The absorption coefficient a(A), the scattering coefficient a(A), and the scattering phase functionp(A, 0), can be expressed as:

The subscript i is used to denote the various radiatively-active components, i.e., air molecules, aerosols, and liquid water and ice cloud particles in the atmosphere; ice crystals and impurities in the snow; brine pockets and air bubbles in the ice; pure water mixed with air bubbles, chlorophyll, inorganic particles, and yellow substance in the ocean. Here A is the wavelength, and © is the scattering angle, which is related to the polar and azimuthal angles through the cosine law of spherical geometry:

cos 0 = cos 0 cos 0' + sin 0 sin 0' cos(^ - </>')

Here denotes the radiation direction prior to scattering and is the radiation direction after scattering.

9.2.4.1 Absorption and Scattering by Atmospheric Molecules and Pure Water

Absorption by molecules in the earth's atmosphere is due to radiatively-active trace gases. For UV radiation and visible light the most important gas is ozone, but oxides of sulfur and nitrogen may (depending on location) have a significant effect on the UV radiation penetration. In the relatively pristine polar regions, we may assume that for UV radiation and visible light, ozone is the only significant absorbing gas in the atmosphere. Thus, the molecular absorption coefficient becomes:

where the subscript m stands for molecules, and an0 and n0^ are the ozone absorption cross section and number density, respectively (Figs. 9.2(a), (b)). At wavelengths longer than 310 nm, the ozone absorption is due to the Huggins bands, whereas the spectrally-broad but weak absorption between 450 and 700 nm is due to the Chappuis bands (Fig. 9.2(a)).

Scattering by molecules in the atmosphere is proportional to the gas density. Thus, the scattering coefficient due to scattering by air molecules is:

where crn Ray (A) ~ A~4 is the Rayleigh scattering cross section, and nm is the total air number density (Fig. 9.2(b)).

Absorption by pure water results from mutual interactions between the intermolecular forces. Calculation of absorption cross sections from first principles is very difficult. Thus, laboratory and in situ measurements (Pegau et al., 1995; Pope and Fry, 1997) become essential for establishing the absorption coefficient anw (A) for pure water (Fig. 9.2(c)).

Wavelength (nm) Wavelength (nm)

Figure 9.2 (a) Absorption cross section of ozone; (b) number density of atmospheric ozone (dashed line) and total air number density (solid line); (c) absorption spectrum for pure ice (dashed line), and for pure water (solid line; (d) chlorophyll-specific absorption spectrum normalized at 440 nm

Wavelength (nm) Wavelength (nm)

Figure 9.2 (a) Absorption cross section of ozone; (b) number density of atmospheric ozone (dashed line) and total air number density (solid line); (c) absorption spectrum for pure ice (dashed line), and for pure water (solid line; (d) chlorophyll-specific absorption spectrum normalized at 440 nm

Scattering in pure water is due to clusters of molecules and ions, much smaller than the wavelength of light, and from an optical point of view, the clusters are uncorrelated (Thomas and Stamnes, 1999). Thus, the wavelength dependence of the scattering cross section for pure water is crnw (A) ~ A~432, similar to the Rayleigh scattering cross section for molecules in the atmosphere. The scattering coefficient for pure water can be approximated by (Morel, 1974):

The phase function for scattering by atmospheric molecules or by pure water Is given by:

For scattering by air molecules (Rayleigh scattering), the parameter x is set equal to 1, while for scattering by pure water, x = 0.835. The latter value of x can be attributed to the anisotropy of the water molecule (Mobley, 1994; Morel and Gentili, 1991).

In addition to molecules, suspended particulate matter in the atmosphere and ocean has a significant impact on the radiative transfer. This particulate matter consists of aerosols (dust, sulfate particles, soot, smoke particles, cloud water droplets, raindrops, ice crystals, etc.) in the atmosphere, snow grains, air bubbles and brine pockets in ice, and hydrosols (suspended particles of organic and inorganic origin) in the water. Thus, to describe radiative transfer in a medium such as the coupled atmosphere-snow-ice-ocean (CASIO) system, we may think of it as being composed of randomly distributed, radiatively-active 'particles' that absorb and scatter radiation.

