I - radiation intensity

Transcription

1 1 Oct Chapter 3: Radiatie Fluxes in Common Land Model 1. Basic Notions Def: Radiation Flux (F) - amount of enery propaatin throuh spatial location per unit area and per unit time [W/m 2 ]. Def: Radiation Intensity in a ien ection (I) - radiation flux per unit solid anle, propaatin throuh spatial location in a ien ection per unit projected area in that ection [W/m 2 Sr]. Fiure 3.1: Radiation Intensity θ Ω - solid anle I - radiation intensity 1/ cos( θ) - projected area F I ( θ) cos( θ) Ω Solid anle, dω sin( θ)dθdϕ ( θ - zenith, ϕ - azimuth) Whole sphere contains solid anle of π. + 2π denotes interation oer the upper and lower hemispheres 2π denotes interation oer the lower hemisphere Flux Up: F I( θ) cos( θ) dω + 2π Flux Down: F I( θ) cos( θ) dω 2π + 2 π F Fiure 3.2: Enery fluxes up and down F 2 π

2 2 2. Net Radiation In this chapter we describe radiation modelin as implemented in Common Land Model (CLM). CLM treats land surface as a system which consist from two radiatiely interactin components: eetation canopy (denoted by V) soil backround (denoted by G) Note that CLM assumes that eetation consist of reen leaes, which can photosynthesize, and other tissues (stems, branches, twis, senescent eetation, etc). Density of eetation is characterized by LAI- Leaf Area Index- one-sided reen leaf area per unit round area in broadleaf canopies as the projected needle leaf area in coniferous canopies SAI- Stem Area Index- area of stems, senescent eetation, per unit round area he sum of LAI and SAI is called Plant Area Index (PAI) CLM models two components of radiation, which interact both with eetation and round: shortwae (solar) lonwae he net radiation at the surface (cf. Fi. 3.3) is Net radiation ( A + AG ) ( L + LG ), where A V and A G denote solar enery fluxes absorbed by eetation and round (not just incomin solar radiation), and L V and L G is the net lonwae radiation from eetation and round Veetation A V L V Ground A G L G Fiure 3.3: Net radiation in CLM. 3. Shortwae Fluxes Modelin of interaction of shortwae radiation with land surface in CLM is based on Radiatie ransfer (R) theory. he most complete formulation of 3D R inoles modelin of radiation intensities in complicated 3D eetation structure. Such leel of details is unnecessary for CLM. Instead simplest ersion of R is used, which is called two-stream approximation. his is sufficient to ealuate at lobal scale key CLM radiation ariables- albedo and FPAR. wostream approximation models up- and downward enery fluxes based on enery conseration law. In this section we introduce basic concepts of CLM s two-stream model.

3 3 Incomin solar radiation can be of two types a) Direct radiation (flux is concentrated in narrow beam). his is a not scattered by atmosphere solar enery flux b) Diffuse radiation (flux is distributed alon anular ections). Anular distribution of the enery is due to atmospheric scatterin Independently from the type of incomin radiation the followin enery conseration law is alid: I R + A V + A G, (1) or A V I R A G, (2) where I - incomin solar enery flux, R - reflected flux, A V - absorptance of eetation, and A G - absorptance of round Direct Incomin Radiation Diffuse Incomin Radiation I R (I ) I R (I ) Veetation A V (I ) Veetation A V (I ) (I α ) α (I ) α (I ) Ground Ground Fiure 3.: CLM modelin of shortwae enery fluxes for ect and fuse incomin radiation.

