Dynamics of the Zonal-Mean, Time-Mean Tropical Circulation

Dynamics of the Zonal-Mean, Time-Mean Tropical Circulation

First consider a hypothetical planet like Earth, but with no continents and no seasons and for which the only friction acting on the atmosphere is at the surface. This planet has an exact nonlinear equilibrium solution for the flow of the atmosphere, characterized by 1. Every column is in radiative-convective equilibrium, 2. Wind vanishes at planet s surface 3. Horizontal pressure gradients balanced by Coriolis accelerations

Hydrostatic balance: p = ρ g z In pressure coordinates: φ = α, p where α = 1 specific volume, ρ φ = gz geopotential

Geostrophic balance on a sphere: p 2 rel φ u = 2 Ω sinθu tan rel a θ a Can be phrased in terms of absolute angular momentum per unit mass: θ ( cos ) M = acosθ Ω a θ + u 2 2 4 4 φ M Ω a cos θ = sinθ θ a 2 cos 3 θ

Eliminate φ using hydrostatic equation: cos p p 2 1 tan θ M α T s* = = 2 2 a θ p θ p θ s* (Thermal wind equation) Take s* = s*( θ ) (convective neutrality) Integrate upward from surface (p=p 0 ), taking u=0 at surface:

2 2 2 2 2 2 s * M = a cos θ Ω a cos θ cotθ( T ) s T θ 2 s * u + 2Ω acosθu+ cotθ( T ) s T = 0. θ u ( T T) s * s 2 Ω sin θ y Implies strongest west-east winds where entropy gradient is strongest, weighted toward equator

Two potential problems with this solution: 1. Not enough angular momentum available for required west-east wind, 2. Equilibrium solution may be unstable

How much angular momentum is needed? At the tropopause, 2 2 2 2 2 2 s * M = a cos θ Ω a cos θ cotθ Ts Tt ( ) How much angular momentum is available? M 2 2 4 eq =Ω a θ s * ( T ) s Tt θ Ω a ( θ ) tanθ 1 cos 2 2 4 cos 2 θ

More generally, we require that angular momentum decrease away from the equator: 2 1 M sinθ θ < 0 1 s * Ω θ + θ θ > sinθ θ θ 2 2 3 2 4 a cos cos cot ( T ) s Tt 0. Let y 2 sec θ T T s * + y > y Ω a y 3 s t 1 0 2 2

What happens when this condition is violated? Overturning circulation must develop Acts to drive actual entropy distribution back toward criticality Postulate constant angular momentum at tropopause: M t = Ωa 2

Integrate thermal wind equations down from tropopause: 2 2 4 2 2 s * M =Ω a + a cos θcotθ T Tt ( ) θ Note: u at surface always < 0 for supercritical entropy distribution

Critical entropy distribution: Ts Tt s * 1 1 = 2 2 1 2 Ω a y 2 y Ts Tt Ts Tt * 1 1 s* = s 1 2 2 2 2 eq y+ Ω a Ω a 2 y Ts Tt * 1 2 1 = 1 cos 2 2 s θ eq + 2 Ω a 2 cos θ

Violation results in large-scale overturning circulation, known as the Hadley Circulation, that transports heat poleward and drives surface entropy gradient back toward its critical value From "The General Circulation of the Tropical Atmosphere and Interactions with Extratropical Latitudes - Vol. 1" by MIT Press. Courtesy of MIT Press. Used with permission.

Simulations by Held and Hou (1980)

Zonal wind at tropopause

The cross-equatorial Hadley Circulation

Numerical simulation using a 2-D primitive equation model with parameterized convection (Pauluis, 2004) Imposed sea surface temperature

Why is there ascent on cold side of equator? Meridional pressure gradient above the PBL: ϕ = V v 2 Ω sinθ u y 1 2 v y 2 2Ωsinθ u

Hydrostatic: ϕ = ϕ R p pbl ( T ) PBL v pbl pbl pbl v = R y y p y pbl p ϕ ϕ p T 2 1 v ppbl Tv = 2Ωsinθ u R 2 y p y pbl

PBL momentum: ϕ V v Fv = 2Ωsinθ u y pbl 2 2 1 v 1 v ppbl Tv Fv + R + 2Ωsinθ u u 2 y 2 y p y pbl F v dominant when pbl V Fv CD v h h< C y ~ D pbl ( ) pbl

Thin, frictionally dominated PBL: V ppbl Tv CD v R + 2Ωsinθ u h p y u constrained by angular momentum conservation: u u min max pbl Ωa = cosθ Ωa 2 = sin θ cosθ 2 2 sin θ sin θ0, ( )

Cold side of equator: V ppbl T CD v R + 2Ω atanθ sin θ sin θ h p y Warm side of equator: v 2 2 2 0 pbl V ppbl Tv CD v R + 2Ω atanθ sin h p y pbl 2 2 θ

p = 50 pbl mb

= 100 ppbl mb

p = pbl 200 mb

Two-D primitive equation model Parameterizations of convection fractional cloudiness radiation surface fluxes Ocean mixed layer energy budget Model forced by annual cycle of solar radiation Available for class projects (see course web page)

Relationship between moist convection and Hadley flow Boundary layer entropy equilibrium hypothesis (Raymond, 1995)

sb h 0= F M + 1 w s s t Mass: ( ( σ ) )( ) s d d b m ( 1 ) M M σ w = w u d d b

F M w s u b s s b m = + (1) Free troposphere heat balance: ( M M w) S = Q, u d cool S c p T θ θ z Convective downdraft: M & ( 1 ) = ε M d p u

ε M = w+ p u Q& cool S (2) Combine (1) and (2) M Note that w u Let wb = γ w 1 ε F Q& = 1 γε p sb sm S 1 F γ s Q& = 1 γε p sb sm S M u > w p s cool cool,

Radiative-convective equilibrium: w=0 F = M s u = Q& s s Q& ( ), cool b m cool Sε p Sε. p Define F ( ) s eq Q& s s ( ) cool b m Sε p

Then w ε p F F s s = 1 γε s s s s ( ) eq ( ) p b m b m eq Surface fluxes: ( * ) 0 F C V s s s k b w > 0 if F s > F eq (s b -s m ) < (s b -s m ) eq Q cool < (Q cool ) eq

Note also that we must have M u 0 so in circumstances under which (1) and (2) yield M u < 0 we take M = u 0, F w = or s s = w s s b s s b m w b m b Q& = cool S F radiative-subsidence balance

Weak Temperature Gradient Approximation (WTG) Sobel and Bretherton, 2000 Ignore time dependence of T above PBL Determine w from aforementioned equations Determine vorticity from w Determine T by inverting balanced flow