A Stochastic Model for Tropical Rainfall. Scott Hottovy and Sam - - PowerPoint PPT Presentation

A Stochastic Model for Tropical Rainfall.

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

United States Naval Academy

ONR DURIP grant N00014-14-1-0251

• S. Hottovy, USNA
• Trop. Rain

Oct 1, 2016

Cloud organization: tropics and midlatitudes

◮ Midlattitudes: Dynamics are quasi-solvable, dominated by

rotation of earth.

◮ Tropics: More random, multi-scale problem.

• S. Hottovy, USNA
• Trop. Rain

Oct 1, 2016

Cloud organization: tropics and midlatitudes

◮ Midlattitudes: Dynamics are quasi-solvable, dominated by

rotation of earth.

◮ Tropics: More random, multi-scale problem.

• S. Hottovy, USNA
• Trop. Rain

Oct 1, 2016

. Column Water Vapor Precipitable water

• S. Hottovy, USNA
• Trop. Rain

Oct 1, 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, USNA
• Trop. Rain

Oct 1, 2016

Motivation

Remove general characteristics of the Power Spectrum?

(from Zhang 2005) (from Wheeler & Kiladis 1999)

• S. Hottovy, USNA
• Trop. Rain

Oct 1, 2016

Motivation

Smooth raw (left) to get a background spectrum (mid.), remove it to get anomalies (right). raw background anomalies

(from Wheeler & Kiladis 1999)

• S. Hottovy, USNA
• Trop. Rain

Oct 1, 2016

Other Models

• 1. Waves

◮ Nonlinear dynamics ◮ Two-way interactions with background state ◮ Khouider & Majda (2006, ...),

Majda & Stechmann (2009, ...), Khouider, Han, Majda, Stechmann (2012), ...

• 2. Stochastic models of convection (stat. phys.)

◮ Stechmann & Neelin (2011),

Stechmann & Neelin (2014), Hottovy & Stechmann (2015), ...

• 3. Both Waves and stochastics

◮ Majda & Khouider (2002),

Majda & Stechmann (2008), Khouider, Majda, Biello (2010), Frenkel, Majda, Khouider (2011, 2012, ...), ...

• S. Hottovy, USNA
• Trop. Rain

Oct 1, 2016

Goals

◮ Develop a model of column water vapor (CWV)

background spectrum on a lattice.

◮ Model the background spectrum of the atmosphere ◮ Model should be simple for fast/cheap simulation ◮ Should have signs of criticality, power laws, spikes in

variance.

◮ from obs. Peters/Neelin 2006, Neelin/Peters/Hales 2009

◮ Hypothesis:

◮ The tropical background state is modeled by turbulent

advection-diffusion of CWV.

• S. Hottovy, USNA
• Trop. Rain

Oct 1, 2016

The model

Hypothesis (q is integrated CWV): ∂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

• b∇2q

Eddy diffusion

• S. Hottovy, USNA
• Trop. Rain

Oct 1, 2016

The model

Hypothesis (q is integrated CWV): ∂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

• b∇2q

Eddy diffusion

Linear stochastic model, discretization of SPDE: ∂q ∂t = F − 1 τ q + D ˙ W + b∇2q

• S. Hottovy, USNA
• Trop. Rain

Oct 1, 2016

Discretization

◮ Size of grid ∆x = ∆y = 5 km for scale of convection ◮ Coarsened to 25 km × 25 km for typical radar footprint

• S. Hottovy, USNA
• Trop. Rain

Oct 1, 2016

Defining precipitation

◮ a site is precipitating when, qi,j(t) > q∗ = 65 mm. ◮ cloud indicator, and precip. rate

σi,j(t) = H(qi,j(t)−q∗), ri,j(t) =

• |F| + qi,j(t)

τ

• σi,j(t).

◮ q comes from a linear equation, σ, r are non-linear!

• S. Hottovy, USNA
• Trop. Rain

Oct 1, 2016

Exact solutions

◮ Fourier transform in space qi,j −

→ ˆ qk are indep. in k.

◮ Makes for very fast/cheap large simulations (106 grid

points < 1 s).

• S. Hottovy, USNA
• Trop. Rain

Oct 1, 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)

E

• |ˆ

q(k, ω)|2 = 1 2 D2

∗

ω2 + c2

k

≈ 1 2 D2

∗

ω2 + ˜ b0|k|2 + τ −2 Space-time “red noise”

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• Trop. Rain

Oct 1, 2016

Precipitation stats

PDF 10-3

0.5 1 1.5 2

Mean Precip. [mm/hr] 0.1 0.2 0.3 0.4 Conditional mean precip. CWV PDF

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)

(figures on right from Peters & Neelin 2006) ◮ peaks concentrated near critical point ◮ shows evidence of (self-organized) criticality

• S. Hottovy, USNA
• Trop. Rain

Oct 1, 2016

Cloud Cluster Size Density

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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, USNA
• Trop. Rain

Oct 1, 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, USNA
• Trop. Rain

Oct 1, 2016

Large τ limit

◮ From connections with GFF, interesting limit of τ → ∞. ◮ Hold mean (Fτ) constant.

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10 Cloud Size Distribution by Area Cloud Area [km2] PDF τ=1 τ=10 τ=96 τ=103 τ=104 Best fit=−1.19

• S. Hottovy, USNA
• Trop. Rain

Oct 1, 2016

Summary

◮ Background spectrum is modeled by turbulent

advection-diffusion of CWV.

◮ Water vapor q(x, y, t) ◮ Linear stochastic model – but nonlinear statistics (of

σ(x, y, t))

◮ Leads to analytic statistics or cheap numerical sampling ◮ Behavior similar to self-organized criticality

• S. Hottovy, USNA
• Trop. Rain

Oct 1, 2016

Summary

◮ Background spectrum is modeled by turbulent

advection-diffusion of CWV.

◮ Water vapor q(x, y, t) ◮ Linear stochastic model – but nonlinear statistics (of

σ(x, y, t))

◮ Leads to analytic statistics or cheap numerical sampling ◮ Behavior similar to self-organized criticality

Thank you for your attention!

(This work was supported by ONR DURIP grant N00014-14-1-0251)

• S. Hottovy, USNA
• Trop. Rain

Oct 1, 2016