Search

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.

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.

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.

With very few exceptions, just about all limited area models (LAMs) used in operational NWP and regional climate modeling use the Davies (Q J R Meteorol Soc 102:405–418, 1976) relaxation lateral boundary conditions (LBCs), even though they make no effort to respect the basic mathematics of the problem. While in the early stages of the primitive equation LAM development in the seventies numerous schemes have been proposed and tested, LAM communities have eventually for the most part settled on the relaxation LBCs with few questions asked. An exception is the Eta model used extensively at NCEP and several other centers, in which the Mesinger (Contrib Atmos Phys 50:200–210, 1977) LBCs are used, designed and based on knowledge available before the introduction of the relaxation scheme. They prescribe variables along the outermost row of grid points only; all of them at the inflow points and one less at the outflow points where the tangential velocity components are extrapolated from inside of the model domain. Additional schemes are in place to suppress separation of gravity-wave solutions on C-subgrids of the model’s E-grid. A recent paper of Veljovic et al. (Meteor Zeitschrift 19:237–246, 2010) included three 32-day forecasts done with both the Eta and the relaxation LBCs and the comparison of some of their verification results. Here we extend this experiment by three additional forecasts to arrive at an ensemble of six members run with both schemes, and present a more complete discussion of results. We in addition show results of one of these forecasts in which the linear change of relaxation coefficients was replaced by the change following the recommendation of Lehmann (Meteorol Atmos Phys 52:1–14, 1993). We feel that the results of our two verification schemes strongly suggest the advantage of the Eta over the conventional relaxation scheme, thereby raising doubts as to the justification for its use.

Upgrades implemented over a number of years in an open source version of the Eta model, posted at the CPTEC web site http://etamodel.cptec.inpe.br/ , are summarized and examples of benefits are shown. The version originates from the NCEP’s Workstation Eta code posted on the NCEP web site http://www.emc.ncep.noaa.gov/mmb/wrkstn_eta , which differs from the NCEP’s latest operational Eta by having the WRF-NMM nonhydrostatic option included. Most of the upgrades made resulted from attention paid to less than satisfactory performance noted in several Eta results, and identification of the reasons for the problem. Others came from simple expectation that including a feature that is physically justified but is missing in the code should help. The most notable of the upgrades are the introduction of the so-called sloping steps, or discretized shaved cells topography; piecewise-linear finite-volume vertical advection of dynamic variables; vapor and hydrometeor loading in the hydrostatic equation, and changes aimed at refining the convection schemes available in the Eta. Several other modifications have to do with the calculation of exchange coefficients, conservation in the vertical diffusion, and diagnostic calculation of 10-m winds. Several examples showing improved performance resulting from the dynamics changes are given. One includes a case of unrealistically low temperatures in several mountain basins generated by a centered vertical advection difference scheme’s unphysical advection from below ground, removed by its replacement with a finite-volume scheme. Another is that of increased katabatic winds in the Terra Nova Bay Antarctica region. Successful forecast of the severe downslope zonda wind case in the lee of the highest peaks of the Andes is also shown, and some of the recent successful verification results of the use of the upgraded model are pointed out. The code is used at numerous places, and along with setup information it is available for outside users at the CPTEC Eta web site given above.

Ever since its introduction to meteorology in the early 1970s, the forward–backward scheme has proven to be a very efficient method of treating gravity waves, with an added bonus of avoiding the time computational mode of the leapfrog scheme. It has been and it is used today in a number of models. When used on a square grid other than the Arakawa C grid, modification is or modifications are available to suppress the noise-generating separation of solutions on elementary C grids. Yet, in spite of a number of papers addressing the scheme and its modification, or modifications, issues remain that have either not been addressed or have been commented upon in a misleading or even in an incorrect way. Specifically, restricting ourselves to the B/E grid does it matter and if so how which of the two equations, momentum and the continuity equation, is integrated forward? Is there just one modification suppressing the separation of solutions, or have there been proposed two modification schemes? Questions made are addressed and a number of misleading statements made are recalled and commented upon. In particular, it is demonstrated that there is no added computational cost in integrating the momentum equation forward, and it is pointed out that this would seem advantageous given the height perturbations excited in the first step following a perturbation at a single height point. Yet, 48-h numerical experiments with a full-physics model show only a barely visible difference between the forecasts done using one and the other equation forward.

