THE SIGMA VS ETA ISSUE
NCEP Environmental Modeling Center, Camp Spring, MD 20746, U.S.A.
Ever since the first tests of the eta coordinate vs sigma have been performed in the mid-eighties, results perhaps strikingly favorable to the eta have been obtained. For a review, see Mesinger (1999). Recently, however, concerns have arisen regarding the eta, primarily because of the difficulties the eta has had with a Wasatch Range downslope windstorm (McDonald et al. 1998, Janjic and DiMego, this volume), and eta's apparent disadvantage compared to sigma in simulating the flow over a small-scale bell mountain (Gallus and Klemp 2000).
In this situation, it seems essential to recall some of the successes of the eta, as well as to consider whether the eta should and/or could be refined compared to its current use in the Eta Model while still keeping its presumed basic strength of quasi-horizontal coordinate surfaces. Some points along these lines will be made here.
An early experiment to which apparently a physical explanation can now be given is that of Black and Janjic (1988). In a sample of thirteen forecasts done with the Eta Model in both its eta and its sigma mode, much greater negative geopotential height errors were obtained in sigma, reminiscent of the strikingly ubiquitous cold bias of atmospheric models. Note that recently "aphysical sources of entropy", addressing here for the most part numerical errors, have been strongly suggested to be the cause of this bias (Johnson 1997). This interpretation is supported by a still more recent study, including numerical experiments, of Egger (1999). The noise identified in the very first Eta Model eta/sigma experiments when the code was switched to sigma (Mesinger et al. 1988), and its tentative explanation as coming from sigma-related numerical errors, are consistent with these suggestions and results.
One should also be reminded that a number of specific mountain-related or topographically forced events that have been noted to be a problem for NCEP sigma system models have been found not to be a problem, or not to that extent to be a problem for the Eta; and that these typical sigma model errors have been reproduced -- or reproduced to a considerable extent -- by the Eta when it was switched to sigma. These events are cold surges in the lee of the Rockies -- with the sigma system models being too slow in propagating the cold air southward; placing of the lows as they form in the lee of the Rockies, very pervasively too far north in sigma; and the placement of a mid-tropospheric cutoff above the U.S. southern Rockies. For some of these results, and references to others, see Mesinger et al. (1977), and Mesinger (1999).
Regarding Wasatch windstorms, a very positive assessment of the Eta performance for the more recent April 1999 case was made by Dunn (http://nimbo.wrh.noaa.gov/Saltlake/projects/990423/case.html). But even though with various elements of the storm, in the words of Dunn "the Eta was superb", its surface winds were "still weaker than the MM5 run at the U. of Utah". Indications are thus compelling that a real disadvantage of the eta is at hand. Reexamination of the eta formulation is thus warranted.
In Mesinger (1984) discretization was an integral part of the definition of the coordinate. This differed from the usual practice of defining the coordinate independently of and prior to discretization. Notice, though, that, using the same definition, the eta coordinate can also be defined prior to discretization. At that point, shape of the topography is arbitrary and at the ground surface air flow through eta coordinate surfaces is permitted. With this more general approach one obtains a system of equation same as in Mesinger (1984) except for an additional term in the pressure tendency equation:
Discretization possibilities now exist also other than that used in the Eta. One of them are the "partial steps" which in Adcroft et al. (1997) ocean experiments performed better than the "full steps". Tripoli and Hoggatt (2000, in preparation), using partial steps -- in their terminology "variably stepped topography" -- present results of an experiment on flow over a mountain in which the simulated flow is extremely similar to the well-known Long's solution for the same problem, largely reminiscent of a downslope windstorm event.
Work on some of these possibilities, and specifically the "partial steps", is in progress at EMC using the current Eta code, with participation also of Dusan Jovic, of the Physics of Weather and Climate Group of the Abdus Salam International Centre for Theoretical Physics, Trieste, Italy.
Adcroft, A., C. Hill, and J. Marshall, 1997: Representation of topography by shaved cells in a height coordinate ocean model. Mon. Wea. Rev., 125, 2293-2315.
Black, T. L., and Janjic, Z. I., 1988: Preliminary forecast results from a step-mountain eta coordinate regional model. 8th Conf. on Numerical Weather Prediction, Baltimore, MD, Amer. Meteor. Soc., 442-447.
Egger, J., 1999: Numerical generation of entropies. Mon. Wea. Rev., 127, 2211-2216.
Gallus, W. A., Jr., and J. B. Klemp, 2000: On the behavior of flow over step orography. Mon. Wea. Rev., 128, (in press).
Johnson, D. R., 1997: "General coldness of climate models" and the Second Law: Implications for modeling the Earth system. J. Climate, 10, 2826-2846.
McDonald, B. E., J. D. Horel, C. J. Stiff, and W. J. Steenburgh, 1998: Observations and simulations of three downslope wind events over the northern Wasatch Mountains. 16th Conf. on Weather Analysis and Forecasting, Phoenix, AZ, Amer. Meteor. Soc., 62-64.
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Mesinger, F., Black, T. L., and Baldwin, M. E., 1997: Impact of resolution and of the eta coordinate on skill of the Eta Model precipitation forecasts. Numerical Methods in Atmospheric and Oceanic Modelling, The André J. Robert Memorial Volume, C. Lin, R. Laprise, and H. Ritchie, Eds., pp. 399-423. Canadian Meteorological and Oceanographic Society/NRC Research Press, Ottawa.