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EART164: PLANETARY ATMOSPHERES Francis Nimmo F.Nimmo EART164 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 EART164 Spring 11 Radiative transfer equations dI I dz dt = ar Optical depth: dz Absorption: Greenhouse effect: 3 T ( ) T 1 2 Radiative Diffusion: 16 T T F ( z) 3 z 4 4 0 Cp Rad. time constant: 1 T0 1/ 4 Teq 2 3 T P g Fsolar (1 A) F.Nimmo EART164 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 EART164 Spring 11 Dynamics at work 13,000 km 30,000 km 24 Jupiter rotations F.Nimmo EART164 Spring 11 Other examples Saturn Venus Titan F.Nimmo EART164 Spring 11 Definitions & Reminders • “Easterly” means “flowing from the east” i.e. an westwards flow. • Eastwards is always in the direction of spin RgT Ideal gas: P Hydrostatic: dP = - g dz N “meridional” y v f R is planetary radius, Rg is gas constant H is scale height R x E u “zonal/ azimuthal” F.Nimmo EART164 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? v 2 LW sin f is a measure of its importance (Rossby number) e.g. Jupiter v~100 m/s, L~10,000km we get ~0.03 so important F.Nimmo EART164 Spring 11 1. Winds F.Nimmo EART164 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 hot cells (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 EART164 Spring 11 Equatorial easterlies (trade winds) F.Nimmo EART164 Spring 11 Zonal Winds Schematic explanation for alternating wind directions. Note that this problem is not understood in detail. F.Nimmo EART164 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 EART164 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 = FE R (1 A) 2 t r 2 t=rotation period, R=planet radius, r=distance (AU) Atmospheric heat capacity = 4R2CpP/g Where’s this from? Extrasolar planet (“hot Jupiter”) So the temp. change relative to background temperature DT gFE (1 A) t 2 T 4PTCp r Small at Venus’ surface (0.4%), larger for Mars (38%) F.Nimmo EART164 Spring 11 Governing equation • Winds are affected primarily by pressure gradients, Coriolis effect, and friction (with the surface, if present): dv 1 P 2W sin f zˆ v F dt • Normally neglect planetary curvature and treat the situation as Cartesian: f =2Wsin f du 1 P fv Fx dt x dv 1 P fu Fy dt y (Units: s-1) u=zonal velocity (xdirection) v=meridional velocity (y-direction) F.Nimmo EART164 Spring 11 Geostrophic balance du 1 P fv Fx dt x • In steady state, neglecting friction we can balance pressure gradients and Coriolis: 1 P Flow is perpendicular to v the pressure gradient! 2 W sin f x L L wind 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 EART164 Spring 11 Rossby number dv 1 P fu dt y • 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 u/t u Ro ~ fu fL • 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 EART164 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?) Rd v prop f 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=(gH)1/2 F.Nimmo EART164 Spring 11 Ekman Layers • Geostrophic flow is influenced by boundaries (e.g. the ground) • The ground exerts a drag on the overlying air du 1 P fv Fx dt x 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 EART164 Spring 11 Ekman Spiral • The effective thickness d of this layer is d W 1/ 2 where W is the rotation angular frequency and is the (effective) viscosity in m2s-1 • The wind direction and magnitude changes with altitude in an Ekman spiral: Actual flow directions Increasing altitude Expected geostrophic flow direction F.Nimmo EART164 Spring 11 Cyclostrophic balance • The centrifugal force (u2/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 T u tanf f 2 Rg F.Nimmo EART164 Spring 11 “Gradient winds” • In some cases both the centrifugal (u2/r) and the Coriolis (2W 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 EART164 Spring 11 Thermal winds • Source of pressure gradients is temperature gradients • If we combine hydrostatic equilibrium (vertical) with geostrophic equilibrium (horizontal) we get: u g T z fT y This is not obvious. The key physical result is that the slopes of constant pressure surfaces get steeper at higher altitudes (see below) P2 z N y cold Small H u(z) hot x P2 P1 P1 cold Large H hot Example: On Earth, mid-latitude easterly winds get stronger with altitude. Why? F.Nimmo EART164 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 y2 u~ W R u R f y u y2 ~ W z RH so u g T gR T ~ z fT y 2WyT y y 4 T T0 exp d 1/ 4 R Hg d 2 W 2 Does this make sense? Latitudinal extent?Venus vs. Earth vs. Mars F.Nimmo EART164 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 u Ro 2 LW sin f du 1 P 2W sin fv Fx dt x u g T z fT y F.Nimmo EART164 Spring 11