1. Introduction
Topography is one of the main sources for atmospheric gravity waves, and simulating their behavior in atmospheric modeling systems is crucial for numerical weather prediction. With the availability of increasing computational resources, atmospheric numerical models are progressing to higher resolutions and are investigating the influences of small-scale terrain on gravity wave dynamics (e.g., Kirshbaum et al. 2007; Sheridan and Vosper 2012). Historically, terrain-following vertical coordinates have been widely used in atmospheric models to represent flow over topography, although it is well known that the terrain influences on the coordinate surfaces can contribute to increased errors in the model numerics, particularly in computing the horizontal pressure gradient terms (Mahrer 1984; Dempsey and Davis 1998). The numerical treatment of real topography is challenging because the terrain may contain significant structure at the smallest scales that are represented in the model, scales for which the model numerics are least accurate. Park et al. (2016, hereafter P16) assessed the influence of terrain smoothing on upper-level turbulence forecasting and documented that the overprediction of upper-air turbulence can be significantly reduced by filtering the high-frequency terrain structure.
To mitigate errors associated with a terrain-following coordinate, there have been a number of studies examining alternative numerical approaches, including alternate numerics to increase the accuracy in calculating the horizontal pressure gradient (cf. Mahrer 1984) and cut-cell techniques to remove terrain influences from the coordinate surfaces (cf. Adcroft et al. 1997; Steppeler et al. 2002). A hybrid terrain-following vertical coordinate provides another approach in reducing numerical errors by more rapidly removing the influences of the terrain on the coordinate surfaces with increasing height, such that they transition to constant height (or pressure) surfaces at mid- and upper levels in the model domain (Simmons and Burridge 1981; Bubnová et al. 1995; Schär et al. 2002; Klemp 2011).
The Advanced Research WRF (WRF-ARW) Model solves the nonhydrostatic equations employing a terrain-following hydrostatic pressure (sigma) vertical coordinate, often characterized as a mass coordinate (Laprise 1992; Klemp et al. 2007; Skamarock et al. 2008). In this study, we formulate a hybrid coordinate for the WRF-ARW following the design in Park et al. (2013) and document its impact in both an idealized and a real-data simulation. For the real-data case, we investigate the role of the hybrid coordinate in the upper-level turbulence forecast analyzed by P16 in assessing the effects of terrain smoothing. Since upper-level turbulence forecasting is quite sensitive to topography and high wind shear at upper levels, this case provides a good test of the dependence of the coordinate formulation on the accuracy of the model numerics over terrain.
Our paper is organized as follows. In section 2, formulation of the hybrid coordinate in the WRF-ARW is introduced, and an idealized test case is presented in section 3 to illustrate the performance of the hybrid coordinate. Section 4 provides an analysis of a real-data forecast for air turbulence and demonstrates that the hybrid vertical coordinate reduces artificial noise (and associated overprediction of turbulence) at mid- and upper levels. The kinetic energy spectra for these simulations provide further confirmation of the beneficial influence of the hybrid coordinate. Section 5 summarizes the results of this study.
2. Formulation of the hybrid coordinate in the WRF mass coordinate
The 
Citation: Monthly Weather Review 147, 3; 10.1175/MWR-D-18-0334.1
In implementing the hybrid mass coordinate in the WRF-ARW, there is little change in the model equations and numerics. Essentially, the only change is that vertically integrating the horizontal divergence in (8) now yields 
Minimum surface pressure 
Citation: Monthly Weather Review 147, 3; 10.1175/MWR-D-18-0334.1
3. An idealized test case
Vertical distribution of coordinate surfaces for (a) the BTF coordinate and (b) the HTF coordinate for the idealized topography (12).
Citation: Monthly Weather Review 147, 3; 10.1175/MWR-D-18-0334.1
Because there is no wind at or below mountain-top level in this test case, the upper-level flow should remain undisturbed as it passes over the mountain. Perturbations of vertical velocity at 5 h are depicted in Fig. 4 and exhibit significant differences. For the BTF simulation in Fig. 4a, stationary waves are excited at upper levels in the domain with maximum vertical velocities reaching about 1 m s−1. In contrast, with the HTF coordinate, the influence of topography is largely removed from the coordinate surfaces in the upper half of the domain, and the perturbations are much weaker, with the maximum vertical velocity less than 0.07 m s−1. This significant reduction of the spurious perturbations is to be expected because for the HTF case, the coordinate surfaces are much more closely aligned with the direction of the flow (see Schär et al. 2002; Shaw and Weller 2016).
