2 The Radiative Transfer Equation

Transcription

1 9 The Radiative Transfer Equation. Radiative transfer without absorption and scattering Free space or homogeneous space I (r,,) I (r,,) r -r d da da Figure.: Following a pencil of radiation in free space from point r to r to show that the radiance is conserved. I (r,,) I (r,,) d r -r da da Energy conservation of stationary radiation means power conservation. Furthermore, in free space (or in a homogeneous medium), rays propagate on straight lines. Therefore dp = I (r, ) n )dda cos d = dp = I (r, ) n )dda cos d (.) Since d = da cos, and d = da cos r r r r, Equation (.) means that I (r,) n ) = I (r, n ) ) or if is an infinitesimal path element along the ray, we can write di = ; also valid for Stokes Vector: di = (.) Equation (.) is the radiative-transfer equation of free space. It is independent of position, and it is valid for all ray directions. Slightly inhomogeneous medium Now we assume that the medium is slightly inhomogeneous, but scattering and absorption are still negligible. Reflection and scattering are negligible if the gradient of the real part of the refractive index is sufficiently small: n' << k, and absorption is negligible if the imaginary part is n"=. Now the rays are no longer straight lines, but they follow the rules of geometric optics (Snell's Law, Fermat's Principle of the shortest path, Eikonal Equation). It can be shown (Mobley 994) that the following quantity is conserved: I = I n' ; or in Stokes Vector form I = I n' (.3) Note that n'=n because n"=. In the more general situation of an anisotropic medium, n' has to be replaced by the ray-refractive index (Bekefi, 966). For illustration and verification of (.3), we investigate the situation of a one-dimensionally inhomogeneous medium where the refractive index decreases in a transition region with increasing height (Figure.).

2 x 3 d n = n'(x 3 ) n > da Figure.: Power conservation for a refracted ray passing from one medium in another through da. Reflection is avoided by a soft transition dn << dx 3 d Power conservation requires dp =dp, thus I (,, )cos d dad = I (n,, )cos d dad (.4) From Snell's law we have sin = n sin. Furthermore, since d = sin d d, d = sin d d, and cos d = d(sin ) = n d(sin ) = n cos d, we get cos d = n cos d (.5) Equations (.4) and (.5) lead to (.3). Equation (.3) also means that the Planck function is not conserved, but the following quantity is: B := B (r,t b ) n'(r) = h 3 = constant (.6) c ( exp(h /k b T b ) ) Since the quantities on the right side are either fundamental constants (h, k b, c ), an independent but fixed variable (), or the brightness temperature T b, it means that T b does not change along the path of propagation. Thus I = B and T b are conserved quantities. This is a first important result, the fundamental theorem of radiometry (Mobley, 994). If the brightness temperature T b did change, it would violate principles of thermodynamics.

3 . Absorbing medium Consider a volume element dv = da, as shown in Figure.3, illuminated by an incident light beam of (normalised) radiance I over an infinitesimal solid angle d. Here is a path element of the beam, and da is the projected area of the volume element. The power lost from the beam by absorption follows from the change of the radiance along the path: di,a = ( I (s + ) I (s)) a = a I (s), where a (/m) is the absorption coefficient. But according to Kirchhoff's Law, in LTE, there is also an equivalent emission term: di,e = ( I (s + ) I (s)) e =+ a B (T(s)). where T(s) is the temperature of the absorbing medium along the path. The total change is the sum di = di,a + di,e = a (s)( B (T(s)) I (s)) and de = a (s) can be regarded as the infinitesimal emissivity over. da I (s,, ) dv=da I (s+,, ) Figure.3: An interacting volume element where absorption and emission take place. Remark: One way to describe the absorption coefficient is by the imaginary part n" of the complex refractive index of the medium a (r) = kn"(r) (/m) (.7) where k = / is the wave number and the wavelength, both in vacuum. This way is used if the absorption can be expressed by the medium along the path. Another way consists of the summation of the absorption cross sections of absorbing particles within the volume element dv, and dividing the sum by dv. The absorption coefficient (unit /m) is the volume density of absorption cross sections. The Radiative Transfer Equation (RTE) considers the changes that occur to I or I along the propagation path s. Budgeting the source and loss terms over the infinitesimal step the resulting differential equation is di (s) = a (s)( B (s) I (s)) (.8) This form of the RTE is called Schwarzschild's Equation. The path dependence of B arises from the dependence on the local temperature T(s). By the use of the normalised quantities, (I and B ), the refractive index does not appear explicitly. In the Rayleigh-Jeans Approximation, the radiative transfer equation simplifies to: dt b = (s) T(s) T a ( (s) b ) (.9)

