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Overview This chapter gives a derivation of the radiative transfer equation (RTE) and energy balance in the atmosphere. Organization This chapter has 4 sections in 51 pages.
This chapter is the backbone of the book. All the machinery for radiative transfer is developed here. Highlights Intensity, flux, energy density, heating rate, extinction, emission, scattering, thermodynamic equilibrium (LTE), blackbody radiation, Kirchhoff’s law, Wien’s law, StefanBoltzmann law, RayleighJeans law, nonthermodynamic equilibrium (NLTE). We will defer NLTE and Approximate Methods to later part of the class. 2.1 Definitions Fig 2.1 Radiance or specific intensity Fig 2.2 Radiative heating Fig 2.3 Planck function (v) and () Intensity (radiance), energy flux, energy density and heating rate I_{} (Wm^{2}hz^{1}sr^{1}), F_{}_{ }(Wm^{2}hz^{1}), u_{} (Jm^{3}hz^{1}), h_{} (Whz^{1}) Integrate over d from 0 to , we have I (Wm^{2}sr^{1}), F (Wm^{2}), u_{ }(Jm^{3}), h (W) I_{}(P,s) = scalar if unpolarized; homogeneous if independent of P, and isotropic if independent of s Polarized light has four Stokes vector (I, Q, U, V). We will defer Stokes vector to Chapter 7. Derivation of the radiative transfer equation (RTE) in separate file. 2.2 Thermal emission B_{}(T) = 2h^{3}/c^{2}[exp(h/kT) 1] (2.37) F_{}(T) = 2_{o}^{1} B_{}(T) d = B_{}(T) F(T) = _{o}^{} B_{}(T) d = T^{4} u_{}(T) = 4B_{}(T)/c = 8h^{3}/c^{3}[exp(h/kT) 1] where = frequency (hz), c = speed of light = 2.998 10^{8}ms^{1},^{ }h = Planck’s constant = 6.626110^{34}Js, k = Boltzmann constant = 1.38110^{23}JK^{1}. Other forms of blackbody radiance are B_{}(T) = 2hc^{2}/^{5}[exp(hc/kT) 1] B^{*}(T) = 2hc^{2}^{*3}/ {exp[hc^{*}/kT] 1} where = c/, ^{*} = /c (in the literature, the wavenumber ^{*} is often written as !). Some authors use f for and for ^{*}. In carrying out the transformations, take care to ensure _{o}^{} Bd^{ }= _{o}^{} B_{}d = _{o}^{} B^{*}d^{* }= T^{4} ^{ } Numerical Example: Referring to Figs. 6.1 and 6.2, let us compute (a) the maximum location and (b) value of the radiance, and (c) its integral. One digit dictionary of physical constants Velocity of light = c = 310^{8} m/s Planck’s constant = h = 710^{34} Js Boltzmann constant = k = 1.410^{23} J/K StefanBoltzmann constant = = 610^{8 }Wm^{2}K^{4}
To compute the position of the maximum of the radiance, use the Wien’s Law in the middle of p.29. T/^{*}_{m }= 0.50995 cm K, or ^{*}_{m} = 1.96 T This gives ^{*}_{m }= 1.96300 600 cm^{1} This value agrees with the top curve in Fig 6.2.
