1. Introduction
The mutual coupling of a forecast model and data assimilation (DA) algorithm in global numerical weather prediction (NWP) systems generates operational analyses that serve as atmospheric initial conditions for the forecast model and verification for objective skill scoring. These systems can also generate historical analyses (so-called reanalyses) that are now used extensively in climate studies and other retrospective research applications. Their growing popularity means that meteorological analyses generally, and reanalyses in particular, are now being studied and postprocessed in considerable detail, bringing to light new issues that are not typically encountered in standard NWP applications.
For example, a series of studies has used reanalysis to compute terms governing vertically integrated budgets of energy, heat, and moisture, to quantify pathways and variability of energy flow through the climate system (see Trenberth and Smith 2009, and references therein). Trenberth et al. (2002) used the method with different reanalyses, gridded both on the model’s intrinsic vertical levels η as well as on reference pressure levels p. Calculations using reanalyses of the National Centers for Environmental Prediction–National Center for Atmospheric Research (NCEP–NCAR; Kalnay et al. 1996) on pressure levels produced energy budget errors that were absent when using pressure-level reanalyses from the European Centre for Medium-Range Weather Forecasts (ECMWF).
The equation governing the vertically integrated total energy budget contains a potential energy divergence term that is proportional to ∇h ⋅ (uΦ), where Φ is geopotential, u is the horizontal wind vector, ∇h = (∂/∂x, ∂/∂y), and thus ∇h ⋅ u is the vertical component of the wind divergence vector, hereafter referred to simply as divergence (Trenberth et al. 2002). Trenberth and Stepaniak (2002) traced the source of errors in their budget calculations to anomalous analyzed divergences at the topmost vertical pressure levels in the stratosphere, which contaminated their vertical integrals of potential energy due to the large stratospheric weighting resulting from the approximately linear growth of Φ with pressure height. These stratospheric divergent wind anomalies clustered preferentially above high steep terrain such as the Andes in NCEP–NCAR reanalyses, but were absent in the ECMWF reanalyses. They attributed this difference to the ECMWF model’s hybrid σ–p coordinate, which transitioned model levels η smoothly from terrain-following surfaces near the ground to isobaric surfaces in the stratosphere. By contrast, the NCEP–NCAR model used a σ (sigma) coordinate (Phillips 1957), which retains terrain-following characteristics throughout its vertical domain.
Their explanation is qualitatively consistent with offline diagnostic calculations and model runs using idealized initial conditions, which both show that hybrid coordinates substantially reduce discretization errors over high terrain, relative to σ coordinates (e.g., Simmons and Burridge 1981; Eckermann 2009). However, this explanation runs counter to objective scoring of 0–5-day forecast experiments using standard metrics such as anomaly height correlations and wind errors, which have revealed largely imperceptible differences in stratospheric skill between σ and hybrid coordinates (Simmons and Strüfing 1983; Eckermann 2009). Koo and Hong (2013) also reported few differences at 0–5 days, but claimed to see bigger impacts on temperature and humidity at longer times, although some of this impact may have been specific to some unique features of their model.
Another possible source of error noted by Trenberth and Stepaniak (2002) was the low (10 hPa) upper boundary, where errors can accrue as a result of the potential for spurious vertical reflection of wave energy and the need for enhanced numerical diffusion at upper levels to absorb this wave energy. It is conceivable that the NCEP–NCAR model had shortcomings in its sponge layer that led to wave reflection from the model lid, particularly given the potential for deep explicitly resolved orographic gravity waves in model forecasts over these mountainous regions (Jiang et al. 2013).
Thus, the origins of the divergence anomalies in NCEP–NCAR reanalyses documented by Trenberth and Stepaniak (2002) remain unclear. Are they common to σ-coordinate systems, or were they peculiar to this particular reanalysis, perhaps resulting from sponge-layer problems or other model errors? Why do they appear so strongly in stratospheric analyses, yet have little or no impact on stratospheric forecast skill? One clue is that previous tests of skill impacts all used archived analyses as initial conditions and verification (Simmons and Strüfing 1983; Eckermann 2009; Koo and Hong 2013), rather than running a fully coupled system that generates both forecasts and analyses. It is therefore conceivable that the DA cycle plays the pivotal role in amplifying these errors in operational systems.
2. NWP system and experiments
We investigate this possibility using the Navy Global Environmental Model (NAVGEM), the Navy’s new operational global NWP system. The inaugural version at NAVGEM transitioned to operations in March 2013. NAVGEM will be described in greater detail elsewhere. Here we focus only on those aspects most relevant to the present study.