4 .3 Including absorption, emission and scattering da I (s,, ) dv=da I (s,',') I (s+,, ) Figure.4: An interacting volume element where absorption, emission and scattering take place. In the final step towar the complete RTE, volume scattering in the volume element dv is included. The losses contain contributions from absorption ( a ) and scattering ( s ), the sum e = a + s being the extinction coefficient ( e ), and the ratio = s = s (.) e a + s is called single-scattering albedo. In analogy to (.8) the RTE now rea di (s) = e (s) I (s) + (s) (.) The source term consists of thermal emission, first term in (.), and of radiation scattered from other directions into the considered ray path, second term in (.): (s) = a (s)b (s) + e(s) p(,,',') I (s,',') d' (.) 4 4 The integral in the second term contains the normalised radiance I at path position s for all directions. The so-called phase function or indicatrix p(,, ', ') describes the transfer of radiance from direction (', ') to (, ). The term "phase function" derives from the lunar phase, i.e. scattering of sunlight by the moon. The phase function is reciprocal: p(',',,) = p(,,','). Energy conservation requires that 4 4 p(,,',') d = s e = (.3) The phase function is also related with the bistatic scattering cross section bi and with the extinction cross section e of the volume element: p(,,',') = p( ) = bi( ) e = bi( ) e (.4) The left-most side of (.4) is expressed in a laboratory system by the spherical i=(',') s=(, ) coordinates with one preferred direction ( = ) whereas the rest refers to the scattering plane with the scattering angle = < (i,s). For μ = cos and μ'= cos', the scattering angle follows from cos = μμ'+ ( μ ) ( μ' )cos( ' ) (.5) In the Rayleigh-Jeans Approximation the RTE is obtained by replacing I by T b, and B by T dt b (s) = e (s) T b (s) + a T(s) + e(s) 4 This is an integro-differential equation for T b. p(,,',') T b (s,',') d' (.6) 4

5 3.4 Formal solution: integral form of the RTE Absorbing, scattering, and emitting medium I = I (s=s,,) I = I (s=s,,) =I (=,,) = I (=,,) s s s+ s -d Figure.5: Integration paths in s and through the medium of radiative interaction. Let us recall that we are looking for an expression to describe how T b or I changes along the path from a starting point at s to an end point at s. First the radiative transfer equation is simplified by making the path variable non dimensional and calling it optical depth. This optical depth, also called opacity (s) at a variable path position s as seen from the end position s is defined by s (s): = (s')' (.7) e s Note that d = e and are in opposite directions. The opacity replaces the geometrical path by an interaction - weighted path. Regions without interaction are not "counted". Furthermore the source function J is defined by J : = e =( )B (T) + 4 With these quantities the radiative transfer equation is simplified to which can also be written as p(,,',') I (s,',') d' (.8) 4 di d I = J (.9) d d e I () ( )= e J (.) s Integration from = to = = e gives the change of exp(-)i in the considered s medium, leading to a formal solution or integral form of the RTE at the output position s : I (s = s ) = I ( = )= I (s )e + e J ()d (.) where I (s ) is the input, and is given by (.7). In the Rayleigh-Jeans Approximation this is Tb ( s ) = Tb ( s) e + ( ) T ( ) + p Tb d e d (,, ', ') (, ', ') ' (.) 4 4 Again, this form is the same for refractive and non-refractive media. It reduces to the scatterfree situation for s = and thus p= =, in which case d= a and Tb ( s ) = Tb ( s) e + T ( ) e d (.3) where is given by (.7), but with e = a.

