The atmosphere can be considered to be a mix of particles, some of which absorb, some of which scatter, and some of which do both. The particles could be molecules, or they could be macroscopic particles of a condensed substance, as in the case of cloud droplets or dust particles. One builds up the absorbing and scattering properties of the atmosphere as a whole from the absorbing and scattering properties of the individual particles. In keeping with the usage in the preceding chapters, we will ultimately characterize the effect of atmospheric composition on radiation in terms of scattering properties per unit mass of atmosphere, just as we did for the absorption coefficient.

Consider a parallel, monochromatic (single-frequency) beam of light with flux F in W/m2 travelling in some specific direction. When the beam encounters a particle of finite extent, a certain

amount of the flux will be absorbed, and a certain amount will be scattered into other angles. The rate at which energy is taken out of the beam by absorption and scattering can be characterized in terms of coefficients with dimensions of area, which are known as cross-sections. The rate of energy absorption is Fxabs, where \abs is the absorption cross-section, and the rate of energy scattering into other directions is Fxsca, where xsca is the scattering cross-section 1. The scattering and absorption cross-sections can be quite different from the actual cross-section area of the object. The cross-sections can be thought of as the cross-section areas of hypothetical equivalent objects which absorb or scatter all light hitting the object while leaving the rest of the beam to pass by undisturbed. The ratio of scattering cross-section to the actual cross-section area of the scatterer is called the scattering efficiency, Qsca. For a spherical particle of radius r, Qsca = xsca/(nr2). The absorption efficiency is defined similarly. For spherical particles, the cross sections are independent of the angle at which the radiation is directed at the particle. For non-spherical particles the cross-sections for an individual particle depend on angle, but the typical physical situation involves scattering off of an ensemble of particles presented with random orientations. In this case, we can average over all orientations and represent the mean scattering or absorption in terms of the cross-section for an equivalent sphere. This approach can break down if particles are not randomly oriented, as can be the case for plate-like ice crystals that become oriented through frictional drag forces as they fall. The single-scattering albedo for a particle is the ratio of flux of the incident beam lost via scattering to net flux lost. Using the notation upo for the single-scattering albedo of an individual particle, we have upo = xsca/(xsca + xabs). Later we will introduce the single-scattering albedo for the medium as a whole. The cross sections for particles or molecules can be measured in the laboratory, and often can be computed from basic physical principles.

Since the radiation fields we will deal with are generally distributed over a range of frequencies and direction, instead of being monochromatic and unidirectional, we will write our equations in terms of the spectral irradiance I introduced in Chapters 3 and 4. Recall that if the spectral irradiance is I((9, 4>), v) at a given point, then Idlldv is the flux of radiation in frequency band dv with directions of travel within a solid angle dl about the direction (9, ^>), which passes through a plane perpendicular to the direction of travel. To apply the results of the preceding paragraph to smoothly distributed radiation, one needs only to substitute Idlldv for the incident flux F.

1The more usual notation for the cross-section is a, but in our subject matter that symbol has been reserved for the Stefan-Boltzman constant. One can think of x as standing for xp^aa-section.

As in previous chapters, we'll make the plane-parallel assumption, and assume that I depends on position only through pressure. Suppose that in the vicinity of some pressure level p there are N scatterers of type i per unit mass of atmosphere, and that each scatterer has mass mj. Suppose that the light impinging on the layer is traveling with angle 0 to the vertical. Then, taking a layer of thickness dp which is small enough that multiple scattering can be neglected, the rate of energy lost by the incident beam due to absorption and due to scattering into different angles is dp dp 1 1

;N ■ (Xabs,i + Xsca,i)IdQ.dv =---qi ■ (-Xabs,i +--Xsca,i)IdQ.dv (5.1)

t\ V/Y abs,i i /\sca,ii~---------ft n V /\.abs,i i g cos 6 g cos 6 mi mi where qi is the mass concentration of the particles in question. From this we can define the absorption coefficient of the substance Ki = Xabs,i/mi which has units of m2/kg. This absorption coefficient is the same quantity we defined in Chapter 4 in connection with gaseous absorption. The additional term in the above equation characterizes the energy lost from the incident beam due to scattering. We won't introduce separate notation for this term since scattering is most commonly characterized in terms of the cross section itself.

