EART 164 PLANETARY ATMOSPHERES Francis Nimmo F Nimmo

EART 164: PLANETARY ATMOSPHERES Francis Nimmo F. Nimmo EART 164 Spring 11

Last Week – Radiative Transfer • • • 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 EART 164 Spring 11

Radiative transfer equations Absorption: Optical depth: Greenhouse effect: Radiative Diffusion: Rad. time constant: F. Nimmo EART 164 Spring 11

Next 2 Weeks – Dynamics • Mostly focused on large-scale, long-term patterns of motion in the atmosphere • What drives them? What do they tell us about conditions within the atmosphere? • Three main topics: – Steady flows (winds) – Boundary layers and turbulence – Waves • See Taylor chapter 8 • Wallace & Hobbs, 2006, chapter 7 also useful • Many of my derivations are going to be simplified! F. Nimmo EART 164 Spring 11

Dynamics at work 13, 000 km 30, 000 km 24 Jupiter rotations F. Nimmo EART 164 Spring 11

Other examples Saturn Venus Titan F. Nimmo EART 164 Spring 11

Definitions & Reminders • “Easterly” means “flowing from the east” i. e. an westwards flow. • Eastwards is always in the direction of spin Ideal gas: Hydrostatic: d. P = - r g dz N “meridional” y v f R is planetary radius, Rg is gas constant H is scale height u R x E “zonal/ azimuthal” F. Nimmo EART 164 Spring 11

Coriolis Effect • Coriolis effect – objects moving on a rotating planet get deflected (e. g. cyclones) • Why? Angular momentum – as an object moves further away from the pole, r increases, so to conserve angular momentum w decreases (it moves backwards relative to Deflection to right the rotation rate) in N hemisphere • Coriolis accel. = - 2 W x v (cross product) = 2 W v sin(f) f is latitude • How important is the Coriolis effect? is a measure of its importance (Rossby number) e. g. Jupiter v~100 m/s, L~10, 000 km we get ~0. 03 so important F. Nimmo EART 164 Spring 11

1. Winds F. Nimmo EART 164 Spring 11

Hadley Cells • Coriolis effect is complicated by fact that parcels of atmosphere rise and fall due to buoyancy (equator is High altitude winds hotter than the poles) Surface winds • The result is that the atmosphere is cold broken up into several Hadley cells hot (see diagram) • How many cells depends on the Rossby number (i. e. rotation rate) Fast rotator e. g. Jupiter Med. rotator e. g. Earth Ro~0. 03 (assumes v=100 m/s) Ro~0. 1 Slow rotator e. g. Venus Ro~50 F. Nimmo EART 164 Spring 11

Equatorial easterlies (trade winds) F. Nimmo EART 164 Spring 11

Zonal Winds Schematic explanation for alternating wind directions. Note that this problem is not understood in detail. F. Nimmo EART 164 Spring 11

Really slow rotators • A sufficiently slowly rotating body will experience DTday-night > DTpole-equator • In this case, you get thermal tides (day-> night) hot cold • Important in the upper atmosphere of Venus • Likely to be important for some exoplanets (“hot Jupiters”) – why? F. Nimmo EART 164 Spring 11

Thermal tides • These are winds which can blow from the hot (sunlit) to the cold (shadowed) side of a planet Solar energy added = t=rotation period, R=planet radius, r=distance (AU) Atmospheric heat capacity = 4 p. R 2 Cp. P/g Where’s this from? Extrasolar planet (“hot Jupiter”) So the temp. change relative to background temperature Small at Venus’ surface (0. 4%), larger for Mars (38%) F. Nimmo EART 164 Spring 11

Governing equation • Winds are affected primarily by pressure gradients, Coriolis effect, and friction (with the surface, if present): • Normally neglect planetary curvature and treat the situation as Cartesian: f =2 Wsin f (Units: s-1) u=zonal velocity (xdirection) v=meridional velocity (y-direction) F. Nimmo EART 164 Spring 11