Vertical velocity at 5 h with the (a) BTF coordinate and (b) HTF coordinate for the idealized test case. Note that contour interval for the HTF case is 10 times smaller than for the BTF case.
Citation: Monthly Weather Review 147, 3; 10.1175/MWR-D-18-0334.1
4. Real-data test case
a. A case on 2 November 2015
To explore the influences of the vertical hybrid coordinate in NWP applications, we have conducted a retrospective simulation for a case over the western United States on 2 November 2015, previously studied by P16. Here, the WRF-ARW is configured as close as possible to the operational NOAA Rapid Refresh (RAP) model, which predicted a huge false alarm of the light intensity of upper-level turbulence that was not supported by numerous null (smooth) observations.
As shown in the 500-hPa analysis in Fig. 5, a large-scale synoptic trough extended from the eastern Pacific through British Columbia at 0000 UTC 2 November 2015. Large gradients of geopotential heights persisted along the southern portion of planetary waves, and strong westerly winds were present in the western part of the Rockies. During the next 24 h, the wind direction changed to southwesterly as the synoptic trough approached the western part of Washington (not shown). For this case, we have run the WRF-ARW forecast over the United States with the same configuration as used for the P16 simulation initialized at 0000 UTC 2 November 2015, but including the hybrid vertical coordinate as specified in (1) with 
Terrain (shading), 500-hPa geopotential height (solid line), and wind vectors at 0000 UTC 2 Nov 2015. Analysis is from GFS, and terrain is from RAP as indicated in Table 1.
Citation: Monthly Weather Review 147, 3; 10.1175/MWR-D-18-0334.1
Summary of experimental configurations with vertical coordinate, interpolation method, and terrain smoothing.
b. Turbulence forecast
To illustrate the influences of terrain smoothing and the hybrid coordinate on the shape of the coordinate surfaces, vertical cross sections of these surfaces are displayed in Fig. 6 along the line AB in Fig. 5. Clearly, the coordinate surfaces with the unfiltered terrain exhibit much more small structure (Fig. 6a) than with the heavily filtered terrain (Fig. 6c). However, regardless of the terrain smoothing, the terrain influence on upper-level coordinate surfaces remains significant. The upper level influences of terrain are essentially eliminated with the hybrid vertical coordinate, as shown in Figs. 6b and 6d for the RAP-HYBRID and SMTH4-HYBRID cases, respectively. Terrain features influencing the coordinate surfaces are rapidly reduced and become almost negligible at levels above 11 km, even with high-frequency topography in RAP-HYBRID.
Vertical-level distributions along line AB in Fig. 5. Test cases for (a) RAP, (b) RAP-HYBRID, (c) SMTH4, and (d) SMTH4-HYBRID are shown. Note that 
Citation: Monthly Weather Review 147, 3; 10.1175/MWR-D-18-0334.1
The 18-h forecast for the calculated eddy dissipation rate (EDR; m2/3 s−1) turbulence and wind speed at 35 000 ft (10 668 m) are displayed in Fig. 7. Here, EDR is derived from the absolute vertical velocity divided by the Richardson number (
EDR (color shadings) turbulence forecasts from (a) RAP, (b) RAP-HYBRID, (c) SMTH1, (d) SMTH1-HYBRID, (e) SMTH4, and (f) SMTH4-HYBRID at 35 000 ft at 1800 UTC 2 Nov. Horizontal wind speed (red lines for 40 and 55 m s−1) and smooth PIREPs corresponding to EDR < 0.05 at ±2 h and ±2500 ft (~762 m) are depicted (black dots). Locations of (e) AB and (f) CD are marked.
