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  • View in gallery

    Wave isochrones and wave activity zones for 15 December 1987 event. Isochrones (times in UTC 15 December) are of troughs of observed waves B (dashed) and C (solid). Times in parentheses indicate that wave is ill defined. For earliest times, waves are weak and positions more approximate. The “L” indicates position of the surface low at the given hour UTC (after Powers and Reed 1993).

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    Model domains for 30/10 experiments. Outer frame is 30-km domain and inner frame is 10-km domain.

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    Sea level pressure fields for hours 3, 6, and 12 of experiment 30KCTRL (0300, 0600, and 1200 UTC 15 December 1987). Contour interval is 1 mb, and frontal positions are indicated. Waves labeled, and wave trough axes analyzed with solid lines. (a) Hour 3, Line AB marks position of cross section in Fig. 6a; (b) hour 6; (c) hour 12.

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    Sea level pressure field for hour 6 of experiment REFL (0600 UTC). Contour interval is 1 mb. Waves labeled and wave trough axes analyzed with solid lines.

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    Sea level pressure field for hour 3 (0300 UTC) of experiment EVR. Contour interval is 1 mb. Waves labeled and wave trough axes analyzed with solid lines. Line AB marks position of cross section in Fig. 6b.

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    Cross section of θ and w through wave 1 in experiments 30KCTRL and EVR at hour 3 (0300 UTC). Cross-sectional lengths are 732 km, and locations are marked as line AB in Figs. 3a and 5; θ is solid line and the interval is 4 K; w is short-dashed line upward; zero contour is the solid boldface line; regions of downward motion bounded by zero contour and negative minima marked; interval is 10 cm s−1. (a) 30KCTRL and (b) EVR.

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    Sea level pressure field for hours 3, 6, 9, and 12 of experiment 10KCTRL (0300, 0600, 0900, and 1200 UTC). Contour interval is 1 mb. Waves labeled and wave trough axes analyzed with solid lines. (a) Hour 3. Area in dashed outline in northern Illinois is that of the 3.3-km grid of experiment 3.3K. Line AB marks position of cross section in Fig. 11. (b) Hour 6. Surface frontal positions marked; “D” and “N” mark locations of soundings in Figs. 8a and 8b, respectively. (c) Hour 9. Surface frontal positions marked. (d) Hour 12.

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    Model soundings from hour 6 of experiment 10KCTRL (0600 UTC). Full barb is 10 kt; pennant is 50 kt. (a) Ducted region sounding. Sounding location marked “D” in Fig. 7b. (b) Nonducted region sounding. Sounding location marked “N” in Fig. 7b.

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    Observational and 10KCTRL surface analyses for 1100 UTC 15 December 1987. Sea level pressure solid; contour interval is 2 mb. (a) Observational analysis. Approximate positions of troughs of waves B and C analyzed with dotted lines. Temperatures and dewpoints are in degrees Celsius. Winds: full barb is 10 kt. (b) 10KCTRL analysis. Wave troughs analyzed with dotted lines.

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    Observed and 10KCTRL 500-mb analyses for 1200 UTC 15 December 1987. Height contours solid; interval is 60 m. Isotachs dashed; contour interval is 10 m s−1. Wind vectors plotted in (b). (a) Observed and (b) 10KCTRL. Coarse grid shown.

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    Cross section of θ, w, and steering-level isotach through wave 1 in experiment 10KCTRL at hour 3 (0300 UTC). Cross-sectional length is 270 km, and location is marked as line AB in Fig. 7a; θ thin solid, interval is 4 K; w dashed upward, bold solid downward; interval is 65 cm s−1; zero contour bold solid and labeled; updraft core maximum is 6.12 m s−1. Heavy long-dashed line represents the steering level wind isotach (28 m s−1), “R” marks the location of the wave ridge, and arrows indicate the sense of vertical motion.

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    Sea level pressure fields for hours 6, 9, and 12 of experiment 3.3K (0600, 0900, and 1200 UTC). Contour interval is 1 mb. Wave trough and ridge axes analyzed with solid and dashed lines, respectively. Gridpoint numbers shown on plot border. Grid position relative to 10-km grid shown in 7a. (a) Hour 6; (b) hour 9; (c) hour 12.

