MET 4430 Physics of Atmospheres Lecture 4 More Radiation 28 SEP-08 OCT 2012

Assignment Reading: Chapter 4 in Houghton, pp Problems: 4.6, 4.10, 4.13

Scattering Obeys Beer’s Law, a Schwarzchild-like equation: With is z reckoned downward from the top of the atmosphere σ is the scattering coefficient and ρ is density Can have other geometry and integration limits

Rayleigh Scattering Particle size, d, is much smaller than wavelength, λ Examples: – Scattering of visible light by air molecules (blue sky) – Scattering of microwaves 3-10 cm by precipitation Intensity, I, scattered in direction θ relative to the incident beam, I 0, at distance R from the scatterer: Here n is the index of refraction. This relation is called the phase function

Rayleigh Phase Function Integrating over all θ yields the loss from the beam: Which differs from (4.2) in Houghton, but is correct

About Rayleigh Scattering Air molecules have no well defined size or index of refraction, so we write Where α is the scattering cross section Because of the λ -4 dependence, blue light scatters more than red, explaining why the sky is blue and sunsets are red. Shadows in distant objects appear blue because of blue light’s scattering into the beam and highlights appear yellow because of blue light’s scattering from the beam The sea is blue because H 2 O had a vibrational mode at or μm that is excited by absorption of red light.

Absorption by Spectral Lines Defined by Quantum-Mechanical resonances with defined energies (or EM wavelengths) – Molecule absorbs a quantum of EM radiation with energy hν – Leaving it in an excited state until … – It emits a quantum radiation with energy hν In a vacuum, the lines are very narrow, but in the Atmosphere they broaden because of: – Pressure (collisions, dominant in the atmosphere) – Temperature (Doppler shift) – Quantum uncertainty hνhν hνhν

Ozone Absorption As before: Where m(O 3 ) is the mixing ratio of ozone and k ν is the absorption coefficient at wavenumber ν = 1/λ The heating is obtained by differentiating along the beam:

Functional Form for Pressure Broadening Where γ is the half-width Pressure variation comes from γ = γ 0 (p/p 0 ) A typical value of γ is 0.1 cm -1 at 1000 mb and 0 o C

Spectrally Integrated Absorption Path of length ℓ with spectral absorption coefficient k ν and density ρ, transmission is: Spectrally integrated absorptance (or equivalent width) is

Weak and Strong Approximations If k ν ρℓ << 1, If k ν ρℓ > 1 near the center of the line, there is a large opaque region in wavenumber space. In this case (ν−ν 0 ) >> 1 on the wings of the distribution and we can neglect γ 2 in the denominator Hint, 1/x  ν; integrate by parts; recognize Gaussian standard form Problem 4.9

Changes With Amount of Absorber

Line Structures strong weak

Transmission Along a Vertical Path Here m is the mixing ratio of absorbing gas, S is the wavenumber-averaged absorption, and ρ is the air density. (weak approximation) Since m is constant for a well-mixed absorber and ρdz = −dz/g:

Strong Approximation Make the Curtis-Godson Approximation p AVE = ½(p 1 + p 2 ) So that the average transmission in a spectral band is Which may be extended to overlapping bands

Combining Lines If we take a Band of wavenumbers Δν wide, containing many non-overlapping bands τ = 1 − ∑W i /Δν Appendix 10 contains tables: – ∑s i (column S, weak approximation) and – ∑(s i γ 0i ) ½ (column R, strong approximaton) – For standard concentrations, suface pressure of , and a range of temperatures, for CO 2 H 2 O and O 3

Wavenumber (cm -1 ) = 1/[Wavelength (μm )x ] Weak (linear) Approximation Strong (square root) Approximation For average atmospheric composition total depth, and standard surface pressure Appendix 10

Integral Transfer Equation Optical depth is: Each slab emits B ν (z) so that its contribution to I is Which integrates to:

Integration Over Frequency Given enough information about bands (i.e., S, γ 0, ν 0 ) it is theoretically possible to perform these integrals, BUT Because of the complexity of molecular spectra for H 2 O, CO 2, O 3, CH 4, … the numerical difficulties are overwhelming. Use Band Approximations instead

Heating and Cooling Rates Upwelling Irradiance from the Plane-Parallel Approximation (πB  B, τ* = (5/3)τ  τ Which, if we neglect downward emission from the layer and absorption of upwelling radiation from below leads to a cooling rate: Primarily due to upward IR emission from CO 2 Cooling to Space Approximation

O 2 + hν  2O O + O 2  O 3 Cooling to Space Solar UV Upwelling IR Doenwelling IR IR Emitted to Space CO 2

Vertical Profile of Radiative Heating Troposphere cools radiatively – Primarily due to IR emission by H 2 O – With help from CO 2 – Some shortwave heating due to VIS absorption by H 2 O Stratosphere is close to radiative equilibrium – UV absorption by O 3 – IR emissinon from CO 2 and H 2 O – And a little by O 3

Global Radiative Balance

Annual Mean Albedo

Annual Mean and Solstice Absorbed Solar Irradiance

Annual Mean Solar, IR & NET Radiation as a Function of Latitude

Summary Beer’s Law and Scattering – Rayleigh scattering and phase function Absorption by spectral lines, such as O 3 Line broadening by temperature (Doppler) and pressure (Lorentz) – Strong and weak approximations – Cutis-Godson approximation Transmission over vertical paths and combining wavenumbers Integral transfer equation Heating and cooling rates – Cooling to Space approximation – Vertical distribution of radiative heating and cooling Global energy budget Meridional distribution of heating and cooling