Hydrostatic and non-hydrostatic dynamics

Hydrostatic equilibrium describes the atmospheric state in which the upward directed pressure gradient force (the decrease of pressure with height) is balanced by the downward-directed gravitational pull of the Earth. On average the Earth’s atmosphere is always close to hydrostatic equilibrium. This has been used to approximate the Euler equations underlying weather prediction models and successfully applied in NWP and climate prediction. Non-hydrostatic dynamical effects start to become important below horizontal scales of about 10km.

The ECMWF IFS model uses a hydrostatic dynamical core for all forecasts.

Dynamical core

The dynamical core of IFS is hydrostatic, two-time-level, semi-implicit, semi-Lagrangian and applies spectral transforms between grid-point space (where the physical parametrizations and advection are calculated) and spectral space. In the vertical the model is discretised using a finite-element scheme. A reduced Gaussian grid is used in the horizontal.

The evolution equations of the IFS are a terrain following mass-based vertical coordinate (Simmons and Burridge, 1981), solving the hydrostatic, shallow-atmosphere equations (Ritchie et al., 1995). The solution procedure uses the spectral transform method. The first spectral transform model was introduced into operations at ECMWF in April 1983. Spectral transforms on the sphere involve discrete spherical harmonics transformations between physical (gridpoint) space and spectral (spherical-harmonics) space. This technique has been successfully combined with semi-implicit time stepping (Robert et al., 1972), where the resulting Helmholtz problem is solved in spectral space, and the unconditional stability of semi-Lagrangian (SL) advection (Temperton et al., 2001), where the only limiting factor on the time step is the magnitude of local truncation errors.

The IFS also has extra configurations available for research experiments that are not used operationally. An example is the dry Eulerian dynamics used for low resolution testing.

The horizontal resolution of IFS has approximately doubled every 8 years, with approximately 9km global grid resolution (and an effective resolution of ~36km) in 2016.

Spectral representation

References

The IFS hydrostatic dynamical core is described in more detail in the following references:

• Hortal, M. (2002). The development and testing of a new two-time-level semi-Lagrangian scheme (SETTLS) in the ECMWF forecast model. Q. J. R. Meteorol. Soc, 128, 1671–1687.
• Hortal, M. and Simmons, A. J. (1991). Use of reduced Gaussian grids in spectral models. Mon. Wea. Rev., 119, 1057–1074.
• Ritchie, H., Temperton, C., Simmons, A., Hortal, M., Davies, T., Dent, D. and Hamrud, M. (1995). Implementation of the semi-Lagrangian method in a high-resolution version of the ECMWF forecast model. Mon. Wea. Rev., 123, 489–514.
• Simmons, A. J. and Burridge, D. M. (1981). An energy and angular momentum conserving vertical finite difference scheme and hybrid vertical coordinates. Mon. Wea. Rev., 109, 758–766.
• Simmons, A. J., Burridge, D. M., Jarraud, M., Girard, C. and Wergen, W. (1989). The ECMWF medium-range prediction models: development of the numerical formulations and the impact of increased resolution. Meteorol. Atmos. Phys., 40, 28–60.
• Temperton, C. (1991). On scalar and vector transform methods for global spectral models. Mon. Wea. Rev., 119, 1303–1307.
• Temperton, C., Hortal, M. and Simmons, A. (2001), A two-time-level semi-Lagrangian global spectral model. Q.J.R. Meteorol. Soc., 127: 111–127.
• Untch, A. and Hortal, M. (2004). A finite-element scheme for the vertical discretisation of the semi-Lagrangian version of the ECMWF forecast model. Q. J. R. Meteorol. Soc., 130, 1505–1530.
• Wedi, N.P. and P.K. Smolarkiewicz (2009). A framework for testing global nonhydrostatic models, Q.J.R. Meteorol. Soc. 135, 469-484.