THE SIGMA VS
ETA ISSUE
Fedor Mesinger
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.
REFERENCES
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.
Mesinger, F., 1984: A
blocking technique for representation of mountains in atmospheric models. Riv.
Meteor. Aeronautica, 44,
195-202.
Mesinger, F., 1999:
Numerical Methods: The Arakawa approach, horizontal grid, global and
limited-area modeling. General Circulation Model Development: Past,
Present and Future, Proc. Symp. in honor of Professor Akio Arakawa, D.
Randall, Ed., Academic Press, (in press).
Mesinger, F., Janjic, Z. I., Nickovic, S., Gavrilov, D., and
Deaven, D. G., 1988: The step-mountain
coordinate: Model description and performance for cases of Alpine lee
cyclogenesis and for a case of an Appalachian redevelopment. Mon.
Wea. Rev., 116, 1493-1518.
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.
_______________________________
E-mail: fedor.mesinger@noaa.gov