Band Transmission and Heating Rates

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1 Band Transmission and Heating Rates Topics: 1 Band transmission 2 Band models 3 Weighting functions 4 Heating rate 5 Broadband fluxes and heating rate profiles Reading: Liou 44,47; Thomas 57,11-13,112 Week 5: September Band Transmission The high spectral detail of molecular absorption lines ( 1 cm 1 ) prevents rapid radiative transfer computations across the spectrum (line-by-line models are very slow) Approximate radiative transfer methods divide the spectrum into spectral bands from 1 to 1 cm 1 wide In each band the Planck function is approximately constant Longwave radiative transfer can then be formulated in terms of mean transmission between levels Hence the need for the mean transmission across a spectral band: T ν Band models give mean transmission for absorber amount u k-distributions are a newer, more flexible method for spectrally averaged radiative transfer Single Line Transmission The effect of many absorption lines on band transmission can be understood by first considering the spectrally integrated transmission of a single line Average absorption across a single line is A ν (u) = 1 T ν = 1 ν (1 ν e k νu )dν Problem with average absorption is that it depends on ν 1

2 Equivalent Width W = νa ν = ν (1 e k νu )dν has units of spectral interval (cm 1 ), is the equivalent width of a fully absorbing (A = 1) rectangular line Schematic diagram illustrating the equivalent width The dotted rectangular area is equal to the hatched area and represents the total energy absorbed in the line [Lenoble, Fig 82] Equivalent Width of Lorentz Line W = 1 exp Suα/π (ν ν ) 2 + α 2 dν = 2παL(x) x = Su 2πα L(x) = xe x [I (x) + I 1 (x)] is the Ladenburg and Reiche function (I and I 1 are modified Bessel functions) A useful approximation is L(x) x[1 + (πx/2) 5/4 ] 2/5 Weak line limit: k ν u 1 W k ν udν = Su Linear in absorber amount and line strength Strong line limit: k ν u 1 W = 2 Suα Line center saturates, absorption increases from expanding width 2

3 Curve of growth describes increase in absorption with absorber amount (a) Curve of growth of a typical spectral line with S = 1 4 cm 1 (g cm 2 ) 1 and α = 6 cm 1 showing the linear and square root regions of growth (b) Actual shapes of the transmission spectrum for different values of absorber amount u = ρl corresponding to the values shown in the arrows in (a) [Houghton, Fig 42] 3

4 Band Models Band models are simple expressions for the mean transmission over a spectral band with many lines Some generally available radiative transfer codes, such as MODTRAN and MDTERP, use band models Theoretical justification: random line spacing for random band models Practical justification: A fit of laboratory or line-by-line transmission data Regular (Elsasser) band model Assume: evenly spaced, identical strength lines Random band models Assume: n randomly spaced lines in ν band ( ν = nδ), lines are independent and have identical shapes, probability density of strength of i th line is p(s i ) Different p(s) give different models, eg Goody or Malkmus Random Band Models Approach: Derive mean transmission by multiplying transmissions of each line at a particular ν, and also integrating over probability distributions of line positions ν i and line strengths S i for each line Assume lines are independent and identically distributed T = n ( ) 1 dν i ds i p(s i ) exp [ us i f(ν ν i )] ν ν i=1 Integral for each line in product is the same: T = { 1 1 } n p(s) (1 exp [ usf(ν)]) dνds ν ν Define mean equivalent width by integrating over prob dist of S: W = p(s) Then the mean transmission is Take limit of n to infinity: ν (1 exp [ usf(ν)]) dν ds T = 1 1 n T = exp 4 W δ W δ n

5 Single line transmission is 1 W ν, but for many random lines it is exponential in mean equivalent width Goody model has exponential line strength distribution: p(s) = 1 S exp( S/ S) for Lorentz line shape with width ᾱ, integral for W gives mean transmission T (u) = exp Su 1 + Su δ ᾱπ Malkmus model has a higher probability of weak lines: 1/2 p(s) 1 S exp( S/ S) for a Lorentz line shape the mean transmission is T (u) = exp πᾱ 2δ Su πᾱ 1/2 The mean transmission as a function of absorber amount is in terms of two parameters, S/δ and ᾱ/δ These parameters depend on pressure and temperature δ is the average line spacing δ = ν/n Weak line limit: Su πᾱ 1 T (u) = exp Su δ 1 Strong line limit of Goody and Malkmus models: Su πᾱ 1 T (u) = exp πᾱ Su δ 5

6 atm 1 atm Goody Random Model for 15 m CO 2 Band Band Mean Absorption cm -1 1 S/ = 7187 cm 2 /g / = Absorber Amount u (g/cm 2 ) Goody random band model absorption plotted as a function of absorber amount for a spectral band across the CO 2 15 µm vibrational band The absorption is higher for the higher pressure 6

