EART164: PLANETARY ATMOSPHERES

Transcription

1 EART164: PLANETARY ATMOSPHERES Francis Nimmo

2 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

3 Radiative transfer equations Absorption: Optical depth: Greenhouse effect: Radiative Diffusion: T di F( z) Rad. time constant: I dt dz = ar T 3 z dz 3 ( ) T 1 2 F C p solar P g T ( 1 A) T T0 1/ 4 T eq

4 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!

5 Dynamics at work 13,000 km 30,000 km 24 Jupiter rotations

6 Other examples Saturn Venus Titan

7 Definitions & Reminders Easterly means flowing from the east i.e. an westwards flow. Eastwards is always in the direction of spin Ideal gas: Hydrostatic: P R g T dp = - g dz R is planetary radius, R g is gas constant H is scale height N y v u meridional f R x E zonal/ azimuthal

8 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 the rotation rate) Coriolis accel. = - 2 W x v (cross product) = 2 W v sin(f) How important is the Coriolis effect? v 2LWsinf f is latitude is a measure of its importance (Rossby number) Deflection to right in N hemisphere e.g. Jupiter v~100 m/s, L~10,000km we get ~0.03 so important

9 1. Winds

10 Hadley Cells Coriolis effect is complicated by fact that parcels of atmosphere rise and fall due to buoyancy (equator is hotter than the poles) hot High altitude winds Surface winds cold Fast rotator e.g. Jupiter The result is that the atmosphere is broken up into several Hadley cells (see diagram) How many cells depends on the Rossby number (i.e. rotation rate) Med. rotator e.g. Earth Slow rotator e.g. Venus Ro~0.03 (assumes v=100 m/s) Ro~0.1 Ro~50

11 Equatorial easterlies (trade winds)

12 Zonal Winds Schematic explanation for alternating wind directions. Note that this problem is not understood in detail.

13 Really slow rotators A sufficiently slowly rotating body will experience DT day-night > DT pole-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?

14 Thermal tides These are winds which can blow from the hot (sunlit) to the cold (shadowed) side of a planet Solar energy added = R t=rotation period, R=planet radius, r=distance (AU) Atmospheric heat capacity = Where s this from? 2 ( 1 A) So the temp. change relative to background temperature DT T ( 1 A) Small at Venus surface (0.4%), larger for Mars (38%) F r E 2 t 4R 2 C p P/g gf 4PTC E p r 2 t Extrasolar planet ( hot Jupiter )

15 dv dt Governing equation Winds are affected primarily by pressure gradients, Coriolis effect, and friction (with the surface, if present): 1 P 2Wsin f zˆ v F Normally neglect planetary curvature and treat the situation as Cartesian: du dt dv dt 1 1 P x P y fv fu F x F y f =2Wsin f (Units: s -1 ) u=zonal velocity (xdirection) v=meridional velocity (y-direction)

16 Geostrophic balance In steady state, neglecting friction we can balance pressure gradients and Coriolis: L wind Coriolis H L v du dt 1 P 2Wsinf x pressure isobars 1 P x fv F x Flow is perpendicular to the pressure gradient! 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?

17 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 Ro u fu t u fl This is called the Rossby number ~ It tells us the importance of the Coriolis effect For small Ro, geostrophy is a good assumption /

18 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 R d is (why?) vprop Rd f Here v prop is the propagation velocity of the particular kind of feature we re interested in Taylor s analysis on p.171 is dimensionally incorrect E.g. gravity waves propagate with v prop =(gh) 1/2

19 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 Coriolis H du dt pressure isobars 1 P x fv F x 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

20 Ekman Spiral The effective thickness d of this layer is d W where W is the rotation angular frequency and is the (effective) viscosity in m 2 s -1 1/ 2 The wind direction and magnitude changes with altitude in an Ekman spiral: Increasing altitude Actual flow directions Expected geostrophic flow direction

21 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 the latitude and planetary radius u If we balance the centrifugal force against the poleward pressure gradient, we get zonal winds with speeds decreasing towards the pole: u 2 f R Rg T tanf f

22 Gradient winds In some cases both the centrifugal (u 2 /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

23 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 N y cold hot z u(z) x Small H This is not obvious. The key physical result is that the slopes of constant pressure surfaces get steeper at higher altitudes (see below) Example: On Earth, mid-latitude easterly winds get stronger with altitude. Why? P 2 P 1 cold hot P 2 P 1 Large H

24 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 R f u y u 2 ~ y 2 R W so u y ~ z RH u ~ g T gr T z ft y 2WyT y W T T exp y 0 d 4 d Latitudinal extent?venus vs. Earth vs. Mars 2 R Hg W 2 1/ 4 Does this make sense?

25 Key Concepts Hadley cell, zonal & meridional circulation Coriolis effect, Rossby number, deformation radius Thermal tides Geostrophic and cyclostrophic balance, gradient winds Thermal winds du dt u Ro 2 L Wsinf 1 P x u g T z ft y 2Wsin fv F x