Parametrization of stochastic effects in an advection–condensation model Yue-Kin Tsang Centre for Geophysical and Astrophysical Fluid Dynamics, Mathematics, University of Exeter Jacques Vanneste (Edinburgh), Geoff Vallis (Exeter) Funded by the EPSRC ReCoVER Network
Atmospheric moisture and climate Earth’s radiation budget: absorption of incoming short-wave radiation generates heat heat carried away by outgoing long-wave radiation (OLR) water vapour is a greenhouse gas that traps OLR 1 2 � q � 2 � q ′ 2 � OLR ∼ − � log [ � q � + q ′ ] � ≈ − log � q � + how fluctuation q ′ is generated? what is the probability distribution of water vapour in the atmosphere?
Condensation of water vapour specific humidity of an air parcel: q = mass of water vapour 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 q s ( y ) as T = T ( y ) , y = latitude (advection on a mid-latitude isentropic surface) or altitude (vertical convection in troposphere)
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 the minimum q s encountered along the way ignore: cloud-scale microphysics, molecular diffusion, . . . etc (Pierrehumbert & Roca, GRL, 1998)
Advection–condensation model Particle formulation: d � X ( t ) = � u d t , d Q ( t ) = ( S − C ) d t air parcel at location � X ( t ) carrying specific humidity Q ( t ) S = moisture source (evaporation) C = condensation sink, in the rapid condensation limit C : Q �→ min [ Q , q s ( � X ) ] saturation profile: q s ( y ) = q 0 exp ( − α y ) Mean-field formulation: ∂ ¯ q ∂ t + � u · ∇ ¯ q = S − C ¯ q ( � x , t ) is treated as a passive scalar field advected by � u
Particle models: previous analytical results 1-D stochastic models: u ∼ spatially uncorrelated random process Pierrehumbert, Brogniez & Roca 2007 : white noise, S = 0 O’Gorman & Schneider 2006 : Ornstein–Uhlenbeck process, S = 0 Sukhatme & Young 2011 : white noise with a boundary source Coherent circulation in the atmosphere Q: response of rainfall patterns to changes in the Hadley cells?
Advection–condensation in cellular flows bounded domain: [ 0 , π ] × [ 0 , π ] , reflective boundaries q s ( y ) = q max exp ( − α y ) : 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 )
Particle formulation √ ψ = − U sin x sin y d X ( t ) = u ( X , Y ) d t + 2 κ d W 1 ( t ) √ 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
PDF of specific humidity – a dry spike √ ψ = − U sin x sin y d X ( t ) = u ( X , Y ) d t + 2 κ d W 1 ( t ) √ 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 0 y = 3 � /4 � 0.5 y = � /2 P ( q ) y = � /4 y � 1 0 0.1 0.2 0.3 q x = π /2 � 1.5 30 P ( q ) 20 10 � 2 0 0 0.2 0.4 0.6 0.8 1 x q
Fokker-Planck equation: solution and diagnostics Steady-state Fokker-Planck equation for P ( x , y , q ) : ǫ − 1 � u · ∇ P = ∇ 2 P , ǫ = κ/ ( UL ) ≪ 1 solve for P ( x , y , q ) by matched asymptotics as ǫ → 0 dry spike: P ( x , y , q ) = δ ( q − q min ) β ( x , y ) /π 2 + F ( x , y , q ) mean moisture input rate: Φ = ǫ − 1 / 2 κ � 8 /π ( q max − q min ) 2 10 mean moisture input rate, � Monte Carlo asymptotics 1 10 ~ � � 1/2 0 10 -1 10 -4 -3 -2 -1 10 10 10 10 � Other diagnostics: horizontal rainfall profile, moisture flux, ...etc, see “Advection–condensation of water vapour in a model of coherent stirring”, Yue-Kin Tsang & Jacques Vanneste (2016)
Mean-field PDE model Weather/climate models represent atmospheric moisture as a coarse-grained field ¯ q ( � x , t ) governed by deterministic PDE Advection–condensation–diffusion: ∂ ¯ q q = κ q ∇ 2 ¯ ∂ t + � u · ∇ ¯ q − C + S κ q : eddy diffusivity representing un-resolved processes boundary source: ¯ q ( x , y = 0 , t ) = q max rapid condensation C : ¯ q ( � x , t ) → min [¯ q ( � x , t ) , q s ( y )]
Mean-field PDE model Weather/climate models represent atmospheric moisture as a coarse-grained field ¯ q ( � x , t ) governed by deterministic PDE Advection–condensation–diffusion: ∂ ¯ q q = κ q ∇ 2 ¯ ∂ t + � u · ∇ ¯ q − C + S κ q : eddy diffusivity representing un-resolved processes boundary source: ¯ q ( x , y = 0 , t ) = q max rapid condensation C : ¯ q ( � x , t ) → min [¯ q ( � x , t ) , q s ( y )]
Why PDE models saturate the domain? The coarse-graining process and the condensation process do not commute: a v e r a g e q s = 2.1 1 2 e 1.7 n s e d n o 2.1 c 1 2 � y 3 � x a v e 2 2 r a g e (c ond e n s e)
Parametrization of condensation ∂ ¯ q q = κ q ∇ 2 ¯ ∂ t + � u · ∇ ¯ ¯ q → C (¯ q , q , q s ) at a grid point ( x , y ) and time t , after advection and diffusion steps ¯ q ( x , y , t ) = q ∗ let’s say imagine there is a distribution P 0 ( q | x , y ) such that � q ∗ = q ′ P 0 ( q ′ | x , y ) d q ′ � q ′ P 1 ( q ′ | x , y ) d q ′ then , ¯ q ( x , y , t + ∆ t ) = before condensation after condensation q s ( y ) q s ( y ) 2 � 0 ( q | x , y ) 1 ( q | x , y ) P P q min q max q min q max q * q *
Test results P 0 ( q | x , y ) : a top hat distribution of width 2 σ as a test, prescribe a constant σ for ¯ q − σ < q s < ¯ q + σ , condensation occurs as: q + σ − q s ] 2 q − [¯ ¯ q → ¯ 4 σ κ q = 0 . 01
Parametrization with dry spike subsidence of dry air parcels is important include a dry spike of amplitude β in P 0 ( q | x , y ) before condensation after condensation q s ( y ) q s ( y ) � � 0 ( q | x , y ) 1 ( q | x , y ) P P q min q max q min q max q * q *
Amplitude of dry spike P ( q min , x , y , t ) = π − 2 β ( x , y ) δ ( q − q min ) ∂β u · ∇ β = κ q ∇ 2 β ∂ t + � β ( x , 0 , t ) = 0 , β ( x , π, t ) = 1
Results with dry spike
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