is somewhat unwieldy, however, so we'll now make a few simplifying assumptions. To keep the algebra simple, in the present discussion we'll assume t* to be very large, so that the lower boundary does not affect the solution. The atmosphere is effectively semi-infinite (a top but no bottom) in this solution. We'll also assume that the asymmetry factor vanishes. The solution begins with taking the derivative of Eq 5.29 and substituting from Eq. 5.28, which gives us d2 (I+ + I-) = —K2(I+ + I-) — 2y^oLq exp( —(t* — t*)/cosZ)) (5.46)

where K is defined by Eq. 5.44 with g set to zero. A particular solution to this equation is

2Y^oLQ cos2 Z

to which we have to add superpositions of the two homogeneous solutions exp(±K(t* — t*)) so as to satisfy the boundary conditions at the top and bottom of the atmosphere. So far the only assumptions we have used are that wo is constant, the thermal emission is neglected, and g = 0. Since the atmosphere is semi-infinite, the only admissible homogeneous solution is aexp(—K(t* — t*)), since the other solution blows up deep in the atmosphere. It remains only to determine a, which is done by applying the condition I- = 0 at the top of the atmosphere. To do this we need I+ — I-. This is obtained from Eq 5.29, which takes on a particularly simple form when g = 0. Using the value of a thus obtained, the net vertical diffuse flux is found to be t t r K 1 + 2ycosZ , , * *,,

I+ — = [ — 2fTK 1 — K2 cos2Zexp(—K(T* — T )) (5 48)

Since I— = 0 at the top of the atmosphere, the albedo is obtained by evaluating this expression at t* = r* and taking the coefficient of the incoming flux Lq cos Z. Thus,

In the absence of scattering, all the incident flux should be absorbed no matter how low the concentration of absorbers, since the atmosphere is assumed infinitely deep. Consistently with this reasoning the above albedo approaches zero as uo ^ 0. For small uo, the albedo increases linearly with uo. It continues to increase monotonically as uo is further increased. In the limit uo ^ 1 where scattering becomes very strong, a ^ 1 and the atmosphere becomes perfectly reflecting; radiation is scattered back to space before it has much opportunity to be absorbed in the atmosphere.

Though strong scattering reduces the opportunity for absorption, it also reduces the depth scale over which the small amount of absorbed radiation is deposited in the atmosphere. The reason is that scattering increases absorption through multiple reflections that increase the path length. To get a better handle on what is going on, we need to examine the vertical profile of the flux as uo ^ 1 while holding the concentration of absorbers fixed. Taking the limit this way would correspond, for example, to looking at ultraviolet absorption as we increase the amount of conservatively scattering cloud particles in an atmosphere while keeping the amount of ultraviolet-absorbing ozone fixed. This is equivalent to writing t* — t* = (k/(1 — omegao))(p/g), if we neglect pressure broadening, since (1 — uo)At gives the absorption in a layer of thickness At . It follows from this expression for optical thickness that all the direct beam flux is converted to diffuse flux in a very thin conversion layer if uo ^ 1. Below the conversion layer all flux is diffuse, and the net vertical flux is then

I+ — I- = —^—~f7 z-t--- exp(—K---)uoLq cos Z

~ */ —. (1 — 2ycosZ)V1 — Uo exp(—2^/YY' h — )Lq cosZ

Hence the flux which manages to penetrate into the atmosphere is absorbed over a layer depth which scales with a/1 — uo, and approaches zero as uo ^ 1. It follows also from Eq. 5.50 that the heating rate d(I+ —1- )/dp remains order unity in this limit, though it becomes concentrated in a thinner and thinner layer near the top of the atmosphere.

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