4 Enery conseration law can be formulated separately for fuse and ect input fluxes. he notations are as follows (cf. Fiure 3.): I and I - incomin ect and fuse solar flux, R (I ) and R (I ) - reflected fuse flux due to input I and I, A V (I ) and A V (I ) - eetation absorbtance due to input I and I, (I ), (I ), (I ) - transmitted ect and fuse fluxes due to input I and I, α and α - round albedos for ect and fuse fluxes. 3.A. ABSORBANCE FOR UNI INCOMING FLUX Unit Direct Incomin Radiation Veetation absorbtance under ect illumination can be deried from enery conseration law (Eq.2). With reference to Fi. 3. we can specify all inoled fluxes, A (I ) 1 R (I ) 1 α (I ) 1 α (I V Unit Diffuse Incomin Radiation [ ] [ ] ) Veetation absorbtance under fuse illumination can be deried similarly, A V (I ) 1 R (I ) [ 1 α ] (I ). Note that simple analytical expression for (I ) can be deried from Radiatie ransfer theory. his is just a Beer s law: (I ) exp( K(L + S) ), where L is a leaf area index (LAI) and S is a stem area index (SAI). K is a coefficient, describin extinction of solar enery, G( µ ) K 1 ω, µ where G is a mean projection of leaf normals in ection θ, µ cos(θ), ω is a leaf albedo. 3.B. ABSORBANCE FOR ARBIRARI INCOMING FLUX Now we consider enery conseration law for arbitrary incomin flux, which consists from ect and fuse components and both of them are not normalized. General Law for otal Enery Absorbed by Veetation and Ground A G I (1 α A ) V I (I A ) + V (I ) + I Consider special case of no eetation. In this case A V 0, A G A V (I [ I (I ) + I (I )] (1 α ) [ I 0 + I 1] (1 α ) I (1 α ) 1+ I (1 α ) + I (1 α ) ).

5 5 Solar Enery Conseration Law Accordin to enery conseration law, Eq. (1), I R V + A V + AG, herefore in the eneral case of ect and fuse incomin radiation we hae: I + I I R (I ) + I R (I ) + A + A. [ ] V G 3.C. ABSORBANCE BY SUNLI AND SHADED LEAVES Now let us assume that leaes and stems absorb all the incomin enery. In this case portion of sunlit eetation (leaes + stems) at the canopy depth (L+S) is defined by transmitted ect radiation (I ) exp( K(L + S)), herefore, total portion of the sunlit eetation in the canopy layer [0; L+S], is f SUNLI L+ S otal shaded area is: f Next, sunlit and shaded area of leaes is, L 1 1 exp( K(L + S)) exp( Kx)dx L + S, K(L + S) L 0 1 SHADED f SUNLI SUNLI SHADED f L, SUNLI f L, SHADED And finally absorbtance of sunlit and shaded leaes is, L A V, SUNLI [ I A(I ) + I A(I )], L + S A V, SHADED 0.

6 6 Oct-13/ wo-stream Radiatie ransfer Model.A. BACKGROUND INFORMAION In preious section we discussed CLM s radiatie transfer fluxes at the boundary of eetation. Gien fluxes enterin eetation (downward incomin radiation) and exitin eetation (transmitted and reflected fluxes) we calculated absorbtance of eetation. In this section we will consider R processes inside of eetation, as described by two-stream approximation in CLM. wo-stream approximation is deried from 1D R equation: µ I(L, Ω) + G( Ω) I(L, Ω) dω Γ( Ω Ω) I(L, Ω ), (1) L AND Boundary conditions where 1D coordinate is specified by Leaf Area Index (L), oriin at the top of canopy, I(L, Ω) - intensity of radiation at depth L and in ection Ω, µ cos(θ), G( Ω) - mean projection of leaf normals in ection Ω : parameter controllin extinction of radiation due to absorbtance by leaes (Fi. 3.5), Γ( Ω Ω) - area scatterin phase function: parameter controllin scatterin of radiation due to leaf reflectance and transmittance (Fi. 3.5) π Radiation extinction (Beam extinction in the ien ection) leaf Ω leaf Ω Radiation scatterin (Beam amplification in the ien ection) Fiure 3.5: Radiation extinction and scatterin Physical meanin of R equation (Eq. 1) is as follows: he radiation intensity in a ien ection, Ω, (erm 1) is chanin due to radiation extinction, caused by leaes absorbtance (erm 2) and scatterin radiation by leaes from other ections, Ω, into this ection, Ω (erm 3). In essence R equation is just an enery conseration law. As a special case of this equation, consider equation for ect radiation. In this case radiation extinction exist due to leaf absorbtance, (erm 2 exist), howeer there are no scatterin inside the ection of ect beam, because this a sinle ection beam (erm 3 does not exist). herefore,