7 Obtaining Band Model Parameters Band model parameters S and ᾱ are derived from absorption line parameters Usually the weak and strong limits of mean equivalent width are fit: S δ = 1 ν n S i i=1 πᾱ S δ = 2 ν n i=1 Si α i (for Goody) The band model parameters are tabulated; for example, Table 175 in Lenoble (1993) has n i=1 S i and n i=1 Si α i for 1 cm 1 bands at three temperatures How well does the Goody random band model work? Comparison of the random band model with laboratory measurements in water vapor bands The left curve is for a pressure of 74 Torr and the right curve is for a pressure of 125 Torr m is the water vapor amount, and m is the amount of water vapor that gives a transmission of 5 at 74 Torr The symbols indicate the particular water vapor vibrational bands from 63 µm to 11 µm [Goody & Yung, Fig 418] 7

8 Inhomogeneous Paths Band models give the transmission for a homogeneous path because band parameters are for one pressure and temperature But band models need to be used for inhomogeneous paths to calculate the transmission between two levels Scaling Approximation: find an equivalent homogeneous path ũ at fixed reference T r, p r that results in the band model having the correct transmission Match optical depth for line wings (centers saturated): ũs(t r )α(t r, p r ) = π(ν ν ) 2 u us(t (u))α(t (u), p(u)) du π(ν ν ) 2 ( ) ( )m p Tr ũ = ρ a dz p r T Integral over height is in terms of integral over absorber amount du = ρ a dz Band model parameters computed for a reference pressure and temperature (say 5 mb, 25 K) Then the density is weighted by p and T m is integrated to get ũ van de Hulst - Curtis - Godson Approximation More accurate band transmission for inhomogeneous paths is obtained with the two-parameter approximation Concept: adjust S and ᾱ in band model to fit weak and strong line limits over inhomogeneous paths Strong line limit - weight line widths according to absorber amount: ᾱ = Often only pressure is used in scaling α: ᾱ = u α(p)du u u α(p, T ) S(T )du u S(T )du = α u p p ρ a dz Weak line limit - temperature effect is in adjusting line strength: S = u S(T )du u du Curtis-Godson approximation does not work well if pressure and absorber amount are not well correlation (eg ozone) 8

9 Thermal Emission Radiative Transfer Revisited Radiative Transfer Equation without scattering (using height coordinate): µ di ν dz = k νρ a (I B ν [T (z)]) I ν (z, µ) is radiance, B ν [T (z)] is Planck function at z, ρ a is absorber density, and k ν is mass absorption coefficient Change in radiance is difference between absorption and emission Integral solution for upwelling radiance assuming a black surface I ν (z, µ) = T ν (z, )I ν (, µ) + z B ν[t (z )] exp Thermal emission RTE using transmission [ 1 ] z k ν ρ a dz k ν ρ a dz /µ µ z Can put RTE in terms of transmission between z and z at angle µ: dt ν (z, z, µ) dz Upwelling radiance [ T ν (z, z, µ) = exp 1 ] z k ν ρ a dz µ z I ν (z, µ) = T ν (z, )B ν [T ()] + Downwelling radiance I ν (z, µ) = z [ = exp 1 ] z k ν ρ a dz kν ρ a µ z µ z B ν [T (z )] B ν[t (z )] Discrete solution using band model transmission Upwelling band integrated radiance at level z i is dt ν (z, z, µ) dz dt ν (z, z, µ) dz dz dz I ν (z i, µ) = T ν (z i, )B ν [T ()]+ j B ν [T j+1/2 ] [T ν (z i, z j, µ) T ν (z i, z j+1, µ)] where T j+1/2 is temperature at midpoint of layer from z j to z j+1, B ν is band integral of Planck function, and T ν (z i, z j, µ) is band model mean transmission from level z i to z j using (scaled) absorber amount u along path at angle µ 9

10 Weighting Functions Weighting functions give the contribution to outgoing radiance from each level Very important for remote sensing, but also useful for understanding IR cooling W ν (z, ) = dt ν (z, ) dz Upwelling radiance is then an integral over the weighting function I ν (, µ) = T ν (, )I ν (, µ) + B ν [T (z)] W ν (z, )dz Weighting function referenced to surface for contribution to downwelling radiance at surface dt W ν (, z) = ν (, z) dz What do weighting functions look like? Schematic for optically thick and optically thin cases from space: Transmission Transmission Weighting function Weight (km -1 ) Transmission Transmission Weighting function Weight (km -1 ) Why do weighting functions look like this? Two factors in weighting function - transmission and extinction W ν (z, ) = e τ(z)/µ k νρ a µ Transmission decreases away from observer, extinction decreases upward 1

