Intensity, flux, energy density, heating rate, extinction, emission, scattering, thermodynamic equilibrium (lte), blackbody radiation, Kirchhoff’s law, Wien’s law, Stefan-Boltzmann law, Rayleigh-Jeans law, non-thermodynamic equilibrium (Nlte)




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НазваниеIntensity, flux, energy density, heating rate, extinction, emission, scattering, thermodynamic equilibrium (lte), blackbody radiation, Kirchhoff’s law, Wien’s law, Stefan-Boltzmann law, Rayleigh-Jeans law, non-thermodynamic equilibrium (Nlte)
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Summary of Chapter 2: Theory of Radiative Transfer


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.


    1. Definition of radiative quantities p.16-27, 12 pages

    2. Thermal emission p.27-43, 17 pages

    3. Integral equations p.43-52, 10 pages

    4. Approximate methods p.52-63, 12 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, Stefan-Boltzmann law, Rayleigh-Jeans law, non-thermodynamic 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-2hz-1sr-1), F (Wm-2hz-1), u (Jm-3hz-1), h (Whz-1)


Integrate over d from 0 to , we have


I (Wm-2sr-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/c2[exp(h/kT) 1] (2.37)


F(T) = 2o1 B(T) d =  B(T)


F(T) = o B(T) d = T4


u(T) = 4B(T)/c = 8h3/c3[exp(h/kT) 1]


where  = frequency (hz), c = speed of light = 2.998 108ms-1, h = Planck’s constant = 6.626110-34Js, k = Boltzmann constant = 1.38110-23JK-1.


Other forms of blackbody radiance are


B(T) = 2hc2/5[exp(hc/kT) 1]


B*(T) = 2hc2*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 Bd = o Bd = o B*d* = T4




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 = 3108 m/s


Planck’s constant = h = 710-34 Js


Boltzmann constant = k = 1.410-23 J/K


Stefan-Boltzmann constant =  = 610-8 Wm-2K-4


  1. Choose T = 300 K (T is same as  in Chapter 2)

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 = 1.96300  600 cm-1


This value agrees with the top curve in Fig 6.2.


  1. Let x = h/kT = hc*/kT since  = c*


At the maximum position,


xm = hc*m/kT =(610-34Js)(3108ms-1)(600cm-1)/(1.410-23 J/K)(300K)


= 2.6 (more accurately, 2.8282)


From the above, we have 1/[exp(xm) 1] = 0.063306  1/16

B(T) at maximum = B*m (T) = 2hc2*m3/ {exp[hc*m /kT] 1} = 2hc2*m3/ {exp(xm) 1}


2hc2 = 2(610-34Js)(3108ms-1)2  10-16 Wm2


*m3 = (600 cm-1)3 = 2.16108 cm-3  2108 (100)4 m-4(cm-1)-1 = 21016 m-4(cm-1)-1


B*m (T) = (10-16 Wm2) (21016 m-4(cm-1)-1)(1/16)  0.15 W/m2 sr cm-1 = 1.510-5 W/cm sr


This value agrees with the top curve in Fig 6.2.


(c) Let us approximate the integral by one-step integration


o Bd  B*m (T) = 30.15 W/m2 sr cm-11000 cm-1 = 450 W/m2


T4 = (610-8 Jm-2K-4)(300 K)4 = 486 W/m2


This checks with the one-step integration.


2.3 Integral equations


Optical path (Fig 2.5)


(1,2) = 12 ev 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 o1 I(,)d

F() = 2 -1o I(,)d

Note that F() and F() defined above are both positive quantities. The net flux is given by


F() = F()  F()


= 2 -11 I(,)d


Substituting in the definitions of F() and F(), we have


F() = 2 o1 I(1,) e-(1-)/ d + 2 1o1 e-(t-)/ J(t,) d dt


= 2 B(T1) o1 e-(1-)/ d + 2 1 B[T(t)]{o1 e-(t-)/ d} dt


= 2 B(T1) o1 e-(1-)/ d + 2 1 B[T(t)]{o1 e-(t-)/ d} dt


= 2 B(T1) E3(1 ) + 2 1 B[T(t)]E2(t)dt (2.107)

where we have assumed that J(t,) = B(T), I(1,) = B(T1), and the exponential integral functions are given by


En() = o1 e-/n-2d = 1 e-xx-n dx (A.6.1)


The two forms can be shown to be equivalent by a change of variable x = 1/. Note also,


dEn/d = En-1() (A.6.2)


The properties of the exponential integral functions are summarized in Goody and Yung, Appendix 6, p. 475-476.


Similarly we have for the downward flux


F() = 2 o B[T(t)]E2(t)dt (2.108)


Collecting all the terms, we have


F() = F()  F() = 2 B(T1)E3(1 ) + 2 1 B[T(t)]E2(t)dt  2o B[T(t)]E2(t)dt


(2.105)


To compute the heating rates, we need to differentiate F()


dF/d = 2B(T1)E2(1 ) 2B(T) + 2 1 B[T(t)]E1(t)dt


dF/d = 2B(T)  2o B[T(t)]E1(t)dt


dF/d = dF/d  dF/d


= 2B(T1)E2(1 ) 4B(T) +2o B[T(t)]E1(t)dt+ 2 1 B[T(t)]E1(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() = 2E3() (6.7)



    1. Approximate methods


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


  1. Derivation of RTE


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: 7114-7134 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: 4963-4967 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: 540-555 Published: OCT 2006


  1. Maxwell/RTE in a semi-coherent medium


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 semi-coherent medium (e.g., sand on a beach) is not. This remains a frontier for research.


  1. Remote Sensing Using Polarized Light


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 quantitative-grade fieldable snapshot imaging spectropolarimeter. OPTICS EXPRESS Volume: 12 Issue: 26 Pages: 6559-6573 Published: DEC 27 2004





Fig 6.1 Gulf of Mexico





Figure 6.2





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