Citation: Monthly Weather Review 147, 3; 10.1175/MWR-D-18-0334.1
Pilot reports (PIREPs) are also shown in Fig. 7. Here, flight observations of null turbulence (EDR < 0.05) are displayed with black dots and asterisks, and light turbulence reports (0.05 < EDR < 0.1) are marked with green asterisks. In the RAP configuration in Fig. 7a, large EDR is predicted over eastern Oregon, northern Idaho, and western Montana associated with cyclonic wind shear in the vicinity of the jet stream. However, with smoother topography, weaker turbulence is predicted in this region as shown in Figs. 7c and 7e, even though simulated jets are almost indistinguishable in the left panels in Fig. 7. Based on pilot reports, light EDR is expected over northeastern Oregon, and thus comparing the left panels in Fig. 7, the SMTH4 results in Fig. 7c provide the closest match. P16 showed that the sensitivity of the mountain-wave structure depended on the terrain features, but they did not explain the reason for the strong sensitivity of turbulence in the northern part of the upper-level jet. To further investigate this behavior, additional experiments with the hybrid vertical coordinate are shown in the right-hand panels in Fig. 7. In the HTF simulations, a large reduction of the turbulence in the jet stream region is evident, as shown in Figs. 7b, 7d, and 7f, and this reduction is largely independent of any terrain smoothing. The significant impact of the HTF coordinate in the absence of terrain smoothing (Figs. 7b vs 7a) is intriguing, as the reduced turbulence occurs in a region where terrain influences are relatively weak. This behavior raises questions as to the influence of the hybrid coordinate on the structure of mountain waves forced by the terrain and the role of the upper-level jet in promoting small-scale disturbances.
To address these questions, we evaluate two different vertical cross sections indicated in Fig. 7: one over the high terrain of the Rocky Mountains (Fig. 8), taken along line AB in Fig. 7e, and one along the region in which RAP predicted strong turbulence (Fig. 9), taken along line CD in Fig. 7f. The left-hand panels in Fig. 8 were shown in P16 and demonstrate the strong sensitivity of mountain waves to the shape of topography. We analyzed this same vertical cross section for simulations including the hybrid vertical coordinate, which are displayed in the right-hand panels of Fig. 8. While the amount of terrain smoothing has a significant influence on the mountain waves at smaller scales, the inclusion of the hybrid coordinate has relatively little influence on the mountain-wave structure (cf. the left and right panels for each terrain profile). There are noticeable small-scale waves in both the RAP and RAP-HYBRID for the unfiltered terrain (Figs. 8a,b), and the maximum/minimum magnitudes of vertical velocities (see brackets in each plot) depend much more on the terrain shape than the vertical coordinate. However, it does not mean that HTF is ineffective for upper-level turbulent forecasting in mountainous regions. As shown in the EDR analysis of Fig. 7b, with HTF, light turbulence over Nevada and Utah is reduced from Fig. 7a and is comparable to that in Figs. 7c and 7e using filtered topography.
Vertical velocity (shading) and theta (contours) along AB in Fig. 7 at 1800 UTC 2 Nov. Cases are from (a) RAP, (b) RAP-HYBRID, (c) SMTH1, (d) SMTH1-HYBRID, (e) SMTH4, and (f) SMTH4-HYBRID. Numbers in the brackets at the top indicate minimum and maximum of θ and vertical velocity in each plot.
Citation: Monthly Weather Review 147, 3; 10.1175/MWR-D-18-0334.1
Vertical cross section for vertical velocity (shading) and level distributions (lines) for (a) RAP, (b) RAP-HYBRID, (c) SMTH1, (d) SMTH1-HYBRID, (e) SMTH4, and (f) SMTH4-HYBRID along CD, which is shown in Fig. 7f. Numbers in the bracket of right top indicate minimum and maximum of vertical velocity in each plot.
Citation: Monthly Weather Review 147, 3; 10.1175/MWR-D-18-0334.1
Along the vertical cross section in Fig. 9, taken along line CD in Fig. 7f, there is no steep terrain, and the shape of the topography is quite similar in the experiments with differing amounts of terrain filtering. However, with the BTF coordinate, there are noticeable differences in the vertical velocity perturbations in the vicinity of the tropopause where the jet stream level winds are strong. In Fig. 9a, the phase of these disturbances do not tilt upwind with height (which would reflect vertical momentum transport), and their horizontal scale is less than 5dx (65 km) in RAP. These unphysical waves appear to result from numerical errors in computing the advection of strong winds near the troposphere along coordinate surfaces that retain small-scale structure due to the underlying terrain. As seen in Fig. 9, either terrain smoothing (Figs. 9c,e) or use of the hybrid coordinate (Fig. 9b) can be effective in reducing these errors. For the BTF coordinate, the minimum and maximum vertical velocities (indicated in brackets) become small as the topographic smoothing is increased, while for the HTF coordinate, the amplitudes are small regardless of the terrain smoothing.
