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S   T   M  -  A  G eo ff rey K. Vallis University of Exeter ICTP lectures, June  

C  Very loosely Ti e large-scale structure of the atmosphere in tropics and extra-tropics.  . General Circulation of the Atmosphere (in brief)  . Ti eory of the Hadley Cell.  . Tropical dynamics • Radiative convective equilibrium. • Moisture, runaway greenhouse and multiple equilibrium  . Scale/intensity of motion in tropics and midlatitudes (’weak temperature gradient’)  . Mid-latitude westerlies  . Ferrel Cell  . Ma tsuno Gill Solution. Short book: h ttp://tiny.cc/Vallis/essence Long book: http://tiny.cc/Vallis/aofd

T HE MOC Summer Winter 10 9 k g s -1 300 200 200 100 0 1 - 50 400 Pressure [hPa] 20 F H H F 0 0 600 0 2 0 0 5 1 -20 - 0 3 -50 800 -100 0 0 1 200 0 1 - -300 1000 80 S 40 S 0 40 N 80 N Latitude N ote strong winter Hadley Cell, and Ferrel Cells.

Z ONAL A VERAGE Z ONAL W IND (a) 30 220 40 20 220 20 210 Z (km) 200 210 220 230 0 10 250 270 -20 (a) Annual mean, zonally-averaged zonal 290 0 wind (heavy contours and shading) and (b) 10 ) Annual mean the zonally-averaged temperature (red, 5 ( U (m/s) thinner contours). (b) Annual mean, 0 zonally averaged zonal winds at the -5 -80 -40 0 40 80 surface. (c) 30 40 210 20 20 230 Z (km) 200 220 0 240 (c) Same except for northern 10 220 230 260 -20 hemisphere winter (DJF). 280 0 (d) 10

T EMPERATURE P ROFILES 80 (a) (b) US standard atmosphere 70 30 global average mesosphere tropics 60 extratropics tropopause-based average Altitude (km) 50 stratopause 20 40 Z (km) 30 stratosphere 10 20 tropopause 10 troposphere 0 0 160 180 200 220 240 260 280 300 200 220 240 260 280 300 Temperature (K) Temperature (K) T emperature profile of US standard atmosphere. Observed profiles.

H ADLEY C ELL G eorge Hadley (  –  ); J. J. Ti omson and William Ferrel (  th century); Lorenz (  , review); Ed Schneider (  ); Held and Hou (  ); Hou (  ) and others. Ti omson (  ) (Brother of Lord Kelvin) Note Pole to equator Hadley Cell, underneath which is a shallow indirect cell, the precursor of the Ferrel Cell,

H ADLEY C ELL O ld View Ti omson (  ), Ferrel (c.  ) Modern(ish) view, (Wallace and Hobbs)

A NGULAR M OMENTUM C ONSERVATION Axis of rotation A ngular momentum conserving wind. m = ( u + Ω a cos ϑ ) a cos ϑ . (  ) If u = 0 at equator then m = Ω a 2 so that Ω a 2 = ( u + Ω a cos ϑ ) a cos ϑ . (  ) and u = Ω a sin 2 ϑ (  ) cos ϑ

M ODERN T HEORY OF H ADLEY C ELL Ti e Hadley Cell cannot extend all the way to the pole on a rotating planet like Earth, for at least two reasons. (It almost can on Venus.) Z onally-Symmetric Equations of Motion ∂ u ∂ t − ( f + ζ ) v + w ∂ u (  ) ∂ z = 0 . Steady solution ( f + ζ ) v = 0 . (  ) Equivalently, on the sphere, ( ϑ = latitude), ∂ϑ + u tan ϑ � � 2 Ω sin ϑ − 1 ∂ u v = 0 (  ) a a Solution: either v = 0 or u = Ω a sin 2 ϑ (  ) cos ϑ . Ti e angular momentum conserving wind.

A NGULAR M OMENTUM C ONSERVATION Tropopause Angular momentum conserving flow Large zonal flow aloft Warm Cool By thermal wind the temperature of the air falls as ascent descent it moves poleward, gets too cold and sinks. 2 Ω sin ϑ ∂ u ∂ z = − 1 ∂ b Frictional return flow (  ) ∂ϑ , a Weak zonal flow at surface Ground where b = g δθ / θ 0 . Equator Subtropics Latitude (Informally, b = temperature. ) R ising air near the equator moves poleward near the tropopause, descending in the subtropics and returning.