At the maximum position, x_{m }= hc^{*}_{m}/kT =(610^{34}Js)(310^{8}ms^{1})(600cm^{1})/(1.410^{23} J/K)(300K) = 2.6 (more accurately, 2.8282) From the above, we have 1/[exp(x_{m}) 1] = 0.063306 1/16 B_{}(T) at maximum = B^{*}_{m}^{ }(T) = 2hc^{2}^{*}_{m}^{3}/ {exp[hc^{*}_{m}^{ }/kT] 1} = 2hc^{2}^{*}_{m}^{3}/ {exp(x_{m}) 1} 2hc^{2} = 2(610^{34}Js)(310^{8}ms^{1})^{2} 10^{16} Wm^{2} ^{*}_{m}^{3} = (600 cm^{1})^{3} = 2.1610^{8} cm^{3} 210^{8} (100)^{4} m^{4}(cm^{1})^{1} = 210^{16} m^{4}(cm^{1})^{1} B^{*}_{m}^{ }(T) = (10^{16} Wm^{2}) (210^{16} m^{4}(cm^{1})^{1})(1/16) 0.15 W/m^{2} sr cm^{1} = 1.510^{5} W/cm sr This value agrees with the top curve in Fig 6.2. (c) Let us approximate the integral by onestep integration _{o}^{} Bd B^{*}_{m}^{ }(T) = 30.15 W/m^{2} sr cm^{1}1000 cm^{1} = 450 W/m^{2} T^{4 }= (610^{8 }Jm^{2}K^{4})(300 K)^{4}_{ }= 486 W/m^{2} This checks with the onestep integration. 2.3 Integral equations Optical path (Fig 2.5) _{}(1,2) = _{1}^{2}_{ }e_{v} ds (2.84) The starting equation is the equation of radiative transfer in a plane parallel atmosphere dI_{}/d_{} = I_{} J_{} (2.93) Note that _{}_{ }= 0 at the top of the atmosphere. This is a deceptively simple equation, because I_{} and J_{} are functions of (_{}, ). There are two complexities that underlie this equation, the boundary conditions and the fact that J_{} may be a function of I_{}. Normally a first order differential equation has only one boundary. Here we have two boundaries, because this is a system of equations in , where can be positive or negative. To appreciate the two boundaries, let us consider the upward beam at . We must integrate from the level _{ }to the bottom at _{ }= _{1 }(at the surface). For simplicity, I will drop all dependence. From (2.93), we have dI/d_{} = I J_{} Move I from RHS to LHS, dI/d_{}I = J_{} Consider the identity e^{}^{/}^{} d(e^{}^{}^{/}^{} I)/d_{} = e^{}^{/}^{} [ (e^{}^{}^{/}^{}/) I+ dI/d_{} e^{}^{}^{/}^{}] = dI/d_{}I Therefore, the equation of radiative transfer becomes e^{}^{/}^{} d(e^{}^{}^{/}^{} I)/d_{} = J Dividing both sides by e^{}^{/}^{}, we have d(e^{}^{}^{/}^{} I)/d_{} = e^{}^{}^{/}^{} J/ We can now integrate both sides from _{ } to_{ }_{1}, we have_{} e^{}^{}^{1/}^{} I(_{1},) e^{}^{}^{/}^{} I(,) = _{}^{}^{1} e^{t/}^{} J(t,) dt/ Note that _{1 }> _{ }and that > 0. Divide both sides by e^{}^{}^{/}^{}, and move the boundary term to the RHS, we have I(,) = e^{(}^{}^{1}^{}^{)/}^{} I(_{1},) + _{}^{}^{1} e^{(t}^{}^{)/}^{} J(t,) dt/ (2.98) I(,) has two terms. One is from the boundary radiance, I(_{1},), attenuated by the transmission. The other is the integral over the source term, J(t,). Note that as t varies from to _{1}, (t) is always positive. To obtain the downward beam at _{}, we integrate d(e^{}^{}^{/}^{} I)/d_{} = e^{}^{}^{/}^{} J/ from the top _{}= 0 to_{ }_{}, e^{}^{}^{/}^{} I(,) I(0,) = _{o}^{} e^{t/}^{} J(t,) dt/ Divide both sides by e^{}^{}^{/}^{}, and move the boundary term to the RHS, we have I(,) = e^{}^{/}^{} I(0,) _{o}^{} e^{(}^{}^{t)/}^{} J(t,) dt/ Remember for downward beam < 0. Therefore, / and is a negative number. Also, (t)/ is negative for the range of t from 0 to . Thus the physical meaning of (8.32.2) is clear. The first firm is from the boundary, attenuated by the transmission. The second term is the contribution from the source term. It is awkward sometimes to keep track of all the minus signs. An alternative to (8.38.2) is to designate the downward beam by a different name, and we have I^{}(,) = I(,) = e^{}^{/}^{} I(0,) + _{o}^{} e^{(}^{}^{t)/}^{} J(t,) dt/ Now is always positive. The usual boundary condition is I(0,) = 0, and we have I^{}(,) = _{o}^{} e^{(}^{}^{t)/}^{} J(t,) dt/ (2.99) For heating calculations we have to obtain energy fluxes by doing angular integration