To fluxes & heating rates: Want to do things like:  Calculate IR forcing due to Greenhouse Gases  Changes in IR forcing due to changes in gas constituents  Calculate instantaneous heating rates due to visible & IR  Interactions of clouds & aerosols with all of the above!  In project 2, you will investigate some of these with an off-the-shelf radiation code.  Idea of this section is to know the qualitative ideas behind the calculations, not to write your own code

Monochromatic Intensity (nonscattering) at wavenumber Intensities (up and down) anywhere in atmosphere Transmittance between layer z1 and z2 Optical depth btn z1 & z2 Contribution from each layer z’ to the intensity at z goes like this. Must sum over all LINES and continuum contribution to get full absorption coefficient at

Must take into account contributions from potentially many gas absorption lines. Multiple calculations to get all layers in atmosphere. But is easily do-able. To integrate over real instrument response functions, may have to average over several closely-spaced values. Monochromatic Intensity (nonscattering) at wavenumber

Monochromatic Fluxes? Can do angular integral by replacing B with πB, and t(z 1,z 2 ) with t F (z 1,z 2 ), which is the monochromatic “flux transmittance”. Showed in book that can get away with “2- stream approximation”: where =~ 3/5, or θ ~ 53°

Broadband fluxes? Needed to calculate forcings (ie cloud forcing, greenhouse gas forcing) Needed to calculate heating rates Thus, needed in all weather and climate models! Must integrate over all wavelengths!! 350 cm cm cm cm -1

How many wavelengths do we need for the broadband calculation? In infrared, need typically 20 to 2000 cm -1. Scales of vibrational “bands” (e.g. CO 2 ν 2 ) is 10s of cm -1 wide. Can roughly approximate Planck function as constant across this scale. Scales of rotational line spacing is ~ 1 cm -1. Scales of rotational line widths (needed to resolve lines: ~ (near surface) to cm -1 (in stratosphere).

Line-by-line: The accurate but REALLY slow method With a modest spacing of cm -1, need ~ 2  10 6 monochromatic RT calculations (assuming 2 stream) just to do IR at one model grid point. Models may have grid points. Calculation must be done every time step (multiple times per day, perhaps every 1-3 hrs). Much too slow for weather/climate models, but can use it for limited testing and validation (e.g. LBLRTM) LW or SW

Monochromatic Flux Reminder where: Flux Weighting Function Monochromatic Flux Transmittance

Another way forward: spectral bands Write as flux contributions from many spectral intervals Assume Planck B is constant over each spectral interval Band-averaged transmittance over spectral interval

Molecular Spectroscopy Absorption Lines; absorption coefficient pressure & temperature dependence Transmission properties of a single fixed p &T Broadband transmission at fixed p &T Empirical -band absorption as a func. of path K-distribution bin by strength Band models statistical treat- ment of lines, distribution of strengths and centers Line-by-line direct sum over all lines, direct integration over frequency and path Transmission models Integral over many lines Integral over single line TOPICS The absorption path length (mass path) Theory of absorption/transmission by a single line (NOT TERRIBLY USEFUL) Spectrally integrated over bands of overlapping lines -Empirical -Band models -K-distribution

Gas Path / Absorber Mass Path Mass mixing ratio Units are kg/m 2.

Two versions of mass path k constant (horizontal) k changing (vertical) k changes in the vertical due only do changes in line widths (which depend on P and T) and line strengths (which depend on T) Mass abs. coeff: m 2 / kg of gas Absorber mass path: kg/m 2 of gas Gas Path / Absorber Mass Path

1 0 transmission W(u) Theory of Frequency-Integrated Absorption by a Single Line In this theory, absorption is typically expressed as an equivalent width (units of frequency, wavenumber, wavelength) and is the absorption by an equivalent, hypothetical square opaque line

1 0 transmission W(u) Theory of Frequency-Integrated Absorption by a Single Line The equivalent width W(u) u Curve of growth Assumes a uniform path (ie non varying P & T) Δν