If there is only one optically active substance i in the atmosphere, we define the optical depth in the vertical direction by the equation dr * 1.1.

dp g mi

Because the absorbing and scattering properties typically depend on wavenumber, the optical depth is generally a function of wavenumber, though we will only append a wavenumber subscript to t* when we wish to call attention specifically to the wavenumber dependence. If there are many types of scatterers and absorbers - which could include particles of a single substance but with different sizes - then we define the optical depth by summing over all species. Thus dr * 1 , 1

where the net absorption coefficient is

We then define the single-scattering albedo for the medium as a whole as

The pair (k,u0) constitutes the basic description of the absorption and scattering properties of the medium. Both are typically functions of wavelength and altitude, and may also directly be functions of pressure and temperature. If the medium consists of only a single type of particle, and the gas in which the particles are suspended neither absorbs nor scatters, then uo = upo. In general, though, the single-scattering albedo of the medium depends on the mix of absorbers and scatterers. For example, an atmosphere may consist of a mix of cloud particles which are perfect scatterers (upo = 1) with a strong greenhouse gas which is an absorber. In this case, uo will go down as the greenhouse gas concentration increases, even if the cloud particle concentration is kept fixed.

Using the definition of optical depth, Eq 5.1 for the rate of energy loss from the beam can be rewritten as simply dl = -Idr*/ cos 6. Since the vertical component of flux is I cos 6, this expression can be recast as an expression for the rate of loss of vertical flux, namely dI cos 0 = —Idr * (5.6)

The proportion of this lost to scattering is while the proportion lost to absorption is 1 — wo. The fate of the energy lost to absorption is different from the fate of that lost due to scattering. The former disappears into the pool of atmospheric heat, whereas energy lost to scattering from one beam reappears as flux in a range of other directions, so we need to keep track of the two loss mechanisms separately. The beam loss in a given direction is offset by two source terms: one due to thermal emission, and one due to scattering from other directions. The thermal emission term is proportional to the Planck function, and can be treated in a fashion similar to that used in deriving the Schwartzschild equations. We'll leave the thermal emission out for now, and concentrate on scattering; the thermal emission term will be put back in in Section 5.5.

To understand better where the scattered flux goes, consider the energy budget for a box of thickness dr* in the vertical, shown from the side in Figure 5.2. Since the radiation field is independent of the horizontal dimensions, the flux entering the box from the side is the same as the flux leaving it from the side, and does not affect the budget. If the base of the box has area A, an amount A ■ I(t*) cos 0 enters the box from the bottom and a somewhat lesser amount A ■ I(0, t* + dr*) cos 0 leaves the box from the top. Taking the difference gives the loss of energy from the beam per unit time, due to scattering and absorption. Using Eq. 5.6 this can be written as simply A ■ I(0, t*)dr*; it doesn't matter whether I is evaluated at dr* or t* + dr* in this expression, since dr* is presumed small. The energy per unit time scattered and restributed into all other directions is then A ■ woI(0, t*)dr*. Now, to write an equation for how the vertical component of flux changes between t * and t * + dr *, we need to find how much flux is added to the direction (0, by scattering from all other directions of radiation impinging on the layer. We can do this by considering the incident radiation one direction at a time, and summing up. Consider a beam of light traveling with direction (0', having radiance I(0',^',t*). The scattering contributed to direction (0, comes from the scatterers in the shaded parallelogram shown in Figure 5.2, which is greater than the amount of scatterer in a rectangular box by a factor of 1/ cos 0. Further, only a proportion of the radiation scattered from the contents of the parallelogram goes into the direction (0, ^>), We will write this proportion as P/4n, where P depends on both the incident and scattered directions. Thus, the radianced contributed to direction (0, by scattering is A ■ (P/4n)woI(0',^',t*)dr*/cos0, and the vertical component of this is obtained by multiplying by cos 0, yielding A ■ (P/4n)woI(0', t* )dr*. This is the vertical flux contributed by scattering, and is added in to the flux leaving the top of the box. The scattering acts as a source of radiation in direction (0, which is added to the right hand side of Eq. 5.6. Dividing out the area of the base of the box, the flux balance for the box becomes dI(0, 4>) cos 0 = —I(0, ^)dr* + — P(0, 0', 4>')I(0', ^')dr* (5.7)

4n if one considers only the flux contributed by scattering of a single direction (0', . To complete the equation, one must integrate over all incident angles (0', To determine the radiation field in its full generality, it is necessary to satisfy the flux balance for each direction of propagation simultaneously. Before proceeding toward that goal, we'll check Eq. 5.7 to verify that the scattered energy is conserved. Applying the control volume sketch to the incident beam direction, we infer that the incident beam traveling in direction (0', deposits energy in the control volume at a rate I(0', ^')dr* (per unit area). A proportion woP/4n of this should show up as an increase in the energy in the box propagating in direction (0, and that is precisely the source term appearing in Eq 5.7 The books are indeed balanced.

