F.Nimmo EART164 Spring 11 EART164: PLANETARY ATMOSPHERES Francis Nimmo

F.Nimmo EART164 Spring 11 Last Week – Clouds & Dust Saturation vapour pressure, Clausius-Clapeyron Moist vs. dry adiabat Cloud albedo effects – do they warm or cool? Giant planet cloud stacks Dust sinking timescale and thermal effects

F.Nimmo EART164 Spring 11 A massive (and complex) subject Important –Radiative transfer dominates upper atmospheres –We use emission/absorption to probe atmospheres We can only scratch the surface: –Black body radiation –Absorption/opacity –Greenhouse effect –Radiative temperature gradients –Radiative time constants This Week – Radiative Transfer

F.Nimmo EART164 Spring 11 Penetration depends on wavelength (other things being equal). Everyday example? Absorption is efficient when particle size exceeds wavelength Example at giant planets Sounding planet atmospheres Increasing wavelength UV ~nbar H 2 absorption Increasing depth vis/NIR ~1 bar Clouds, aerosols Thermal IR ~few bar (variable) CH 4 etc radio ~10 bar NH 3

F.Nimmo EART164 Spring 11 Absorption vs. Emission warm cold  warm cold  Absorption vs. emission tells us about vertical temperature structure This is useful e.g. for exoplanets Lissauer & DePater Section 4.3 Jupiter spectrum

F.Nimmo EART164 Spring 11 Intensity (I ) is the rate at which energy having a wavelength between and +  passes through a solid angle (W/m m -2 sr -1 ) Intensity is constant along a ray travelling through space The Planck function (see next slide) is expressed as intensity Definitions I Radiative flux (F ) is the net rate at which energy having a wavelength between and  passes through unit area in a particular direction (W/m m -2 sr -1 ) Think of it as adding up all the I components travelling in a particular direction For an isotropic radiation field F =0 + + I I I FF

F.Nimmo EART164 Spring 11 max in cm e.g. Sun T=6000 K  max =0.5  m Mars T=250 K max =12  m  =5.7x10 -8 in SI units Black body basics 1. Planck function (intensity): Defined in terms of frequency or wavelength. Upwards (half-hemisphere) flux is 2  B 2. Wavelength & frequency: 3. Wien’s law: 4. Stefan-Boltzmann law

F.Nimmo EART164 Spring 11 The total absorption depends on  and , and how they vary with z. The optical depth  is a dimensionless measure of the total absorption over a distance h: Optical depth, absorption, opacity I I-  I zz  I=-I   z  =absorption coefft. (kg -1 m 2 )  =density (kg m -3 ) I = intensity You can show (how?) that I=I 0 exp(-  ) So the optical depth tells you how many factors of e the incident light has been reduced by over the distance d. Large  = light mostly absorbed.

F.Nimmo EART164 Spring 11 For a purely absorptive atmosphere, we have Source function If the atmosphere is emitting as a black body (and is in local thermal equilibrium) then we get In the latter case, the changeover from the incident light to the local source happens at roughly  =1 (as expected)

F.Nimmo EART164 Spring 11 The absorption coefficient  is also called the opacity* You can think of it representing the surface area of absorbers of a given mass Opacity depends strongly on temperature, phase (e.g. are clouds present?) and composition, as well as wavelength For gases, opacity is often between – m 2 kg -1 (higher at high temperatures) For solid spheres of radius r and solid density  s Absorption & opacity *Really, the Rosseland mean opacity because we’re integrating over all wavelengths Here  is mass of solids/unit volume h is total thickness Example: Mars dust storms Optical depth 1 would be a smoggy day in LA (This is the same as Patrick’s equation)

F.Nimmo EART164 Spring 11 Radiative Transfer for Non-Experts* Black body radiation at a point is isotropic But if there is a temperature gradient, there will still be a net flux (upwards – downwards radiation) For an atmosphere in equilibrium, the flux gradient must be zero everywhere (otherwise it would be heating/cooling) HOT COLD Net flux F We can write the relationship between the net flux and the upwards-downwards radiation as follows: See Lissauer and DePater Annoying geometrical factor * That would be me

F.Nimmo EART164 Spring 11 We can then derive (very useful!): Radiative Diffusion If we assume that  is constant and cheat a bit, we get Strictly speaking  is Rosseland mean opacity But this means we can treat radiation transfer as a heat diffusion problem – big simplification

F.Nimmo EART164 Spring 11 Radiative Diffusion - Example Earth’s mesosphere has a temperature of about 230 K, a temperature gradient of 3 K/km and an elevation of about 70km. Roughly what is the density? What opacity would be required to transmit the solar flux of ~10 3 Wm -2 ?

F.Nimmo EART164 Spring 11 “Two-stream” approximation – very like we discussed in Week 2 Atmosphere heated from below (no downwards flux at top of atmosphere) Greenhouse effect Net flux F Surface T s T0T0 T1T1 Upwards flux F          Downwards flux F       Downwards flux      Upwards flux F       Net flux F This gives us two useful results we’ve seen before: It also means the effective depth of radiation is at  =2/3

F.Nimmo EART164 Spring 11 Greenhouse effect EarthMars T eq (K) T 0 (K) T s (K) Inferred  Fraction transmitted A consequence of this model is that the surface is hotter than air immediately above it. We can derive the surface temperature T s :

F.Nimmo EART164 Spring 11 Atmosphere can transfer heat depending on opacity and temperature gradient Competition with convection... Convection vs. Conduction Does this equation make sense? Radiation dominates (low optical depth) Convection dominates (high optical depth)  crit Whichever is smaller wins -dT/dz ad -dT/dz rad

F.Nimmo EART164 Spring 11 Radiative time constant Atmospheric heat capacity (per m 2 ): Radiative flux: Time constant: E.g. for Earth time constant is ~ 1 month For Mars time constant is a few days

F.Nimmo EART164 Spring 11 Key Concepts Black body radiation, Planck function, Wien’s law Absorption, emission, opacity, optical depth Intensity, flux Radiative diffusion, convection vs. conduction Greenhouse effect Radiative time constant

F.Nimmo EART164 Spring 11 Key equations Absorption: Optical depth: Radiative Diffusion: Rad. time constant: Greenhouse effect:

F.Nimmo EART164 Spring 11 End of lecture

F.Nimmo EART164 Spring 11 Simplified Structure dl Transmitted Absorbed Scattered Emitted I What does “optical depth” mean? Simplest example with no emission What are units of absorption?

F.Nimmo EART164 Spring 11 LTE – emiss = absorb Useful! What are applications? Mathematical trickery!

F.Nimmo EART164 Spring 11 Simplified Structure dl Transmitted Absorbed+ Re-radiated Scattering neglected What does “optical depth” mean? Approx expression, actually ¾ What happens as tau-> 0? What are units of absorption?