The forecast model’s dynamical core uses the hydrostatic three-time level semi-Lagrangian (SL) semi-implicit formulation of Ritchie et al. (1995). Terrain is specified as the mean elevation within each Gaussian grid box, as derived from the Global Land One-km Base Elevation (GLOBE) model. These values are then smoothed spectrally with a Lanczos filter to reduce terrain variance and spectral ringing near the truncation scale (see appendix A of Hogan and Rosmond 1991). Estimates of the terrain variance and azimuthal anisotropy within each grid box are used to parameterize subgrid-scale orographic gravity wave and flow-blocking drag (Hogan et al. 2003). NAVGEM also parameterizes turbulent form drag due to subgrid-scale orographic roughness and includes a four-soil/vegetation land surface model (Hogan 2007) that interacts with radiation and vertical mixing parameterizations within the boundary layer. In the model’s top two layers, shown in orange in Fig. 1, scale-dependent divergence, vorticity, and thermal damping are enhanced to absorb resolved waves and reduce wave reflection from the model’s rigid lid.
NAVGEM interface pressures around a 34.5°N latitude circle using the (a) hybrid and (b) sigma coordinate based on same profile of sea level L60 layer thicknesses. (c) Sea level L60 layer thicknesses plotted as pressure height thicknesses for a scale height of 7 km. The green layer in (a) is the lowest isobaric interface pressure of 85 hPa, orange curves show interface levels of the heavily diffused upper-level sponge layers, and the red line shows the rigid upper boundary at ptop = 0.04 hPa.
Citation: Monthly Weather Review 142, 1; 10.1175/MWR-D-13-00203.1
For its DA, NAVGEM uses the Naval Research Laboratory Atmospheric Variational DA System–Accelerated Representer (NAVDAS-AR: Xu et al. 2005; Rosmond and Xu 2006). NAVDAS-AR is a four-dimensional variational data assimilation (4DVAR) algorithm that incorporates a fast radiative transfer model and a variational bias correction scheme to assimilate temperature, ozone, and water vapor information from tropospheric and stratospheric radiances acquired by a variety of operational satellite remote sensors.
We ran two forecast-assimilation experiments with NAVGEM at T359L60, starting at 0000 UTC 5 October 2011 and continuing to the end of January 2012. The runs were identical apart from a change in the SL model’s vertical coordinate. As illustrated in Fig. 1c, each experiment used the same L60 profile of sea level model layer thicknesses up to a top interface level of ptop = 0.04 hPa. However, the changes in the pressure 
3. Results
a. Mean divergence fields
Figure 2 plots maps of monthly mean analyzed divergence for November 2011 at ~50 hPa over the Americas, Greenland, Africa, and the Gulf states, using the analyzed fields on the model’s native Gaussian grid and η levels. The HYB fields (top row) are mostly featureless, with peak values of approximately ±10−5 s−1. Weak enhancements result mostly from residual levels of explicitly resolved orographic gravity wave activity that survive monthly averaging. For example, the weak HYB activity above and downsteam of the southern Andes is due to resolved orographic gravity waves that often propagated into the stratosphere (see later Fig. 4), as is often observed here in November (e.g., Eckermann and Preusse 1999).
Maps of time-mean analyzed divergence for November 2011 on the NAVGEM model grid at ~50 hPa over different landmasses for the (top) HYB and (bottom) SIG experiments. Note the lower dynamic range of the color scale (far right) for the HYB experiment relative to SIG. Maximum and minimum values (10−6 s−1) are shown at the bottom left of each panel. Orange curves show five cross sections profiled in Fig. 3.
Citation: Monthly Weather Review 142, 1; 10.1175/MWR-D-13-00203.1
By contrast, the SIG experiment (bottom row) shows divergence structures with peak amplitudes ~5–15 times larger that are tied strongly to the underlying orography of the Andes, Sierra Madres, the Greenland Ice Shelf, the Ethiopian and Kenyan Highlands, the Sarawat Mountains along the west coast of Yemen and Saudi Arabia, the Zagreb Mountains in Iran, and the Himalayas. This anomalous stratospheric divergence structure directly over steep mountainous regions in South America and Africa is very similar to that found by Trenberth and Stepaniak (2002) in the NCEP–NCAR reanalyses (see their Figs. 5 and 6).