Figure 5.2: A scattering control volume, showing the flux added in the direction (0, due to scattering of an incident beam with direction (0', Only the contribution from the slab of thickness dr* is considered. The incident beam illuminates the entire slab, but only the scatterers in the shaded parallelogram contribute to scattered radiation in the direction (0, ^>). The solid squiggly line represents scattered radiation, and the dashed squiggly line represents the vertical component of the scattered flux. The vertical straight,dashed arrows give the vertical component of the flux in the (0, direction, and show how it changes as the slab is traversed. Flux is lost from the (0, beam owing to absorption and scattering. The flux lost due to scattering shows up as scattererd radiation in all other directions; these are not shown in the diagram.

Figure 5.2: A scattering control volume, showing the flux added in the direction (0, due to scattering of an incident beam with direction (0', Only the contribution from the slab of thickness dr* is considered. The incident beam illuminates the entire slab, but only the scatterers in the shaded parallelogram contribute to scattered radiation in the direction (0, ^>). The solid squiggly line represents scattered radiation, and the dashed squiggly line represents the vertical component of the scattered flux. The vertical straight,dashed arrows give the vertical component of the flux in the (0, direction, and show how it changes as the slab is traversed. Flux is lost from the (0, beam owing to absorption and scattering. The flux lost due to scattering shows up as scattererd radiation in all other directions; these are not shown in the diagram.

It is worth thinking quite hard about Figure 5.2, because the cosine terms that appear in such computations - and which are the source of most of the difficulties in writing two-stream approximations - can be quite confusing. The cosine weights play two quite different roles. In one guise, they express the number of scatterers or absorbers encountered along a slanted path, but in another guise they represent the projection of the flux on the vertical direction. Most confusion can be resolved by thinking hard about the energy budget of the control volume.

The quantity P introduced in Figure 5.2 is called the phase function, and describes how the scattered radiation is distributed over directions. For spherically symmetric scatterers, the phase function depends only on the angle © between the incident beam and a scattered beam (as depicted in Fig. 5.1). The phase function is usually expressed as a function of cos©. If fi[ is the unit vector in the direction of propagation of the incident beam, and fsca is the unit vector in the direction of propagation of some scattered radiation, then where 0 and $ are the direction angles of the incident beam and 0' and are the angles of the scattered beam under consideration. The phase function for the medium as a whole can be determined from the phase functions of the individual particles doing the scattering - remember that from and dr* we already know the amount of energy scattered out of a beam, so the phase function only needs to tell us how that energy is distributed amongst directions. The phase function for an individual particle is defined in such a way that the scattered flux within an element of solid angle d£l' near direction (0', $') is XsCaI(0, 4>)P(cos©(0, 0', 4>'))d£l' / 4n. P is normalized such that J" PdiY = 4n, so that integrating the scattered flux over all solid angles yields xscaI. Note further that where solid angle integrals without limits specified explicitly denote integration over the entire sphere. The final equality is a matter of definition and the other two equalities follow because one is free to rotate the coordinate system so as to define the angles with respect to any chosen axis, if one is integrating over the entire sphere. Isotropic scattering, in which the scattered radiation is distributed uniformly over all angles, is defined by P = 1.

If the scatterers in the atmosphere are all identical particles, then the phase function for the medium is the same as the phase function for an individual particle. If the phase functions differ from one particle to another, then the phase function for the medium is simply the average of the individual particle phase functions, weighted compatibly with Eq 5.3. The averaging is particularly important when the particles are non-spherical. Though the phase function for any individual particle is not a function of cos © alone, the particles are generally oriented in random directions, and the average phase function for an ensemble of randomly oriented particles acts like the phase function for an equivalent sphere.