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    High-pass-filtered observed barogram for DBQ for period 0020–1220 UTC 15 December 1987. Ordinate in millibars; abscissa lists time in hours from 2120 UTC 14 December 1987. Periods of less than 3 h retained. Location of DBQ shown in Fig. 1.

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    Sea level pressure fields for hours 17.33–18.33 (1720–1820 UTC 9 May 1979) of simulation of 9 May 1979 event. Model grid size (10 km) and physics as in 10KCTRL. Contour interval is 1 mb. Gravity wave troughs (T) and ridges (R) labeled and marked by solid and dashed lines, respectively. (a) Hour 17.33; (b) hour 17.67; (c) hour 18; (d) hour 18.33.

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    Bandpass-filtered observed barograms for DEC, MMO, and NBU for period 2120–2320 UTC 14–15 December 1987. Ordinate in millibars; abscissa in hours from 2120 UTC 14 December. Peaks of observed waves A, B, and C labeled. Filter retains periods of about 30 min–3 h. Locations of DEC, MMO, and NBU shown in Fig. 1.

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    Observed barograms for JLN and SGF for period 2000–0600 UTC 14–15 December 1987. Locations of JLN and SGF shown in Fig. 1. Ordinate in millibars, with 8 mb added to all SGF values for plotting purposes. Abscissa in hours from 2000 UTC 14 December. Arrows over SGF trace indicate approximate period of wave C.

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    Sea level pressure field for hours 4, 6, and 8 of experiment RD (0400, 0600, and 0800 UTC). Contour interval is 1 mb. Waves labeled and trough axes analyzed with solid lines. (a) Hour 4; (b) hour 6; (c) hour 8.

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    Sea level pressure fields for hours 10 and 12 of experiment RM (1000 and 1200 UTC). Contour interval is 1 mb. Waves labeled and wave trough axes analyzed with solid lines. (a) Hour 10; (b) hour 12.

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    Sea level pressure field for hours 3 and 6 (0300 and 0600 UTC) of experiment NLC. Contour interval is 1 mb. Waves labeled and wave trough axes analyzed with solid lines. (a) Hour 3; (b) hour 6.

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    Sea level pressure field for hours 3 and 12 of experiment KS (0300 and 1200 UTC). Contour interval is 1 mb. Waves labeled and wave trough axes analyzed with solid lines in (a). (a) Hour 3; (b) hour 12.

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    Observed precipitation (cm) for 0000–1200 UTC 15 December 1987. Contour interval is 1 cm except for first plotted contour, which is 0.5 cm. Circles locate precipitation recording stations.

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    Total precipitation (mm) on fine grid for experiments 10KCTRL, KS, and KFS for hours 0–12. Contour interval is 10 mm except for first plotted contour, which is 5 mm. Gridpoint numbers shown on plot borders. (a) 10KCTRL; (b) KS; (c) KFS.

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    Parameterized precipitation (mm) on fine grid for experiments 10KCTRL, KS, and KFS for hours 0–12. Contour interval is 5 mm except for the first and second plotted contours, which are 0.1 and 0.5 mm, respectively. Gridpoint numbers shown on plot borders. (a) 10KCTRL; (b) KS; (c) KFS.

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    Sea level pressure field for hours 3 and 12 of experiment KFS (0300 and 1200 UTC). Contour interval is 1 mb. Selected waves labeled and wave trough axes analyzed with solid lines. (a) Hour 3; (b) hour 12.

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    Observed, 10KCTRL, and 30KCTRL high-pass-filtered barograms for SPI. Abscissa in hours from 0400 UTC 15 December 1987; total period shown corresponds to 0400–1200 UTC 15 December. Ordinate in millibars. (a) Observed; (b) 10KCTRL; (c) 30KCTRL.

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    Observed, 10KCTRL, and 30KCTRL smoothed ACF and ME spectra for SPI. ACF spectra solid; ME spectra dashed. Lag (M) and pole (P) ratios shown, and numbers of points in the records are 97 (observed), 99 (10KCTRL), and 91 (30KCTRL); 90% confidence intervals marked with dotted lines. (a) Observed spectra; M = 48, P = 15. (b) 10KCTRL spectra; M = 49, P = 15. (c) 30KCTRL spectra; M = 45, P = 14.

