Advection-condensation of water vapor with coherent stirring: a stochastic approach Yue-Kin Tsang Centre for Geophysical and Astrophysical Fluid Dynamics University of Exeter Jacques Vanneste School of Mathematics, University of Edinburgh
Condensation of water vapour specific humidity of an air parcel: q = mass of water vapor total air mass saturation specific humidity, q s ( T ) when q > q s , condensation occurs excessive moisture precipitates out, q → q s q s ( T ) decreases with temperature T T decreases with latitude ⇒ q s position dependent T 2 < T 1 q s ( T 1 ) q s ( T 2 )
Atmospheric moisture and climate Earth’s radiation budget: absorption of incoming shortwave radiation generates heat heat carried away by outgoing longwave radiation (OLR) water vapor is a greenhouse gas that traps OLR OLR ∼ − � log q � 1 2 � q ′ 2 � OLR ∼ − � log[ � q � + q ′ ] � ≈ − log � q � + 2 � q � how fluctuation q ′ is generated? what is the probability distribution of water vapor in the atmosphere?
Advection-condensation paradigm Large-scale advection + condensation → reproduce (leading-order) observed humidity distribution Observation Simulation – velocity and q s field from observation – trace parcel trajectories backward to the lower boundary layer (source) – track condensation along the way ignore: cloud-scale microphysics, molecular diffusion, . . . (Pierrehumbert & Roca, GRL, 1998)
Advection-condensation model PDE formualtion: ∂q ∂t + � u · ∇ q = S − C q is treated as a passive scalar advected by a prescribed � u Particle formulation: d � X ( t ) = � u d t , d Q ( t ) = ( S − C )d t air parcel at location � X carrying specific humidity Q S = moisture source (evaporation) C = condensation sink, in the rapid condensation limit C : q ( � x, t ) �→ min [ q ( � x, t ) , q s ( � x ) ] saturation profile: q s ( y ) = q 0 exp( − αy ) y = latitude (advection on a midlatitude isentropic surface) or altitude (vertical convection in troposphere)
Previous analytical results 1D stochastic models: u ∼ spatially uncorrelated random process Pierrehumbert, Brogniez & Roca 2007 : white noise, S = 0 O’Gorman & Schneider 2006 : Ornstein-Uhlenbeck process, S = 0 0.7 0.6 Random Walk with Barrier Diffusion Condensation Fit, q = Aexp(–B (Dt) 1/2 ) 0.5 0.4 q 0.3 0.2 0.1 0 0 1 2 3 4 5 Dt F IG . 2. Mean specific humidity vs meridional distance for initial F IGURE 6.8. Decay of ensemble mean specific humidity at y = 0 . 5 value problem. Moisture distributions are shown after the evolu- tion times T at which L ( T ) � 4 L s in each case. Solid lines are for the bounded random walk with a barrier at y = 0 . The thin Sukhatme & Young 2011 : white noise with a boundary source x 10 4 2000 2 PDF from ERA 1.5 1500 1 0.5 h(r) 0 1000 5 months 10 500 100 80 60 0 0.2 0.4 0.6 0.8 1 40 20 0 r RH
Coherent circulation in the atmosphere moist, warm air rises near the equator poleward transport in the upper troposphere subsidence in the subtropics ( ∼ 30 ◦ N and 30 ◦ S) transport towards the equator in the lower troposphere Q: response of rainfall patterns to changes in the Hadley cells?
Steady-state problem bounded domain: [0 , π ] × [0 , π ] , reflective B.C. q s ( y ) = q max exp( − αy ) : q s (0) = q max and q s ( π ) = q min resetting source: Q = q max if particle hits y = 0 π y 0 0 π x cellular flow: ψ = − U sin( x ) sin( y ) ; ( u, v ) = ( − ψ y , ψ x )
Stochastic system with source √ d X ( t ) = u ( X, Y ) d t + 2 κ d W 1 ( t ) ψ = − U sin x sin y √ u = − ψ y d Y ( t ) = v ( X, Y ) d t + 2 κ d W 2 ( t ) v = ψ x d Q ( t ) = [ S ( Y ) − C ( Q, Y )]d t log 10 q at time = 0.0 0 3 2.5 −0.5 2 U = 1 y −1 κ = 10 − 2 1.5 1 −1.5 0.5 −2 0 0 0.5 1 1.5 2 2.5 3 x
Source boundary layer 0 x = π /2 30 −0.5 20 P ( q ) y −1 10 ✛ ✘ −1.5 0 0 0.2 0.4 0.8 1 ✚ ✙ 0.6 q −2 x Bimodal distribution : layer consists mainly of either: Q = q min from upstream of the flow and diffuse in from the domain interior Q ≈ q max from the resetting source particles with Q ≈ q max spreading into the domain as x increases
Condensation boundary layer ✛ ✘ 0 y = 3 π /4 −0.5 y = π /2 P ( q ) y −1 y = π /4 −1.5 0 0.1 0.2 0.3 ✚ ✙ q −2 x moist particles move up into region of low q s ( y ) at some fixed height y 1 : mainly consists of Q = q min (diffuse in from the interior) and Q = q s ( y 1 ) — Bimodal distribution condensation ⇒ localized rainfall over a narrow O ( ǫ 1 / 2 ) region
Interior region ✬ ✩ 0 0.04 −0.5 0.03 mean q y U = 1 −1 U = 5 U = 20 0.02 ✫ ✪ −1.5 0.01 q min 0 50 100 time −2 x a homogeneous region of very dry air Q ≈ q min is created in the domain interior the vortex "shields" the source from the interior interior effectively undergoing stochastic drying
Steady-state problem Steady-state Fokker-Planck equation for P ( x, y, q ) : ǫ − 1 � u · ∇ P − ∂ q [( S − C ) P ] = ∇ 2 P , ǫ = κ/ ( UL ) ≪ 1 Rapid condensation limit : P ( x, y, q ) � = 0 � for x, y ∈ [0 , π ] and q ∈ [ q min , q s ( y )] C = 0 Resetting source at bottom boundary : P ( x, y = 0 , q ) = π − 1 δ ( q − q max ) At the top boundary : P ( x, y = π, q ) = π − 1 δ ( q − q min ) Hence, ǫ − 1 � u · ∇ P = ∇ 2 P which predicts a boundary layer of thickness O ( ǫ 1 / 2 )
Matched asymptotics 1. domain interior, to leading-order: P 0 = π − 2 δ ( q − q min ) 2. source boundary layer: P 0 = G ( x, y ) δ ( q − q min ) + H ( q, x, y ) 3. condensation boundary layer: P 0 = G ( x, y ) δ ( q − q min ) + [ π − 2 − G ( x, y )] δ ( q − q s ( y )) In the O ( ǫ 1 / 2 ) boundary layers, introducing coordinates (Childress 1979): � ζ = ǫ − 1 / 2 ψ and σ = |∇ ψ | d l , l = arclength Equation for G ( σ, ζ ) reduces to: ∂ σ G = ∂ ζζ G
Mean moisture input rate Φ � 8 κ Φ = ǫ − 1 / 2 π ( q max − q min ) , ǫ = κ/ ( UL ) 2 10 Monte Carlo mean moisture input rate, Φ asymptotics 1 10 ~ ε −1/2 0 10 -1 10 -4 -3 -2 -1 10 10 10 10 ε Other diagnostics: horizontal rainfall profile, vertical moisture flux, . . . etc
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