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ECMWF Governing Equations 4 Slide 1 Governing Equations IV by Nils Wedi (room 007; ext. 2657) Thanks to Anton Beljaars

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ECMWF Governing Equations 4 Slide 2 Introduction Nonhydrostatic model NH - IFS Physics - Dynamics coupling

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ECMWF Governing Equations 4 Slide 3 Introduction – A history Resolution increases of the deterministic 10-day medium-range Integrated Forecast System (IFS) over ~25 years at ECMWF: 1987: T 106 (~125km) 1991: T 213 (~63km) 1998: T L 319 (~63km) 2000: T L 511 (~39km) 2006: T L 799 (~25km) 2010: T L 1279 (~16km) 2015?: T L 2047 (~10km) 2020-???: (~1-10km) Non-hydrostatic, cloud-permitting, substan- tially different cloud-microphysics and turbulence parametrization, substantially different dynamics-physics interaction ?

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ECMWF Governing Equations 4 Slide 4 Ultra-high resolution global IFS simulations T L 0799 (~ 25km) >> 843,490 points per field/level T L 1279 (~ 16km) >> 2,140,702 points per field/level T L 2047 (~ 10km) >> 5,447,118 points per field/level T L 3999 (~ 5km) >> 20,696,844 points per field/level (world record for spectral model ?!)

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ECMWF Governing Equations 4 Slide 5 Orography – T1279 Max global altitude = 6503m Alps

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ECMWF Governing Equations 4 Slide 6 Orography - T3999 Alps Max global altitude = 7185m

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ECMWF Governing Equations 4 Slide 7 H T L 3999 NH T L 3999 Computational Cost at T L 3999 hydrostatic vs. non-hydrostatic IFS

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ECMWF Governing Equations 4 Slide 8 Nonhydrostatic IFS (NH-IFS) Bubnova et al. (1995); Benard et al. (2004), Benard et al. (2005), Benard et al. (2009), Wedi and Smolarkiewicz (2009), Wedi et al. (2009) Arpégé/ALADIN/Arome/HIRLAM/ECMWF nonhydrostatic dynamical core, which was developed by Météo-France and their ALADIN partners and later incorporated into the ECMWF model and adopted by HIRLAM.

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ECMWF Governing Equations 4 Slide 9 Vertical coordinate with coordinate transformation coefficient hybrid vertical coordinate Simmons and Burridge (1981) Prognostic surface pressure tendency: Denotes hydrostatic pressure in the context of a shallow, vertically unbounded planetary atmosphere.

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ECMWF Governing Equations 4 Slide 10 Two new prognostic variables in the nonhydrostatic formulation Nonhydrostatic pressure departure vertical divergence Three-dimensional divergence writes With residual residual Define also:

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ECMWF Governing Equations 4 Slide 11 NH-IFS prognostic equations Physics

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ECMWF Governing Equations 4 Slide 12 Diagnostic relations With

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ECMWF Governing Equations 4 Slide 13 Auxiliary diagnostic relations

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ECMWF Governing Equations 4 Slide 14 Numerical solution Advection via a two-time-level semi-Lagrangian numerical technique as before. Semi-implicit procedure with two reference states with respect to gravity and acoustic waves, respectively. The resulting Helmholtz equation is more complicated but can still be solved (subject to some constraints on the vertical discretization) with a direct spectral method as before. (Benard et al 2004,2005)

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ECMWF Governing Equations 4 Slide 15 Hierarchy of test cases Acoustic waves Gravity waves Planetary waves Convective motion Idealized dry atmospheric variability and mean states Idealized moist atmospheric variability and mean states Seasonal climate, intraseasonal variability Medium-range forecast performance at hydrostatic scales High-resolution forecasts at nonhydrostatic scales

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ECMWF Governing Equations 4 Slide 16 Spherical acoustic wave explicit implicit analytic NH-IFS horizontal vertical C ~ 340m/s

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ECMWF Governing Equations 4 Slide 17 Orographic gravity waves H - IFS

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ECMWF Governing Equations 4 Slide 18 Orographic gravity waves – NH - IFS

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ECMWF Governing Equations 4 Slide 19 Scores T L 1279 L91 ~ 16 km NH H

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ECMWF Governing Equations 4 Slide 20 Physics – Dynamics coupling Physics, parametrization: the mathematical procedure describing the statistical effect of subgrid-scale processes on the mean flow expressed in terms of large scale parameters, processes are typically: vertical diffusion, orography, cloud processes, convection, radiation Dynamics: computation of all the other terms of the Navier- Stokes equations (eg. in IFS: semi-Lagrangian advection) The Physics in IFS is currently formulated inherently hydrostatic, because the parametrizations are formulated as independent vertical columns on given pressure levels and pressure is NOT changed directly as a result of sub-gridscale interactions ! The boundaries between Physics and Dynamics are a moving target …

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ECMWF Governing Equations 4 Slide 21 Different scales involved NH-effects visible

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ECMWF Governing Equations 4 Slide 22 Single timestep in two-time-level-scheme

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ECMWF Governing Equations 4 Slide 23 Cost partition of a single time-step Note: Increase in CPU time substantial if the time step is reduced for the physics only.

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ECMWF Governing Equations 4 Slide 24 dynamics-physics coupling

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ECMWF Governing Equations 4 Slide 25 Noise in the operational forecast eliminated through modified coupling

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ECMWF Governing Equations 4 Slide 26 Wrong equilibrium ?

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ECMWF Governing Equations 4 Slide 27 Compute D+P(T) independant

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ECMWF Governing Equations 4 Slide 28 Compute P(D,T)

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ECMWF Governing Equations 4 Slide 29 Sequential vs. parallel split of 2 processes vdif + dynamics parallel split sequential split A. Beljaars

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ECMWF Governing Equations 4 Slide 30 Negative tracer concentration – Vertical diffusion Negative tracer concentrations noticed despite a quasi- monotone advection scheme (Anton Beljaars)

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ECMWF Governing Equations 4 Slide 31 Physics-Dynamics coupling Vertical diffusion Single-layer problem (Kalnay and Kanamitsu, 1988) dynamics positive definite

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ECMWF Governing Equations 4 Slide 32 Physics-Dynamics coupling Vertical diffusion Two-layer problem Not positive definite depends on !!! dynamics positive definite

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(D+P) t+ t D t+ t (D+P) t = 1.5 = 1 Anton Beljaars Negative tracer concentration with over-implicit formulation

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