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 q + q ′ = “resolved” + “sub-grid-scale” Decompose: q = ¯ ∂ ¯ q + ¯ ∂ t = − (¯ u ¯ q ) x − (¯ v ¯ q ) y S − ( u ′ q ′ ) x − ( v ′ q ′ ) y � �� � � �� � τ q + D ˙ − 1 b ∇ 2 q W Eddy diffusion Damping + Forcing S. Hottovy, USNA Trop. Rain Oct 1, 2016

The model Hypothesis ( q is integrated CWV): ∂ q ∂ t + ( uq ) x + ( vq ) y = S q + q ′ = “resolved” + “sub-grid-scale” Decompose: q = ¯ ∂ ¯ q + ¯ ∂ t = − (¯ u ¯ q ) x − (¯ v ¯ q ) y S − ( u ′ q ′ ) x − ( v ′ q ′ ) y � �� � � �� � τ q + D ˙ − 1 b ∇ 2 q W Eddy diffusion Damping + Forcing Linear stochastic model, discretization of SPDE: ∂ q ∂ t = F − 1 τ q + D ˙ W + b ∇ 2 q 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, q i , j ( t ) > q ∗ = 65 mm. ◮ cloud indicator, and precip. rate � � | F | + q i , j ( t ) σ i , j ( t ) = H ( q i , j ( t ) − q ∗ ) , r i , j ( t ) = σ i , j ( t ) . τ ◮ 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 q i , j − → ˆ q k are indep. in k . ◮ Makes for very fast/cheap large simulations (10 6 grid points < 1 s). S. Hottovy, USNA Trop. Rain Oct 1, 2016

Power Spectrum PSD of CWV 0.7 Frequency (cpd) 0.6 0.5 0.4 0.3 0.2 0.1 −10 0 10 Wavenumber k (2 π /40000 km) (from Wheeler & Kiladis 1999) 6 6.5 7 D 2 D 2 = 1 ≈ 1 � q ( k , ω ) | 2 � ∗ ∗ E | ˆ ω 2 + c 2 ω 2 + ˜ b 0 | k | 2 + τ − 2 2 2 k Space-time “red noise” S. Hottovy, USNA Trop. Rain Oct 1, 2016

Precipitation stats 0.4 Mean Precip. [mm/hr] Conditional mean precip. 0.3 CWV PDF 2 PDF 10 -3 1.5 0.2 1 0.1 0.5 0 0 90 10 4 10 3 80 10 2 Variance × L 2 10 1 70 Precipitation variance × L 0.42 10 0 10 –1 60 10 –2 50 10 –3 10 –4 30 40 50 60 70 40 w (mm) 30 L = 2 L = 1 20 L = 0.5 L = 0.25 10 0 50 55 60 65 70 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 (from Wood & Field 2011) Cloud Size Distribution by Area 0 10 − 1 10 − 2 10 PDF − 3 10 − 4 10 − 5 10 Cloud pdf Best fit − 1.528 − 6 10 1 2 3 4 10 10 10 10 Cloud Area [km 2 ] 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 ◮ Edwards-Wilkinson model ◮ Parameters: (1995) (1D Stochastic Heat Equation) D 2 ∗ ↔ k B T ◮ Model for 1D random b ↔ J growth of a surface. F ↔ H = ∂ 2 q ( x , t ) dq ( x , t ) + ˙ W ( x , t ) b : promotes spatial regularity ∂ x 2 dt τ − 1 : promotes temporal regularity (1D, τ → ∞ limit) F : promotes a shift in the spatial average S. Hottovy, USNA Trop. Rain Oct 1, 2016

Large τ limit ◮ From connections with GFF, interesting limit of τ → ∞ . ◮ Hold mean ( F τ ) constant. Cloud Size Distribution by Area 0 10 τ =1 τ =10 −1 10 τ =96 τ =10 3 τ =10 4 −2 10 Best fit=−1.19 PDF −3 10 −4 10 −5 10 −6 10 1 2 3 4 5 6 10 10 10 10 10 10 Cloud Area [km 2 ] 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