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    Observed, 10KCTRL, and 30KCTRL smoothed ACVF spectra for SPI. Observed spectrum solid, 10KCTRL spectrum dashed, and 30KCTRL spectrum dotted. Lag ratios are as in Figs. 26a–c for each record source.

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    Observed, 10KCTRL, and 30KCTRL smoothed ME spectra for RFD and ORD. Arrows indicate significant peaks. Observed spectrum solid, 10KCTRL spectrum dashed, and 30KCTRL spectrum dotted. Pole ratios are as in Figs. 26a–c for each record source. (a) RFD and (b) ORD.

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Numerical Model Simulations of a Mesoscale Gravity Wave Event: Sensitivity Tests and Spectral Analyses

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  • 1 National Center for Atmospheric Research, Boulder, Colorado
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Abstract

This study presents numerical model experiments and spectral investigations involving a mesoscale gravity wave event. Its purposes are to determine the sensitivity of mesoscale gravity wave simulation to model configuration and physics and to evaluate spectrally both an observed and simulated wave episode. The case is the large-amplitude wave event of 15 December 1987 in the central United States, and the model employed is the Pennsylvania State University–National Center for Atmospheric Research Mesoscale Model 5 (MM5). The primary MM5 configuration features a nested domain with 10-km horizontal resolution, 41 σ-level vertical resolution, nonhydrostatic physics, a radiative upper boundary condition, an explicit moist process scheme with ice physics, and the Grell cumulus parameterization. Experiments are performed to investigate the effects on wave simulation of upper boundary condition, hydrostatics, vertical and horizontal resolution, and moist physics.

From the sensitivity tests it is found that wave development and maintenance are insensitive to the upper boundary condition, that wave simulation is insensitive to hydrostatic/nonhydrostatic differences at 10-km horizontal resolution, and that wave production and structure are insensitive to vertical resolution. With respect to horizontal resolution, an expanded wave scale spectrum and shorter minimum wavelengths appear as grid size is decreased. With respect to moist physics, latent heating is found to be necessary for model wave development, and model wave production and strength are, to an extent, sensitive to the moist process package employed. Elevated convection is the model wave forcing mechanism, and, at a given grid size, wave response varies with the degree to which such convection is explicit.

Statistical analyses consisting of high-pass filtering and power spectrum analysis of observed and model surface pressure data are performed. The filtering analyses indicate more realistic simulated wave activity as horizontal resolution is increased. The spectral analyses uncover bimodal distributions in both the observed and model spectra and show that the model, in general, does not overproduce mesoscale gravity wave energy. With respect to the model alone, the spectral analyses reveal that as grid size is decreased the significant spectral frequencies shift upward while both the average power in the mesoscale wave band and spatial variability of the power in such band decrease.

Corresponding author address: Dr. Jordan G. Powers, MMM Division, NCAR, P.O. Box 3000, Boulder, CO 80307-3000.

Email: powers@ncar.ucar.edu

Abstract

This study presents numerical model experiments and spectral investigations involving a mesoscale gravity wave event. Its purposes are to determine the sensitivity of mesoscale gravity wave simulation to model configuration and physics and to evaluate spectrally both an observed and simulated wave episode. The case is the large-amplitude wave event of 15 December 1987 in the central United States, and the model employed is the Pennsylvania State University–National Center for Atmospheric Research Mesoscale Model 5 (MM5). The primary MM5 configuration features a nested domain with 10-km horizontal resolution, 41 σ-level vertical resolution, nonhydrostatic physics, a radiative upper boundary condition, an explicit moist process scheme with ice physics, and the Grell cumulus parameterization. Experiments are performed to investigate the effects on wave simulation of upper boundary condition, hydrostatics, vertical and horizontal resolution, and moist physics.

From the sensitivity tests it is found that wave development and maintenance are insensitive to the upper boundary condition, that wave simulation is insensitive to hydrostatic/nonhydrostatic differences at 10-km horizontal resolution, and that wave production and structure are insensitive to vertical resolution. With respect to horizontal resolution, an expanded wave scale spectrum and shorter minimum wavelengths appear as grid size is decreased. With respect to moist physics, latent heating is found to be necessary for model wave development, and model wave production and strength are, to an extent, sensitive to the moist process package employed. Elevated convection is the model wave forcing mechanism, and, at a given grid size, wave response varies with the degree to which such convection is explicit.