Geostrophic balance • In steady state, neglecting friction we can balance pressure gradients and Coriolis: Flow is perpendicular to the pressure gradient! L wind L pressure Coriolis H isobars • The result is that winds flow along isobars and will form cyclones or anti-cyclones • What are wind speeds on Earth? • How do they change with latitude? F. Nimmo EART 164 Spring 11

Rossby number • For geostrophy to apply, the first term on the LHS must be small compared to the second • Assuming u~v and taking the ratio we get • This is called the Rossby number • It tells us the importance of the Coriolis effect • For small Ro, geostrophy is a good assumption F. Nimmo EART 164 Spring 11

Rossby deformation radius • Short distance flows travel parallel to pressure gradient • Long distance flows are curved because of the Coriolis effect (geostrophy dominates when Ro<1) • The deformation radius is the changeover distance • It controls the characteristic scale of features such as weather fronts • At its simplest, the deformation radius Rd is (why? ) Taylor’s analysis on p. 171 is dimensionally incorrect • Here vprop is the propagation velocity of the particular kind of feature we’re interested in • E. g. gravity waves propagate with vprop=(g. H)1/2 F. Nimmo EART 164 Spring 11

Ekman Layers • Geostrophic flow is influenced by boundaries (e. g. the ground) • The ground exerts a drag on the overlying air with drag no drag pressure Coriolis H isobars • This drag deflects the air in a near-surface layer known as the boundary layer (to the left of the predicted direction in the northern hemisphere) • The velocity is zero at the surface F. Nimmo EART 164 Spring 11

Ekman Spiral • The effective thickness d of this layer is where W is the rotation angular frequency and n is the (effective) viscosity in m 2 s-1 • The wind direction and magnitude changes with altitude in an Ekman spiral: Actual flow directions Increasing altitude Expected geostrophic flow direction F. Nimmo EART 164 Spring 11

Cyclostrophic balance • The centrifugal force (u 2/r) arises when an air packet follows a curved trajectory. This is different from the Coriolis force, which is due to moving on a rotating body. • Normally we ignore the centrifugal force, but on slow rotators (e. g. Venus) it can be important • E. g. zonal winds follow a curved trajectory determined by u the latitude and planetary radius R • If we balance the centrifugal force against the poleward pressure gradient, we get zonal winds with speeds decreasing towards the pole: f F. Nimmo EART 164 Spring 11

“Gradient winds” • In some cases both the centrifugal (u 2/r) and the Coriolis (2 W x u) accelerations may be important • The combined accelerations are then balanced by the pressure gradient • Depending on the flow direction, these gradient winds can be either stronger or weaker than pure geostrophic winds Insert diagram here Wallace & Hobbs Ch. 7 F. Nimmo EART 164 Spring 11

Thermal winds • Source of pressure gradients is temperature gradients • If we combine hydrostatic equilibrium (vertical) with geostrophic equilibrium (horizontal) we get: This is not obvious. The key physical result is that the slopes of constant pressure surfaces get steeper at higher altitudes (see below) P 2 z N y cold Small H u(z) hot x P 2 P 1 cold Large H hot Example: On Earth, mid-latitude easterly winds get stronger with altitude. Why? F. Nimmo EART 164 Spring 11

Mars dynamics example • Combining thermal winds and angular momentum conservation (slightly different approach to Taylor) • Angular momentum: zonal velocity increases polewards • Thermal wind: zonal velocity increases with altitude so u R f y Does this make sense? Latitudinal extent? Venus vs. Earth vs. Mars F. Nimmo EART 164 Spring 11

Key Concepts • • • Hadley cell, zonal & meridional circulation Coriolis effect, Rossby number, deformation radius Thermal tides Geostrophic and cyclostrophic balance, gradient winds Thermal winds F. Nimmo EART 164 Spring 11