Frequency-Integrated Absorption by a Single Line (contd) Two basic limits (for convenience set 0 =0): 1. Weak line limit (linear) ie W(u) is linear in u

Frequency-Integrated Absorption by a Single Line (contd) 2. Strong line limit (square root) Consider the line wings of a pressure-broadened line ie W(u) is goes as sqrt in u

Frequency-Integrated Absorption by a Single Line (summary) Curve of growth Weak line limit occurs as line centers fill in (u small) Strong line limit occurs as wings broaden out (u big)

The Empirical Approach to Broadband Absorption favored before the advent of computing power and availability of spectroscopic data bases – also favored approach of GCM model Parameterizations in 70s & 80s – also useful for estimating bulk solar absorption Examples of empirical broad-band relations Provided the absorptions are (spectrally) independent

Typical values of column ozone DU~ 0.3 cm NTP) Example#1 of the UV broad-band curve of growth

Band Models Treat spectral interval as containing either regularly spaced or randomly occurring lines Typically assume homogeneous P & T (like on a horizontal path) – no pressure broadening! Later on relax this with additional approaches. Not always very accurate (not using real spectroscopy! Essentially fit 2 parameters to spectroscopy)

Band Models Equally spaced lines by δ Lines all have same strength S y=α L /δ is the “grayness parameter” Randomly spaced lines, average line spacing is δ Lines have distribution of strengths: Malkmus Model

Malkmus Band model: Parameter fits  For different bands, you run a line-by- line reference code and fit parameters to the transmittance as a function of u. Parameters for the Random/Malkmus model shown (taken from Stephens notes)

Inhomogenous Paths – HCG Approximation (van de Hulst – Curtis – Godson) p2p2 p1p1

 k k j -dk/2 k j +dk/2 The k-Distribution Method – more modern approach f(k j ) is the fraction of the interval  where k j -dk/2<k < k j +dk/2

Vs. Frequency Sorted lowest-to-highest Inhomogenous Paths – “Correlated-K method”

Vs. Frequency Sorted lowest-to-highest Inhomogenous Paths – “Correlated-K method”

Local Heating Rate – depends on the rate of change of the net flux because an increase in F net (z) with increasing z implies a cooling. Note the sign convention Petty uses!

Flux equations (SW or LW!) Band-averaged transmittance over spectral interval Recall Band-Averaged Transmittance

Difference to get NET Flux at level z Could combine, but we will leave separate! Now just differentiate w.r.t. z…

Differentiate w.r.t. z to get Heating Rate due to spectral band Δν i

Heating Rate due to spectral band Δν i (Form 1) Flux from surface Flux from TOA Flux from layers below Flux from layers above Emission Downwards

Heating Rate due to spectral band Δν i (Form 2) Coupling with Surface Coupling with Space (LW: Cooling; SW: Heating) Coupling with layers above Coupling with layers below

Fluxes & heating rates

Shortwave Heating Terms C+D = 0 because there is no Shortwave emission in atmosphere. Term (B) dominates (absorption from TOA). H2O & Ozone are the dominant gases. Their density profiles, SZA and clouds determine the heating rate profiles H2O dominates in troposphere; Ozone dominates in stratosphere.

Heating rates decline when sun is lower in sky! Implies a strong diurnal and seasonal cycle.

Longwave Heating&Cooling Depends upon both gas profiles AND temperature profile Term (B) dominates: “Cooling-to- Space approximation” H2O, CO2 dominant; Ozone important in stratosphere. CO2 has mild heating right at tropopause; strong cooling above as “cooling-to-space” term kicks in. H2O Dominant in lower atmosphere; two peaks due to two absorption features (18-25 micron pure rotation feature & 5-8 micron vibration feature)

From BUGSrad! Tropical atmosphere In stratosphere, LW & SW nearly balance (over a full diurnal cycle!) In troposphere, other heating terms from sensible & Latent heating balance the stronger LW cooling. Can easily see the little “CO 2 Peak” at the tropopause

Project 2 Use Online tool to explore LW & SW heating rates and effects due to adding clouds & gases. Write-up to explain what you found.