PHY2505 - Lecture 10

1. PHY2505 - Lecture 10 Infrared radiation in a cloudy atmosphere: approximations

2. For climate modelling including clouds, approximations are required to compute the multiple scattering or diffuse term of RTE efficiently Multiple scattering can make an important contribution to the solution – it’s the reason for changes in the uniformity of scattered radiation – in the visible intensity of the sky, or the brightness of clouds The most “natural” approximation is called the Successive Order of Scattering approximation or SOS Approximations RTE Multiple scattering absorption

3. Underlying idea: Individual scattered beams can be added to form the total intensity Strategy: we sum singly, doubly, triply scattered radiances each In calculated separately using I n-1 Disadvantage: Can require many orders of scattering to reach a converging solution. No easy way of determining how many are required for a particular problem Successive Orders of Scattering (SOS) For nth order: n n n n n-1

4. Underlying idea: Because radiation flux and heating rates are angular-averaged properties, one can expect that details of the angular variation of intensity are not very important for the predictions of these quantities. Strategy: Introduce an “effective” angular averaged intensity/flux (stream) so the RTE reduces to two coupled ordinary differential equations. Disadvantages of the two-stream approximations: Two-steam methods provide acceptable accuracy but over a restricted range of the parameters. There is no a priory method to estimate the accuracy, so one needs to use the “exact” method to obtain an accurate solution which can be used to estimate the accuracy of two-stream solutions. Advantages of the two-stream approximations: Two-stream approximations are computationally efficient (therefore they are often used in climate models) and often sufficiently accurate. How do we best determine the “effective” intensity (i.e., the effective scattering angle)? Two stream approximation

5. One possible strategy: define as the intensity-weighted angular means For isotropic radiation field, this gives A problem with this approach is that we don’t know a priori the angular distribution of scattering.. Better approximation is to use the Guassian quadrature approximation Choosing the effective scattering angle

6. Gaussian quadrature applied to any function f(m) gives where aj are the weights defined as andmjare the zeros of the even-order Legendre polynomialsP2n(m) Gaussian quadrature

7. The first 4 approximations are given in Liou: Gaussian quadrature m=1/Ö3

8. Using the first approximation, m1=1/Ö3the phase function can be expanded to give where + for upwards and – for downwards directions and g is the asymmetry factor introduced in the Mie scattering section – the first moment of the phase function – our weighted effective scattering angle The integral in our RTE is now Phase function expansion

9. If we now express the RTE in terms of upwards and downwards intensities we get Where This is the two stream approximation, so called because we are evaluating the RTE for two effective scattering angles, m1 and –m1 - an upwards and downwards stream Two stream approximation

10. Analytical solutions The final step is to parameterize the Planck emission function as the Planck functions at top and bottom of the layer respectively The solutions to our equations are [see Liou p305-306 for method] where and K and H can be found for particular boundary conditions..

11. Flux densities From the expressions for radiances, we can use these to give diffuse fluxes A better approximation for flux calculations is to use four streams: two streams in the upward direction and two streams in the downward direction where the first two Gaussian quadrature values are used: m1, m2

12. Eddington’s approximation Underlying idea: Truncation of expanded series Strategy: Expand both Intensities and Phase function in terms of Legendre Polynomials For N=1 Following a similar method of solution, this gives expressions for flux densities:

13. Comparison of Eddingtons and Two stream Approximations

14. SUMMARY • Methods have been presented to calculate radiative transfer in the infrared • ABSORPTION: • To deal with absorption we use line-by-line methods for highest accuracy. • Band models incorporating methods like the correlated K approximation are still widely used (MODTRAN) • Inhomogenuities in the atmospheric path are approximated using the Curtis-Godson method • SCATTERING: • Diffuse radiation due to multiple scattering is important in the thermal infrared • The multiple scattering problem can be solved exactly • Approximations that are commonly used for climate modelling are the two/four stream method and Eddington’s approximation

15. Results from model computations • Recall our expression for infrared cooling rate: • where net flux at a height z is given by • We are going to look at results from model computations for three cases: • How do individual spectroscopic bands effect cooling? (CO2) • How well does the Correlated-K method approximate Line-by-line calculations? (clear atmosphere) • What results do we get for low, middle and high cloud for the two/four stream approximation?

16. CASE 1: How do individual spectroscopic bands effect cooling? • 12C16O2 • Line-by-line computation • Cooling to space approximation • Fundamental is major contributer in the lower atmosphere • Hot bands are significant in the upper stratosphere

17. CASE 2: Comparison of LBL and Correlated K code LBL CKD error • Cooling by water vapour dominates in lower atmosphere < 20km • CO2 dominates above 20km (different scale) • O3 heating can be seen between 20-30km • CKD error errors are large in regions of low density: thin layers far away can effect local cooling rates

18. CASE 3: Computation of fluxes in clear and cloudy atmospheres • LBL + 2-4 stream approximation • Mid-Lat Summer (sub-arctic winter) • Significant cooling/heating in region of cloud – shows importance.. • Largest effect at mid-level cloud • At low levels the difference in temperature between surface and cloud base not large • At high level, there is a lower water content (higher ice) leading to reduced effect • Cloud effect directly related to altitude and physical composition