7 7 µ I (L, Ω0 ) + G( Ω0 ) I (L, Ω0 ) 0, L Solution of this equation is exponent: G( Ω0 ) I (L, Ω0 ) exp L (2) µ ( Ω0 ) Next, consider deriation of the two stream equation for radiation fluxes. Equation for fluxes can be deried fro Eq. (1) for ectional intensities by anular interation of intensities. Recall that, radiation fluxes in upward and downward ections are defined throuh radiation intensities as follows, F I( θ) µ ( θ) dω F + 2π 2π I( θ) µ ( θ) dω o derie two stream equation one should specify assumptions/approximations. Simplest ersion of two-stream can be deried for horizontal leaes. In this case after interation of the (Eq.1) we will et: F (L) ρ F (L) + ( τ 1) F (L), (3) L F (L) ρ F (L) + ( τ 1) F (L), L AND Boundary conditions, where: F (L) and F (L) - total radiation fluxes up and down (ect+fuse) L - leaf area index - leaf reflectance (Fi. 3.6) ρ - leaf transmittance (Fi.3.6) τ transmittance 0.2 Fiure 3.6: ypical spectra of leaf transmittance and reflectance from field measurements Reflectance reflectance absorptance ransmittance Waelenth, nm

8 8 Physical meanin of two stream equations (Eq. 3) is as follows. Consider the firsts equation. Chanes in the downward flux are due to downward scatterin of upward flux (the ain term) and attenuation of the downward flux caused by reflectance and absorbtance (loss term). Similar meanin has the second equation..b. CLMS WO-SREAM Version of the two stream equation used in CLM is more sophisticated. It is formulated for arbitrary leaf normal orientation, takes into account leaf area index (LAI), stem area index (SAI), mixture of eetation with etc. Dickinson (1983) formulated two stream equations which are currently used in CLM. CLM s two stream equations for fuse radiatie fluxes are as follows: µ F (L) ω β F [1 (1 β) ω] F (L) + ω µ K (1 β0 ) exp( K (L + S)), () (L + S) µ F (L) ω β F [1 (1 β) ω] F (L) + ω µ K β0 exp( K (L + S)), (L + S) source due to ect beam AND Boundary conditions (as specified by incomin flux at the top and round albedos at the bottom) where: F (L) and F (L) - fuse radiation fluxes up and down per unit incident flux L and S - Leaf Area Index (LAI) and Stem Area Index (SAI) K G / µ - optical depth of ect beam per unit L+S µ cos(θ) G( θ) - mean projection of leaf normals in ection θ. µ - aerae inerse K per unit L+S ω- effectie albedo of scatterin elements (leaes, stems, and ). Note for eetation elements albedo includes not only reflectance component but also transmittance β and β0 - upscatter parameters for fuse and ect radiation, respectiely. hose parameters are used as coefficients for ω to account for complex effects of scatterin. Note that CLM s two stream has similar structure to simple two-steam equation for horizontal leaes. he ference is that CLM equations are formulated for fuse radiation only and therefore hae additional source term, which is due to the enery input from ect radiation (compare source term with Eq. 2) Physical meanin of CLMs two stream equation (Eq.) is similar to two stream equation for horizontal leaes (Eq. 3): Chanes in the downward flux are due to downward scatterin of upward flux (the ain term) and attenuation of the downward flux caused by reflectance and absorbtance (loss term) and due to downward scatterin of the ect radiation (ain term).

9 9 Gien boundary condition from soil, CLMs two steam equations can be soled to ealuate the followin fluxes per unit incident flux at VIS ( nm) and NIR (>700 nm) bands: Absorbed by eetation Reflected by eetation ransmitted throuh eetation.c. BOUNDARY CONDIIONS Boundary conditions (as specified by incomin flux at the top and round albedos at the bottom): F (L + S 0) F Where α is a round albedo..d. OPICAL PARAMEERS F (L + S L total + Stotal ) αf, Next we describe optical parameters of the CLM s two stream accordin to Sellers (1985): Mean projection of leaes normals in ection θ, G-function is G ( µ ) a + b µ Where 2 a b (1 2a) And is the deiation of leaf anles from a random distribution: + 1 for horizontal leaes 0 for randoml leaes 1 for ertical leaes Note that CLM can simulate any cases of leaf normal orientations in between aboe extremes. Aerae inerse optical depth 1 1 µ dµ K( µ ) G( Leaf and Stem optical properties µ µ dµ ) 1 b 1 a b a + b ln a CLM assumes that eetation consist of reen leaes and stems. Leaf (stem) optical properties leaf stem are characterized by leaf (stem) reflectance, ρ ( ρ ), and leaf (stem) transmittance leaf stem τ ( τ ) respectiely. Veetation optical properties are characterized by weihed aerae of leaf and stem optical properties e leaf leaf stem stem ρ ρ w + ρ w, e leaf leaf stem stem τ τ w + τ w,

