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Advection-condensation of water vapor with coherent stirring: a - - PowerPoint PPT Presentation
Advection-condensation of water vapor with coherent stirring: a - - PowerPoint PPT Presentation
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
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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 OLR ∼ − log[q + q′] ≈ − log q +
1 2 q
2 q′2
how fluctuation q′ is generated? what is the probability distribution of water vapor in the atmosphere?
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Advection-condensation paradigm
Large-scale advection + condensation → reproduce (leading-order) observed humidity distribution
Observation Simulation – velocity and qs 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)
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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 dt , dQ(t) = (S − C)dt 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) , qs( x) ] saturation profile: qs(y) = q0 exp(−αy) y = latitude (advection on a midlatitude isentropic surface) or altitude (vertical convection in troposphere)
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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 0.5 0.4 0.3 q 0.2 0.1 1 2 Dt Random Walk with Barrier Diffusion Condensation Fit, q = Aexp(–B (Dt)1/2) 3 4 5
FIGURE 6.8. Decay of ensemble mean specific humidity at y = 0.5 for the bounded random walk with a barrier at y = 0. The thin
- FIG. 2. Mean specific humidity vs meridional distance for initial
value problem. Moisture distributions are shown after the evolu- tion times T at which L(T) 4Ls in each case. Solid lines are
Sukhatme & Young 2011: white noise with a boundary source
0.2 0.4 0.6 0.8 1 500 1000 1500 2000 r h(r) 5 10 20 40 60 80 100 0.5 1 1.5 2 x 104 RH months PDF from ERA
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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?
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Steady-state problem
bounded domain: [0, π] × [0, π], reflective B.C.
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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Stochastic system with source 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
U = 1 κ = 10−2
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
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Source boundary layer
x y
−2 −1.5 −1 −0.5
0.2 0.4 0.6 0.8 1
q
10 20 30
P(q) x = π/2
✛ ✚ ✘ ✙ Bimodal distribution: layer consists mainly of either:
Q = qmin from upstream of the flow and diffuse in from the domain interior Q ≈ qmax from the resetting source particles with Q ≈ qmax spreading into the domain as x increases
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Condensation boundary layer
x y
−2 −1.5 −1 −0.5
y = 3π/4 P(q) y = π/2
0.1 0.2 0.3
q y = π/4
✛ ✚ ✘ ✙
moist particles move up into region of low qs(y) at some fixed height y1: mainly consists of Q = qmin (diffuse in from the interior) and Q = qs(y1) — Bimodal distribution condensation ⇒ localized rainfall over a narrow O(ǫ1/2) region
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Interior region
x y
−2 −1.5 −1 −0.5
50 100
time
0.01 0.02 0.03 0.04
mean q U = 1 U = 5 U = 20
qmin
✬ ✫ ✩ ✪
a homogeneous region of very dry air Q ≈ qmin is created in the domain interior the vortex "shields" the source from the interior interior effectively undergoing stochastic drying
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Steady-state problem
Steady-state Fokker-Planck equation for P(x, y, q): ǫ−1 u · ∇P − ∂q[(S − C)P] = ∇2P , ǫ = κ/(UL) ≪ 1
Rapid condensation limit:
P(x, y, q) = 0 C = 0
- for x, y ∈ [0, π] and q ∈ [qmin, qs(y)]
Resetting source at bottom boundary:
P(x, y = 0, q) = π−1δ(q − qmax)
At the top boundary: P(x, y = π, q) = π−1δ(q − qmin)
Hence, ǫ−1 u · ∇P = ∇2P which predicts a boundary layer of thickness O(ǫ1/2)
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Matched asymptotics
- 1. domain interior, to leading-order:
P0 = π−2δ(q − qmin)
- 2. source boundary layer:
P0 = G(x, y) δ(q − qmin) + H(q, x, y)
- 3. condensation boundary layer:
P0 = G(x, y) δ(q − qmin) + [π−2 − G(x, y)] δ(q − qs(y)) In the O(ǫ1/2) boundary layers, introducing coordinates (Childress 1979): ζ = ǫ−1/2ψ and σ =
- |∇ψ| dl ,
l = arclength Equation for G(σ, ζ) reduces to: ∂σG = ∂ζζG
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Mean moisture input rate Φ
Φ = ǫ−1/2
- 8κ
π (qmax − qmin) , ǫ = κ/(UL)
10
- 4
10
- 3
10
- 2
10
- 1
ε
10
- 1
10 10
1
10
2