Radiative Transfer Multiple scattering: two stream approach 2 N. Kämpfer non Institute of Applied Physics University of Bern 28. Oct. 24
Outline non non
Interpretation of some specific cases Semi-infinite cloud Aim: Gain insight into how ω and g influence reflection and absorption properties of clouds Assumption: Exclude influences from below cloud assume a semi-infinite cloud. Cloud top at τ =. Total opacity of cloud τ A photon incident on top has essentially no chance of emerging from the bottom before either getting absorbed or getting scattered back up through cloud top Examples of such clouds: Thick atmospheres of e.g. Venus or Jupiter Deep ocean We obtain for intensities: I (τ) = I r e Γτ I (τ) = I e Γτ non and for the albedo, i.e. the ratio of reflected to incident flux: Albedo= πi () πi () = r = ωg ω ωg+ ω
Semi-infinite cloud Discussion Albedo= πi () πi () = r = ωg ω ωg+ ω It is now clear why r is called albedo If ω = then r = regardless of g as long as g < If there is no absorption, then any photon entering on top of a must eventually emerge on top again If g = then r = regardless of ω as long as ω < Every photon that is scattered continues traveling in exactly same direction unrealistic More realistic is g = 5 for clouds in solar band Clouds are almost not absorbing, i.e. ω non Scattering properties of water spheres
pro memoria: Scattering properties of spheres Q ext Extinction Efficiency 4 m =.33 3.5 m =.33 +.i m =.33 +.i 3 2.5 2.5.5.5 5 5 2 25 3 35 4 45 5 5 5 2 25 3 35 4 45 5 Size parameter x=2!r/" Size parameter x=2!r/" Single Scatter Albedo Scattering Asymmetry Parameter non # g Q abs Absorption Efficiency 2 m =.33 +.i m =.33 +.i m =.33 +.i.5 m =.33 +.i m =.33 +.i m =.33 +.i m =.33 +.i m =.33 +.i 5 5 2 25 3 35 4 45 5 Size parameter x=2!r/" copied from Petty, Atmospheric radiation m =.33 m =.33 +.i m =.33 +.i m =.33 +.i 5 5 2 25 3 35 4 45 5 Size parameter x=2!r/"
pro memoria: ω and co-albedo for water and ice spheres ω (a) Single Scatter Albedo - Water vs Ice.7.5.3. Water, r = 2 µm Ice, r = 2 µm Wavelength λ [µm] (c) Single Scatter Albedo - Cloud Droplets -ω.. (b) Single Scatter Co-Albedo - Water vs Ice.. Water, r = 2 µm Ice r = 2 µm e-5.5.5 2 2.5 3 Wavelength λ [µm] (d) Single Scatter Co-Albedo - Cloud Droplets non ω.7.5.3 r = 5 µm r = µm r = 2 µm. Wavelength λ [µm] copied from Petty, Atmospheric radiation -ω.... r = 5 µm r = µm r = 2 µm e-5.5.5 2 2.5 3 Wavelength λ [µm]
Semi-infinite cloud 8r.7.5.3.. Albedo of a Semi-Infinite Cloud g= g=5 9 absorptivity is a = r because t = for a ω = 99 and g = 5 absorptivity = 5% contradiction?? ω is the measure for a single scattering event. Chance of photon is very low to be absorbed in any single extinction event, but multiple scattering enhances probability For what value of τ can a cloud be viewed as effectively semi-infinite? 99 ω 999 9999 99999 non
Interpretation of some specific cases Non, ω = Consider cloud with finite optical depth given by τ Assume that scattering is conservative, i.e. ω = This is realistic for most clouds over the visible spectrum and absorption is negligible General solution: With the same boundary conditions, i.e. I () = I and I (τ ) = non I (τ) = I ( g)(τ τ) + ( g)τ I (τ) = I [ + ( g)(τ τ)] + ( g)τ reflectivity: r = I () I () = ( g)τ + ( g)τ = t transmittance: t = I (τ ) I () = + ( g)τ r + t = because absorption is zero as ω = Note: τ can be large and still have significant transmission. Example τ = t =.6
Interpretation of some specific cases Non, ω = reflectivity: r = I () I () = ( g)τ + ( g)τ = t transmittance: t = I (τ ) I () = + ( g)τ Question: How depends reflectivity on wavelength? Differentiate equation for reflectivity r, i.e. dr dλ = dr dτ dτ dλ For τ dr dλ i.e. essentially independent of wavelength. Multiple scattering is washing out spectral dependence. Clouds are white. For τ dr dλ dτ dλ i.e. essentially reflectivity of individual scatterers non
Interpretation of some specific cases Non The diffuse downward irradiance beneath a cloud is light that has been scattered, i.e. the difference between the total downward irradiance F and the unscattered downward one, F u. D = F F u = F t F e τ non copied from C.Bohren, Fundamentals of Atmospheric radiation Clouds increase the diffuse downward radiation - but only up to a point. With increasing optical thickness τ clouds eventually reduce the downward radiation to less than what it would be from a clear sky Clouds are both givers and takers of light (C.Bohren)
.7.5.3.. Albedo of a Semi-Infinite Cloud g= g=5 9 Interpretation of general case Absorbing cloud, ω < and arbitrary τ From the general two stream equations, it follows for t and r: Transmittance t = r 2 e Γτ r 2 e Γτ ω= ω=99 ω=9 ω= Albedo r = r [eγτ e Γτ ] e Γτ r 2 e Γτ ω= ω=99 ω=9 ω= non 2 3 4 5 6 7 8 9 τ* 2 3 4 5 6 7 8 9 Transmittance, g = 5 Albedo Even slight decreases in ω can lead to pronounced decreases in both t and r and thus an increase in a = t r 8r Asymptotic value of r is r link to τ* 99 999 9999 99999
Interpretation of general case Absorbing cloud, ω < and arbitrary τ Absorptance a = r t Absorptance ω=99 ω=9 ω= non 2 3 4 5 6 7 8 9 copied from G.Petty, Atmospheric radiation τ*
8r Semi-infinite cloud Albedo of a Semi-Infinite Cloud.7.5.3. g= g=5 non. 9 99 ω 999 9999 99999 absorptivity is a = r because t = for a ω = 99 and g = 5 absorptivity = 5% contradiction?? ω is the measure for a single scattering event. Chance of photon is very low to be absorbed in any single extinction event, but multiple scattering enhances probability For what value of τ can a cloud be viewed as effectively semi-infinite; in the sense that further increases in τ don t significantly change the cloud s overall radiative properties.