A considerable number of authors presented experiments in which degradation of large scale circulation occurred in regional climate integrations when large-scale nudging was not used (e.g., VON STORCH et al., 2000; BINER et al., 2000; ROCKEL et al., 2008; SANCHEZ-GOMEZ et al., 2008; ALEXANDRU et al., 2009; among others). We here show an earlier 9-member ensemble result of the June-August precipitation difference over the contiguous United States between the "flood year" of 1993 and the "drought year" of 1988, in which the Eta model nested in the COLA AGCM gave a rather accurate depiction of the analyzed difference, even though the driver AGCM failed in doing so to the extent of having a minimum in the area where the maximum ought to be. It is suggested that this could hardly have been possible without an RCM's improvement in the large scales of the driver AGCM. We further revisit the issue by comparing the large scale skill of the Eta RCM against that of a global ECMWF 32-day ensemble forecast used as its driver. Another issue we are looking into is that of the lateral boundary condition (LBC) scheme. The question we ask is whether the almost universally used but somewhat costly relaxation scheme is necessary for a desirable RCM performance? We address this by running the Eta in two versions differing in the lateral boundary scheme used. One of these is the traditional relaxation scheme and the other is the Eta model scheme in which information is used at the outermost boundary only and not all variables are prescribed at the outflow boundary. The skills of these two sets of RCM forecasts are compared against each other and also against that of their driver. A novelty in our experiments is the verification used. In order to test the large scale skill we are looking at the forecast position accuracy of the strongest winds at the jet stream level, which we have taken as 250 hPa. We do this by calculating bias adjusted equitable threat scores (Mesinger 2008) and frequency bias scores for wind speeds greater than a chosen wind speed threshold, with the ECMWF analyses used as truth. We also calculate a traditional RMS difference between the forecast and analyzed winds at this same level. Our results show the Eta RCM skill in forecasting large scales with no interior nudging to be just about the same as and usually even slightly higher than that of the driver model. As to the LBC impact, no disadvantage compared to relaxation was seen from using the Eta scheme, in spite of its requiring information from the outermost RCM boundary only.

Bias-adjusted threat and equitable threat scores were designed to account for the effects of placement errors in assessing the performance of under- or overbiased forecasts. These bias-adjusted performance measures exhibit bias sensitivity. The critical performance ratio (CPR) is the minimum fraction of added forecasts that are correct for a performance measure to indicate improvement if bias is increased. In the opposite case, the CPR is the maximum fraction of removed forecasts that are correct for a performance measure to indicate improvement if bias is decreased. The CPR is derived here for the bias-adjusted threat and equitable threat scores to quantify bias sensitivity relative to several other measures of performance including conventional threat and equitable threat scores. The CPR for a bias-adjusted equitable threat score may indicate the likelihood of preserving or increasing the conventional equitable threat score if forecasts are bias corrected based on past performance.

This study investigates the capability of the dynamic downscaling method (DDM) in a North American regional climate study using the Eta/Simplified Simple Biosphere (SSiB) Regional Climate Model (RCM). The main objective is to understand whether the Eta/SSiB RCM is capable of simulating North American regional climate features, mainly precipitation, at different scales under imposed boundary conditions. The summer of 1998 was selected for this study and the summers of 1993 and 1995 were used to confirm the 1998 results. The observed precipitation, NCEP–NCAR Global Reanalysis (NNGR), and North American Regional Reanalysis (NARR) were used for evaluation of the model’s simulations and/or as lateral boundary conditions (LBCs). A spectral analysis was applied to quantitatively examine the RCM’s downscaling ability at different scales. The simulations indicated that choice of domain size, LBCs, and grid spacing were crucial for the DDM. Several tests with different domain sizes indicated that the model in the North American climate simulation was particularly sensitive to its southern boundary position because of the importance of moisture transport by the southerly low-level jet (LLJ) in summer precipitation. Among these tests, only the RCM with 32-km resolution and NNGR LBC or with 80-km resolution and NARR LBC, in conjunction with appropriate domain sizes, was able to properly simulate precipitation and other atmospheric variables—especially humidity over the southeastern United States—during all three summer months—and produce a better spectral power distribution than that associated with the imposed LBC (for the 32-km case) and retain spectral power for large wavelengths (for the 80-km case). The analysis suggests that there might be strong atmospheric components of high-frequency variability over the Gulf of Mexico and the southeastern United States.