We assume that the optical properties of particles (i.e., aerosols, cloud particles, snow grains, brine pockets, etc.) can be approximated by those of spheres. Then, the absorption and scattering coefficients and the scattering phase function can be written as:

J n(r)dr where Qa(r) or Qa(r) is defined as the ratio of the absorption or scattering cross section for a spherical particle of radius r to the geometrical cross section nr2, n(r) is the particle size distribution, and r is the radius of each individual particle. For a specific value of r, we can compute Qa (r), Qa (r), and pp (2,0, r) using Lorenz-Mie theory, but evaluation of Eqs. (9.7) - (9.9) requires knowledge of the r =

particle size distribution n(r), which is usually unknown.

Hamre et al. (2004) showed that Eqs. (9.7) - (9.9) can be considerably simplified by making the following assumptions:

• The particle distribution is characterized by an effective radius

| m"n(r )r 3drj j" °"n(r )r 2dr, which obviates the need for an integration over r.

• The particles are weakly absorbing, so that

Qa (r) ~----[mrel - (mrel " 1)3'2] , where mi p is the imaginarY part

U mrel ,p of the refractive index of the particle, A is the wavelength in vacuum, and mrel = mr / mr med is the ratio of the real part of the refractive index of the particle (mr p) to that of the surrounding medium (mr med).

• The particles are large compared to the wavelength (2nr / A» 1), which implies Qa (r) = 2.

• The scattering phase function can be represented by the one parameter Henyey-Greenstein phase function, which depends only on the asymmetry

1 f1

ap (A) = a(A) ■ — • [1 - (mr2d -1)3/2 ] f (9.10)

mrel

where a(A) is the absorption coefficient for the material of which the particle is

composed, and fV =— nr ne, where ne is the number of particles per unit volume with radius r .

9.2.4.3 Optical Properties of the Ocean

To represent the optical properties of the water beneath the ice, we may use the model of Morel (1991), updated by Morel and Maritorena (2001), according to which the absorption and scattering coefficients and the scattering phase function are given by:

where the subscript w denotes pure water, the subscript C denotes chlorophyll-a related absorption and scattering, and the subscript Y stands for yellow substance. The absorption coefficient for pure water was discussed above (see Fig. 9.2(c)), and ac (A) = 0.064AC (A)C065

where AC (A) is the normalized absorption spectrum for the chlorophyll-a related absorption displayed in Fig. 9.2(d), and C(mg • m~3) is the chlorophyll-a concentration. The absorption by yellow substance is expressed as:

where aY (440) = 0.2[aw (440) + 0.064AC (440)C065], and the chlorophyll-a related scattering coefficient is given by:

^550

Scattering by pure water was discussed in Section 9.2.4.1. 9.2.4.4 Definitions of Irradiance and Radiance

The spectral net irradiance Fv is defined as the net energy d3E crossing a surface element dA (with unit normal n) per unit time and per unit frequency (Thomas and Stamnes, 1999):

d3 E

v dAdtdv

Since the irradiance is positive if the energy flow is into the hemisphere centered on the direction n and negative if the flow is into the opposite hemisphere, we may define spectral hemispherical irradiances F* = d3E+ /dAdtdv and F~ = d3E~/dAdtdv . Thus, the spectral net irradiance becomes Fv= F^- F~ , and

Consider any small subset of the energy d4 E that flows within a solid angle dm around the direction ¿2 in the time interval dt and within the frequency range dv . If this subset of radiation has passed through a surface element dA (with unit normal n ), then the energy per unit area per unit solid angle, per unit frequency, and per unit time, defines the spectral radiance Iv :

d4 F

cosOdAdtdmdv

Here d is the angle between the surface normal n and the direction of propagation Ù. It is clear from these definitions that F* = J do cos 9IV and

F~ = -1 docos#Iv. Thus, the spectral net irradiance can be expressed as:

Finally, we define the mean intensity as: I = (1/4 n)J d®Iv, which is simply the radiance averaged over the sphere.