If one divides Eq 5.7 by dr * and integrates over all incident directions (0', $') the equation for the vertical component of the flux due to radiation traveling in direction (0, is found to be where cos © is given in terms of (0, 0', $-4>') by Eq. 5.8. Thermal emission would add an additional source term B(v, T(t*)) to the right hand side, but we shall leave that out for now. This is the full equation whose solutions give the radiation field. The integral couples together all directions of cos © = fi[ • nsca = cos 0 cos 0' + sin 0 sin 0' cos(^ — 4>')

propagation; if one approximated the integral by a sum over 100 angles, for example, the equation would be the equivalent of solving a system of 100 coupled ordinary differential equations. While, with modern computers, this is not so overwhelming a task as it once might have seemed, it is still intractible in typical climate calculations, where one is doing the calculation for each of a large array of wavenumbers, at each time step of a radiative-convective model, and perhaps for each latitude and longitude grid point in a general circulation model as well. Moreover, it is always helpful to have a simplified form in hand if one's goal is understanding and not merely computing a number. Hence, our emphasis will be on reduction of the equation to an approximate set of equations for two streams of radiation, which may be thought of as the upward and downward streams. In this section we'll derive some exact constraints, which will be used to obtain two-stream closures of the problem in Section 5.5.

We first need to define the upward and downward fluxes, which are

The fluxes are defined in such a way that both are positive numbers. Given that dQ can be written as d cos 0 • d^ it is convenient to write all the fluxes as a function of cos 0, as we have done here. Henceforth we shall use Q+ and Q- as shorthand for integral over the upward or downward hemisphere, respectively. With these definitions, the net vertical flux (positive upward) is I+ — I_ = JIcos 0dQ, the integral being taken over the full sphere.

Solar radiation enters the top of the atmosphere in the form of a nearly parallel beam of radiation, characterized by an essentially unique angle of propagation. It is gradually converted by scattering into radiation that is continuously distributed over angles. Because the incoming solar radiation has an angular distribution concentrated on a single direction of propagation, it is useful to divide the radiation up into a direct beam component propagating exactly in this direction, and a diffuse component, travelling over all angles. You can see the Sun as a sharply defined disk in clear sky, which shows that the direct-beam solar radiation isn't completely converted into diffuse radiation by scattering, except perhaps in heavily cloudy conditions. To define the direct beam flux, let Lq be the solar constant and Z be the angle between the vertical and the line pointing toward the Sun; Z is called the zenith angle. By convention, the zenith angle is defined as the angle of the vector pointing toward the Sun, rather than the direction of the rays coming from the Sun. Thus, if 0dir is the angle of the direct-beam radiation in our usual angle coordinate system, the zenith angle is Z = n — 0dir. The azimuth angle of the direct beam radiation ^dir is defined in the usual coordinate system.

Now, since the direct beam flux is concentrated in a single direction, there is essentially zero probability of any scattered flux contributing back into the exact direct beam direction. That would be like the exactly hitting an infinitesimal dot on a dartboard. Therefore, flux is scattered out of the direct beam but is never added into it, and the direct beam decays exponentially. Making use of the slant path, the direct beam flux is then Lq exp( — (r^ — t*)/ cos Z). We rewrite the flux as the sum of a diffuse component and the direct beam:

I (cos 0, = Idiff (cos 0, + Lq exp( —(t^ — t *)/cos Z )6(0 — (n — Z — ^¿r) (5.12)

where Idiff is the diffuse flux and 6 is the Dirac delta-function. From now on, for economy of notation we'll drop the "diff" subscript on the diffuse radiation and simply write I for the diffuse component. In typical situations, the top-of-atmosphere boundary condition states that the radiance of all downward-directed angles of the diffuse component must vanish.

Substituting into Eq 5.10, the equation for the diffuse flux becomes —I (cos 6, <) cos 6 = - I (cos 6, <) + — f P (cos 0)I (cos 6', <')dQ'

+ Lq P(cos 0(- cos Z, cos 6,< - <dir) exp (-(t* - T*)/ cos Z) 4n

The scattering from the direct beam acts as a source term for the diffuse radiation. Integrating over all angles yields the following exact expression for the net vertical diffuse flux dd* (I+ - I_) = -(1 - -o )J I (cos 6',<')dfi' + -oLe exp (-(t* - t*)/cos Z) (5.14)

since / P(cos0)dQ = 4n. In this expression, I+ and I- now represent just the diffuse part of the flux. Conservative scattering - that is, scattering without absorption - is defined by —o = 1. For conservative scattering the first term on the right hand side of Eq. 5.14 vanishes. Integrating the direct beam term with respect to t* just multiplies it by cos Z, whence we are left with the result that I+ -I- - Lq cos Z exp(-(t* - t*)/cos Z) is a constant. Thus, for conservative scattering the sum of the direct beam vertical flux - which is negative because it is downward - with the diffuse flux is independent of height. As the direct beam is depleted, the flux lost goes completely into the diffuse component. This is as it should be, because, in conservative scattering, the flux lost has no place else to go.