6 .5 The Flux Equation 4 First it is noted that the path derivative in the RTE can be written as di (r, n ) ) = n ) I (r, n ) ) = ( n ) I (r, n ) )) (.4) where n ) is the unit vector in the direction of the path s. Integrating (.4) over direction gives the flux divergence F where F correspon to (.5), but for the normalised radiance I. Integrating the right-hand side of the RTE (.-.) gives F = e w + a 4B + e 4 4 I (',') 4 p(,,',')dd' (.5) ( ) and The integrals are eliminated with the normalisation (.3): e w + a 4B + s w with the introduction of w = I d = cu 4. Then we get: F = a ( ) (.6) w + 4B This is the net source (+), sink (-) of radiative power per unit volume element and per unit frequency interval at the given location. The equation states that F changes by the absorption coefficient only. This means that in a conservative medium, defined by a =, the flux is free of divergence: F =. This is also true for a > in thermodynamic equilibrium where the two terms in the bracket of (.6) cancel. Integration of (.6) over frequency gives the rate of change of radiation energy density which is related to the local cooling /heating rate by radiation. ( ) F = a w + 4B T d = c p t (.7) where is the mass density and c p the specific heat at constant pressure in the given volume element. This is the continuity equation for the balance of radiation and heat energy (expansion and internal). In the special case where the absorption coefficient is constant over frequency, we can directly express the emission term by the temperature: F = a w (k b T) 4 5h 3 (.8) c Problem Compute the radiative heat loss and cooling rate at night at the top of the atmosphere for T =, 5, 7K, if we assume Equation (.8) to apply and if a =./km. Use c p =5J /kg /K (dry air) and =.4kg /m 3 (typical value at km altitude. Furthermore, assume that w = 4 5 (k b T) 4 (radiation comes mainly from the lower hemisphere). 5h 3 c

7 5.6 Plane-parallel medium Radiation in a 3-dimensional medium is difficult to handle. Simpler are -dimensional media where the parameters depend on one spatial coordinate, only. We will concentrate on planeparallel media to mimic situations close to the surface of a planet (Figure.6). The medium parameters depend on z (or z ) only. Top at z=z, z = z s = z /μ; where μ = cos (.9) z Bottom z=z, z = x Figure.6: Geometry in a plane-parallel atmosphere with height variable z. The path is at an angle with respect to the z axis. An azimuth angle () measured from the horizontal x axis is used to orient the path around the z axis. A first ray path s in direction (, ) is shown in Figure.6. Another ray path s ' in a direction ( ', ') may be defined similarly by s'= z /μ', where μ ' = cos '. By the convention of Chandrasekhar (96), we understand μ>, writing μ to express downwelling rays. Two rays may interact through scattering through the phase function, given by (.4), depending also on z (or z ). The scattering angle is determined by (.5). Eliminating s by z and μ, the radiative transfer equation rea μ di (z,μ,) dz and with the introduction of the zenith optical depth z = e (z) I (z,μ,) + (z,μ,) (.3) z z : = (z')dz' (.3) e z thus = / μ, giving (see also Chandrasekhar, 96, p. ) z μ di d z = I J (.3) with the formal solutions for upwelling and for downwelling radiation at z (or z ): I ( z,+μ,)= I (,μ,)exp z + exp z' z J ( z ',+μ) d z' (.33) μ μ μ I ( z,μ,)= I (,μ,)exp z z + exp ' z z J ( z ',μ) d z ' (.34) μ μ μ where = z (z = z ) is the zenith opacity of the total layer, and the source function J is, according to (.8), given by J : = =( )B (T) + p( z,μ,,',') I ( z,μ',') d' (.35) e 4 The escaping radiances are solutions at the boundaries ( z =, ) of the layer: 4 z

8 6 I (,+μ,)= I (,μ,)exp + exp z' J ( z ',+μ) d z' (.36) μ μ μ I (,μ,)= I (,μ,)exp + exp ' z J ( z ',μ) d z ' (.37) μ μ μ and respective expressions apply to the Rayleigh-Jeans Approximations. Figure.7: A text written on a wire grid is illuminated by a lamp from below and towar a wall in the background. The shadow shows the word "invisibile", whereas the text in the foreground rea "visibile". Explain this contradiction, using the RTE.