Orange lines in Fig. 2 mark selected cross sections, along which vertical profiles of the mean analyzed divergence structure are plotted in Fig. 3 for each experiment. The largest mean values in the HYB experiment are confined to the troposphere where the coordinate has terrain-following features. Since divergences are an indirect proxy for vertical motion, we note that their dipole structure near the surface over the Andes and Greenland, for example, is consistent with upward and downward motion across coordinate surfaces as flow is diverted either over or around the obstacle on the upslope, then reverts to a return flow on the downslope. This dipole structure attenuates with height in response to both atmospheric stratification and the rapid flattening of hybrid-coordinate surfaces with height. In the SIG experiments, however, while similar low-level dipole structure is also seen, there is additional structure above that oscillates with height and extends strongly through the entire atmospheric column right up to the model top.
Longitude–pressure (height) cross sections of time-mean analyzed divergence for November 2011 along the five trajectories shown in Fig. 2, for the (top) HYB and (bottom) SIG experiments. Note the different dynamic range of the color scale beneath each plot pairing. Maximum and minimum values (10−6 s−1) in the stratosphere from 100 to 0.1 hPa are shown at the top of each panel.
Citation: Monthly Weather Review 142, 1; 10.1175/MWR-D-13-00203.1
b. Skill impacts
Figure 4 provides an example of how these anomalous deep divergence structures above steep terrain in the SIG experiments can degrade analysis skill. The top panels of Fig. 4 show maps of analyzed 15-hPa divergences from the HYB and SIG experiments, which both appear to capture plane orographic gravity wave structure emanating from the Andes below. However, corresponding longitude–height cross sections at ~30°S in the panels beneath show that only the HYB analysis contains a realistically tilted wave phase structure throughout the troposphere and stratosphere, whereas the SIG cross section is dominated by standing-wave-like anomalies like those seen in Fig. 3. The potential for sigma-coordinate errors in orographic gravity wave solutions has been noted before in idealized simulations (e.g., Klemp et al. 2003).
Maps of analyzed divergence at 15 hPa at 0000 UTC 18 Nov 2011 over the Andes for the (a) HYB and (b) SIG experiments. Dotted orange line shows same Andes cross section at 30°S from earlier figures, along which the (c),(d) longitude–height cross section of divergence is plotted. Maximum and minimum stratospheric values are given beneath each panel.
Citation: Monthly Weather Review 142, 1; 10.1175/MWR-D-13-00203.1
Daily 0–120-h forecasts from each experiment were used to quantify forecast skill over the entire analysis period, excluding the initial “spinup” period from 5 October to 7 November. The top panels of Fig. 5 plot the root-mean-square (rms) temperature errors versus forecast hour at 100 and 50 hPa in the Northern Hemisphere, using radiosondes as verification. Panels below show corresponding time series of these rms temperature errors at 120 h. Persistent and statistically significant differences are evident, with the HYB experiment consistently outperforming the SIG experiment. Other results (not shown) reveal improved 50- and 100-hPa skill in the tropics, in other fields such as winds, and when using self-analysis rather than radiosondes as verification. The increased 50-hPa errors in both experiments in January coincided with a stratospheric sudden warming, which is known to degrade stratospheric skill relative to more dynamically quiescent periods (Lahoz 1999).
Plots of mean rms temperature error (with respect to radiosondes) vs forecast hour, averaged from 8 Nov 2011 to 31 Jan 2012 at (a) 100 and (b) 50 hPa for experiments SIG (green) and HYB (purple). The solid line is the mean and the shaded area denotes standard deviation. Corresponding time series of 120-h forecast errors at (c) 100 and (d) 50 hPa.
Citation: Monthly Weather Review 142, 1; 10.1175/MWR-D-13-00203.1
4. Discussion
a. Divergence anomalies
Stratospheric divergence anomalies in our SIG analyses reproduce the salient aspects of “pathologies” in NCEP–NCAR reanalyses documented by Trenberth and Stepaniak (2002), and the order of magnitude reductions in anomaly amplitudes in our HYB experiment definitively prove their hypothesis that these anomalies were produced by the σ-coordinate model used in the NCEP–NCAR system at the time. Our findings are consistent with the hypothesis that they arise, at least in part, from discretization errors in the computation of pressure gradient forces on model layers tilted by steep underlying terrain (Kurihara 1968; Corby et al. 1972; Gary 1973; Simmons and Burridge 1981; Mesinger 1982), which, because of the generally sharp change in temperature gradient at the tropopause, can amplify considerably in the stratosphere (Kurihara 1968; Simmons and Burridge 1981; Eckermann 2009). For example, the order of magnitude error reductions between the HYB and SIG experiment, as well as near-constant error amplitudes in the stratosphere, are in excellent agreement with earlier error predictions for hybrid and sigma coordinates from both offline diagnostic calculations as well as model nature runs using idealized initial conditions (see Figs. 13 and 14 of Eckermann 2009).
