Despite its successful performance compared to other U.S. NWS/NMC and later NCEP/EMC operational models, the Eta model was essentially frozen in 2002. This decision was made primarily due to the problem with step topography identified by Gallus and Klemp (2000); see the remarks by DiMego cited in Mesinger and Veljovic (2017, hereafter MV17). However, after about three years of EMC efforts dedicated to its planned replacement by NMM/GSI, the results of the EMC four-plus-month “parallel” test in 2006 showed verification scores favoring the Eta system (see MV17, Fig. 4).Even so, the Gallus–Klemp problem needed to be addressed. This was accomplished by changing from step topography to “sloping steps,” now generally referred to as cut-cells (see MV17, Fig. 7). Regardless, the Eta continued to perform well; see the results of Veljovic et al. (2010), Mesinger and Veljovic (2013), and MV17, Fig. 11, the last of which discussed further in Mesinger and Veljovic (2020). In this later experiment, the Eta ensemble members driven by ECMWF (EC) members, although having about the same resolution for the first 10 days, achieved better scores for 250 hPa winds stronger than 45 m s⁻¹ than their EC drivers.A puzzling byproduct of that experiment was that the Eta ensemble, when switched to use sigma, still achieved these 250 hPa wind scores better than their driver members, although to a lesser extent. The reasons for these results are addressed. Unique features of the Eta contributing to this advantage and addressed include its “fairly well-posed” (McDonald 2003) lateral boundary conditions (LBCs), specification of topography without smoothing, and the introduction of finite-volume vertical and slantwise advection.New results presented here for the impact of the Eta LBCs show an advantage of the Eta LBCs over the ubiquitously used relaxation LBCs are giving more than twice as many times more accurate verifications of the upper tropospheric winds stronger than 45 m s⁻¹ than the relaxation LBCs. And a more accurate average equitable threat score adjusted to unit bias, ETSa.The introduction of finite-volume vertical and slantwise advection compared to previous Lorenz-Arakawa centered finite-difference schemes enabled a more accurate placement of the foehn-type warming in the lee of Andes, in a challenging zonda downslope windstorm. In that experiment very steep topography of the Andes was used as generally done in the Eta without smoothing, customary in terrain-following models.

An experiment reported in Mesinger and Veljovic (JMSJ 2020) and at the preceding EGU General Assembly, showed an advantage of the Eta over its driver ECMWF ensemble members in placing 250 hPa jet stream winds east of the Rockies. Verifications subsequent to 2020 confirmed this advantage. A byproduct of that experiment was that of the Eta ensemble switched to use sigma, Eta/sigma, also achieving 250 hPa wind speed scores better than their driver members, although to a lesser extent. It follows that the Eta must include feature or features additional to the eta coordinate responsible for this advantage over the ECMWF.An experiment we have done strongly suggests that the van Leer type finite-volume vertical advection of the Eta, implemented in 2007, may be a significant contributor to this advantage. In that experiment, having replaced a centered finite-difference Lorenz-Arakawa scheme, this finite-volume scheme enabled a successful simulation of an intense downslope windstorm in the lee of the Andes.Another likely and perhaps unique feature of the Eta contributing to that advantage is its sophisticated representation of topography, designed to arrive at the most realistic grid-cell values with no smoothing (Mesinger and Veljovic, MAAP 2017).While apparently a widespread opinion is that it is a disadvantage of terrain intersecting coordinates that “vertical resolution in the boundary layer becomes reduced at mountain tops as model grids are typically vertically stretched at higher altitudes (Thuburn, 10.1007/978-3-642-11640-7 2011),” a comprehensive 2006 NCEP parallel test gave the opposite result. With seemingly equal PBL schemes, the Eta showed a higher surface layer accuracy over high topography than the NMM, using a hybrid terrain-following system (Mesinger, BLM 2023).Hundreds of thousands of the Eta forecasts and experiments performed demonstrate that the relaxation lateral boundary condition, almost universally used in regional climate models (RCMs), in addition to conflicting with the properties of the basic equations used, is unnecessary. Similarly, so-called large scale or spectral nudging, frequently applied in RCMs, based on an ill-founded belief, should only be detrimental if possible numerical issues of the limited area model used are addressed. Note that this is confirmed by the Eta vs ECMWF results we refer to above.Even so, to have large scales of a nested model ensemble members most times more accurate than those of their driver members, surely requires not only the absence of detrimental techniques, but also the use of a lateral boundary condition (LBC) scheme that is not inducing major errors. The scheme of the Eta is at the outflow points of the boundary prescribing one less condition than at the inflow points (e.g., Mesinger and Veljovic, MAAP 2013), and has for that reason been referred to by McDonald (MWR 2003) as one of “fairly well-posed” schemes.