c. Kinetic energy spectra
The kinetic energy spectra for all of the simulations discussed above are summarized in Fig. 10. As described by Skamarock (2004) and Errico (1985), we calculated the one-dimensional spectra, averaging over the 12–24-h integrations at 1-h intervals beginning at 1200 UTC 2 November. The one-dimensional Fourier transforms are calculated along isobaric levels, and results at 500, 300, and 100 hPa are depicted in the top, middle, and bottom panels of Fig. 10, respectively. The left-hand panels in Fig. 10 show spectra from the BTF simulations, and the right-hand panels are for the HTF simulations. Here, we focus on the higher wavenumbers in the simulations, corresponding to length scales less than 500 km. The overall behavior of these spectra is similar to those previously analyzed using WRF (e.g., Skamarock 2004; Waite and Snyder 2013). However, for the simulations presented here, some noticeable differences in the high-wavenumber tails of the spectra are apparent in Fig. 10. The BTF simulation with no terrain smoothing (RAP) shows an unphysical buildup of energy at small scales less than 4dx at all isobaric levels. As Skamarock (2004) indicated, energy buildup in small scales can be caused by insufficient diffusion near the grid scale. However, in this case, the artificial energy at small scales appears to be caused by the influences of small-scale terrain structure on the coordinate surfaces.
One-dimensional energy spectra during 1500–2100 UTC 2 Nov from (left) original WRF BTF vertical coordinate and (right) HTF vertical coordinate. Hourly predictions averaged over 12–24 h are used for the calculations computing the spectra at (a),(b) 500, (c),(d) 300, and (e),(f) 100 hPa. Note that only the smaller-scale wavelengths (≤500 km) are displayed to provide better clarity for the small-scale behavior.
Citation: Monthly Weather Review 147, 3; 10.1175/MWR-D-18-0334.1
With filtered topography (SMTH1 and SMTH4), the energy buildup at small scales is effectively removed in both the BTF and HTF simulations, as shown in Fig. 10. For the hybrid coordinate with no terrain smoothing (RAP-HYBRID), while this small-scale energy buildup is apparent at 500 hPa (Fig. 10b), it is not present at higher levels (Figs. 10d,f), where the terrain influences on the coordinate surfaces have been largely removed. This behavior suggests that the artificially large energy at the smallest scales in the absence of terrain smoothing is due to small-scale terrain influences on the coordinate surfaces rather than the direct forcing by small-scale features in the unfiltered terrain. This interpretation is also consistent with the results shown in Fig. 9, where small-scale perturbations are apparently due to numerical errors arising as strong horizontal winds are advected along coordinate surfaces that are perturbed by small-scale terrain influences.
d. Sensitivity test for 
The HTF simulations discussed above were obtained with 
(top) As in Fig. 7, but for RAP-HYBRID with (a) 
Citation: Monthly Weather Review 147, 3; 10.1175/MWR-D-18-0334.1
5. Summary
We have formulated a hybrid terrain-following sigma-pressure vertical coordinate for the nonhydrostatic WRF-ARW Model and evaluated it for an idealized test case and for a real-data case for an upper-air turbulence forecast. This hybrid formulation allows the terrain influences on the coordinate surfaces to be removed more rapidly with height than in the basic terrain-following sigma coordinate. The formulation is designed to transition more aggressively toward a pressure coordinate with increasing altitude, to limit the collapse of the vertical grid increments just above high terrain, and to smoothly transition to a pure pressure coordinate at a user-specified level. For the idealized test case, we demonstrate that in BTF coordinate simulations, significant artificial perturbations can arise as the advection of upper-level winds is computed along coordinate surfaces that are perturbed by residual influences of the terrain in the sigma coordinate formulation. This artificial behavior largely disappears with the HTF coordinate, which removes the terrain influences from the coordinate surfaces more rapidly with increasing altitude [as also documented by Schär et al. (2002) and Shaw and Weller (2016)].
