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Parametrization of stochastic effects in an advectioncondensation - - PowerPoint PPT Presentation
Parametrization of stochastic effects in an advectioncondensation - - PowerPoint PPT Presentation
Parametrization of stochastic effects in an advectioncondensation model Yue-Kin Tsang Centre for Geophysical and Astrophysical Fluid Dynamics, Mathematics, University of Exeter Jacques Vanneste (Edinburgh), Geoff Vallis (Exeter) Funded by
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Condensation of water vapour specific humidity of an air parcel: q = mass of water vapour total air mass saturation specific humidity, qs(T)
when q > qs, condensation occurs excessive moisture precipitates out, q → qs qs(T) decreases with temperature T qs(y) as T = T(y), y = latitude (advection on a mid-latitude isentropic surface) or altitude (vertical convection in troposphere)
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Advection–condensation paradigm
Large-scale advection + condensation → reproduce (leading-order) observed humidity distribution
- bservation
simulation velocity and qs field from observation trace parcel trajectories backward to the lower boundary layer (source) track the minimum qs encountered along the way ignore: cloud-scale microphysics, molecular diffusion, . . . etc
(Pierrehumbert & Roca, GRL, 1998)
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Advection–condensation model Particle formulation: d X(t) = u dt , dQ(t) = (S − C)dt 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 , qs( X) ] saturation profile: qs(y) = q0 exp(−αy)
Mean-field formulation: ∂¯ q ∂t + u · ∇¯ q = S − C ¯ q( x, t) is treated as a passive scalar field advected by u
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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?
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Advection–condensation in cellular flows bounded domain: [0, π] × [0, π], reflective boundaries qs(y) = qmax exp(−αy): qs(0) = qmax and qs(π) = qmin resetting source: Q = qmax if particle hits y = 0 π π x y cellular flow: ψ = −U sin(x) sin(y);
(u, v) = (−ψy, ψx)
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Particle formulation dX(t) = u(X, Y) dt + √ 2κ dW1(t) dY(t) = v(X, Y) dt + √ 2κ dW2(t) dQ(t) = [S(Y) − C(Q, Y)]dt
ψ = −U sin x sin y u = −ψy v = ψx
0.5 1 1.5 2 2.5 3 0.5 1 1.5 2 2.5 3
x y
log10 q at time = 0.0 −2 −1.5 −1 −0.5
U = 1 κ = 10−2
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PDF of specific humidity – a dry spike dX(t) = u(X, Y) dt + √ 2κ dW1(t) dY(t) = v(X, Y) dt + √ 2κ dW2(t) dQ(t) = [S(Y) − C(Q, Y)]dt
ψ = −U sin x sin y u = −ψy v = ψx
x y
2 1.5 1 0.5
y = 3/4 P(q) y = /2
0.1 0.2 0.3
q y = /4
0.2 0.4 0.6 0.8 1
q
10 20 30
P(q) x = π/2
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Fokker-Planck equation: solution and diagnostics
Steady-state Fokker-Planck equation for P(x, y, q): ǫ−1 u · ∇P = ∇2P , ǫ = κ/(UL) ≪ 1 solve for P(x, y, q) by matched asymptotics as ǫ → 0 dry spike: P(x, y, q) = δ(q − qmin)β(x, y)/π2 + F(x, y, q) mean moisture input rate: Φ = ǫ−1/2κ
- 8/π(qmax − qmin)
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mean moisture input rate, Monte Carlo asymptotics ~ 1/2 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)
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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 ∂t + u · ∇¯ q = κq∇2¯ q − C + S
κq: eddy diffusivity representing un-resolved processes boundary source: ¯ q(x, y = 0, t) = qmax rapid condensation C : ¯ q( x, t) → min[¯ q( x, t), qs(y)]
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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 ∂t + u · ∇¯ q = κq∇2¯ q − C + S
κq: eddy diffusivity representing un-resolved processes boundary source: ¯ q(x, y = 0, t) = qmax rapid condensation C : ¯ q( x, t) → min[¯ q( x, t), qs(y)]
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Why PDE models saturate the domain? The coarse-graining process and the condensation process do not commute:
2 1 3 1 2.1 2 2 2 1.7
qs = 2.1
x y
c
- n
d e n s e a v e r a g e a v era g e (c ond e nse)
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Parametrization of condensation
∂¯ q ∂t + u · ∇¯ q = κq∇2¯ q, ¯ q → C(¯ q, qs) 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 P0(q|x, y) such that q∗ =
- q′P0(q′|x, y) dq′
then, ¯ q(x, y, t + ∆t) =
- q′P1(q′|x, y) dq′
P
0 (q | x,y)
before condensation P
1 (q | x,y)
after condensation qs(y) q* qmin qmax qs(y) qmax qmin q* 2
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Test results P0(q|x, y): a top hat distribution of width 2σ as a test, prescribe a constant σ for ¯ q − σ < qs < ¯ q + σ, condensation occurs as: ¯ q → ¯ q − [¯ q + σ − qs]2 4σ κq = 0.01
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Parametrization with dry spike subsidence of dry air parcels is important include a dry spike of amplitude β in P0(q|x, y)
P
0 (q | x,y)
before condensation P
1 (q | x,y)
after condensation qs(y) q* qmin qmax qs(y) qmax qmin q*
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Amplitude of dry spike
P(qmin, x, y, t) = π−2β(x, y)δ(q − qmin) ∂β ∂t + u · ∇β = κq∇2β β(x, 0, t) = 0 , β(x, π, t) = 1
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