Eq. 5.14 provides the first of the two constraints needed to derive the two-stream approximations. The second constraint is provided by multiplying Eq 5.13 by a function H(cos 6) which is antisymmetric between the upward and downward hemispheres, and then performing the angle integral. The rationale for multiplying by an antisymmetric function is that we already know something about I+ - I- from the first constraint, and weighting by an antisymmetric functions gives us some information about I+ +I-. Multiplying by H and carrying out the angle integral, we get

-d* JI (cos 6, <)H (cos 6) cos 6dQ = -J IH (cos 6)dQ + -o J G(cos 6')I (cos 6', <' )dQ'

where

G(cos 6') = -1 J H(cos 6)P(cos 0(cos 6, cos 6', cos <))dQ (5.16)

We were free to replace cos(< - <') in this expression by cos <, since the integral is taken over all angles < and so a constant shift of azimuth angle does not change the value of the integral. Sinced H is assumed antisymmetric, the function G(cos 6') characterizes the up-down asymmetry of scattering of a beam coming in with angle 6'. The symmetry properties of cos0 imply that G(- cos 6') = -G(cos 6').

Exercise 5.2.1 Derive the claimed antisymmetry property of G.

If the phase function satisfies P(cos0) = P(-cos0) the scattering is said to be symmetric. For symmetric scatterers, there is no difference between scattering in the forward and backward directions. From Eq. 5.8 it follows that cos 0(cos 6, cos 6', <) = - cos 0(- cos 6, cos 6', < + n). The antisymmetry of H(cos 6) then implies that G vanishes if P is symmetric, since the contribution to the integral from (cos6,<) cancels the contribution from (-cos6,phi + n). For symmetric scattering, Eq. 5.15 takes on a particularly simple form, since both terms proportional to -o vanish. The physical content of this result is that symmetric scattering does not directly affect the asymmetric component of the diffuse radiation, since equal amounts are scattered into the upward and downward directions.

When the scattering isn't symmetric, the terms involving G do not vanish, and we need a way to characterize the asymmetry of the phase function. The most common measure of asymmetry is the cosine-weighted average of the phase function

1 r1

2 Jcose=-i which goes simply by the name of the asymmetry factor. The asymmetry factor vanishes for symmetric scattering. All radiation is backscattered in the limit g = —1, as if the scattering particles were little mirrors. When g = 1 there is no back-scatter at all, and all rays continue in the forward direction, though their direction of travel is altered by the particles, much as if they were little lenses.

The asymmetry factor g characterized forward-backward scattering asymmetry relative to the direction of travel of the incident beam, but some tedious manipulations with Eq. 5.8 allow one to show that the same factor characterizes cosine-weighted asymmetry in the upward-downward direction, regardless of the direction of the incident beam. Specifically, if the incident beam has direction (^',0'), then

-1 J P(cos ©(cos 0, cos 0', cos cos 0dQ = g cos 0' (5.18)

where d^ = d^-d cos 0 as usual. This leads to a particularly tidy result if we choose H(cos 0) = cos 0 in Eq. 5.15, since then G(cos 0') = gcos 0' and the antisymmetric projection of the scattering equation becomes d

— I (cos 0,^)cos2 0dn = —(1 — wog)(I+ — I_) + woLegcos Z exp( —(t^ — t*)/cos Z) (5.19)

The integral appearing on the left hand side is sensitive only to the symmetric component of the radiance field. In order to obtain a two-stream closure, it is necessary to express the integral in terms of I+ + I_, which requires making an assumption about the angular distribution of the radiation. The same assumption applied to the right hand side of Eq. 5.14 allows one to estimate J IdQ in terms of I+ + I_. The different forms of two-stream approximations we shall encounter correspond to different assumptions about the angular distribution of radiance.