of the radiances. Again, we will drop the index . F^{}() = 2 _{o}^{1} I(,)d F^{}() = 2 _{1}^{o} I(,)d Note that F^{}() and F^{}() defined above are both positive quantities. The net flux is given by F() = F^{}() F^{}() = 2 _{1}^{1} I(,)d Substituting in the definitions of F^{}() and F^{}(), we have F^{}() = 2 _{o}^{1} I(_{1},) e^{(}^{}^{1}^{}^{)/}^{} d + 2 _{}^{}^{1}_{o}^{1} e^{(t}^{}^{)/}^{} J(t,) d dt = 2 B(T_{1}) _{o}^{1} e^{(}^{}^{1}^{}^{)/}^{} d + 2 _{}^{}^{1} B[T(t)]{_{o}^{1} e^{(t}^{}^{)/}^{} d} dt = 2 B(T_{1}) _{o}^{1} e^{(}^{}^{1}^{}^{)/}^{} d + 2 _{}^{}^{1} B[T(t)]{_{o}^{1} e^{(t}^{}^{)/}^{} d} dt = 2 B(T_{1}) E_{3}(_{1} ) + 2 _{}^{}^{1} B[T(t)]E_{2}(t)dt (2.107) where we have assumed that J(t,) = B(T), I(_{1},) = B(T_{1}), and the exponential integral functions are given by E_{n}() = _{o}^{1} e^{}^{}^{/}^{}^{n2}d = _{1}^{∞} e^{}^{}^{x}x^{n }dx (A.6.1) The two forms can be shown to be equivalent by a change of variable x = 1/. Note also, dE_{n}/d = E_{n1}() (A.6.2) The properties of the exponential integral functions are summarized in Goody and Yung, Appendix 6, p. 475476. Similarly we have for the downward flux F^{}() = 2 _{o}^{} B[T(t)]E_{2}(t)dt (2.108) Collecting all the terms, we have F() = F^{}() F^{}() = 2 B(T_{1})E_{3}(_{1} ) + 2 _{}^{}^{1} B[T(t)]E_{2}(t)dt 2_{o}^{} B[T(t)]E_{2}(t)dt (2.105) To compute the heating rates, we need to differentiate F() dF^{}/d = 2B(T_{1})E_{2}(_{1} ) 2B(T) + 2 _{}^{}^{1} B[T(t)]E_{1}(t)dt dF^{}/d = 2B(T) 2_{o}^{} B[T(t)]E_{1}(t)dt dF/d = dF^{}/d dF^{}/d = 2B(T_{1})E_{2}(_{1} ) 4B(T) +2_{o}^{} B[T(t)]E_{1}(t)dt+ 2 _{}^{}^{1} B[T(t)]E_{1}(t)dt This is essentially (2.111). There is a factor 2 missing on the RHS of (2.111). The above expression can be further reduced to a more elegant form. Define a flux transmission function ^{f}() = 2E_{3}() (6.7)
We will return to approximate methods in Chapter 8. Here, it is sufficient to note that at the core (see, e.g., (2.151), RTE is a diffusion equation). Food for Thought
There are at least three ways to derive RTE. First method: Equations (2.15) to (2.17) give a heuristic derivation based on the continuity equation. A more general derivation is given in Radiation in the Atmosphere: A Course in Theoretical Meteorology by Wilford Zdunkowski, Thomas Trautmann, and Andreas Bott (Hardcover – April 16, 2007) Second method: RTE is a special case of the Boltzmann equation for transport. Third method: Application of Maxwell’s equations to a random medium Mishchenko, MI, Vector radiative transfer equation for arbitrarily shaped and arbitrarily oriented particles: a microphysical derivation from statistical electromagnetics. APPLIED OPTICS Volume: 41 Issue: 33 Pages: 71147134 Published: NOV 20 2002 Mishchenko, MI, Microphysical approach to polarized radiative transfer: extension to the case of an external observation point. APPLIED OPTICS Volume: 42 Issue: 24 Pages: 49634967 Published: AUG 20 2003 Mishchenko, MI, Maxwell's equations, radiative transfer, and coherent backscattering: A general perspective. JOURNAL OF QUANTITATIVE SPECTROSCOPY & RADIATIVE TRANSFER Volume: 101 Issue: 3 Special Issue: SI Pages: 540555 Published: OCT 2006
While the interaction of light with a smooth surface (totally coherent) or a cloud of random particles in the atmosphere (totally incoherent) is understood, that for a semicoherent medium (e.g., sand on a beach) is not. This remains a frontier for research.
The technology has just matured. We can easily measure polarization up to 1x10^{5} without using any rotating parts. Jones, SH; Iannarilli, FJ; Kebabian, PL, Realization of quantitativegrade fieldable snapshot imaging spectropolarimeter. OPTICS EXPRESS Volume: 12 Issue: 26 Pages: 65596573 Published: DEC 27 2004 Fig 6.1 Gulf of Mexico Figure 6.2 