A Stochastic Model for the Background Spectrum of Tropical - PowerPoint PPT Presentation

A Stochastic Model for the Background Spectrum of Tropical Convection Scott Hottovy and Samuel N. Stechmann shottovy@math.wisc.edu University of Wisconsin-Madison AMS 32nd Conference on Hurricanes and Tropical Meteorology, April 18, 2016 S. Hottovy, UW Stochastic model for background spectrum April 18, 2016

Motivation Remove general characteristics of the Power Spectrum? (from Zhang 2005) (from Wheeler & Kiladis 1999) S. Hottovy, UW Stochastic model for background spectrum April 18, 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, UW Stochastic model for background spectrum April 18, 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, UW Stochastic model for background spectrum April 18, 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, UW Stochastic model for background spectrum April 18, 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, UW Stochastic model for background spectrum April 18, 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, UW Stochastic model for background spectrum April 18, 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, UW Stochastic model for background spectrum April 18, 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, UW Stochastic model for background spectrum April 18, 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, UW Stochastic model for background spectrum April 18, 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 q ( k , ω ) | 2 ] = 1 ≈ 1 ∗ ∗ E [ | ˆ ω 2 + c 2 ω 2 + ˜ b 0 | k | 2 + τ − 2 2 2 k Space-time “red noise” S. Hottovy, UW Stochastic model for background spectrum April 18, 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, UW Stochastic model for background spectrum April 18, 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, UW Stochastic model for background spectrum April 18, 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, UW Stochastic model for background spectrum April 18, 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, UW Stochastic model for background spectrum April 18, 2016