Incentive for writing a limited area weather prediction model stemmed from the author’s several years stay at the University of California in Los Angeles, at the end of the sixties. Exposed to what he refers to as the Akio Arakawa approach, having had an idea for a scheme that was an improvement to what Arakawa was using, and being aware of the importance of topography for the weather of the country he was to continue his career in, led in 1973 to his first limited area 3D code, the forerunner of what was to become the Eta model. Refinements and enhancements introduced by the author in subsequent years and of the collaborator he acquired, Zaviša Janjić, resulted in the code that when installed at the then U.S. National Meteorological Center, attracted attention. Hallmarks of the model were Mesinger’s eta vertical coordinate, and Janjić’s transformation of the Arakawa horizontal advection scheme to the model’s semi-staggered B/E grid. In 1993 the Eta became the primary regional forecasting model of the U.S. Weather Bureau, and in 1998 its precipitation accuracy of 24-48 h forecasts became higher across all intensity thresholds than that of its predecessor, the Nested Grid Model (NGM) for its 00-24 h forecasts. Lately, the Eta is extensively used also as a regional climate model (RCM), mostly over the South American domain, and in near-real time as a tool for the North American Regional Reanalysis (NARR), run by the U.S. National Centers for Environmental Prediction/Climate Prediction Center. Several later unique numerical refinements of the Eta addressing problems noticed are summarized in a “before and after” fashion, and results are mentioned of its ensemble skill compared to that of its highly acclaimed driver European Centre for Medium-Range Weather Forecasts (ECMWF) model.

An experiment reported in Mesinger and Veljovic (JMSJ 2020) showed anadvantage of the Eta over its driver ECMWF ensemble members in placing 250 hPa jetstream winds during a period of an upper tropospheric trough crossing the Rockies. Abyproduct of that experiment was that of the Eta ensemble switched to use sigma,Eta/sigma, also achieving 250 hPa wind speed scores better than their driver members,although to a lesser extent. Nevertheless, it follows that the Eta must include feature orfeatures additional to the eta coordinate responsible for this advantage over theECMWF.An experiment we have done strongly suggests that the van Leer type verticaladvection of the Eta, implemented in 2007, is a significant contributor to this advantage.In this experiment, having replaced a centered finite-difference Lorenz-Arakawa schemethis finite-volume scheme enabled a successful simulation of an intense downslopewindstorm in the lee of the Andes.While apparently a widespread opinion is that it is a disadvantage of terrainintersecting coordinates that “vertical resolution in the boundary layer becomes reducedat mountain tops as model grids are typically vertically stretched at higher altitudes,” avery comprehensive 2006 NCEP parallel test gave just the opposite result. Withseemingly equal ABL schemes, the Eta showed a higher surface layer accuracy overhigh topography than the NMM, using a hybrid terrain-following system (Mesinger, BLM2022).Hundreds of thousands of the Eta forecasts and experiments performeddemonstrate that the relaxation lateral boundary conditions almost universally used inregional climate modeling (RCM)–in addition to conflicting with the properties of thebasic equations used–are unnecessary. Similarly, frequently applied in RCMs so-calledlarge scale or spectral nudging, being based on an ill-founded belief, should only bedetrimental if possible numerical issues of the limited area model used are addressed.Note that this is confirmed by the results we refer to above.

This article describes a teaching strategy that synergizes computing and management, aimed at the running of complex projects in industry and academia, in the areas of civil engineering, physics, geosciences, and a number of other related fields. The course derived from this strategy includes four parts: (a) Computing with a selected set of modern paradigms-the stress is on Control Flow and Data Flow computing paradigms, but paradigms conditionally referred to as Energy Flow and Diffusion Flow are also covered; (b) Project management that is holistic-the stress is on the wide plethora of issues spanning from the preparation of project proposals, all the way to incorporation activities to follow after the completion of a successful project; (c) Examples from past research and development experiences-the stress is on experiences of leading experts from academia and industry; (d) Student projects that stimulate creativity-the stress is on methods that educators could use to induce and accelerate the creativity of students in general. Finally, the article ends with selected pearls of wisdom that could be treated as suggestions for further elaboration.