Statistical analyses consisting of high-pass filtering and power spectrum analysis of observed and model surface pressure data are performed. The filtering analyses indicate more realistic simulated wave activity as horizontal resolution is increased. The spectral analyses uncover bimodal distributions in both the observed and model spectra and show that the model, in general, does not overproduce mesoscale gravity wave energy. With respect to the model alone, the spectral analyses reveal that as grid size is decreased the significant spectral frequencies shift upward while both the average power in the mesoscale wave band and spatial variability of the power in such band decrease.

Corresponding author address: Dr. Jordan G. Powers, MMM Division, NCAR, P.O. Box 3000, Boulder, CO 80307-3000.

Email: powers@ncar.ucar.edu

1. Introduction

Mesoscale gravity waves are tropospheric gravity waves with horizontal wavelengths and periods typically in the ranges of 50–350 km and 0.5–3.5 h, respectively. Although the corpus of mesoscale gravity wave studies has relied almost exclusively upon observational data, an exception has been the study of Powers and Reed (1993) (hereafter PR), the first published work to use a numerical forecast model for the express purpose of investigating a mesoscale gravity wave event. Powers and Reed presented observational analyses and model simulations of the large-amplitude wave event of 15 December 1987 in the central United States. This case had been studied previously in observational investigations by Schneider (1990) and Marwitz and Toth (1993).

The study of mesoscale gravity waves with mesoscale forecast models is a fairly new methodology. Works other than PR that have analyzed such waves in numerical models are those of Zhang and Fritsch (1988), Cram et al. (1992), Pokrandt et al. (1993, 1994), and Kaplan et al. (1994). Zhang and Fritsch (1988) reported a 250–300-km wave produced by the Pennsylvania State University–National Center for Atmospheric Research (NCAR) Mesoscale Model 4 (MM4) in a simulation of a mesoscale convective system (MCS). Cram et al. (1992) analyzed a squall line simulated by the Colorado State University RAMS (Regional Amospheric Modeling System) model that was associated with a gravity wave of mesoscale dimension. Besides PR, modeling works expressly focusing on mesoscale gravity wave events are described in conference preprints by Pokrandt et al. (1993, 1994) and Kaplan et al. (1994). The reports of Pokrandt et al. (1993, 1994) summarize results of simulations of the 15 December 1987 case with the University of Wisconsin Nonhydrostatic Modeling System. Using grid sizes down to 16.6 km, Pokrandt et al. (1993) described some success in the event’s simulation, with the generation of a mesoscale gravity wave with a length of about 200 km and a phase speed of 50 m s−1. Convection (grid resolved) was identified as the likely wave source. Pokrandt et al. (1994) investigated the development mechanisms of the observed waves of the December 1987 event. They hypothesized that the waves began as the rising branch of a Sawyer–Eliassen circulation. Kaplan et al. (1994) described simulations of the wave event of 11 July 1981 in eastern Montana (see Koch and Golus 1988). Using the hydrostatic GMASS model with a 16-km/8-km nested grid setup, they offered that a complex process of geostrophic adjustment was responsible for waves seen in one of their simulations.

Powers and Reed (1993) performed an observational analysis of the 15 December 1987 case followed by model experiments designed to investigate wave simulation and development. Their observational analysis found the three most prominent waves of the event to have had wavelengths of 100–160 km, phase speeds of approximately 30 m s−1, and double amplitudes (filtered) of up to 7 mb. While an effective wave duct was found, evidence of convection suggested that wave-CISK (conditional instability of the second kind) could have contributed to wave maintenance and amplification. As for wave genesis, the evidence allowed each of convection, shearing instability, and geostrophic adjustment as source mechanisms. This was in accord with the prior study of Schneider (1990), which found that the evidence supported convection, shear, and geostrophic adjustment; and the study of Marwitz and Toth (1993), which found that the evidence supported convection and shear.

The modeling of PR was done with the Pennsylvania State University–NCAR MM4 with a 30-km grid size. The MM4 produced mesoscale gravity waves and reasonably simulated wave structure, length, amplitude, and linear extent. The three analyzed waves in the PR control run had wavelengths of 150–200 km, average phase speeds of 38–56 m s−1, and double amplitudes (unfiltered) of 4–13 mb. The modeling results indicated convective processes as both the model and actual wave genesis mechanism, although shearing instability remained a possibility. Although an experiment removing the latent heat of condensation and displaying a lack of wave development was considered to discount geostrophic adjustment as the cause of the model or observed waves, it may be argued that its possibility as a wave source would remain, as the absence of latent heating could have sufficiently altered the upper-level jet in the model and thus reduced any associated imbalances. The possibility of such a limitation on the noted PR experiment is recognized.