For other forms of H the asymmetry function G(cos 0') has more complicated behavior that is not so simple to characterize. The other form of H we shall have occasion to deal with is f1 for cos 0 > 0, H(cos 0)= , f 0 < 0 (5.20)

I —1 for cos 0 < 0, which is used to derive the hemispherically-isotropic form of the two-stream equations. This choice is convenient because the left hand side of Eq. 5.15 reduces to the derivative of I+ + I_, but it is inconvenient because G no longer has a simple cosinusoidal dependence on the incident angle. One could simply compute G from the phase function for the medium and use this to form the weights in the scattering equation, but given the inaccuracies we already accept in reducing the problem to two streams, it is hardly worth the effort. Instead, we will approximate G as having a cosinusoidal dependance as it does in the previous case. This approximation is exact if the phase function has the form P = 1 — 3 b + a cos © + b cos2 ©, and one can add a third and fourth order term without very seriously compromising the representation. By carrying out the integral defining the asymmetry factor, we find that g = 1 a. Then, evaluating G for the assumed form of phase function we find G = 1 a cos 0' = | gcos 0'. With this result the antisymmetric scattering equation projection becomes

(I+ + I_) = —(1 — 3g)(I+ — I-) + ^qlq3gcos Zexp ( —(t^ — t*)/cos Z) (5.21)

The right hand side becomes precisely the same as Eq. 5.19 if we redefine the asymmetry factor to be 3g. Eq. 5.21 is already written in terms of the upward and downward stream, and needs no further approximation in order to be used to derive a two-stream approximation. To complete the derivation of the hemispherically isotropic two-stream equation, one need only write the integral f IdQ appearing in Eq. 5.14 in terms of I+ +1_ using the assumed angular distribution of radiance. If I is assumed hemispherically isotropic in forward and backward directions separately, then this integral is in fact 2(I+ + I_), which completes the closure of the problem.

There is one last basic quantity we need to define, namely the index of refraction, which characterizes the effect of a medium on the propagation of electromagnetic radiation. It will turn out that the index of refraction amounts to an alternate way of representing the information already present in the scattering and absorption cross-sections. For a broad class of materials - including all that are of significance in planetary climate - the propagation of electromagnetic radiation in the material is described by equations that are identical to Maxwell's electromagnetic equations, save for a change in the constant that determines the speed of propagation (the "speed of light"). In particular, the equations remain linear, so that the superposition of any two solutions to the wave equations is also a solution, allowing complicated solutions to be built up from solutions of more elementary form. The reduction in speed of light in a medium comes about because the electric field of an imposed wave induces a dipole moment in the molecules making up the material, which in turn gives rise to an electric field which modifies that of the imposed wave. The equations remain linear because the induced dipole moment for non-exotic materials is simply proportional to the imposed electric field. When the medium is nonabsorbing, the ratio of the speed in a vacuum to the speed in the medium is a real number known as the index of refraction.

The physical import of the index of refraction is that, at a discontinuity in the index such as occurs at the surface of a cloud particle suspended in an atmosphere, the jump in the propagation speed leads to partial reflection of light hitting the interface, and deflection (refraction) of the transmitted light relative to the original direction of travel. The larger the jump in the index of refraction, the larger is the reflection and refraction. To a considerable extent, the refraction of light upon hitting an interface can be understood in terms of a particle viewpoint. If one represents a parallel beam of light as a set of parallel streams of particles all moving at speed ci in the outer medium, then if the streams hit an interface with a medium where the speed it c2 < ci, then the streams that hit first will be slowed down first, meaning that the wave front will tilt and the direction of propagation of the beam will be deflected toward the normal, as shown in Figure 5.3. The classic analogy is with a column of soldiers marching in line, who encounter the edge of a muddy field which slows the rate of march. If ©i is the angle of incidence relative to the normal to the interface, and ©2 is the angle of the refracted beam on the other side of the interface, then the deflection due to the change in speed is described by Snell's Law, which states c2 sin ©i = ci sin ©2, or equivalently sin©2 = (ni/n2)sin©i. Now, if a beam is traveling within the medium at angle ©2 and exits into a medium with lower index of refraction (e.g. glass to air), then the angle of the exiting beam is given by sin©i = (n2/ni)sin©2; hence, the beam is deflected away from the normal, as indicated in the sketch. At such an interface, if (n2/ni)sin©2 > 1 then there is no transmitted beam and the ray is refracted so much that it is totally reflected back into the medium

Figure 5.3: Refraction of a beam of light at an interface between a medium with index of refraction ni and a medium with index of refraction n2. In the sketch, n2 > n1 so the speed of light is slower in the medium than in the surroundings, as is the case for glass or water in air.