With probably no exception, in atmospheric numerical models, a high vertical resolution is used close to the surface, with gradually reduced resolution higher up. This seems an obvious choice given the importance and complexity of processes close to the ground, and the cost of using a high near-surface resolution throughout the model atmosphere. But there are disadvantages involved that deserve attention. One is that the performance of numerical schemes is generally better for uniform resolution, in particular when the finite-volume approach is used. Another is that with the usual terrain-following vertical coordinate, horizontal flow across high topography will be subject to severe resolution changes encountering the topography. An unintended experiment of the impact of these disadvantages is a by-product of the so-called "parallel" run of two models at the U.S. National Centers for Environmental Prediction in 2006, when the operational Eta model was compared against its intended replacement, the NMM model. In that four+ month experiment the Eta model more accurately forecast 10-m wind speed and 2-m temperatures over the mostly high topography of the western United States than the NMM, despite its much poorer vertical resolution over that area and not too different physical parametrizations. It is suggested that the severe NMM grid cell resolution change of horizontal flow encountering high topography with terrain-following coordinates is the main cause of this result.

Almost universally, in Regional Climate Modeling (RCM) integrations, Davies’ relaxation lateral boundary conditions are applied. They force variables in a number of rows around the boundary to conform to the driver global model values, completely at the boundary, and less and less toward the inside of the integration domain. Very often, in addition, investigators apply so-called large scale or spectral nudging inside the domain, forcing the integration variables not to depart much from those of the driver model. It is pointed out that there is no scientific basis for these two practices. So why are they used? In particular for the former of these two, it is suggested that reasons must be either a belief that this is a practice RCM should follow, or a technique to address numerical issues of the limited area model used, or a combination of the two. For the latter, a belief only. Examples are shown that, in the absence of these two stratagems, the limited area model can improve on large scales inside its domain. This demonstrates that their use, aimed to force variables inside the domain not to depart much from the driver model data, should be detrimental, if possible numerical issues of the model used were to be remedied.

While the terrain-following (sigma) system of representing topography in atmospheric models has been dominant for about the last 60 years, already half a century ago problems using the system were reported in areas of steep topography. A number of schemes had been proposed to address these problems. However, when topography steepness exceeds a given limit all these schemes except the vertical interpolation of the pressure gradient begin to use model information that for physical reasons they should not use. A radical departure from the system was that of the step-topography eta; but its attractiveness was reduced by the discovery of the corner separation problem. The shaved-cell scheme, nowadays referred to as cut-cell, was free of that problem, and was tested subsequently in idealized as well as real case experiments with encouraging results. The eta discretization has lately been refined to make it also a cut-cell scheme. Another method referred to usually as immersed boundary method enabling treatment of terrain as complex as urban landscape came from computational fluid dynamics. It was made available coupled to the atmospheric Weather Research and Forecasting model. Results of recent experiments of the cut-cell Eta driven by European Centre for Medium-Range Weather Forecasts (ECMWF) ensemble members are analyzed. In these experiments, all cut-cell Eta members achieved better verification scores with respect to 250 hPa wind speed than their ECMWF driver members. This occurred when an upper-tropospheric trough was crossing the Rocky Mountains barrier. These results are considerably less favorable for the Eta when switched to use sigma, i.e., Eta/sigma, pointing to the benefits of using topography intersecting as opposed to terrain-following systems. But even so the Eta/sigma shows an advantage over its driver members, suggesting that its other features deserve attention.

Abstract The astonishing development of computer technology since the mid-20th century has been accompanied by a corresponding proliferation in the numerical methods that have been developed to improve the simulation of atmospheric flows. This article reviews some of the numerical developments concern the ongoing improvements of weather forecasting and climate simulation models. Early computers were single-processor machines with severely limited memory capacity and computational speed, requiring simplified representations of the atmospheric equations and low resolution. As the hardware evolved and memory and speed increased, it became feasible to accommodate more complete representations of the dynamic and physical atmospheric processes. These more faithful representations of the so-called primitive equations included dynamic modes that are not necessarily of meteorological significance, which in turn led to additional computational challenges. Understanding which problems required attention and how they should be addressed was not a straightforward and unique process, and it resulted in the variety of approaches that are summarized in this article. At about the turn of the century, the most dramatic developments in hardware were the inauguration of the era of massively parallel computers, together with the vast increase in the amount of rapidly accessible memory that the new architectures provided. These advances and opportunities have demanded a thorough reassessment of the numerical methods that are most successfully adapted to this new computational environment. This article combines a survey of the important historical landmarks together with a somewhat speculative review of methods that, at the time of writing, seem to hold out the promise of further advancing the art and science of atmospheric numerical modeling.