10 10 Where L w leaf L + S w stem S L + S Gien transmittance and reflectances of eetated elements, its albedo is defined as, e e e ω ρ + τ. Refer to able 1 for eetation optical properties utilized by CLM. ρ leaf VIS ρ leaf NIR ρ stem VIS ρ stem NIR τ leaf VIS τ leaf NIR τ stem VIS τ stem NIR able 3.1: Optical properties of ferent eetation types (Plant Functional ypes) in CLM. Veetation and Snow optical properties CLM also accounts for possible the mixture of eetation and, as a weihted aerae of both to calculate effectie albedo of mixture: e wet wet ω ω ( 1 f ) + ω f, where f wet is the wet fraction of the canopy. When temperature of the leaes is lower than freezin point, CLM assumes that canopy is coer with.

11 11 Upscatterin term for fuse and ect radiation CLM describes complex scatterin effects, which can not be accounted for by usin only effectie albedo, ω. Instead, upscatterin term, β, is used in couple with ω. In the case of eetation only, upscatterin coefficient can be specified as follows. For fuse radiation upscatterin coefficient is defined as: 2 e e 1 e e e e 1+ ω β ρ + τ + ( ρ τ ). 2 2 Consider special case of horizontal leaes. In this case 1, and ω β ρ, ρ 1 (1 β) ω 1 (1 ) ω 1 ω + ρ 1 τ. ω herefore CLM s two-stream (Eq.) become identical with Eq. (3) for horizontal leaes. his proides additional means to understand meanin of the upscatterin coefficients- as they were introduced to account for more complex features of radiation scatterin, arisin due to arbitrary orientation of leaf normal compared to simple case of horizontal leaes with sinle ection of orientation of leaf normals. For ect radiation upscatterin coefficient is defined as: e e e 1+ µ K ω µ G( µ ) ω β0 µ µ d K 2 µ G ( µ ) + µ G( µ ).E. SOLUION OF HE CLM S WO-SREAM Finally, solution of the CLM s two-stream equation has the followin eneral form: F A exp[ K(L + S) ] + B exp[ η(l + S) ] + C exp[ η(l + S) ], F A exp[ K(L + S) ] + B exp[ η(l + S) ] + C exp[ η(l + S) ], where A, A, B, B, C, C, and η are alebraic combination of coefficients of CLMS s two-stream equation as specified in CLM tech notes (WWW1). Physical meanin of each term of solution on the riht-hand side is as follows: contribution from ect flux is ien by the first term, contributions from fuse downward and upward fluxes are ien by second and third terms, respectiely. 1 0

12 12 Oct Boundary Conditions- Ground Albedo Ground surface below eetation may sinificantly influence radiatie transfer processes especially for the sparse eetation. CLM discriminates between two components of the round: Soil Snow Key round ariable for the purpose of modelin shortwae fluxes is albedo, as it seres to specify boundary conditions for two-stream model. CLM models albedo as function of: Ground type (soil and ) Flux type (ect and fuse) Waelenth (VIS and NIR) otal round ect ( α ) and fuse ( α ) albedos are modeled in CLM as a weihted combination of soil and albedos, α ( λ) α ( λ) (1 f ) + α ( λ) f, where λ is a waelenth. Coefficient soil soil ( λ) f α ( λ) α (1 f ) + α ( λ) f, f is the fraction of the round coered with, z 10 z + z. Aboe, z is the depth of (in meters), z [m] is the momentum rouhness lenth for soil. Estimation of the fraction is an approximation. CLM assumes that if effectie rouhness lenth (10 ) of soil is comparable to depth, wind can blow away and expose bare soil ( albedos. f 5.A. SOIL ALBEDO z 0 0 <1). Below we proide details on the CLM s parameterization of soil and Notion of soil is quite broadly defined in CLM and includes laciers, lakes, wetlands, and soil itself. In eneral, such surfaces posses relatiely stable in time characteristics, in contrast to rapidly chanin properties of. Characteristics of albedos for ferent types of soil types are ien below. Glaciers albedos soil soil soil ( λ NIR) αsoil ( λ NIR) α ( λ VIS) α ( λ VIS) 0.80, α Glaciers ect and fuse albedos are equal. Albedo at VIS is hiher than at NIR band (in contrast to eetation).