During the last somewhat more than a year of the operational running of the Eta Model at the U.S. National Meteorological Center (NMC) a considerable body of statistics has been accumulated demonstrating the model's significantly increased accuracy in forecasting precipitation compared to that of NMC's Nested Grid Model (NGM). The model has shown a smaller but just as consistent advantage in skill against that of NMC's global spectral model. An obvious question is whether there are design features of the Eta Model which could be identified as responsible for this generally improved performance. One feature on which we have data is spatial resolution. In an experiment we performed, a sample of 148 forecasts was formed from each of four models: three versions of the Eta Model differing in resolution only, and the NGM. The Eta Model of that time had achieved a substantial advantage over the NGM with no increase in resolution at all of our eight precipitation categories except at the lightest “rain/n...

Following a few historical remarks, approximations used in formulating the “dynamics” of limited-area and variable resolution atmospheric forecasting models are reviewed. Particular attention is given to current efforts to relax or remove the hydrostatic approximation. Turning to numerical methods used in discretizing the equations, an attempt is made to record recent work and to clarify the motivation for the various approaches being followed by different modeling centers. Topics commented upon include: semi-Lagrangian methods, numerical formulation of nonhydrostatic models, resolution, the eta (step-mountain) vs sigma or isentropic/sigma vertical coordinate, choice of the vertical grid, numerics of the propagation of gravity waves, and the box-average vs pointsample treatment of predicted variables. It is finally pointed out that the extraordinary diversity of roads being taken shows that a lot remains to be discovered as to what possible rewards may be found in exploring one or the other of the principles underlying the methods being developed.

Since 9 June 1993, the eta coordinate regional model has been run twice daily at the National Centers for Environmental Prediction (NCEP, previously the National Meteorological Center) as the NCEP's “early” operational model. Its performance is regularly monitored in a variety of ways, with particular attention given to precipitation forecasts. Throughout this period, the eta model has demonstrated significantly increased accuracy in forecasting daily precipitation amounts compared to NCEP's Nested Grid Model (NGM). The model has shown a smaller but equally consistent advantage in skill against that of NCEP's global spectral model. Precipitation scores of these three operational models for the 6-month period March–August 1995 are presented. This interval is chosen because the 6-month-long periods September–February and March–August have been used in previous model comparisons and because an upgraded version of the eta model, run at 48-km resolution, was also regularly executed twice daily during the March...

A model using shallow-water equations with an Arakawa-type scheme for momentum terms is tested on a quasi-uniform geometry on the sphere, derived by a spherical expansion of the inscribed cube based on the gnomonic projection. Thereby, a quasi-homogeneous distribution of grid points is achieved, and a global finite-difference model is designed which does not require Fourier filtering or suffer from the burden of redundant computational points at high latitudes. Difficulties resulting from the directional discontinuity of the coordinate lines crossing the edges of the expanded cube are almost completely eliminated by using the Arakawa B-grid, so that only scalar points are placed along the edges. An alternative approach is developed based on numerical orthogonalization of the grid whereby, inter alia, the directional discontinuity at the edges is avoided at the cost of some accumulation of points in the vicinity of the vertices of the cube. In the customary Rossby-Haurwitz wave-4 tests, both approaches are shown to converge to a visually indistinguishable solution as the resolution is increased. However, with the orthogonalized, conformal grid, convergence towards the asymptotic solution was substantially faster.