In considering the impact of the hybrid coordinate in real-date simulations, we do not attempt here to address the influences on overall forecast accuracy. Rather, we evaluate a specific case for an upper-air turbulence forecast that was previously studied by P16 in assessing the influences of terrain smoothing on the resulting turbulence forecast. As P16 demonstrated, this terrain smoothing can remove the artificial amplification of upper-level disturbances in WRF-ARW simulations that may arise when small scales in the terrain are not filtered. We extend the analyses in that study by demonstrating that the overprediction of upper-level turbulence can also be reduced by using the HTF coordinate, even in the absence of any terrain smoothing. In the EDR turbulence analysis, with RAP configurations, the EDR is overestimated in regions where the upper-level jet is strong and exhibits strong horizontal wind shear. The disturbances that produce these large EDR values in BTF simulations appear to result from numerical errors in representing strong advection along irregular coordinate surfaces that are present due to the influence of small-scale terrain features. With the HTF coordinate, these errors are much reduced as the coordinate surfaces are nearly constant pressure surfaces in the vicinity of the jet. These simulations suggest that either terrain smoothing or a hybrid vertical coordinate may be effective in reducing the overestimation of upper-level waves that contribute to false alarm predictions of air turbulence.
We also conducted simulations (not discussed above) to see if larger internal diffusion is effective in reducing these artificial disturbances that arise with the BTF coordinate. For one test, we switched the calculation of horizontal advection from fifth to third order (increasing the implicit diffusion), and in another, we turned on an explicit sixth-order horizontal filter [as configured and implemented by Knievel et al. (2007)]. Although these filters should increase the dissipation at small scales, they did not noticeably reduce the disturbances contributing to the overprediction of turbulence. The ineffectiveness of these filters may be due in part to the fact that they are computed along coordinate surfaces instead of along constant height or pressure surfaces and therefore may be adversely affected by small-scale terrain influences on the coordinate surfaces.
In evaluating the model kinetic energy, we find that with the BTF coordinate and no terrain smoothing (the RAP case) unphysical energy accumulation is significant at high wavenumbers at all levels. With filtered topography, both the SMTH1 and SMTH4 cases exhibit no artificial accumulation of energy at high wavenumbers. In RAP-HYBRID, since the HTF coordinate transitions more rapidly to constant pressure coordinate surfaces, unphysical buildup of energy is not observed at the upper levels. The reduction in the noise that contributes to the overprediction of turbulence is somewhat sensitive to how rapidly the hybrid coordinate transitions to a pressure coordinate, as determined by the parameter 
Acknowledgments
(SHP) This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST; 2018R1A2B6008078) and (in part) by the Yonsei University Future-leading Research Initiative of 2018-22-0021. (JBK) Funding for the NCAR portion of this research was provided through support from the National Science Foundation under Cooperative Support Agreement AGS-0856145 and from the NOAA NCAR IDIQ T0006, distributed through the Developmental Testbed Center. NCAR is sponsored by the National Science Foundation.
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, https://doi.org/10.1175/1520-0493(1997)125<2293:ROTBSC>2.0.CO;2.
Bubnová, R., G. Hello, P. Bénard, and J.-F. Geleyn, 1995: Integration of the fully elastic equations cast in the hydrostatic pressure terrain-following coordinate in the framework of the ARPEGE/Aladin NWP system. Mon. Wea. Rev., 123, 515–535, https://doi.org/10.1175/1520-0493(1995)123<0515:IOTFEE>2.0.CO;2.
Dempsey, D., and C. Davis, 1998: Error analysis and tests of pressure-gradient force schemes in a nonhydrostatic, mesoscale model. Preprints, 12th Conf. on Numerical Weather Prediction, Phoenix, AZ, Amer. Meteor. Soc., 236–239.
Errico, R. M., 1985: Spectra computed from a limited area grid. Mon. Wea. Rev., 113, 1554–1562, https://doi.org/10.1175/1520-0493(1985)113<1554:SCFALA>2.0.CO;2.
Kim, J.-H., R. Sharman, M. Strahan, J. W. Scheck, C. Bartholomew, J. C. H. Cheung, P. Buchanan, and N. Gait, 2018: Improvements in nonconvective aviation turbulence prediction for the World Area Forecast System (WAFS). Bull. Amer. Meteor. Soc., 99, 2295–2311, https://doi.org/10.1175/BAMS-D-17-0117.1.
Kirshbaum, D. J., R. Rotunno, and G. H. Bryan, 2007: The spacing of orographic rainbands triggered by small-scale topography. J. Atmos. Sci., 64, 4222–4245, https://doi.org/10.1175/2007JAS2335.1.