Sensitivity experiments in PR showed that (i) the model waves did not stem from initialization imbalances (adiabatic nonlinear normal mode initialization was used), (ii) model wave production was not limited to the early hours of a simulation, (iii) model mesoscale waves could be produced using grid sizes up to 45 km, and (iv) model wave development did not occur when latent heating was removed. Model output analyses revealed that a wave duct that was effective with respect to the observed waves did cover the wave activity area, but that the model duct was not consistently effective with respect to all of the analyzed model waves. The analyses also showed that elevated convection travelling with the model waves represented a continual forcing mechanism, and such analyses (and observations) suggested convection’s potential as a maintenance/reinforcement mechanism for the observed waves. The limitations of the simulations, stated by PR, were (i) the reflective nature of the upper boundary, (ii) the 30-km grid size, and (iii) the undetermined wave sensitivity to moist process schemes. The reproducibility of model success in simulating other cases was also unknown.

The purposes of the current study are to resolve the modeling issues raised in PR, to investigate modeling issues that have developed since and to perform comparative spectral analyses of observed and simulated wave events. The testbed for the MM5 experiments is the 15 December 1987 event. Powers (1994) has explored the issue of the MM5’s ability to simulate different mesoscale gravity wave events through simulations of the events of 9 May 1979 in the central United States described by Stobie et al. (1983) and of 27 February 1984 in the southeastern United States described by Bosart and Seimon (1988). It has been found that the model can reproduce mesoscale gravity wave activity in the regions and during the periods that such was observed in these cases, and thus the MM5’s ability to produce mesoscale gravity waves is not limited to simulations of the 15 December 1987 event.

Section 2 of this paper recaps the 15 December 1987 wave event. Section 3 describes the mesoscale model and the experiments performed. Section 4 presents the modeling results, while section 5 presents spectral analyses of the observational and model data. Section 6 concludes with a summary of the findings.

2. Synopsis of wave event

The wave event of 15 December 1987 is now summarized, and to avoid undue repetition of other published accounts the reader is referred to Schneider (1990) and PR for complete details. The wave activity occurred in the synoptic context of explosive cyclogenesis over land. At 0000 UTC 15 December (PR, Fig. 1a) a surface low of approximately 1000 mb was organizing in southern Arkansas. It intensified while tracking northward through the Mississippi Valley and by 1200 UTC exhibited a central pressure of 980 mb in north-central Illinois (PR, Fig. 1c). Peak intensity and a depth of about 980 mb were maintained through 1800 UTC as the low center moved slowly north-northeastward. Figure 1 shows selected positions of the synoptic low.

The event’s wave activity consisted of a long period of waves of varying mesoscales, with many of such disturbances having amplitudes over 1 mb. The main area of wave activity stretched from southwestern Missouri across Illinois and into Wisconsin and Michigan (Fig. 1). The three strongest waves were denoted A, B, and C in PR; were best defined from 0500 to 1300 UTC; and were identified and tracked through the bandpass filtering and cross correlation of barogram data. These waves had wavelengths of 100–160 km, phase speeds of 30 m s−1, and maximum double amplitudes (derived from filtered barograms) of nearly 7 mb (Table 1). In this paper, the wave “core” area shall refer to the region in which the amplitudes of waves A–C were, in general, maximized: that is, central to northern Illinois.

Figure 1 shows the isochrones for waves B and C, the two strongest and last waves in the event. It should be noted that isochrones such as these are simplifications of the wave activity. They reflect only two of the event’s numerous waves and do not reflect the variations and nonuniformity in wave strength, length, and coherence along wave fronts or over time, which the data show (Powers 1994).

The wave activity area lay mostly northwest to north of the surface warm and occluded fronts (PR, Figs. 1a–c, 4; see also Fig. 9a, discussed below). A distinct lower-tropospheric stable layer (duct) through which the waves propagated was associated with the warm frontal surface aloft. The waves moved in the direction of the generally south-southwesterly mid- to upper-level flow. The synoptic setting and wave movement were in most respects typical of those for mesoscale gravity wave events (see Uccellini and Koch 1987).