Some differences are evident between the SIG anomalies reported here and those documented by Trenberth and Stepaniak (2002). Their upper-level divergence anomalies had a zigzag vertical structure with a grid-scale wavelength of 2Δη. The SIG structures in Fig. 3 have both shallow and deep vertical wavelengths, with no systematic 2Δη structure. The NCEP–NCAR anomalies grew appreciably in amplitude with height, being small near 100 hPa but large at 10 hPa. Our anomalies extend to the surface and persist with similar amplitude through the full depth of the stratosphere and lower mesosphere. Trenberth et al. (2002) intimated that errors in their climate diagnostics were greater in the pressure-level data relative to the data on η layers. Intercomparisons reveal largely identical anomalies on η and p levels in our SIG and HYB experiments. As one example, Fig. 6 plots divergence cross sections across the Andes at 8°S from pressure-level fields, revealing the same shallow divergence anomaly structure in the SIG experiment evident in the corresponding η-level data in Fig. 3. This suggests additional errors associated with the low 10-hPa upper boundary in the NCEP–NCAR system played a role in producing those other error properties.
Longitude–pressure (height) cross sections of time-mean analyzed 0000 UTC divergence, regridded at 1° × 1° onto reference pressure levels, at 8°S across the Andes, for (a),(b) November 2011 and (c),(d) December 2011 from (left) the HYB experiment and (right) the SIG experiment, replicating a similar presentation for NCEP–NCAR reanalyses by Trenberth and Stepaniak (2002) and for the η-level fields in Fig. 3.
Citation: Monthly Weather Review 142, 1; 10.1175/MWR-D-13-00203.1
These persistent stratospheric divergence errors in the SIG analyses grow through the forecast-assimilation update cycle. Errors originate as small discretization errors in 0–9-h forecasts that in turn act as backgrounds for the DA algorithm. Similar errors also arise here in the tangent-linear model (TLM) used in the 4DVAR algorithm, though this is not relevant to the 3DVAR-based NCEP–NCAR results. Ideally, observational increments would correct these background errors, but do not because stratospheric wind observations are sparse, coming only from scattered radiosondes extending no higher than ~10 hPa. These sparse wind data lack the necessary geographical coverage and horizontal resolution to correct small-scale divergent wind errors. Likewise, temperature increments produce only a balanced rotational wind correction on larger horizontal scales. Thus, small-scale divergent wind errors in the backgrounds appear largely unaltered in the output analyses, then pass as initial conditions to the forecast model, which amplifies the errors further in the next 0–9-h forecast. Over the October 2011–January 2012 period of this run, this continuous cycling allows these errors to grow much as they would in a 4-month nature run of the forecast model alone.
b. Skill impacts
Many global NWP and climate models now use hybrid coordinates: indeed, the ECMWF NWP system transitioned from sigma to hybrid coordinates 30 years ago. This conveys the impression that hybrid-coordinate models clearly and consistently outperform their sigma-coordinate counterparts. Certainly idealized diagnostic and modeling studies have shown repeatedly that hybrid coordinates should reduce errors relative to sigma coordinates (Kurihara 1968; Simmons and Burridge 1981; Eckermann 2009). Yet objective evidence of positive skill impacts in operational systems has proven elusive. For example, the ECMWF’s original move from hybrid to sigma coordinates occurred despite no clear evidence of positive skill impacts (Simmons and Strüfing 1983), a finding replicated in more recent forecasting assessments with other NWP models (Eckermann 2009; Koo and Hong 2013). Mesinger et al. (1988) review similarly equivocal findings with mesoscale NWP models. This has had practical implications. To implement a model change, most operational centers now require that the change produces improvements in objective skill scores that exceed some minimum threshold. Since hybrid coordinates generally show neutral skill impacts, many NWP systems, notably mesoscale models, have persisted with sigma coordinates (e.g., Dudhia 1993; Skamarock et al. 2008).
The results from our experiments help to clarify this apparent skill paradox. Previous forecasting experiments that found little difference in skill between hybrid and sigma forecasts used archived analyses from a static operational system as both initial conditions for the forecasts and as the verifying analysis (Simmons and Strüfing 1983; Eckermann 2009; Koo and Hong 2013). Here we compared forecast skill using the separate independent analyses and forecasts generated in each experiment. These have revealed for the first time statistically significant differences in stratospheric skill scores, with the HYB experiment consistently outperforming the SIG experiment (see Fig. 5). Since the only difference between our two experiments was the vertical coordinate (see Fig. 1), Fig. 5 provides objective proof that the improved error suppression of hybrid coordinates leads to improved stratospheric forecast skill in NAVGEM.
