The CMC 15km Global Deterministic Prediction System with the Yin-Yang grid.
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1 The CMC 15km Global Deterministic Prediction System with the Yin-Yang grid. Abdessamad Qaddouri and Vivian Lee Environment Canada The 2012 Workshop on the Solution of Partial Differential Equations on the Sphere September 24-28, 2012, Cambridge, UK
2 Outline Introduction and motivation Model problem and method Numerical results : statistical comparison of 5 day forecasts and observations Ensemble-Variational assimilation Global LatLon ( ) ( 15km at 49 o north) Global Yin-Yang 2 ( ) ( 15km at equator)
3 Motivation (Kageyama and Sato, Geochem. Geophys. Geosystems 2004) Composite grid Pole free Each piece is LatLon Numerical schemes on Lat/Lon adapted to Yin-Yang Schwarz method easily implemented ( 2 way-coupling of 2 LAM models)
4 Model problem (Girard et al CMC report) The hydrostatic primitives equations (HPEs) in log-p coordinate d dt [ dv h dt Bs + ln + fk V h + RT ζ Bs + ζ φ = F H, (1) [ ] d d dt ln κ dt (Bs) + ζ = F T, (2) ( 1 + B ζ s Some Operators and Fields d dt = t + 1 cos 2 θ ( ) T T)] ( U + ζ V h + ζ ζ + ζ = 0, (3) T T ( ζ φ /RT ) = 0, (4) (ζ + Bs) λ + V cos θ + ζ θ ζ, ζ V h = 1 ( ) U cos 2 + cos θ V, θ λ θ U = u cos θ, V = v cos θ, a a ζ = a ( 1 cos θ a 2 λ, cos θ ) a 2, θ )
5 Vertical coordinate ζ = ln p Bs ; B = Λ r ; s = ln(p surf /10 5 ) Λ = ( (ζ ζ top )/(ζ surf ζ top ) ) ;0 r = r max (r max r min ) Λ 30 log-hydro-pressure hybrid terrain-following vert. coord. (lid 0.1mb) Below 200 hpa.left : variable r(r max = 100, r min = 2) ; Right : constant r = 4.5
6 Solution with iterative Schwarz Method Domain = 2 overlapping sub-domains (YIN/YANG) Solve HPEs equations iteratively on Sub-domains exchange variables at interfaces : Cubic Lagrange interpolation On each sub-domain : same local solver with the same time step 1. The 2 time level semi-lagrangian method with an implicit time discretization. 2. Finite differences on horizontal Arakawa-C grid and on vertical Charney- Phillips grid
7 Spatial discretization (Girard et al CMC report) Vertical finite differences on Charney-Phillips grid d dt dv h + fk V dt h + RT ζ ζ Bs + ζ φ = 0, (5) ( ) [ d T d dt ln ( ) ζ T κ B s + ζ] = 0, (6) dt [ ( Bs + ln 1 + δζ B ζ s )] + ζ V h + δ ζ ζ + ζ ζ = 0, (7) T T + δ ζ ( ζ φ /RT ) δ ζ (ζ + Bs) = 0, (8)
8 Finite differences on Arakawa C grid du dt fv θ,λ + RT ζ,λ 1 a 2 λ Bs + 1 a 2 λ φ = 0 dv dt + fuλ,θ + RT ζ,θ cos θ a 2 θ Bs + cos θ a 2 θ φ = 0
9 Temporal discretization ( Côté and Staniforth 1988 Mon. Wea. Rev. ; Yeh at al Mon. Wea. Rev.) On each subdomain for each prognostic variable F df dt + G = 0 (9) Time discretization and weighted G terms along trajectory F F + [( 12 ] t + ɛ)g + (12 ɛ)g = 0. (10) Approximate solution for a trajectory calculation dr dt = V d h (r, ζ, t) dζ dt = ζ(r, ζ, t) 2 r dt 2 = rv2 h a 2 d 2 ζ = 0, (11) dt2
10 Boundary conditions vertical : Horizontal : Dirichlet type ζ = 0 at ζ = ζ surf, ζ top. (12)
11 3D Elliptic boundary value problem on Yin-Yang grid (Qaddouri et al Appl. Numer. Math. ; Qaddouri and Lee 2011 QJMRS) Linear set of equations is reduced to EBVP ζ P + γ κτ 2 RT (δ2 ζ + δ ζ ζ )P = AP = R E, (13) where P = φ + RT Bs with the following top and bottom boundary conditions : δ ζ P top = C top, (δ ζ P bot + κp ζ bot ) = C bot. R E is the right-hand side, C top and C bot are constants and a 2 ζ = 1 2 cos 2 θ λ sin θ cos2 θ sin θ. (14) Iterative solution AP (1),k = R (1) E on Ω 1, AP (2),k = R (2) E on Ω 2, B (1) 1 P (1),k = B (1) 1 P (2),k 1, on δω 1, B (2) 1 P (2),k = B (2) 1 P (1),k 1, on δω 2
12 Digital Filter and High-order diffusion Diabatic digital filter 1 with a 6 hour span. Scale selective 2 6 applied to variables(u,v,w, ζ) 1. Fillion, L., H. L. Mitchel, H. R. Ritchie and A. N. Staniforth, 1995 : The impact of a digital filter finalization technique in a global data assimilation system, Tellus, 47A, Qaddouri A.,and V.Lee 2008 : Solution of the implicit formulation of high order diffusion for the Canadian Atmospheric GEM model. Proc Spring Simulation Multiconf., High Performance Computing Symp., J.A. Hamilton, Jr. et al. (eds), Soc. For Modeling and Simulation Internat., Ottawa, Canada, 2008, pp
