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Governing Equations III. by Nils Wedi (room 007; ext. 2657). Thanks to Anton Beljaars. Introduction. Mass-based hydrostatic and nonhydrostatic models The IFS equations Physics - Dynamics coupling. Introduction.

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## Governing Equations III

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**Governing Equations III**by Nils Wedi (room 007; ext. 2657) Thanks to Anton Beljaars**Introduction**• Mass-based hydrostatic and nonhydrostatic models • The IFS equations • Physics - Dynamics coupling**Introduction**• Resolution increases of the deterministic 10-day medium-range Integrated Forecast System (IFS) over ~28 years at ECMWF and possible future projections**NH**H**Max global altitude = 6503m**Orography – T1279 (16km) Alps**Max global altitude = 7185m**Orography - T3999 (5km) Alps**Max global altitude = 8060m**Orography - T7999 (2.5km) Alps**Pressure based formulationsHydrostatic**Hydrostatic equations in pressure coordinates**Pressure based formulationsHistorical NH**(Miller (1974); Miller and White (1984))**Pressure based formulations**(Rõõm et. al (2001), and references therein) developed within the HIRLAM group**Pressure based formulationsMass-coordinate**Define ‘mass-based coordinate’ coordinate: Laprise (1992) ‘hydrostatic pressure’ in a vertically unbounded shallow atmosphere By definition monotonic with respect to geometrical height relates toRõõm et. al (2001):**Pressure based formulations**Laprise (1992) Momentum equation Thermodynamic equation Continuity equation with**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); Yessad and Wedi (2011) • 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 HARMONIE.**Vertical coordinate**hybrid vertical coordinate Simmons and Burridge (1981) Denotes hydrostatic pressure in the context of a shallow, vertically unbounded planetary atmosphere. Prognostic surface pressure tendency: with coordinate transformation coefficient**Non-hydrostatic shallow atmosphere (NHS)**Distinction between hydrostatic pressure and total pressure ‘Nonhydrostatic pressure departure’ Introduce: Here hydrostatic pressure follows from the prognostic surface pressure equation as before ! Note that the geopotential is derived from**NHS – continued …**‘Physics’ Projecting on temperature and horizontal velocities only, quasi-anelastic coupling…**NHS – For the solution we need the three-dimensional**divergence horizontal divergence ‘vertical divergence’ ‘X-term residual’ Arises due to the formulation of divergence in time-dependent curvilinear coordinates !**NHS – continued …**This requires additional boundary conditions for w: The associated linear system, used in the semi-implicit solution procedure, is formulated in d4 to ensure stability: Hence explicit conversions between w and d4 are required. An alternative formulation with a prognostic equation for d4 exists and is used in the Météo-France AROME model.**Deep atmosphere formulations**• Quasi-hydrostatic system (QHE) following White and Bromley (1995). • Nonhydrostatic deep atmosphere formulation (NHD) following Wood and Staniforth (2003). See Yessad and Wedi (2011) for more details.**Numerical solution**• Two-time-level, semi-implicit, semi-Lagrangian. • Semi-implicit procedure with two reference states, with respect to gravity and acoustic waves, respectively. • The resulting Helmholtz equation can be solved (subject to some constraints on the vertical discretization) with a directspectral method, that is, a mathematical separation of the horizontal and vertical part of the linear problem in spectral space, with the remainder representing at most a pentadiagonal problem of dimension NLEV2. Non-linear residuals are treated explicitly (or iteratively implicitly)! (Robert, 1972; Bénardet al 2004,2005,2010)**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**Spherical acoustic wave**analytic vertical horizontal NH-IFS explicit implicit**“Scores”**TL1279 L91 ~ 16 km NH H**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” …**Different scales involved**NH-effects visible**Computational Cost: 10 day forecast of thenonhydrostatic IFS**NH IFS TL3999 L91 (5 km) on IBM Power7 TSTEP=180s, 3.1s/iteration Using 1024 tasks x16 OpenMP threads 10 day forecast ~ 4 hours for this configuration**Noise in the operational forecasteliminated through modified**coupling**A. Beljaars**Sequential vs. parallel split of 2 processesvdif + dynamics parallel split sequential split

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