13 13 Frozen lakes and wetlands albedos α soil ( λ VIS) αsoil ( λ VIS) 0.60 α soil ( λ NIR) αsoil ( λ NIR) 0.0 Characteristics of frozen lakes and wetlands albedos are quite similar to those of laciers. he ferences in the alues are due to ferent ae and structure of ice and due to mixture of eetation and ice. Unfrozen lakes and wetlands albedos 0.05 α soil ( λ) αsoil ( λ) µ here is no distinction between ect and fuse and VIS and NIR albedos for unfrozen lakes and wetlands. Note the dependance on cosine of Solar Zenith Anle, µ. Soil albedos Soil albedos are aryin and specified as function of color class (soil brihtness): α λ) α ( λ) ( α ( λ) + d) α ( λ), soil ( soil sat dry where d , Parameter d depends on olumetric water content of the soil top layer as modeled in CLM. Albedos for saturated and dry soils are ien in able 3.2 below. able 3.2: Dry and saturated soil albedos

14 1 5.B. SNOW ALBEDO Accordin to obserations albedo, is decreasin as ae of is increasin. Also, ect albedo is hiher than fuse albedo. Parameterization of the albedo accounts for this. Snow albedo for fuse illumination where, F ae,0 (,0 [ 1 C( λ) F ] α ( λ) α ( λ), α λ) is the albedo of fresh for SZA<60 0 and C( λ) is an empirical constant. accounts for reduction of albedo with ae (due to increasin rain size, t, soot content) for SZA<60 Ae of, τ 0, F ae ae τ, is incremented in the CLM runs at each time step as follows τ0 (r1 + r2 τ 0, for M Where t - model time step, [s] 1x10-6, [s -1 ] τ 0 + r 3 ) t, for 0 < M > 800 M -mass of water, [k/m r 1 - represents rain rowth due to apor fusion r 2 - represents effects at freezin point r 3 - represents effects t and soot 2 ] 800 CLM assumes that a fall of 10 [k/m 2 ] of liquid water equialent restores ae to that of fresh. Howeer, typical precipitation in one model step is lower, and therefore CLM correct for the partial ae reduction as follows, τ n+ 1 ( τ n + τ n+ 1 n [ 0.1 (M M )] )1 where τ is the ae at the current time step, is the ae at the preious time step, M n+ 1 n+ 1 M n Snow albedo for ect beam n τ is the chane in the mass of water due to fall. α CLM deries fuse albedo,, from ect beam albedo,, by applyin correction for hih Solar Zenith Anles (SZA), α ( λ) α ( λ) + 0. f ( µ ) [1 α ( λ)],, α

15 15 where function f ( µ ) is a correction factor, which account for the obsered increase of ect beam albedo for SZA>60 0, as compared to fuse albedo, 1+ 1/ b 1 0,if SZA 60 f ( µ ) 1 + µ 2 b b 0 0,if SZA < 60 Parameter b controls sensitiity of correction factor to SZA ariations. Accordin to obserations, b Examples of wo-stream Simulations Examples of two-stream simulations are ien for random leaes. Simulations were based on Stochastic R model (Shabano et al., 2000) run in the two-stream model. Some minor discrepancies may be found with CLM s two-stream. Fi. 3.7 shows ariation of the surface albedo as function of LAI of eetation and soil optical properties. In the case of no eetation albedo of the total system is equialent to the soil albedo. ypically, round albedo at VIS and NIR waelenths are linearly related (cf. soil line). If soil albedo is constant, the albedo of total system is chanin as function of LAI alon trajectories, called soil isolines. If LAI is lare independently from soil albedos, all trajectories conere to a sinle point in VIS-NIR spectral space. If LAI is constant, total albedo aries alon LAI isolines. Fiure 3.8 shows impact of Solar Zenith Anle (SZA) on dependance of albedo, absorptance and transmittance from LAI. Fiure 3.7: Simulated Albedo in Red-NIR spectral space. Input flux: ect radiation only.

