Stochastic modelling and parametrization of atmospheric moisture - PowerPoint PPT Presentation

Stochastic modelling and parametrization of atmospheric moisture transport Yue-Kin Tsang Centre for Geophysical and Astrophysical Fluid Dynamics Mathematics, University of Exeter Jacques Vanneste (University of Edinburgh) Geoff Vallis (University of Exeter)

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 probability distribution of water vapor in the atmosphere? T 2 < T 1 q s ( T 1 ) q s ( T 2 )

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 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( − αy ) y = latitude (advection on a midlatitude isentropic surface) or altitude (vertical convection in troposphere) Mean-field formualtion: ∂q ∂t + � u · ∇ q = S − C q ( � x, t ) is treated as a passive scalar field advected by � u

Particle models: 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 −0.5 0.04 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 )

Solution and diagnostics solve P ( x, y, q ) by matched asymptotics as ǫ → 0 dry peak: 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 � 8 κ/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

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

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

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 )]

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? q ) = τ − 1 C PDE (¯ c (¯ q − q s ) H (¯ q − q s ) , H : Heaviside step function u · ∇ P + ∂ q [( S − C ) P ] = κ b ∇ 2 P Fokker-Planck: ∂ t P + � � q max q ( x, y, t ) = π 2 q ′ P ( x, y, q ′ , t )d q ′ ¯ q min � q max ( q ′ − q s ) P ( x, y, q ′ , t )d q ¯ C = π 2 q s ( y ) condensation and averaging do not commute a v e r a g e q s = 2.1 1 2 1.7 c ond e n s e 2.1 1 2 � y 3 � x a v e r a g e 2 2 (c ond e n s e)

Parametrization of condensation ∂ ¯ q q = κ q ∇ 2 ¯ ∂t + � u · ∇ ¯ q → C (¯ ¯ q, q s ) q, at a grid point ( x, y ) and time t , after advection and diffusion steps let’s say q ( x, y, t ) = q ∗ ¯ imagine there is a distribution P 0 ( q | x, y ) such that � q ′ P 0 ( q ′ | x, y ) d q ′ 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 σ 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 peak subsidence of dry air parcels is important include a dry peak of amplitude β in P 0 ( q | x, y ) before condensation after condensation q s ( y ) q s ( y ) β β 0 ( q | x , y ) 1 ( q | x , y ) P P q min q max q min q max q * q *

Amplitude of dry peak β ( x, y ) = π 2 ρ ( x, y ) , P ( q min , x, y, t ) = δ ( q − q min ) ρ ( x, y ) ∂ρ u · ∇ ρ = κ q ∇ 2 ρ ∂t + � ρ ( x, π, t ) = π − 2 ρ ( x, 0 , t ) = 0 ,

Results with dry peak