In this note two major points are made. First, pressure advection needs to be carried within the adjustment step of split schemes if conservation of energy in the transformation between kinetic and potential energy is to be made possible. Second, in contrast to several recently published views, additive split schemes, without incremental addition of the advection contributions, are not necessarily noisy, given that the presented scheme is not. an analysis in support of the first point and evidence in support of the second are presented. Finally, several additional remarks regarding the ratio of the adjustment and advection time steps and energy conservation in split-explicit models are made.

The benefits of assimilation of precipitation data had been demonstrated in diabetic initialization and nudging-type experiments some years ago. In four-dimensional variational (4DVAR) data assimilation, however, the precipitation data have not yet been used. To correctly assimilate the precipitation data by the 4DVAR technique, the problems related to the first-order discontinuities in the “full-physics” forecast model should be solved first. To address this problem in the full-physics regional NMC eta forecast model, a modified, more continuous version of the Beta-Miller cumulus convection scheme is defined and examined as a possible solution. The 4DVAR data assimilation experiments ate performed using the conventional data (in this case, analyses of T, ps, u, v, and q) and the precipitation data (the analysis of 24-h accumulated precipitation). The full-physics NMC eta model and the adjoint model with convective processes are used in the experiments. The control variable of the minimization pr...

It is suggested that there are two major problems with the “standard” methods of reducing pressure to sea level based on the surface temperature or the lowest-layer(s) temperature of a numerical model. The first is that using air temperatures above elevated terrain for reducing pressure to sea level is in conflict with the presumed objective of the reduction. The authors take this to be the derivation of a pressure field appropriate to sea level that to the extent possible maintains the shape of the constant-elevation isobars and reflects the changes in the horizontal of the magnitudes of horizontal pressure gradients, as these exist at the ground surface. The other problem is that evidence is emerging showing that with the increasing realism in the representation of mountains in numerical models the performance of the standard reduction methods is about to deteriorate to the point of becoming unacceptable. Fortunately, as proposed earlier by the first author, an alternative exists that is both s...

Expectations various regional climate modelers have expressed as to the impact on large scales are recalled. While some authors do mention the possibility of improvement also at large scales (e.g., Giorgi, J Phys IV France 139:101–118, 2006), the majority clearly accepts the view of “downscaling” as an effort in which the driver global model large scales are hoped to be preserved as much as possible and only small scales improved compared to those of the driver model. Many authors find it even desirable to use the so-called “large-scale nudging” in order to help achieve this objective. Mesinger et al. (Limited area predictability: can “upscaling” also take place? Research activities in atmospheric and oceanic modeling, WMO, Geneva, CAS/JSC WGNE Rep. No. 32, 5.30-5.31, 2002; see also Mesinger, The Eta model: design, history, performance, what lessons have we learned? In: Symposium on the 50th anniversary of operational numerical weather prediction, University of Maryland, College Park, MD, 14–17 June 2004, Preprints CD-ROM, 20pp, 2004) have however argued that various NWP results of the Eta model at NCEP strongly suggest that improvements in the large scales of the global driver model have been taking place more often than not. In addition, there was a four-month nine-member ensemble result of Fennessy and Altshuler in the early 2000s, published recently (Veljovic et al. Meteorol Z 19:237–246, 2010), in which an RCM achieved a dramatic improvement over its driver AGCM in hindcasting the precipitation difference over the central United States between the “flood year” of 1993 and the “drought year” of 1988; which we do not believe could have been possible without a significant improvement in the large scales. If this indeed is so and could be generalized, then large-scale nudging would not only be unnecessary but may also be harmful to the result. It could however be that this holds for some models while not for others. In that case, why so is a question of obvious importance. Given however that claims have even been made that improvements in large scales in regional climate modeling may be impossible for any models, hard evidence of specific large-scale improvements achieved are desirable. The preceding and additional points are discussed as well as more detail given, summarizing the results of perhaps the first comprehensive direct tests of the issue (Veljovic et al. Meteorol Z 19:237–246, 2010). Additional results are shown regarding the impact of the choice of the lateral boundary conditions (LBC) scheme, pointing to the advantage of the Eta (Mesinger, Contrib Atmos Phys 50:200–210, 1977) over the conventional and costlier relaxation scheme. As to the large scales question posed, the results summarized show that driving the Eta by ECMWF 32-day ensemble members the driver model large scales tended to be improved more often than not, giving support for our tenet that improving large scales as well in RCM efforts is possible. We furthermore argue that pursuing this objective should be beneficial for the improvement in smaller scales as well.