This finding indicates that, like the divergent wind anomalies in the SIG analyses, these increased stratospheric errors in the SIG forecasts grow through the forecast-assimilation update cycle. This study has focused solely on small-scale discretization errors in divergence over localized steep terrain. It seems unlikely that these anomalies alone can fully explain the hemispheric skill impacts evident in Fig. 5. Additional sources of error growth through the update cycle may also contribute, such as large-scale advection and constituent transport errors (Schär et al. 2002; Hoinka and Zängl 2004), and possible upscale growth of these errors with time, perhaps through flow-on error growth in both resolved and parameterized orographic gravity wave drag (see, e.g., Fig. 4) that can have well-known impacts on stratospheric prediction skill (e.g., Shutts and Vosper 2011).
The skill impacts found here have implications for both hybrid- and sigma-coordinate models. While the sigma coordinate is unique, there is no single hybrid coordinate and indeed a continuum of possible choices is available. Simmons and Burridge (1981), and more recently Eckermann (2009), have shown that different hybrid σ–p coordinates have very different error characteristics, with some having much better error suppression properties than others. Yet, as summarized in Figs. 17 and 18 of Eckermann (2009), a wide variety of hybrid coordinates is implemented in current global NWP and climate models, with little apparent rationale for most choices. Given that Fig. 5 shows that the error growth properties of a vertical coordinate can have statistically significant impacts on skill, it suggests that greater attention be paid in the future to implementing a hybrid coordinate with the best objective error suppression characteristics for a given modeling application. Such an initiative would parallel similar ongoing efforts in the mesoscale NWP community (Schär et al. 2002; Klemp 2011).
5. Summary
Analyses generated by an operational NWP system using a σ (sigma) vertical coordinate in the forecast model revealed deep stratospheric divergence anomalies above high steep terrain. These anomalies were essentially absent from the stratospheric analysis when repeating these runs using a hybrid σ–p (sigma–pressure) vertical coordinate, confirming the σ-coordinate origins of similar anomalies in NCEP–NCAR reanalyses (Trenberth and Stepaniak 2002).
These σ-coordinate anomalies originate as discretization errors in the computation of pressure gradient forces within model layers that are tilted by steep underlying terrain. They grow as 0–9-h forecast backgrounds pass through successive update cycles without significant correction by observational DA increments, due to sparse observations and the short horizontal scale of this divergent error structure. This pathway for error growth explains why previous forecasting experiments using archived analysis found little difference in skill between sigma and hybrid-coordinate forecasts.
The anomalous stratospheric divergent wind structure in our σ-coordinate analysis did not reproduce all aspects of the structure in NCEP–NCAR reanalyses documented by Trenberth and Stepaniak (2002). In particular, we find no preferred 2Δη vertical scale, no large growth with height through the stratosphere, and no obvious differences in error amplitudes between η or pressure levels. These unsimulated features likely point to additional sponge-layer-related anomalies near 10 hPa in the NCEP–NCAR reanalyses.
These σ-coordinate errors also negatively impact explicitly resolved orographic gravity waves in the analysis.
Relative to the σ-coordinate experiment, the hybrid-coordinate experiment yields statistically significant improvements in stratospheric skill scores over a 3-month forecasting and analysis interval (from November 2011 through January 2012).
Our results should further motivate ongoing efforts to implement specific types of hybrid vertical coordinates in NWP and climate models that most effectively suppress the formation and growth of deep discretization errors over high steep terrain (e.g., Eckermann 2009).
Acknowledgments
These NAVGEM runs were facilitated by a grant of computer time from the DoD High Performance Computing Modernization Program at the Navy DoD Supercomputing Resource Center (DSRC), and we thank our NRL colleagues Jim Ridout, Ben Ruston, Tim Whitcomb, and Karl Hoppel for helping us to run NAVGEM on Navy DSRC systems. Helpful comments on earlier drafts by Jim Doyle, Carolyn Reynolds, and two anonymous reviewers are gratefully acknowledged. This work was supported by the Office of Naval Research (ONR) through the NRL 6.1 work unit “The Boundary Paradox” and the ONR Departmental Research Initiative “Prediction of Seasonal and Intraseasonal Oscillations.”
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