- a phenomenon known as total internal reflection. In reality, there is always some partial reflection at an interface. Partial reflection, as well as many other phenomena we shall encounter, depends on the wave nature of light as described by Maxwell's equations, and cannot be captured by the "corpuscular" viewpoint. This was as much of a groundbreaking conceptual challenge for early optical theorists as blackbody radiation was for investigators presiding over the dawn of quantum theory.

The concept of index of refraction can be extended to absorbing media. Suppose that a plane wave propagating through the medium has spatial dependence exp(2nikx), where x is the distance measured in the direction of propagation. Then, the expression for the speed of the wave in terms of its frequency and wavenumber becomes

kf where c is the speed of light in a vacuum. Thus k = (v/c)n. Note that v/c is the vacuum wavenumber we have been using all along to characterize radiation. For real n, k is the wavenumber in the medium, which is larger than the vacuum wavenumber by a factor of n. If we allow n to be complex, its imaginary part characterizes the absorption properties of the medium. To see this, write kn + iki =- ur + i-ni (5.23)

Since the wave has spatial dependence exp(2nikx) = exp(wnikrx) exp(—2nkjx), the coefficient 2nkj = 2nuI(v/c) gives the attenuation of the light by absorption, per unit distance travelled. Note that because of the factor v/c, the quantity 2nnI gives the attenuation of the beam after it has traveled by a distance equal to one wavelength of the light. Hence nI = 1 corresponds to an extremely strong absorption. Visible light traveling through such a medium, for example, would be almost completely absorbed by the time it had traveled one pm.

The absorption coefficient kI is proportional to the absorption cross section per unit mass we introduced in Chapter 4, and which reappeared above in the context of absorption by particles. If the density of the medium is p, the corresponding mass absorption coefficient is k = 2nkI/p =

Thermal-IR |
Near-IR |
Solar |
UV-B | |

Liquid water |
1.40 |
1.31 |
1.33 |
1.43 |

Water ice |
1.53 |
1.29 |
1.31 |
1.39 |

CO2 ice |
1.45 |
1.40 |
1.41 |
1.54 |

Liquid CH4 |
1.28 |
1.27 |
1.27 |
1.49 |

H2SO4 38% |
1.56 |
1.36 |
1.38 |
1.53 |

H2SO4 81% |
1.41 |
1.51 |
1.44 |
1.58 |

Table 5.1: Real part of the index of refraction for selected condensed substances. Thermal-IR data is at 600 cm-1, Near-IR is at 6000 cm-1, Solar at 17000 cm-1 (.59 pm), and UV-B at 50000 cm-1 (.2 pm). Liquid water data was taken at a temperature of 293K, water ice at 273K, CO2 ice at 100K, liquid methane at 112K and H2SO4 at approximately 270K. The percentage concentrations given for the latter are in weight percent.

Table 5.1: Real part of the index of refraction for selected condensed substances. Thermal-IR data is at 600 cm-1, Near-IR is at 6000 cm-1, Solar at 17000 cm-1 (.59 pm), and UV-B at 50000 cm-1 (.2 pm). Liquid water data was taken at a temperature of 293K, water ice at 273K, CO2 ice at 100K, liquid methane at 112K and H2SO4 at approximately 270K. The percentage concentrations given for the latter are in weight percent.

n ■ (v/c)/p. In mks units, this quantity has units ofm2/kg, and is thus an absorption cross section per unit mass.