T HERMODYNAMICS F orcing via a thermal relaxation to radiative equilibrium temperature θ E : � y � 2 θ E = θ E 0 − ∆θ (  ) . a Actual temperature from thermal wind: − 1 ∂ b ∂ϑ = 2 Ω sin ϑ ∂ u (  ) ∂ z , a where b = g δθ / θ 0 . Gives: sin 3 ϑ ∂ϑ = − 2 Ω 2 a 1 ∂θ (  ) cos ϑ , a θ 0 gH and θ = θ ( 0 ) − θ 0 Ω 2 y 4 (  ) 2 gH a 2 , Actual temperature cannot fall below radiative equilibrium temperature, so extent of Hadley Cell is: � 1 / 2 � 2 ∆θ gH ϑ M = y M = (  ) Ω 2 a 2 θ 0 a

T HERMODYNAMIC B UDGET 310 – Hadley Cell is thermodynamically self-contained. 300 – Average forcing temperature = Averaged 290 solution temperature 280 – Equal area construction gives latitude of edge of Hadley Cell: 270 � 1 / 2 � 1 / 2 � 5 � 5 gH ∆θ H 260 ϑ H = = a 3 R T (Ang. mom) θ 0 Ω 2 a 2 3 250 T (Rad equil) R ≡ gH ∆θ H 240 θ 0 Ω 2 a 2 , 0 10 20 30 40 50 Latitude Ti ermal Rossby number.

I DEAL H ADLEY C ELL S OLUTION W inds Temperature 300 310 u (spherical) 300 250 u (Cartesian) Zonal wind [m/s] Temperature [K] 290 200 280 150 270 100 260 T (Ang. mom) 50 250 T (Rad equil) 0 240 0 10 20 30 40 50 0 10 20 30 40 50 Latitude Latitude Blue: Temperature of the angular momentum conserving wind Red: Radiative equilibrium temperature.

H ADLEY C ELL S TRENGTH w ∂θ ∂ z ≈ θ E 0 − θ H θ E 0 − θ gives (  ) w ≈ . τ θ 0 ∆ V τ From solution: θ E 0 − θ = 5 R ∆θ (  ) 18 τ . τ Ti e vertical velocity is then given by w ≈ 5 R ∆θ H H (  ) . 18 τ∆θ V Transform to a streamfunction: Ψ ∼ R 3 / 2 aH ∆ H ∝ ( ∆θ H ) 5 / 2 , (  ) τ∆ V Strength proportional to gradient of radiative-equilibrium meridional temperature gradient.

T HE M OIST H ADLEY C ELL Ti e Hadley Cell is not ‘driven’ by convection, or by moisture — but moisture is important! Temperature: • Temperature of solution (red line —) dry forcing temperature, moist forcing temperature, unaltered. solution • Moist forcing ( θ ∗ E · · · · · · ) di ff ers more from Temperature equilibrium temperature than does dry solution • So circulation is stronger. • Moisture is enhancing, not causing, the Hadley Cell. (Also changes the static stability.) Latitude

������������������ �������� �� �� �� �� �� � �� �������������� �� �� �� �� � ����������� ����������� ���������������� U H OH ! B aroclinic Instability Ti e shear is so large it becomes unstable. ( Ti e traditional view of Hadley Cell termination.) Use quasi-geostrophic theory:: Critical shear for instability: d = H 2 N 2 a ∆ U C = 1 4 β L 2 (  ) 8 Ω y 2 , Ang mom solution: ∆ U M = Ω y 2 (  ) 2 a , Cross-over latitude (scaling, not exact): � N H � 1 / 2 ϑ C = y C (  ) = , a 2 Ω a On Earth, Hadley Cell is inhibited by baroclinic instability. Not so on Venus.

E DDY EFFECTS B aroclinic Instability @ @y u 0 v 0 = 0 Flow satisfies: − ( f + ζ ) v = − ∂ ∂ y u ′ v ′ . (  ) Rossby waves Rossby waves Baroclinic zone u 0 v 0 > 0 u 0 v 0 < 0 Edge of the Hadley cell where v = 0 � � HADLEY u ′ v ′ and thus ∂ y = 0 . CELL z Need not be exactly at onset of baroclinic instability. y Equator Mid-latitudes

N UMERICAL S IMULATIONS Z onally symmetric and  D C ourtesy C. Walker cf., Walker and Schneider (  ) ��� ��� ��� �� �� � �� �� �� �� � �� �� Zonally symmetric  D  D simulations have a narrower, stronger Hadley Cell.

N UMERICAL S IMULATIONS Z onally symmetric and  D �� �� �� �� � � ������������������������������������������������������������������������ ������������������������������������������������������������������������ Zonally symmetric  D

N UMERICAL S IMULATIONS ��� ��� ��� �� �� � �� �� �� �� � �� �� �� �� �� �� � � ������������������������������������������������������������������������ ������������������������������������������������������������������������

S EASONAL C YCLE I ncoming Solar Various Obliquities Top of atmosphere incoming solar radiation Annual mean Solstice

T HE S EASONAL H ADLEY C ELL  . Hadley Cell is not centered o ff the equator.  . Strong winter cell. � � �������� ������ ��� � ������ ��� � ������ � ������� �