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: dq i , j = F − 1 � τ q i , j + D ∗ ˙ W i , j − b ( q ( i , j ) − q ( i ′ , j ′ ) ) dt � ( i , j ) , ( i ′ , j ′ ) � Similar to Stochastic PDE: ∂ q ∂ t = F − 1 τ q + D ∗ ˙ W + b 0 ∇ 2 q S. Hottovy, UW Spatiotemporal stochastic model for precip. Jan 7, 2016

Model with Spatial Variability Linear stochastic model: dq i , j = F − 1 � τ q i , j + D ∗ ˙ W i , j − b ( q ( i , j ) − q ( i ′ , j ′ ) ) dt � ( i , j ) , ( i ′ , j ′ ) � Similar to Stochastic PDE: ∂ q ∂ t = F − 1 τ q + D ∗ ˙ W + b 0 ∇ 2 q Relation to atmospheric dynamics: ∂ q ∂ t + ( uq ) x + ( vq ) y = S q + q ′ = “resolved” + “sub-grid-scale” Decompose: q = ¯ ∂ ¯ q + ¯ S − ( u ′ q ′ ) x − ( v ′ q ′ ) y ∂ t = − (¯ u ¯ q ) x − (¯ v ¯ q ) y � �� � � �� � τ q + D ∗ ˙ b 0 ∇ 2 q − 1 W Eddy diffusion Damping + Forcing S. Hottovy, UW Spatiotemporal stochastic model for precip. Jan 7, 2016

Stationary PDF in Fourier Space � −| ˆ � q k | 2 q k ) = 1 Gaussian: ρ (ˆ exp D 2 Z k ∗ / c k Independent ˆ q k , ˆ q k ′ ⇒ 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 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) 6 6.5 7 (from Wheeler & Kiladis 1999) D 2 D 2 q ( k , ω ) | 2 = 1 ≈ 1 ∗ ∗ | ˆ ω 2 + c 2 ω 2 + ˜ b 0 | k | 2 + τ − 2 2 2 k S. Hottovy, UW Spatiotemporal stochastic model for precip. Jan 7, 2016

Mean Precipitation Conditional Mean Precip. [mm/hr] 0.25 |F| σ 0.2 (q−q * ) σ /tau 0.15 [|F|+(q−q * )/tau] σ β =0.62 0.1 β =0.23 0.05 0 55 60 65 70 75 q M [mm] i,j Power law fits near a critical point. S. Hottovy, UW Spatiotemporal stochastic model for precip. Jan 7, 2016

Variance 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) Peak near critical value. S. Hottovy, UW Spatiotemporal stochastic model for precip. Jan 7, 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 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 ◮ 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, 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