EART164: PLANETARY ATMOSPHERES

1. EART164: PLANETARY ATMOSPHERES Francis Nimmo

2. 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

3. This Week – Radiative Transfer • 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

4. Sounding planet atmospheres • Penetration depends on wavelength (other things being equal). Everyday example? • Absorption is efficient when particle size exceeds wavelength • Example at giant planets Increasing wavelength radio Thermal IR vis/NIR UV ~nbar H2 absorption Increasing depth ~1 bar Clouds, aerosols ~few bar (variable) CH4 etc ~10 bar NH3

5. Absorption vs. Emission I I • Absorption vs. emission tells us about vertical temperature structure • This is useful e.g. for exoplanets l l cold warm warm cold Jupiter spectrum Lissauer & DePater Section 4.3

6. Definitions Il • Intensity (Il) is the rate at which energy having a wavelength between l and l+Dl 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 • Radiative flux (Fl) is the net rate at which energy having a wavelength between l and l+Dl passes through unit area in a particular direction (W/m m-2 sr-1) • Think of it as adding up all the Ilcomponents travelling in a particular direction • For an isotropic radiation field Fl=0 Fl = + + Il Il Il

7. Black body basics 1. Planck function (intensity): Defined in terms of frequency or wavelength. Upwards (half-hemisphere) flux is 2pBn 2. Wavelength & frequency: 3. Wien’s law: lmax in cm e.g. Sun T=6000 K lmax=0.5 mm Mars T=250 K lmax=12 mm 4. Stefan-Boltzmann law s=5.7x10-8 in SI units

8. Optical depth, absorption, opacity I-DI a=absorption coefft. (kg-1 m2) r=density (kg m-3) I = intensity • The total absorption depends on r and a, and how they vary with z. • The optical depth t is a dimensionless measure of the total absorption over a distance h: Dz DI=-IarDz I • You can show (how?) that I=I0exp(-t) • So the optical depth tells you how many factors of e the incident light has been reduced by over the distance d. • Large t = light mostly absorbed.

9. Source function • For a purely absorptive atmosphere, we have • 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 t=1 (as expected)

10. Absorption & opacity • The absorption coefficient a 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 10-2 – 10-5 m2 kg-1 (higher at high temperatures) • For solid spheres of radius r and solid density rs (This is the same as Patrick’s equation) • Example: Mars dust storms Optical depth 1 would be a smoggy day in LA *Really, the Rosseland mean opacity because we’re integrating over all wavelengths Here r is mass of solids/unit volume h is total thickness

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) COLD We can write the relationship between the net flux and the upwards-downwards radiation as follows: Net flux Fn Annoying geometrical factor HOT * That would be me See Lissauer and DePater 3.3.1

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

13. 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 ~103 Wm-2?

14. Greenhouse effect • “Two-stream” approximation – very like we discussed in Week 2 • Atmosphere heated from below (no downwards flux at top of atmosphere) • This gives us two useful results we’ve seen before: Upwards flux Fn +2pBn (T0)=4pBn(T0) T0 Downwards flux Fn -2pBn (T0)=0 Net flux Fn T1 Downwards flux 2pBn (T1) Net flux Fn It also means the effective depth of radiation is at t =2/3 Upwards flux Fn +2pBn (T1) Surface Ts

15. Greenhouse effect A consequence of this model is that the surface is hotter than air immediately above it. We can derive the surface temperature Ts:

16. Convection vs. Conduction • Atmosphere can transfer heat depending on opacity and temperature gradient • Competition with convection . . . Whichever is smaller wins -dT/dzad Radiation dominates (low optical depth) -dT/dzrad rcrit Does this equation make sense? Convection dominates (high optical depth)

17. Radiative time constant Atmospheric heat capacity (per m2): Radiative flux: Time constant: E.g. for Earth time constant is ~ 1 month For Mars time constant is a few days

18. 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

19. Key equations Absorption: Optical depth: Greenhouse effect: Radiative Diffusion: Rad. time constant:

20. End of lecture

21. Simplified Structure What does “optical depth” mean? dl Transmitted In Absorbed Scattered Emitted Simplest example with no emission What are units of absorption?

22. LTE – emiss = absorb Mathematical trickery! Useful!What are applications?

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