ECMWF tests new numerical scheme for vertical grid

Filip Váňa, Michail Diamantakis


ECMWF has worked with experts in its Member States to test a new numerical scheme for calculations over the vertical grid used in its Integrated Forecasting System (IFS). The results are very encouraging. The IFS employs a spectral method to solve the equations describing atmospheric dynamics in the horizontal and a finite element method to solve them in the vertical. The spherical harmonics of the spectral method cannot easily be used in the vertical due to a lack of periodicity and the use of irregularly spaced levels, varying from tens of metres near the ground to several kilometres near the model top. Instead, the vertical discretisation makes use of piecewise continuous functions as base functions. This method, known as finite element discretisation (hereafter VFE for vertical finite element) was implemented in the IFS in 2002.

Motivation for a new scheme

The IFS currently uses a hydrostatic model, which means that it implicitly assumes the upward-directed pressure gradient force (the decrease of pressure with height) is balanced by the (nearly) downward-directed gravitational pull of the Earth. This works well at the resolutions used operationally in the IFS for both forecasting and data assimilation. If in the future the IFS is to run at higher resolutions, it may become necessary to use the more general nonhydrostatic (NH) dynamics, and this, among other aspects, requires the existing VFE used in the IFS to be modified. One of the requirements for a new vertical scheme is that it is sufficiently similar in the hydrostatic and NH version of the model so that the existing hydrostatic data assimilation system can easily be used to derive initial conditions for the NH model formulation. Other reasons for a new scheme are the need to enhance flexibility in the chosen accuracy in the vertical and robustness with reduced precision.

Forecasts using the new VFE. The charts show maximum 10 m wind speed for a 10-day forecast of Hurricane Dorian, starting from 12 UTC on 31 August 2019, at a grid spacing of 5 km, for every 12 hours starting 12 hours into the forecast. The left-hand panel shows the forecast using the hydrostatic IFS with the new VFE and the right-hand panel the forecast using the NH-IFS with the new VFE. Both forecasts were run in single precision, using the same time step of 240 s and, where applicable, identical settings for model dynamics.

Successful collaboration

A team of scientists in ECMWF’s Member States, led by Jozef Vivoda from the Slovak Hydrometeorological Institute (SHMI) and Petra Smolíková from the Czech Hydrometeorological Institute (CHMI), have developed a new formulation of the VFE scheme which works equally well with hydrostatic and NH dynamics in a stable and robust manner. ECMWF was keen to follow up on this exciting new development as soon as the new code became available from its European partners from the limited-area community and Météo-France. The leading VFE developer, Jozef Vivoda, visited ECMWF for two months earlier this year to adapt the new version of VFE and to ensure that it works successfully in the IFS. During the visit, it was shown that the new scheme offers several benefits for the IFS at the same computational cost. First, it makes it possible to increase the order of the base functions, implying higher accuracy in the vertical discretisation. Second, the new VFE is less sensitive to the model’s vertical resolution and numerical precision. This makes the scheme more robust for the use of single precision calculations in the IFS. A third benefit is that NH dynamics can be modelled using the same vertical discretisation as in the modelling of hydrostatic dynamics. This means that any differences in results from NH simulations can be attributed more directly to the switch from the hydrostatic approximation to the more general NH approach rather than to differences in the numerical discretisation.

We have recently run a 10‑day forecast at a grid spacing of 5 km with the new NH-VFE for Hurricane Dorian and compared it with the equivalent hydrostatic forecast using the new VFE. As the figure shows, the forecasts are remarkably similar. They also agree very well with the observed hurricane track, indeed better than the equivalent HRES forecast.


Based on very encouraging results in first tests, it is anticipated that the new VFE will replace the existing scheme once it has been extended to and carefully tested for all configurations used at ECMWF. Research visits are a great way to continue the successful research cooperation with the (nonhydrostatic) limited-area model community, and we encourage applications as part of ECMWF’s short-term visitor programme.