An earlier attempt to estimate the effect of hail suppression by silver iodide seeding in eastern parts of Yugoslavia, based on hail-frequency data at stations having professional observers, is extended here. Hail-frequency data only are considered, rather than the hail- and the ice pellet-frequency data taken together. The period of the data is extended from 37 to 40 years. A statistical analysis of the probability of the observed result being obtained by chance is made, based on the permutation test; a sensitivity test of the possible observer-subjectivity effect is done; and several tests of and corrections for any climate and observing practices change are also made. The ratio of the average hail frequency during the seeding activities in the area of the station and the average frequency before these activities shows a reduction in the hail frequency by about 25%. A synthetic histogram of the frequency ratios resulting from 10 000 random permutations (station by station) of the observed frequ...

Impact on forecast accuracy of the choice of the step-mountain (“eta”) vs. the traditional sigma coordinate is examined by using a model which can be run as either an eta or sigma system model via a simple switch in its code. This is done by (a) synoptic examination of the differences in a set of three forecasts, and (b) comparison of precipitation skill scores in a set of nine consecutive forecasts; each of these sets of forecasts was performed using one and then the other of the two systems. Both efforts indicate that greater accuracy is achnieved in the step-mountain coordinate forecasts. The three forecasts examined from the synoptic point of view are also compared against forecasts of another sigma system model. The two sigma models are found to have errors of an almost identical pattern (two cases) or to have errors which seemed to have the same basic cause.

Among the wide variety of performance measures available for the assessment of skill of deterministic precipitation forecasts, the equitable threat score (ETS) might well be the one used most frequently. It is typically used in conjunction with the bias score. However, apart from its mathematical definition the meaning of the ETS is not clear. It has been pointed out (Mason, 1989; Hamill, 1999) that forecasts with a larger bias tend to have a higher ETS. Even so, the present author has not seen this having been accounted for in any of numerous papers that in recent years have used the ETS along with bias "as a measure of forecast accuracy". A method to adjust the threat score (TS) or the ETS so as to arrive at their values that correspond to unit bias in order to show the model's or forecaster's accuracy in extit{placing} precipitation has been proposed earlier by the present author (Mesinger and Brill, the so-called dH/dF method). A serious deficiency however has since been noted with the dH/dF method in that the hypothetical function that it arrives at to interpolate or extrapolate the observed value of hits to unit bias can have values of hits greater than forecast when the forecast area tends to zero. Another method is proposed here based on the assumption that the increase in hits per unit increase in false alarms is proportional to the yet unhit area. This new method removes the deficiency of the dH/dF method. Examples of its performance for 12 months of forecasts by three NCEP operational models are given.

This chapter focuses on some of the issues of the Arakawa approach in numerical methods, as an illustration of the variety of considerations that are made and opportunities that present themselves. Some of the recent ideas and results in addressing the horizontal grid problems are also reviewed. Propagation of gravity waves forced at individual grid points on a semi-staggered B/E grid is explained in the chapter. A synopsis of the results of the Eta model, viewed in comparison with other models of NMC is considered. Some of the recent Eta model results offer guidance regarding a number of central issues of the limited-area modeling strategy. The Eta model is a limited-area model with the numerical formulation designed following the Arakawa principles. There are specific features of the Eta numerics or setup that could be identified as making a notable contribution to its advantage. The prospects of constructing well-behaved global finite-difference models using the expanded cube approach are also explained in the chapter.

Several planetary boundary layer and turbulence transport parameterization advancements are reported upon, with the majority of them discovered by looking for a solution to a problem noticed in the course of running an operational NWP model. A surprising degree of sensitivity was repeatedly discovered to solutions found, with the refinements resulting in substantial improvements in the realism of forecasting a variety of actual atmospheric events. These improvements came from a reformulation of the Mellor-Yamada level 2 parameterization of surface fluxes via a change in the associated length scale; from a number of refinements in the specification of the Mellor-Yamada level 2.5 master length scale; and from the discovery of the reason for and improved handling of its realizability problem. Examples of the impact are shown. A refinement is summarized of the explicit molecular sublayer parameterization of Janjic, used over water as a constant-gradient approximation of the Liu, Katsaros and Businger relations. The refinement is arrived at by application of a momentum flux data summary and a suggestion by Brutsaert.