16 16 Fiure 3.8: Impact of Solar Zenith Anle on simulated Albedo, Absorptance and ransmittance as function of LAI at Red and NIR waelenths. Simulations with SZA0 0 are shown in black, while simulation with SZA60 0, are in shown in red. Input flux: ect radiation only. Ground albedo (soil) was set to 0.1. Assinment 01: CLM s two-stream equations can be soled analytically (CLM s tech notes, pp.25-27). Write a code which implements the solution for upward F (L) and downward F (L) fuse fluxes. Input flux- ect radiation only. Study two eetation classes (plant functional types, PF): Broadleaf Deciduous rees temperate (BD temperate) and Crop 1. For each eetation type derie solution at VIS and NIR waelenth. For optical properties of eetation refer to able 3.1. For round albedos refer to able 3.2 (use dry dark soil pattern #8). Present plots of F (L) as function of: 1) LAI for SZA15 0 and 2) SZA (Solar Zenith Anle) for LAI3. Compare the ferences between two eetation types

17 17 Oct-25/ Lonwae Fluxes 7.A. BACKGROUND INFORMAION Def: Black body - ideal object which absorb all the incomin radiation. Def: hermal equilibrium- such a state of the system, that incomin and outoin radiation fluxes are is in balance (equal) and temperature of the system is not chanin throuh the time. Def: Absorptiity of surface, a, is the fraction of incident radiation absorbed by a surface. For black body a1. Def: Emissiity of a surface, ε, is the ratio of the actual radiation emitted by a surface to that amount emitted by a black body. For black body ε 1 Kirchhoff Law - In the state of thermal equilibrium absorptiity, a, is equal to emissiity, ε. Plank Equation - Spectral radiation flux emitted by black body is described by Plank equation, 8πhc L( λ ), 5 λ [exp(hc / kλ) 1] where λ - waelenth, [ µ m ], L( λ) - spectral radiation flux, [W m -2 1 m µ ], - temperature of the object, [K], h Plank constant, 6.63x10-3, [J] c - speed of liht in acuum, x10 8, [ms -1 ] k - Boltzman constant, 1.381x10-23 [J K -1 ], π Stefan-Boltzman Law: Enery flux emitted by unit area of a plane surface of black body into a hemisphere is proportional to the fourth power of its absolute temperature, where L is enery flux [W m -2 ], is a temperature [K], σ L σ, is a Stefan-Boltzman constant, 5.67x10-8 [W m -2 K -1 ] Stefan-Boltzman law can be deried from Plank equation by interation oer emitted spectrum, 8πhc L L( λ)dλl dλ σ λ [exp(hc / kλ) 1]

18 18 ypical terrestrial objects (soil, eetation, water,, ice) hae such rane of temperatures (around 300K) such that emitted radiation is confined mostly in the thermal rane of waelenths (3-100 µ m ). In contrast, temperature of Sun is much hiher (6000K) and therefore maximum emitted radiation occupies VIS-NIR waelenth band ( nm). Schematic plot for lonwae enery fluxes for typical terrestrial object is ien in Fi ( 1 ε) L Scattered flux L Incomin flux ε L Absorbed flux ε σ Emitted flux Fiure 3.9: Lonwae Fluxes- Incomin, Scattered (Reflected/ransmitted), and Emitted. 7.B. VEGEAION AND GROUND FLUXES Consider propaation of lonwae radiation fluxes (heat transfer) in the two-component system (Fi 3.10): Veetation Ground he system is illuminated from the top with incomin lonwae radiation from the atmosphere. In the followin discussion round seres as a bottom boundary for lon wae radiation in eetation canopy. Analysis of the more complex processes of heat transfer deep into the round (thermal conductiity) is beyond the scope of this discussion. Note also, that CLM describes system not in equilibrium- temperature of eetation and round may chane from preious ( n ) to the next step ( n+1 ) in the CLM run. Neertheless CLM assumes that absorptiity is equal to emissiity. Veetation Fluxes he Downward flux from eetation is (1 ε ) Latm +ε L, (5) and the upward flux from eetation is L where - downward flux from atmosphere L atm + (1 ε ) Latm + (1 ε ) L +ε, (6)