Klemp, J. B., 2011: A terrain-following coordinate with smoothed coordinate surfaces. Mon. Wea. Rev., 139, 2163–2169, https://doi.org/10.1175/MWR-D-10-05046.1.
Klemp, J. B., W. C. Skamarock, and J. Dudhia, 2007: Conservative split-explicit time integration methods for the compressible nonhydrostatic equations. Mon. Wea. Rev., 135, 2897–2913, https://doi.org/10.1175/MWR3440.1.
Knievel, J. C., G. H. Bryan, and J. P. Hacker, 2007: Explicit numerical diffusion in the WRF Model. Mon. Wea. Rev., 135, 3808–3824, https://doi.org/10.1175/2007MWR2100.1.
Laprise, R., 1992: The Euler equations of motion with hydrostatic pressure as an independent coordinate. Mon. Wea. Rev., 120, 197–207, https://doi.org/10.1175/1520-0493(1992)120<0197:TEEOMW>2.0.CO;2.
Mahrer, Y., 1984: An improved numerical approximation of the horizontal gradients in a terrain-following coordinate system. Mon. Wea. Rev., 112, 918–922, https://doi.org/10.1175/1520-0493(1984)112<0918:AINAOT>2.0.CO;2.
Park, S.-H., W. C. Skamarock, J. B. Klemp, L. D. Fowler, and M. G. Duda, 2013: Evaluation of global atmospheric solvers using extensions of the Jablonowski and Williamson baroclinic wave test case. Mon. Wea. Rev., 141, 3116–3129, https://doi.org/10.1175/MWR-D-12-00096.1.
Park, S.-H., J.-H. Kim, R. D. Sharman, and J. B. Klemp, 2016: Update of upper level turbulence forecast by reducing unphysical components of topography in the numerical weather prediction model. Geophys. Res. Lett., 43, 7718–7724, https://doi.org/10.1002/2016GL069446.
Schär, C., D. Leuenberger, O. Fuhrer, D. Lüthi, and C. Girard, 2002: A new terrain-following vertical coordinate formulation for atmospheric prediction models. Mon. Wea. Rev., 130, 2459–2480, https://doi.org/10.1175/1520-0493(2002)130<2459:ANTFVC>2.0.CO;2.
Sharman, R. D., and J. M. Pearson, 2017: Prediction of energy dissipation rates for aviation turbulence. Part I: Forecasting nonconvective turbulence. J. Appl. Meteor. Climatol., 56, 317–337, https://doi.org/10.1175/JAMC-D-16-0205.1.
Shaw, J., and H. Weller, 2016: Comparison of terrain-following and cut-cell grids using a nonhydrostatic model. Mon. Wea. Rev., 144, 2085–2099, https://doi.org/10.1175/MWR-D-15-0226.1.
Sheridan, P., and S. Vosper, 2012: High-resolution simulations of lee waves and downslope winds over the Sierra Nevada during T-REX IOP 6. J. Appl. Meteor. Climatol., 51, 1333–1352, https://doi.org/10.1175/JAMC-D-11-0207.1.
Simmons, A. J., and D. M. Burridge, 1981: An energy and angular-momentum conserving finite-difference scheme and hybrid vertical coordinates. Mon. Wea. Rev., 109, 758–766, https://doi.org/10.1175/1520-0493(1981)109<0758:AEAAMC>2.0.CO;2.
Skamarock, W. C., 2004: Evaluating mesoscale NWP models using kinetic energy spectra. Mon. Wea. Rev., 132, 3019–3032, https://doi.org/10.1175/MWR2830.1.
Skamarock, W. C., and Coauthors, 2008: A description of the Advanced Research WRF version 3. NCAR Tech. Note NCAR/TN-475+STR, 113 pp., https://doi.org/10.5065/D68S4MVH.
Steppeler, J., H.-W. Bitzer, M. Minotte, and L. Bonaventura, 2002: Nonhydrostatic atmospheric modeling using a z-coordinate representation. Mon. Wea. Rev., 130, 2143–2149, https://doi.org/10.1175/1520-0493(2002)130<2143:NAMUAZ>2.0.CO;2.
Waite, M. L., and C. Snyder, 2013: Mesoscale energy spectra of moist baroclinic waves. J. Atmos. Sci., 70, 1242–1256, https://doi.org/10.1175/JAS-D-11-0347.1.