‘Source-sink’ shallow water integrations are performed, using two limited area models with the Coriolis terms included. One of the models was defined on the staggered C grid, and the other on the semi-staggered E/B grid. For each grid, experiments were performed with grid sizes of 250, 125 and 62.5 km. A source and a sink, of constant intensity, were placed symmetrically in the central part of the domain. They covered, for the three resolutions considered, areas of one, four and sixteen grid points, respectively. At the sink ‘point’, after 24 hours, substantial differences in the depth of the water existed between the lowest resolution experiments on the two grids. The differences were very much reduced in the two 125 km experiments, and were virtually absent in the highest resolution, 62.5 km, experiments. With increasing resolution, the depths at the sink on the two grids converged with about equal rapidity towards a value in between the two lowest resolution values. As to be expected on the basis of the geostrophic adjustment theory when applied to the spatially discretized systems, the depth on the low resolution C grid solution was found to be an underestimate, and that on the E grid an overestimate, of the true value. It is demonstrated that the rate of convergence on the E grid can be improved by increasing the weight of the modification introduced to cope with the grid separation problem. An implication of the results obtained in the simple experiments reported here is that forcing schemes should be considered which would avoid forcing at single grid points. Where appropriate, simultaneous forcing at several neighbouring points may be an attractive alternative approach.

The minimum subsystem of the linearized finite-difference shallow water equations was found within which the instability first noticed with the ECMWF energy and enstrophy conserving scheme (Hollingsworth and Kallberg 1979; Hollingsworth et al. 1983) still occurs. This subsystem consists of the Coriolis and the advection terms.

For use in a model on the semi-staggered E (in the Arakawa notation) grid, a number of conserving schemes for the horizontal advection are developed and analyzed. For the rotation terms of the momentum advection, the second-order enstrophy and energy-conserving scheme of Janjic (1977) is generalized to conserve energy in case of divergent flow. A family of analogs of the Arakawa (1966) fourth-order scheme is obtained following a transformation of its component Jacobians. For the kinetic energy advection terms, a fourth- (or approximately fourth) order scheme is developed which maintains the total kinetic energy and, in addition, makes no contribution to the change in the finite-difference vorticity. For the resulting both second- and fourth-order momentum advection scheme, a modification is pointed out which avoids the non-cancellation of terms considered recently by Hollingsworth and Kallberg (1979), and shown to lead to a linear instability of a zonally uniform inertia-gravity wave. Finally, a ...

The problem of the pressure gradient force error in the case of the terrain-following (sigma) coordinate does not appear to have a solution. The problem is not one of truncation error in the calculation of space derivatives involved. Thus, with temperature profiles resulting in large errors, an increase in vertical resolution may not reduce and is even likely to increase the error. Therefore, an approach abandoning the sigma system has been proposed. It involves the use of “step” mountains with coordinate surfaces prescribed to remain at fixed elevations at places where they touch (and define) or intersect the ground surface. Thus, the coordinate surfaces are quasi-horizontal, and the sigma system problem is not present. At the same time, the simplicity of the sigma system is maintained. In this paper, design of the model (“silhouette” averaged) mountains, properties of the wall boundary condition, and the scheme for calculation of the potential to kinetic energy conversion are presented. For an ...

The problem of the calculation of the pressure gradient force in sigma (terrain following) coordinate models is reviewed. It is pointed out that one of the well known schemes, the scheme of Corby et al. (1972), does not result in a convergent difference analogue of the pressure gradient force. Some other schemes, such as the one of Burridge and Haseler (1977) used at the European Centre for Medium Range Weather Forecasts, have a convergence condition, requiring thickness of sigma layers greater than (or equal to) a given threshold value. The two-term error problem of these analogues is reexamined. A numerical example is given, showing that, in accordance with the convergence condition, an increase in vertical resolution will generally not result in a reduction of the error.

