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, qs(T) when q > qs, condensation occurs excessive moisture precipitates out, q → qs qs(T) decreases with temperature T probability distribution of water vapor in the atmosphere? qs(T1) qs(T2) T2 < T1

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)

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) 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 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

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.

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)

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

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

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

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

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)

Solution and diagnostics

solve P(x, y, q) by matched asymptotics as ǫ → 0 dry peak: P(x, y, q) = δ(q − qmin)β(x, y)/π2 + F(x, y, q) mean moisture input rate: 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

mean moisture input rate, Φ Monte Carlo asymptotics ~ε−1/2

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 ∂t + u · ∇¯ q = κq∇2¯ q − C + S κq: 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 ∂t + u · ∇¯ q = κq∇2¯ q − C + S κq: eddy diffusivity representing un-resolved processes boundary source: ¯ q(x, y = 0, t) = qmax

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)]

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)]

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)]

Why PDE models saturate the domain?

CPDE(¯ q) = τ −1

c (¯

q − qs)H(¯ q − qs), H: Heaviside step function Fokker-Planck: ∂tP + u · ∇P + ∂q[(S − C)P] = κb∇2P

¯ q(x, y, t) = π2 qmax

qmin

q′P(x, y, q′, t)dq′ ¯ C = π2 qmax

qs(y)

(q′ − qs)P(x, y, q′, t)dq

condensation and averaging do not commute

2 1 3 1 2.1 2 2 2 1.7

qs = 2.1

x y

c ond e nse a v era g e a v era g e (c ond e nse)

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* 2σ

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

Parametrization with dry peak

subsidence of dry air parcels is important include a dry peak 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* β β

Amplitude of dry peak

β(x, y) = π2ρ(x, y) , P(qmin, x, y, t) = δ(q − qmin)ρ(x, y) ∂ρ ∂t + u · ∇ρ = κq∇2ρ ρ(x, 0, t) = 0 , ρ(x, π, t) = π−2

Results with dry peak