13 Physical parametrizations in the GDPS Physical Process Description Reference Boundary layer turbulence Orographic blocking Orographic gravity wave drag Radiative transfer Non-orographic wave drag Methane oxidation gravity Grid-scale microphysics Deep convection Turbulent kinetic energy (TKE) 1.5- order closure scheme Low-level drag based on effective height, slope and eccentricity of subgrid-scale mountains Upper-level drag due to breaking of mountain waves, based on the local Froude number Correlated k-distribution for gaseous transmission Doppler-spread scheme with prescribed source spectrum Simple estimate of stratospheric humidification due to methane oxidation Single prognostic variable for cloud water/ice Based on convective inhibition and release of convective available potential energy (CAPE) Belair et al. 1999, Bougeault and Lacarrere 1989 Zadra et al. 2003, Lott and Miller 1997 McFarlane 1987 Li and Barker 2005 Hines 1997a,b Charron et al Sundqvist 1978, Pudykiewicz et al Kain and Fritsch 1990, 1993 Shallow convection Kuo-type closure (Kuo transient) Belair et al Interactions between Surface, Biosphere Land-surface interactions and Atmosphere (ISBA) scheme, multi-layer with 10 prognostic variables Noilhan and Planton 1989, Belair et al. 2003a,b
14 Two-way coupling 2 LAMs Parallel implementation
15 Communication Pattern
16 Geophysical fields are independently calculated on the Yin and Yang grids : ME filtered topography. MG land fraction (land-sea mask). VF vegetation fraction per class (26 classes). VG class number of dominant vegetation type. Z0,ZP (effective) roughness length for momentum. etc...
17 filtered topography on Yin and Yang Subgrids.
18 Numerical results Objective evaluation of 5 days forecasts against observations. Verification is done for a set of 42 winter and 42 summer integrations initialized with analysis of Two configurations with the same model : Yin-Yang : 2 ( L) LatLon : ( L).
19 North America objective evaluation for 44 summer cases (120hr forecasts).
20 North America objective evaluation for 44 winter cases (120hr forecasts).
21 Europe objective evaluation for 44 summer cases (120hr forecasts).
22 Europe objective evaluation for 44 winter cases (120hr forecasts).
23 World objective evaluation for 44 summer cases (120hr forecasts).
24 World objective evaluation for 44 winter cases (120 hr forecasts).
25
26
27
28 Timings(seconds) on IBM-P7 for 505(480+25) time steps (approx 26% of operational run) (10 day forecast = 1945( ) time steps) 1 GEM LATLON GEM Yin-Yang (1728x1335x80L) 2 x (2047x683x80L) pts pts PE topology NpxXNpyXOpenMP (4x44x16)(44 nodes) 2x(20x30x2) (38 nodes) Total time DYNSTEP ADW SOL BAC PHYSTEP HORDIFF (implicit) 27 (explicit) VSPNG 2 38 (implicit) 8 (explicit) OUTDYN OUTPHY 15 5 NESTBCS N/A model timestep = 450sec
29 GEM Yin-Yang Timings(seconds) on IBM-P7 for 505(480+25) time steps Number of Nodes PE topology NpxXNpyXOpenMP 2 x (20x30x2) 2 x (20x30x4) 2 x (38x30x2) TOTAL DYNSTEP ADW SOL 233 (FFT) 199 (FFT) 1018 (Iterative) 1 BAC PHYSTEP HORDIFF VSPNG OUTDYN OUTPHY NESTBCS : FGMRES with Block-Jacobi preconditioner
30 Ensemble-Variational assimilation : En-Var Buehner et al Mon. Wea. Rev. Introduction En-Var approach is currently being tested to replace 4D-Var En-Var is a hybrid approach using variational assimilation with the already available EnKF 4D ensemble covariances by making use of the 4D ensembles, En-Var performs a 4D analysis without the need of the tangent-linear and adjoint of forecast model Consequently, it is more computationally efficient ( 10% 20% of the computational resources) and easier to maintain/adapt than 4D-Var Summary of Preliminary Results : En-Var produces similar quality forecasts as 4D-Var below 20hPa in the extra-tropics and is, significantly improved in the tropics for above 20hPa, scores are similar to 3D-Var but worse than 4D-Var ; we could potentially improve this by raising the EnKF model top from 2hPa to 0.1hPa
31
32 Thank you
The CMC 15km Global Deterministic Prediction System with the Yin-Yang grid.
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