9.2.4.5 Absorption, Scattering, and Extinction by Molecules and Particles

A beam of light incident on a thin layer with thickness ds of radiatively-active matter (gases and/or particles) is attenuated so that the differential loss in radiance is proportional to the incident light: dIv=-k(v,s) Ivds , where k(v,s) is called the extinction or attenuation coefficient. Integration yields

where 12 denotes the propagation direction of the beam. The dimensionless extinction (or attenuation) optical path (or opacity) along the path s is given by f>s rs (v) = J ds' k (v, s'). Attenuation of a light beam can be caused by either absorption or scattering. The extinction (or attenuation) optical path of a mixture of scattering/ absorbing molecules and particles is defined as the sum of the scattering and absorption optical paths, rs (v) = rsc (v) + ra (v), where rsc (v) = ^ J ds'ai (v, s')

i and ra(v) = ^ J As'at (v,s'). The sum is over all optically active species (each i denoted by the subscript i), and <Ji (v, s) and at (v, s) [m-1] are the scattering and absorption coefficients, and ki (v, s) = ^-(v, s) + ^-(v, s) is the extinction or attenuation coefficient. These are defined as cr;(v, s) = ani (v)nj (s) and ^-(v, s) =

an,i (V)ni (s), where ni [m-3] is the concentration and ani (v) and crni (v) [m2] are the absorption and scattering cross sections of the ith optically active species (molecule or particle).

If we are interested primarily in energy transfer, rather than the directional dependence of the radiation, it is sufficient to work with the azimuthally-averaged radiance Iv( z, u), where z denotes the level in the medium (height in the atmosphere or depth in the ocean), and u = cos^, d being the polar angle. It is convenient to split the radiation field into two parts: (1) the direct solar beam, which is exponentially attenuated upon passage through the atmosphere and ocean, and (2) the diffuse or scattered radiation.

According to Eq. (9.16), the penetration of the direct solar beam through the atmosphere may be written as Isol(s,f2) = Fs S(£2 - Qf)e~Ts(v). Here f20 = (#„,$,) is a unit vector in the direction of the incident solar beam, where ju0 = cos#0, and 0O is the solar zenith angle as illustrated in Fig. 9.3. Fs is the solar irradiance (normal to the solar beam direction ¿2 0) incident at the top of the atmosphere. In plane-parallel geometry, the vertical optical depth is defined in terms of the optical path rs (v) as follows:

i or dr(v, z) = -k (v, z)dz = ; (z) + an, i (z)]ni (z)dz dr(v, z) = -k (v, z)dz = ; (z) + an, i (z)]ni (z)dz

Ocean

Figure 9.3 Schematic illustration of two adjacent media with a flat interface, such as the atmosphere overlying a calm ocean. Because the real part of the refractive index (mr) of the atmosphere ( matm r » 1 ) is different from that of the ocean (mocn r » 1.33), radiation distributed over 2n sr in the atmosphere will be confined to a cone less than 2n sr in the ocean (region II). Radiation in the ocean within region I will be totally reflected when striking the interface from below (Thomas and Stamnes, 1999)

Ocean

Figure 9.3 Schematic illustration of two adjacent media with a flat interface, such as the atmosphere overlying a calm ocean. Because the real part of the refractive index (mr) of the atmosphere ( matm r » 1 ) is different from that of the ocean (mocn r » 1.33), radiation distributed over 2n sr in the atmosphere will be confined to a cone less than 2n sr in the ocean (region II). Radiation in the ocean within region I will be totally reflected when striking the interface from below (Thomas and Stamnes, 1999)

so that for a plane-parallel medium the direct component becomes:

Here u = cos0, d is the polar angle of the observation direction, ju=\u |, and 0 denotes the azimuth angle of the observation direction.

In a stratified medium, the optical properties vary only in the vertical direction, and the transfer of diffuse radiation through such a medium is described by:

u d/C^w) =_k(z)j(z, u) + ff(z)I f1 du' p(z, u', u)I(z, u') + S\z, u) (9.18) dz 2 J-1

where I (z, u) is the azimuthally-averaged diffuse radiance. The term on the left side in Eq. (9.18) is the change in the diffuse radiance I (z, u) along a slant path dz/u. The first term on the right side is the loss of radiation out of the beam due to extinction, and the second term is the gain due to multiple scattering. The third term, S'(z, u) defined below, is the solar pseudo-source, proportional to the attenuated solar beam, which 'drives' the diffuse radiation. Using the non-dimensional optical depth, dr( z) = - k (z)d z as the independent variable, we may rewrite Eq. (9.18) as follows:

u d 1 fou) = i(T,u) _ £(£) f1 du' p(r,u',u) I(r,u') - S*(r,u) (9.19)

where the single-scattering albedo is defined as a(r(z)) = a(z)/ k(z).

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