The real index of refraction for some common cloud-forming substances is given in Table 5.1. The index of refraction for these and similar substances lies approximately in the range 1.25 to 1.5, and is only weakly dependent on wavenumber; data also shows the index of refraction to depend only weakly on temperature. The weak dependence of index of refraction on wavelength does give rise to a number of readily observable phenomena, such as separation of colors by a prism or the droplets that give rise to rainbows, but such phenomena, beautiful as they are, are of little importance to planetary energy balance. The one exception to the typically gradual variation of the real index of refraction occurs near spectrally localized absorption features; the real index also has strong variations in the vicinity of such points. In considering the scattering of light by particles suspenden in an atmospheric gas, the index of refraction of the gas can generally be set to unity without much loss of accuracy. A vacuum has n =1, and gases at most densities we'll consider are not much different. Specifically, for a gas n - 1 is proportional to the density. At 293K and 1bar, Earth air has an index of refraction of 1.0003 in the visible spectrum. CO2 in the same conditions has an index of 1.0004, and even at the 90 bar surface pressure of Venus has an index of only 1.016. The resultant refraction by the atmospheric gas can be useful in determining the properties of an atmosphere through observations of refraction from the visible through radio spectrum, but it has little effect on scattering by cloud particles.

Insofar as the real index of refraction goes, it would appear that it matters very little what substance a cloud is made of. The minor differences seen in Table 5.1 are far less important than the effects of cloud particle size and the mass of condensed substance in a cloud. The absorption properties, on the other hand, vary substantially from one substance to another, and these can have profound consequences for the effect of clouds on the planetary energy budget. The behavior of the imaginary index for liquid water, water ice, and CO2 ice is shown in Figure 5.4. Water and waterice clouds are nearly transparent throughout most of the solar spectrum; for these substances, nj is less than 10-6 for wavenumbers between 10000cm-1 and 48000cm-1 (wavelengths between 1 pm and .2 pm, though the absorption increases sharply as one moves into the far ultraviolet. In the thermal infrared spectrum, however, water and water-ice are very good absorbers, having nj in excess of .1 between wavenumbers of 50 and 1000 cm-1. Such a large value of nj implies that most thermal infrared flux would be absorbed when passing through a cloud particle having a diameter of 10 pm. For this reason, infrared scattering by water and water-ice clouds can be safely neglected, such clouds being treated as pure absorbers and emitters of infrared. This is not the case for clouds made of CO2 ice (important on Early Mars and perhaps Snowball Earth) or liquid CH4

(important on Titan). CO2 ice clouds are still quite transparent in the solar spectrum, apart from strong absorption in the far ultraviolet. In constrast to water clouds, however, they are largely transparent to thermal infrared. For CO2 ice clouds, n/ is under 10_4 between 1000 and 2000 cm-1, and even between 500 and 1000 cm_1 n/ is generally below .01 except for a strong,narrow absorption feature near 600 cm_1. Likewise, liquid methane has n/ well under .001 between 10 and 1200 cm_1. In both cases the infrared scattering effect of clouds can have an important effect on the OLR, leading to a novel form of greenhouse effect. Concentrated sulfuric acid, which makes up aerosols on Earth and the clouds of Venus, is quite transparent for wavenumbers larger than 4000 cm_1 but the imaginary index of refraction increases greatly at smaller wavenumbers, and in the thermal infrared sulfuric acid absorbs nearly as well as water. Nonetheless, the scattering by sulfuric acid clouds has a significant effect on the OLR of Venus, because Venus is so hot that it has considerable thermal emission at wavenumbers greater than 4000cm_1.

The strongly reflecting character of sulfate aerosols explains the volcanic cooling of the troposphere seen in the temperature time series of Figure 1.17, but what accounts for the accompanying stratospheric warming? Since the aerosols are largely transparent in the visible and solar near-infrared, the answer must lie in the the thermal infrared effect. This seems paradoxical, since we already know that increasing the infrared opacity of the stratosphere by adding CO2 cools the stratosphere. The resolution to this paradox is found in the difference in absorption spectrum between CO2 and the aerosols. CO2 absorbs and emits very selectively and we saw in Chapter 4 that this leads to a stratospheric temperature that is considerably colder than the grey-body skin temperature. Sulfate aerosols, in contrast, act much more like a grey body. Therefore, they raise the stratospheric temperature towards the grey-body skin temperature. Any aerosol that absorbs broadly in the thermal infrared should behave similarly.

As a general rule of thumb, typical cloud-forming condensates tend to be very transparent in the visible and near-ultraviolet and quite transparent in the near-infrared, but vary considerably in their absorption properties in the thermal infrared. Most substances - whether gaseous or condensed - are very good absorbers in the very shortwave part of the ultraviolet spectrum, with wavelengths below .1 ^m. For this reason, this part of the UV spectrum is often referred to as "vacuum UV," because it is essentially only present in the hard vacuum of outer space.

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