To determine the effect of switching between the eta and the sigma coordinate in numerical weather prediction involving topography, five sets of tests were performed. The eta version did better in all of them particularly with precipitation scores and more accurate placement of storms. However, a problem of flow separation in the lee of the bell-shaped topography discovered by Gallus and Klemp seemed to many to suggest the eta coordinate to be ill suited for high-resolution models. Flow separation is shown not to occur following a refinement of the eta discretization. Trying to identify a primary cause of the improvement in 250 hPa winds previously demonstrated in Eta ensemble members over their ECMWF driver members, ten of the Eta members were run switched to sigma. At a critical time, the Eta members in eta mode showed a tendency for more accurate tilt of a 250 hPa trough than the members run in sigma mode. The experiment was rerun for a more recent and higher resolution ECMWF ensemble, and for an increased number of members. The advantage of the Eta over ECMWF is seen again, even though this time, the Eta resolution during the first 10 days of the experiment was about the same as that of driver members. Rerunning the Eta ensemble switched to sigma showed an advantage in the Eta/eta 250 hPa wind scores used, again associated with an upper-air trough's movement across the Rockies. Better positioning of lee lows ahead of these troughs using Eta/eta is suggested to be making significant contributions to its better precipitation scores. Implications of experiments done for regional climate modeling are discussed as well.

The design of the Eta model goes back to early 1970s, when its original dynamical core was designed following the philosophy of Akio Arakawa of emulating important properties of the atmospheric governing equations. The core’s later major features were invented and implemented in the mid-1980s. Once a comprehensive physics package was added, the model became operational as a regional NWP model in the United States in 1993. Its use for regional climate projections followed later, for the South American region and then for a regional reanalysis over the North American region. Summary of the model’s dynamical core is given, followed by that of its physics package. Results of experiments revealing the model’s ability to generate added value even at large scales when run as a regional climate model (RCM) are summarized. The Eta model is applied on various climate scales seamlessly, from subseasonal, seasonal to multidecadal, from coarse 40 km up to high 5 km resolution. Examples of applications to various socioe‐ conomic sectors, such as for hydropower management, crop yield forecasts, environ‐ mental and forest conservation, urban areas management, assessment of natural disaster risks, etc., are given. The Eta RCM capability to reproduce extreme climatic values is pointed out.

Over the years as many as five times documented tests were done comparing the Eta model against the same code but switched to use sigma and in all of them the eta version did better. Among these results, better precipitation scores, and more accurate placement of storms, stand out. A possibility that these results came because the Eta precipitation schemes were “tuned” to work best with the eta would seem to have been eliminated by the results of the parallel test comparing the Eta/EDAS system against the NMM-WRF/GSI system during the 5+ months of 2006. In this parallel the operational Eta although “frozen” for considerable time achieved better precipitation scores than the NMM-WRF that used more advanced data assimilation system, the more so the further one moved away from the data assimilation time. A weakness that received extraordinary notoriety of flow separation in the lee of the Witch of Agnesi topography, is shown to have been removed with the latest refinement of the sloping steps eta discretization. Among results presented in Veljovic et al. (Meteor. Z., 2010) were those of an experiment in which 26 Eta ensemble members driven by an ECMWF 32-day ensemble mostly had better scores in placing strong 250 hPa winds than their driver members. Trying to identify the primary cause of this perhaps surprising result 10 of the Eta members were driven by switching the vertical coordinate to sigma. While no obvious impact on 250 hPa wind scores stood out, a tendency was seen for more accurate tilt of the 250 hPa trough of the eta compared to sigma members. To test the sensitivity to resolution and also to check on the robustness of this Eta vs ECMWF result to the choice of the period a 10-member Eta experiment was rerun for a more recent ECMWF ensemble, one initialized 4 October 2012, when its resolution was higher than of that used previously. The advantage of the Eta members more frequently than not is seen again, even though this time the resolution of the Eta during the first 10 days of the experiment was about the same as that of the driver ECMWF members. Rerunning the Eta ensemble with the code switched to sigma this time however an advantage of the Eta/eta over the Eta/sigma is seen, quite considerable during the early 2-6 day period of the experiment when a deep upper-air trough was moving across the Rockies.