A Lagrangean-type numerical forecasting method is developed in which the computational (grid) points are advected by the wind and the necessary space derivatives (in the pressure gradient terms, for example) are computed using the values of the variables at all the computation points that at the particular moment are within a prescribed distance of the point for which the computation is done. In this way, the forecasting problem reduces to solving the ordinary differential equations of motion and thermodynamics for each computation point, instead of solving the partial differential equations in the Eulerian or classical Lagrangean way. The method has some advantages over the conventional Eulerian scheme: simplicity (there are no advection terms), lack of computational dispersion in the advection terms and therefore better simulation of atmospheric advection and deformation effects, very little inconvenience due to the spherical shape of the earth, and the possibility for a variable space resoluti...

Experimental study of the problem of the forced adjustment of the wind field to the height field with the Mintz-Arakawa two-level atmospheric general circulation model. In all but one of the experiments, the height field was assumed to be perfectly observed at 6-hr intervals, over a time period of one day or less, and from these height data the vector wind field was computed by forced dynamical adjustment. In one experiment, the temperature alone was prescribed. The winds computed in these experiments were compared with the 'control' winds of the general circulation simulation. The best agreement between the computed and the control winds was obtained when the time-differencing scheme in the governing finite-difference equations of motion had a large rate of damping of high-frequency motions. This damping rate also determined the optimum fraction and frequency of restoration of the height (or temperature) fields. With strong damping, total restoration at every time step gave the most rapid rate of wind error reduction and the smallest asymptotic limit of the wind error.

A numerical experiment is performed with the purpose of investigating the behavior of the trajectories of a very large number of constant-volume particles. Practical significance is given to this problem by the possibility of using superpressured, constant-volume balloons for routine upper air observations. The computational scheme of the experiment is described with emphasis on some aspects of trajectory computations. Thirty-day diagnostic trajectories are computed for two levels using the total velocity components, and for one level using only the nondivergent ones, and the resulting spacing of the trajectory points is discussed. Theory of the distances to the nearest among a large number of points is developed and applied for the statistical description of the results. Histograms of distances from the constant-volume particles as well as from random points in space to the nearest neighboring constant-volume particle are computed, and compared with the frequency function of those distances for ...

Vertical interpolation of the data from the constant-pressure to the constant-density surfaces is discussed, and the results of an experiment given, wherein the data are interpolated to a constant-density surface using two different sets of the constant-pressure data. Comparison of the two results gives an estimate of the order of magnitude of the errors inherent in the interpolation.

A method has been developed to inherently prevent the false two-grid-interval wave in height field, which is, to the author's knowledge, at present permitted as a stationary solution to the gravity-wave part of all space-centered schemes used to solve the primitive equations in numerical forecasting and simulation experiments. The proposed method preserves space- and time-centered differencing and second-order accuracy. It involves introduction of auxiliary velocity points midway between the neighboring height grid points. Velocity components at these auxiliary points are assumed to have space-averaged values at the beginning of a considered time step; acceleration contributions are evaluated and added to these initial values to obtain components at the middle or at the end of the time step. Resulting velocity components are then used for a more accurate calculation of the divergence term in the continuity equation. The procedure leads to schemes which propagate single-grid-point height perturbations to all the existing height grid points, and thus do not permit formation of a false high wave number noise in the forecast fields. For the Heun, leapfrog, and implicit time differencing schemes an analysis is performed of the effect that this procedure has on the properties of the scheme. For the Heun scheme, the procedure results in a reduction of the instability of the scheme, and in a reduction of the truncation error. For the leapfrog scheme slight instability is produced; however, values of the amplification factors, and a numerical example, indicate that this modified leapfrog scheme can still successfully be used for integration of the primitive equations. For the implicit scheme, the procedure preserves the unconditional stability of the scheme, and also reduces the truncation error. Finally, a numerical example is shown, giving a dramatic illustration of the advantages of the proposed method. DOI: 10.1111/j.2153-3490.1973.tb00629.x