11 Transmission Weighting function GHz 5496 GHz 5374 GHz 53 GHz 4 35 MSU 4 MSU 3 MSU 2 MSU 1 nadir viewing MLS atmos Transmission Weight (km -1 ) Transmission profiles and weighting functions referenced to space for the Microwave Sounding Unit on the NOAA polar orbiting satellites An upwelling Earth radiance spectrum measured by the Infrared Interferometer Spectrometer aboard the Nimbus 4 satellite Planck radiance curves are also shown [Liou, 1992; Fig 21] 11

12 Heating Rates Calculating broadband longwave fluxes requires three integrals: 1) over height in RTE, 2) over angle to get flux, 3) over spectrum Net flux - net power per area passing through level F net = F F = 2π 1 1 I(µ)µdµ Heating (or cooling) from broadband net flux convergence: dt dt = 1 df net rad ρc p dz Heating from absorption, cooling from emission = g C p df net dp Consider a layer: flux entering = F bot + F top flux leaving = F bot + Ftop Net flux convergence = absorbed - emitted = entering - leaving Example: Longwave radiative cooling at night US standard atmosphere, to 1 km layer (fluxes calculated with MODTRAN) F 1 = 25 W/m 2 F = 39 W/m 2 entering = 64 F = 285 W/m 2 F 1 = 375 W/m 2 leaving = 66 Net flux converged = 2 W/m 2 dt/dt = 1 F net ρc p z = 2 J s 1 m 2 (117 kg/m 3 )(14 J kg 1 K 1 )(1 m) dt/dt = K/s = 15 K/day Net flux convergence is usually computed by differencing discrete layer fluxes May get numerical errors from differencing almost equal up and down fluxes in opaque atmospheres Curtis matrix approach used for Venus 12

13 Exchange of Flux between Layers Derive net flux convergence from radiative transfer equation to understand sources of heating/cooling at a level The spectral net flux convergence may be obtained from RTE solution by taking d/dz of F F : df ν,net dz = πb ν () T ν f (, z) z + z πb ν(z ) 2 Tν f(z, z) dz z z + πb ν (z ) 2 Tν f(z, z ) dz z z z T f ν (z, z ) is flux transmission between levels z and z Add and subtract in two terms like πb ν (z) z 2 Tν f(z, z ) z z dz = πb ν (z) T ν f (z, ) z Final result for exchange of flux between layers form of net flux divergence: df ν,net dz (z) = πb ν (z) T ν f (z, ) z (, z) z π[b ν(z ) B ν (z)] 2 Tν f (z, z) dz z z + π[b ν () B ν (z)] T ν f + + z z π[b ν (z ) B ν (z)] 2 Tν f (z, z ) dz z z First term is cooling to space - often a good approximation Cooling to space is simply the weighting function times the Planck flux Second term is exchange of energy with surface Important if large temperature difference and transmission is high Integrals are exchange between levels below and above Role of second derivative of transmission can be understood using 2 T f ν (z, z) z z β is emissivity/absorptivity of thin layer β(z )β(z)t f ν (z, z) 13

14 See how well cooling to space works: Cooling to space compared to total cooling The solid lines include all terms in the heating equation; the dotted line is calculated from the cooling to space approximation The vertical scale is ln(p(z)/p sfc ) (a) CO 2 15 µm band for a midlatitude temperature profile (b) O 3 96 µm band for a tropical temperature profile (c)h 2 O for a tropical temperature profile with wet and dry stratospheres (d) H 2 O for a tropical temperature profile (e) H 2 O for a mid-winter, arctic temperature profile [from Rodgers and Walshaw (1966)] [Goody & Yung, Fig 614] 14

15 Alternative Method for Heating Rates Can derive net flux convergence by integrating RTE over dµdφ 2π { 1 1 µ di dz = β abs [I(µ) B] df net } dµdφ = 4πβ abs (Ī dz B) where Ī is the mean radiance and β abs is volume absorption coefficient Divergence of flux - cooling from Planck emission, Convergence of flux - heating from absorption of radiation MDTERP MDTERP (Maryland Terrestial Radiation Package) is a narrow-band longwave radiative transfer model with a graphical user interface developed by Robert Ellingson and Ezra Takara at the University of Maryland 15

16 Broadband Fluxes and Heating Rates from MDTERP MDTERP profiles for midlatitude summer atmosphere Longwave Fluxes Upwelling Downwelling Net Longwave Cooling Rate Flux (W/m 2 ) Cooling Rate (K/day) MDTERP profiles for midlatitude winter atmosphere Longwave Fluxes Upwelling Downwelling Net Longwave Cooling Rate Flux (W/m 2 ) Cooling Rate (K/day) 16

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