Interpretation of some specific cases Semi-infinite cloud as an approximation Equivalent to τ is t. In reality t > but we can choose t very small, say t. Solve t = τ* r 2 e Γτ r 2 e Γτ for τ Minimum Thickness of a Semi-Infinite Cloud g=5 g= ω.3.7.5 Water, r = 2 µm Ice, r = 2 µm Wavelength λ [µm] (c) Single Scatter Albedo - Cloud Droplets -ω.. e-5.5 non clouds over non black (d) Single Scat... W. 9 99 ω 999 9999 99999 999999.3. copied from G.Petty, Atmospheric radiation r = 5 µm r = µm r = 2 µm Wavelength λ [µm]. e-5.5 W
. Interpretation of some specific cases Semi-infinite cloud as an approximation τ* Minimum Thickness of a Semi-Infinite Cloud. 9 99 ω 999 9999 g=5 g= 99999 999999 ω (c) Single Scatter Albedo - Cloud Droplets.7.5.3. Water, r = 2 µm Ice, r = 2 µm Wavelength λ [µm] r = 5 µm r = µm r = 2 µm Wavelength λ [µm] A strongly may be semi-infinite for rather small τ, while a strongly scattering cloud continues to transmit at least % until τ reaches a value of several hundred relatively thin water clouds may be treated as opaque in thermal IR for which ω, while the same clouds may be far from opaque in the visible, despite having roughly the same τ in both bands -ω. e-5 (d) Single Scatter Co-Albedo - Clo... Water, r = 2 µm Ice r = 2 µm.5.5 2 Wavelength λ [µm] non. r = 5 µm r = µm r = 2 µm e-5.5.5 2 Wavelength λ [µm]
Clouds over non black s Assume a non black below cloud boundary conditions will change: I (τ ) = r sfc I (τ ) Radiation transmitted through cloud will be reflected on some will penetrate the cloud but some will be scattered down thus increasing radiation on ad infinitum r = r + r sfct 2 r sfc r and t = t r sfc r non copied from Petty, Atmospheric radiation Net result: increase in the total downward flux on and increase of albedo at cloud top Making ground more reflective makes sky brighter above!
m m m cm cm mm. mm µm µm.3.5.7.5 2 3 4 5 7 5 2 3 5 Wavelength [µm] From G.Petty: A first course in atmospheric radiation From Bohren & Clothiaux: Fundamentals of atmospheric radiation Water ( o C) Ice (-5 o C) Introduction Literature Manifestations Flux and intensity Blackbody Planck Wien, Stefan-Boltzmann Kirchho N and Basic relations Kramers-Kronig Mixtures Lorentz model Debye Applications Clouds Multiple scattering can greatly amplify photon paths A cloud is equivalent (as far as attenuation is concerned) to a slab of water of thickness L 3( g) k ad where L is cloud liquid path, d drop diameter and the absorption coefficient k a m. Also check viewgraph about optical thickness of a cloud layer, τ 75, L 5mm Amplification factor of order of! Bottoms of very thick clouds can have a bluish cast non Example: Penetration depth of radiation in water and ice Depth [m] Radiation Penetration Depth in Water and Ice
Snow holes and crevasses Example of medium where absorption is dominant non
Wet sand Sand has some color slightly absorbing, ω The relative index of refraction m for sand grains in air is different to sand grains in water. From Mie theory it follows that g increases when m of the particle and the surrounding medium move closer together, i.e. more forward scattering. Relative change of r with g is given by r r g = ω ( g) ωg Greatest relative change in r occurs for g close to Decreased reflectivity upon wetting is a consequence of single scattering peaking more sharply in the forward direction non...and why is the foam bright?
Broken beer bottles Beer bottles have a color. Pulverizing them decreases the glass particles... non...and why are clouds white?,
Radiative Transfer is Fun non