Numerical Design Issues as seen from the Eta Experience: Review and Outlook

Doug Lilly is one of the very few pioneers instrumental in getting the NWP to start moving ahead in the early sixties, and one of the key people in tracing for us the road ahead ever since. Numerical design, storm-scale weather, and some of the events due to topography, are among the many issues he had focused on, endowing us with fundamental knowledge as a result. Principles Doug advanced at the very early times, along with Arakawa, are the cornerstones of the operational Eta model, and its close relative the NMM model. Other Eta numerical features, and in particular the quasi-horizontal vertical coordinate to deal with the problems of topography, were introduced into the Eta predecessor model later, but were advanced following the same general principles: to design schemes which will agree as much as possible with the role the terms addressed have in the atmospheric governing equations. What evidence do we have of the impact on model performance, and can we tell what is a promising way to move ahead?

Choice of the vertical coordinate for NWP and climate models remains an unsettled subject, and is the one I will focus on first. Motivation for the eta coordinate is recalled. Following compelling results favoring the eta with step-topography discretization from its inception in the mid-eighties until the late-nineties, difficulties the eta had with a downslope windstorm became known. This, and Gallus and Klemp 2D experiments and arguments, led to a widespread view that the eta is "ill suited for high resolution prediction models". Accordingly, a number of recent development efforts are using sigma or sigma-like coordinates. But is giving up on the quasi-horizontal coordinates justified? The sigma coordinate problem – frequently misrepresented as that of “two large pressure-gradient force terms ...” – is summarized. It is not a problem that goes away with increased resolution.

Monitoring the performance of the operational Eta as its forecast time kept being extended (to 84 h in April 2000) and resolution increased offers a glimpse at relevant information. Thus: Can one notice that the Eta skill relative to that of its driver model, GFS, at extended forecast times reflects the inflow of the less accurate LBC data of the GFS forecast of 6 h ago? Two attempts to identify this loss of skill were made and no loss was noticed. It is hypothesized that the eta coordinate is a good candidate for a major role in this resilience of the Eta to show the impact of the advection of the less accurate LB data. This is motivated by the skill the large domain 48-km Eta has shown in the mid- to later nineties in comparison to a smaller domain 29-km Eta in spite of its considerable handicap of using 12-h old lateral boundary information compared to the current LB information of the 29-km Eta. Improved largest scales when using a large domain seem the only credible explanation. Eta strength at extended forecast times, relative to that of the GFS, in placing the centers of major storms east of the Rockies is consistent with this hypothesis.

But what about the notorious eta downslope windstorm problem? A simple explanation of the step-topography downslope windstorm problem is offered. Refinement of the eta discretization, based on this explanation, is summarized. It can be viewed as a discretized version of the Adcroft et al. “shaved cells” method. It is shown that the refined, “sloping-steps” eta discretization appears to remove the problem as it is illustrated by the Gallus-Klemp experiment, while preserving the strong eta conservation features, and even improving on a momentum conservation problem of the step-topography. Discretization developed is an add-on to the traditional step-topography Eta code, and preserves the simplicity of the code.

Placement of precipitation over topography is another aspect that has been claimed as being negatively affected by the choice of the eta. Availability of three model QPF scores over the NMM domains offers an opportunity to compare the performance of the Eta against two NCEP sigma system models, in the eastern United States (“East”), with no major topography, against that in the western United States (“West”), where topography is dominant. It is seen that in the East the GFS is doing best, with the Eta and the NMM being somewhat behind and similar; the Eta just a little better. In the West, the Eta is clearly doing best, in particular at the heaviest precipitation categories.

As to another numerical design issue on which the Eta experience offers guidance, the impact of formal accuracy may stand out. In past comprehensive comparisons the Eta had done considerably better than two competing models with higher formal accuracies. There is at least one another published result of no benefit from higher accuracy schemes in a full-physics NWP model. A suggestion is reiterated that the inconsistency in treating dynamics, with grid point values looked at as point-samples of smooth functions, and physical parameterizations, with grid point values considered to represent grid-box averages, is an excellent candidate for the main reason. The conservation approach such as that of the Eta is less affected by this inconsistency. Strictly finite-volume schemes are a way out of the difficulty, and/or moving away from single grid box or single-column parameterizations. The Eta being approximately a finite-volume model seems likely to have contributed significantly to the success it had enjoyed and enjoys.

In conclusion, comments are offered regarding possibilities for a further significant increase in the deterministic synoptic to mesoscale NWP skill several days ahead.