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

2. Introduction
Nonhydrostatic model NH - IFS
Physics - Dynamics coupling

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: 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 ?

4. 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 ?!)

5. Max global altitude = 6503m
Orography – T1279
Alps

6. Max global altitude = 7185m
Orography - T3999
Alps

7. Computational Cost at TL3999 hydrostatic vs. non-hydrostatic IFS
H TL3999
NH TL3999

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.

9. 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

10. Two new prognostic variables in the nonhydrostatic formulation
pressure departure’
‘vertical divergence’
Define also:
With residual residual
Three-dimensional divergence writes

11. NH-IFS prognostic equations
‘Physics’

12. Diagnostic relations
With

13. Auxiliary diagnostic relations

14. 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)

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

16. Spherical acoustic wave
analytic
explicit
NH-IFS
horizontal
vertical
C ~ 340m/s
implicit

17. Orographic gravity waves H - IFS

18. Orographic gravity waves – NH - IFS

19. “Scores”
TL1279 L91 ~ 16 km NH H

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” …

21. Different scales involved
NH-effects visible

22. Single timestep in two-time-level-scheme

23. Cost partition of a single time-step
Note: Increase in CPU time substantial if the time step is reduced for the ‘physics’ only.

24. dynamics-physics coupling

25. Noise in the operational forecast eliminated through modified coupling

26. Wrong equilibrium ?

27. Compute D+P(T) independant

28. Compute P(D,T)

29. Sequential vs. parallel split of 2 processes vdif + dynamics
A. Beljaars
parallel split
sequential split

30. Negative tracer concentration – Vertical diffusion
Negative tracer concentrations noticed despite a quasi- monotone advection scheme
(Anton Beljaars)

31. 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

32. Physics-Dynamics coupling Vertical diffusion
Two-layer
problem
Not positive
definite
depends on  !!!
dynamics positive
definite

33.  = 1.5  = 1 (D+P)t (D+P)t+t Dt+t Negative tracer concentration
with over-implicit formulation
 = 1
Anton Beljaars