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**Governing Equations IV**

by Nils Wedi (room 007; ext. 2657) Thanks to Anton Beljaars

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Introduction Nonhydrostatic model NH - IFS Physics - Dynamics coupling

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**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: TL319 (~63km) 2000: TL511 (~39km) 2006: TL799 (~25km) 2010: TL1279 (~16km) 2015?: TL2047 (~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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**Ultra-high resolution global IFS simulations**

TL0799 (~ 25km) >> ,490 points per field/level TL1279 (~ 16km) >> 2,140,702 points per field/level TL2047 (~ 10km) >> 5,447,118 points per field/level TL3999 (~ 5km) >> 20,696,844 points per field/level (world record for spectral model ?!)

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Max global altitude = 6503m Orography – T1279 Alps

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Max global altitude = 7185m Orography - T3999 Alps

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**Computational Cost at TL3999 hydrostatic vs. non-hydrostatic IFS**

H TL3999 NH TL3999

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**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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**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

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**Two new prognostic variables in the nonhydrostatic formulation**

pressure departure’ ‘vertical divergence’ Define also: With residual residual Three-dimensional divergence writes

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**NH-IFS prognostic equations**

‘Physics’

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Diagnostic relations With

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**Auxiliary diagnostic relations**

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**Numerical solution (Benard et al 2004,2005)**

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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**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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**Spherical acoustic wave**

analytic explicit NH-IFS horizontal vertical C ~ 340m/s implicit

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**Orographic gravity waves H - IFS**

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**Orographic gravity waves – NH - IFS**

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“Scores” TL1279 L91 ~ 16 km NH H

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**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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**Different scales involved**

NH-effects visible

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**Single timestep in two-time-level-scheme**

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**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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**dynamics-physics coupling**

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**Noise in the operational forecast eliminated through modified coupling**

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Wrong equilibrium ?

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**Compute D+P(T) independant**

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Compute P(D,T)

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**Sequential vs. parallel split of 2 processes vdif + dynamics**

A. Beljaars parallel split sequential split

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**Negative tracer concentration – Vertical diffusion**

Negative tracer concentrations noticed despite a quasi- monotone advection scheme (Anton Beljaars)

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**Physics-Dynamics coupling Vertical diffusion**

(Kalnay and Kanamitsu, 1988) Single-layer problem Greater 0 with initial condition greater 0, dynamics positive monotone, alpha > 0 dynamics positive definite

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**Physics-Dynamics coupling Vertical diffusion**

Two-layer problem Not positive definite depends on !!! dynamics positive definite

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** = 1.5 = 1 (D+P)t (D+P)t+t Dt+t Negative tracer concentration**

with over-implicit formulation = 1 Anton Beljaars

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