A spatiotemporal stochastic model for tropical precipitation and - - PowerPoint PPT Presentation

A spatiotemporal stochastic model for tropical precipitation and water vapor dynamics.

Scott Hottovy and Sam Stechmann (UW) shottovy@math.wisc.edu

University of Wisconsin

ONR DURIP grant N00014-14-1-0251

• S. Hottovy, UW

Spatiotemporal stochastic model for precip. Jan 7, 2016

Cloud organization: tropics and midlatitudes

◮ Midlattitudes: Dynamics are quasi-solvable, dominated by

rotation of earth.

◮ Tropics: More random, multi-scale problem.

• S. Hottovy, UW

Spatiotemporal stochastic model for precip. Jan 7, 2016

Cloud organization: tropics and midlatitudes

◮ Midlattitudes: Dynamics are quasi-solvable, dominated by

rotation of earth.

◮ Tropics: More random, multi-scale problem.

• S. Hottovy, UW

Spatiotemporal stochastic model for precip. Jan 7, 2016

. Column Water Vapor Precipitable water

• S. Hottovy, UW

Spatiotemporal stochastic model for precip. Jan 7, 2016

Multiscale clouds and waves

Precipitation Spectral Power

(of Fourier transform in space & time)

from Wheeler & Kiladis 1999 2000–2001 (from Zhang 2005)

• S. Hottovy, UW

Spatiotemporal stochastic model for precip. Jan 7, 2016

Smoothing the PSD

from Lin et al 2006

To accentuate the features of the raw data. Smooth it and remove a Background Spectrum.

• S. Hottovy, UW

Spatiotemporal stochastic model for precip. Jan 7, 2016

Smoothing the PSD

from Lin et al 2006

To accentuate the features of the raw data. Smooth it and remove a Background Spectrum.

◮ Goal: Create a model of background spec. to aid in

understanding of waves.

• S. Hottovy, UW

Spatiotemporal stochastic model for precip. Jan 7, 2016

Model with Spatial Variability

Linear stochastic model: dqi,j dt = F − 1 τ qi,j + D∗ ˙ Wi,j − b

• (i,j),(i′,j′)

(q(i,j) − q(i′,j′)) Similar to Stochastic PDE: ∂q ∂t = F − 1 τ q + D∗ ˙ W + b0∇2q

• S. Hottovy, UW

Spatiotemporal stochastic model for precip. Jan 7, 2016

Model with Spatial Variability

Linear stochastic model: dqi,j dt = F − 1 τ qi,j + D∗ ˙ Wi,j − b

• (i,j),(i′,j′)

(q(i,j) − q(i′,j′)) Similar to Stochastic PDE: ∂q ∂t = F − 1 τ q + D∗ ˙ W + b0∇2q Relation to atmospheric dynamics: ∂q ∂t + (uq)x + (vq)y = S Decompose: q = ¯ q + q′ = “resolved” + “sub-grid-scale” ∂¯ q ∂t = −(¯ u¯ q)x − (¯ v ¯ q)y

• − 1

τ q + D∗ ˙

W

Damping + Forcing

+ ¯ S −(u′q′)x − (v ′q′)y

• b0∇2q

Eddy diffusion

• S. Hottovy, UW

Spatiotemporal stochastic model for precip. Jan 7, 2016

Stationary PDF in Fourier Space

Gaussian: ρ(ˆ qk) = 1 Zk exp −|ˆ qk|2 D2

∗/ck

• Independent ˆ

qk, ˆ qk′ ⇒ cheap to sample the pdf σ = H(q − q∗) Precip = ( 1

τ q + |F|)σ

• S. Hottovy, UW

Spatiotemporal stochastic model for precip. Jan 7, 2016

Power Spectrum

Wavenumber k (2π/40000 km) Frequency (cpd) PSD of CWV −10 10 0.1 0.2 0.3 0.4 0.5 0.6 0.7 6 6.5 7

(from Wheeler & Kiladis 1999)

|ˆ q(k, ω)|2 = 1 2 D2

∗

ω2 + c2

k

≈ 1 2 D2

∗

ω2 + ˜ b0|k|2 + τ −2

• S. Hottovy, UW

Spatiotemporal stochastic model for precip. Jan 7, 2016

Mean Precipitation

55 60 65 70 75 0.05 0.1 0.15 0.2 0.25 Conditional Mean Precip. [mm/hr] qM

i,j

[mm] |F|σ (q−q*)σ/tau [|F|+(q−q*)/tau]σ β =0.62 β =0.23

Power law fits near a critical point.

• S. Hottovy, UW

Spatiotemporal stochastic model for precip. Jan 7, 2016

Variance

80 70 60 50 40 30 20 10 90 Precipitation variance × L0.42 103 102 101 100 10–1 10–2 10–3 104 10–4 Variance × L2 40 50 60 w (mm) 30 70 L = 2 L = 1 L = 0.5 L = 0.25 55 60 65 70 50 75 w (mm)

Peak near critical value.

• S. Hottovy, UW

Spatiotemporal stochastic model for precip. Jan 7, 2016

Cloud Cluster Size Density

10

1

10

2

10

3

10

4

10

−6

10

−5

10

−4

10

−3

10

−2

10

−1

10 Cloud Size Distribution by Area Cloud Area [km2] PDF Cloud pdf Best fit−1.528 (from Wood & Field 2011)

Exponent prediction from WF11: 1.66±0.06 or 1.87±0.06

• S. Hottovy, UW

Spatiotemporal stochastic model for precip. Jan 7, 2016

Connection with stat-phys models

◮ Why does a simple linear model have evidence of

criticality?

◮ with 2-D Ising Model ◮ Parameters:

D2

∗ ↔ kBT

b ↔ J F ↔ H

b: promotes spatial regularity τ −1: promotes temporal regularity F: promotes a shift in the spatial average

◮ Edwards-Wilkinson model

(1995) (1D Stochastic Heat Equation)

◮ Model for 1D random

growth of a surface. dq(x, t) dt = ∂2q(x, t) ∂x2 + ˙ W (x, t) (1D, τ → ∞ limit)

• S. Hottovy, UW

Spatiotemporal stochastic model for precip. Jan 7, 2016

Summary

◮ Linear stochastic model – but nonlinear statistics (of

σ(x, y, t))

◮ Simple model captures obs. very well ◮ Related to atmospheric evolution equations ◮ Behavior similar to phase transition and self-organized

criticality

◮ Related to classic statistical physics models

Statistical physics provides useful organizing principles for understanding a complex system References:

[1] Hottovy, S., & Stechmann, S. N. (2015). A Spatiotemporal Stochastic Model for Tropical Precipitation and Water Vapor Dynamics. Journal of the Atmospheric Sciences, 72(12), 4721-4738.

• S. Hottovy, UW

Spatiotemporal stochastic model for precip. Jan 7, 2016