Lateral Heat Transport 6 4 Ocean Transport Atmosphere Transport Total Transport petaWatts 2 0 -2 -4 -6 -90 -60 -30 0 30 60 Latitude Heat Transport by Oceans and Atmosphere 90 Steady Flow: ∇ •  Frad kˆ + Fconv kˆ + ρ VE  = 0,   where 1 2 E ≡ c pT + gz + Lv q + | V | 2 Integrate from surface to top of atmosphere: ∇ • ρ VE + FradTOA − ( Frad + Fconv ) surface = 0 What causes lateral enthalpy transport by atmosphere? 1: Large-scale, quasi-steady overturning motion in the Tropics, 2: Eddies with horizontal dimensions of ~ 3000 km in middle and high latitudes 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 ρ ≡ specific volume, φ = gz ≡ geopotential Horizontal force balance in inertial reference frame: ∂p du = −α , ∂x dt ∂p dv = −α ∂y dt Rotating reference frame of Earth: dv ∂p u = −α − sin θ dt ∂y r u = Ωa cos θ + urel , r = a cos θ 2 u dv ∂p 2 θ θ θ θ cos sin 2 sin tan a u → = −α −Ω − Ω − rel dt a ∂y 2 rel Bracketed term absorbed into defintion of gravity: 2 urel ∂p dv tan θ = −α − 2Ω sin θ urel − ∂y dt a ∂p ≅ −α − 2Ω sin θ urel ∂y ∂p ≡ −α − furel , where f ≡ 2Ω sin θ ∂y Phrase in pressure coordinates: ∂p ∂p ∂p dp = dx + dy + dz = 0 ∂x ∂y ∂z  ∂z   ∂φ  ∂p ∂p  ∂z  → = −   = ρg   = ρ   ∂y ∂z  ∂y  p  ∂y  p  ∂y  p  ∂φ  dv = −   − furel dt  ∂y  p Force balance:  ∂φ    = − furel  ∂y  p Geostrophic balance  ∂φ    = − furel  ∂y  p  ∂φ    = −α  ∂p  Eliminate φ : ∂u  ∂α  R  ∂T  = f  =   ∂p  ∂y  p p  ∂y  p Thermal wind Zonal wind increases with altitude if temperature decreases toward pole Moist adiabatic atmosphere: s* = constant α = α ( s*, p )  ∂α   ∂α  ∂s * →  =   ∂y  p  ∂s *  p ∂y  ∂α   ∂T  Maxwell :    =  ∂s *  p  ∂p  s* ∂u  ∂T  ∂s * → f =  ∂p  ∂p  s* ∂y Integrate from surface to tropopause, taking u=0 at surface: ∂sb ∂s * = − (Ts − TT ) fuT = − (Ts − TT ) ∂y ∂y uT Ts − TT ) ∂sb ( =− 2Ω 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 Angular momentum per unit mass: M = a cos θ ( Ωa cos θ + u ) At tropopause: M T = a cos θ ( Ωa cos θ + uT ) Ts − TT ) ∂sb   ( = a cos θ  Ωa cos θ −  2Ωa sin θ ∂θ   Maximum possible value of M is its resting value at equator: M max = Ωa 2 We require that M ≤ M max : ∂sb 2Ω a 2 sin θ tan θ >− Ts − TT ∂θ 2 2 Violated in much of Tropics 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 Concept of eddy fluxes: ∇ • ρ VE + FradTOA − ( Frad + Fconv ) surface = 0 ρ V = { ρ V} + ρ V ', E = { E} + E ', 1 where { X } ≡ 2π ∫ 2π 0 X dλ → ∇ • { ρ V ' E '} + { ρ V}{ E} + FradTOA − ( Frad + Fconv ) surface = 0   Eddy heat fluxes not efficient enough to prevent temperature gradients from developing Fluxes broadly down-gradient, but not related in a simple way to temperature gradients