19 19 L - downward flux from eetation L - upward flux from round L + - upward flux from eetation ε - emissiity ( absorptiity) of eetation - temperature of the eetation Note that due to radiatie couplin of eetation and round the upward flux from eetation includes the round component (this explains the subscript notation, +). L atm L + Veetation NE L L Ground NE L L Fiure 3.10: Lonwae Fluxes in Veetation + Ground system. Ground Fluxes For the purpose of modelin lonwae radiation, CLM implements only upward fluxes from the round (no downward fluxes are calculated), where - upward flux from round L L (1 ε ) L +ε, (7) L - downward flux from eetation (ealuated from eetation fluxes)

20 20 ε - emissiity ( absorptiity) of round - temperature of round Note, that accordin to Eq. (7) round upward flux does not depend ectly on the flux from atmosphere ( ). he round is allowed to interact with atmosphere only inectly throuh L atm eetation (throuh L ). herefore, CLM assumes no ect streamin of atmospheric flux throuh aps in the eetation. Emissiity of Veetation and Ground in CLM Durin runs, CLM uses the followin alues of eetation and round emissiity. Emissiity of round is: 0.96 for soil, 0.97 for laciers, ε 0.96 for wetlands, 0.97 for. he emissiity of eetation is calculated as, L + S ε 1 exp, µ where L is LAI, S is SAI, and µ is aerae inerse optical depth for lonwae radiation. emperature and Fluxes increments Durin CLM runs temperature of the eetation and round is updated incrementally. emperature Updates results in correspondin updates emitted fluxes for eetation and round, ε ε n + 1 n n 3 n + 1 n ( ) ε ( ) + ε ( ) ( ) n + 1 n n 3 n + 1 n ( ) ε ( ) + ε ( ) ( ), (8). (9) Note that formulas aboe are obtained accordin to standard aylor expansion technique, f () f ( + ) f (t) +, and f () ε, f () 3 ε n + 1 n.,

21 21 otal Upward Flux from Veetation and Ground Consider Eqs. (5)-(7) for eetation and round fluxes and note interdependencies. Upward flux from total system of eetation + round (, Eq. (6)) depends on upward flux from L + round ( L, Eq. (7)), which in tern depends on the downward flux from eetation (, Eq. (5)). System of Eqs. (5)-(7) can be soled for L + in terms of known input ariables, Latm,,, ε, ε. For the purpose of representation of solution in terms of CLM s incremental temperature chanes we will use Equations (8)-(9) as well in the followin deriations. Consider two cases. In the simplest case of no eetation, (1 ε L + ) L L atm +ε (1 ε ) L atm +ε n n 3 n + 1 n ( ) + ε ( ) ( ) In the second case, round is coered by eetation, and Eq. (6) is alid, (1 ε L ) L + atm (1 ε + (1 ε ) L ) L atm +ε + (1 ε ) L +ε he unknown L can be deried by combinin Eqs. (5) and (7), ( 1 ε + (1 ε ( 1 ε ) (1 ε ) ε L ) (1 ε ) L n n 3 n + 1 n ( ) + ε ( ) ( ) +ε [(1 ε ) Latm +ε ] + ε n n ) Latm + (1 ε ) ε ( ) + ε ( ) + n 3 n + 1 n n 3 n + 1 n ( ) ( ) + ε ( ) ( ) Finally, substitutin the deried expression in the aboe equation for L + (A) (1 ε ) L + atm (B) ( 1 ε ) (1 ε ) (1 ε ) L + + atm (C) + ( 1 ε ) (1 ε ) ε ( ) + (D) + ( 1 ε ) ε ( ) + (E) + ε ( ) + n n n n 3 n + 1 n + ε ( ) n 3 n + 1 ) (1 ε ) ε ( (F) ( ) + L + (G) + (1 ε ( ) ) + n + 1 (H) + (1 ε ) ε ( ) ( ) Note that each term in solution has clear physical meanin, n 3 n. n.., we will et,. L

22 22 erm (A) - atmospheric flux reflected by eetation back to atmosphere erm (B) - atmospheric flux transmitted throuh eetation to round, reflected by round back to eetation and transmitted by eetation back to atmosphere erm (C) - flux, emitted by eetation to the round, reflected by round back to eetation and transmitted to atmosphere erm (D) - flux, emitted by round, and transmitted throuh eetation to atmosphere erm (E) - flux, emitted by eetation to atmosphere erm (F) increase/decrease in the flux (due to increase/decrease in eetation temperature), that is emitted by eetation to atmosphere erm (G) increase/decrease in the flux (due to increase/decrease in eetation temperature), that is emitted by eetation to the round, reflected by round back to eetation and transmitted throuh eetation back to atmosphere erm (H) increase/decrease in the flux (due to increase/decrease in round temperature), that is emitted by round and transmitted throuh eetation to the atmosphere. 7.C. NE FLUXES Next, we ealuate net fluxes separately for round and for eetation. Net flux for eetation/round is calculated as a ference between outoin (emitted) and incomin (absorbed) fluxes. wo cases should be considered for round net fluxes: with and without eetation. In the case of no eetation, NE OU IN L L L ε ε L, where first tem on the riht (loss) corresponds to emitted flux from round and second term (ain) corresponds to absorbed flux from atmosphere. In the second case, when round is coered by eetation, net round flux is, NE OU IN atm L L L ε ε L, where first tem on the riht (loss) corresponds to emitted flux from round and second term (ain) corresponds to absorbed flux from eetation. Note, for both cases, the deried net round flux seres as lonwae radiation forcin to calculate soil temperature. Next, consider net radiation flux for eetation, L NE L OU L IN [2 ε (1 ε ε ε ε [ 1+ (1 ε ) (1 ε )] Latm (10) First term on the riht (loss) is most complex and will be considered last. Second term (ain) corresponds to absorption of emitted by round flux. he third term (ain) corresponds to absorption of atmospheric flux and consists of two sub-terms. Firsts sub-term corresponds to ect absorption of incomin atmospheric flux. Second sub-term corresponds to absorption of atmospheric flux transmitted throuh eetation, reflected by round back to eetation. First tem (loss) is the most complex and can be ealuated as follows. Exitin from eetation flux has three components (A, B, C, refer to Fi. 3.11). Component A corresponds to ect emission of eetation, )] ε

23 23 A ε, Component B corresponds to emission of eetation flux down toward round and absorbtance by round, B ε ε Component C corresponds to emission of eetation flux down toward eetation, reflection by eetation and transmission back to atmosphere, B (1 ε ) (1 ε Now, the sum of A, B and C corresponds to total emission (exitin flux) and can be ealuated as follows A + B + C ε [ 1+ ε [ 1+ ε [ 1+ ε + ε ε + (1 ε + (1 ε + 1 ε ε ) (1 ε ) (1 ε ) ε + (1 ε (1 ε )] ε )] ε )] ε ) (1 ε [ 2 ε (1 ε )] ε, hus, A+B+C corresponds to the first term of Eq. (10). ) ε A ε σ C (1 ε )(1 ε ) ε σ Veetation ε σ ( 1 ε ) ε σ Ground B ε ε σ Fiure 3.11: Enery loss by eetation due to interaction with round.

24 2 Assinment 02: It is well known that atmosphere acts as a blanket for Earth: it transmits incomin shortwae solar radiation to the Earth surface and traps outoin terrestrial lon-wae radiation. his constitutes the mechanism of the reen house effect - temperature of the Earth surface is hiher in presence of atmosphere compared to case of no atmosphere. Demonstrate this effect with enery balance equations. Calculate surface temperature with and without atmosphere. Make the followin assumptions: wo-layer atmosphere, temperature of the top layer is 1 255K and temperature of the lower layer is 2 303K ( 1 < 2 ). Incomin solar shortwae flux, absorbed by errestrial Ecosystem, S20 [Wm -2 ] Atmosphere is a black body with respect to lon wae radiation- it absorb all the incomin lonwae radiation is absorbed Atmosphere is completely transparent to shortwae radiation Earth surface is a black body both for shortwae solar and lonwae atmospheric radiation No enery ain/loss in each layer (outoin radiation is equal to incomin radiation) References CLM ech notes (pp. 37-0), Dickinson, R.E. (1983) Land surface processes and climate-surface albedos and enery balance. Adanced Geophysics, 25, Sellers, P.J. (1985). Canopy reflectance, photosynthesis and transpiration. International Journal of Remote Sensin: Shabano, N.V., Knyazikhin, Y., Baret, F., Myneni, R.B. (2000). Stochastic Modelin of Radiation Reime in Discontinuous Veetation Canopies. Remote Sensin of Enironment, 7(1):