3. Model description and experiments performed

a. Model description

The model used is the nonhydrostatic, primitive equation Pennsylvania State University–NCAR Mesoscale Model 5 (MM5). For a full treatment of the MM5, the reader may consult Dudhia (1993), Grell et al. (1995), and Haagenson et al. (1994).

The MM5 is used with horizontal grid sizes of 30, 10, and 3.3 km. The primary setup consists of a 30-km coarse mesh containing a 10-km fine mesh covering the area of observed wave activity (Fig. 2). For all 30-km/10-km (hereafter 30/10) experiments, 41 σ levels (defined below) from the ground to 100 mb provide vertical resolution. In addition to the 30/10 experiments, three experiments have been performed using a single 30-km grid only. Two of the single 30-km grid experiments have 24 σ levels, and one has 49 σ levels.

In the MM5 the basic variables of p, T, and ρ are defined by a constant reference state and perturbations from it (Grell et al. 1995). For example, pressure is defined by
px, y, z, tp0zpx, y, z, t
where p0 is the reference, or base state, pressure and p′ is the deviation from this. In this setting, pressure perturbation p′ is the predicted variable.
The MM5 uses a terrain-following σ vertical coordinate defined as
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where ps and pt are the reference state surface and model top pressures (Grell et al. 1995). The pressure p at a grid point is thus given by
ppsptσptp

On the upper boundary a radiative condition simulates the transmission of vertically propagating gravity wave energy (Klemp and Durran 1983; Bougeault 1983). On the lateral boundaries, relaxation or “nudging” conditions blend the model-predicted values along the flanks with those from a large-scale analysis (Grell et al. 1995).

The initialization technique of removal of the mean divergence in a column is employed. This reduces external gravity wave modes and noise early in the forecast (Washington and Baumhefner 1975). In PR, the use of adiabatic implicit normal mode initialization, which balances initial mass and wind fields to remove unrealistic gravity wave noise (Errico and Bates 1988), was tested and was found to have no significant impact on mesoscale gravity wave simulation.

The data are initialized by running the model for a 12-h period, with the hour 12 forecast becoming the first guess field for the sensitivity test runs. This first guess field is objectively reanalyzed with upper-air and surface observations (Manning and Haagenson 1992) as well as data for the surface, 850-, 700-, 500-, 300-, 200-, and 100-mb levels obtained from subjective analyses. The inclusion of the latter data, or “bogussing” (see Manning and Haagenson 1992), is done for conformity with the procedure used in the PR experiments, and the data were added in the modeling of PR in an attempt to capture better the advance of the surface cold front.

The model employs an explicit scheme for grid-resolved precipitation and a cumulus parameterization for subgrid-scale moist processes. The explicit scheme is based on that of Hsie et al. (1984) and includes ice physics (Dudhia 1989). The primary convective scheme is that of Grell (1993). The explicit scheme has prognostic equations for water vapor, cloud water, and rain water, and it operates when a grid point is saturated. In the ice physics scheme, cloud water and rainwater below 0°C are considered to be cloud ice and snow, respectively. The explicit scheme parameterizes these processes: condensation of water vapor into cloud water (ice), accretion of rain and snow, evaporation (sublimation) of rain (snow), initiation of ice crystals, and sublimation and deposition of cloud ice (Grell et al. 1995).

The Grell cumulus parameterization, which was developed from the Arakawa–Schubert scheme (Arakawa and Schubert 1974), relates the amount of convection to the rate of destabilization by the larger-scale environment. Its trigger function involves a check of the lifting necessary to bring a parcel to its LFC (level of free convection), beginning first with a parcel at the level of maximum moist static energy (see Bresch 1994). The scheme proceeds if the lifting depth is less than or equal to 50 mb. The Grell scheme includes the effects of moist convective-scale downdrafts and assumes a single cloud updraft and downdraft per grid box (Grell et al. 1995). In two other experiments, the Kuo (Grell et al. 1995) and Kain–Fritsch (Kain and Fritsch 1993) parameterizations substitute for the Grell scheme. In the Kuo scheme, the triggering of convection depends on the vertically integrated moisture convergence (Grell et al. 1995). In the Kain–Fritsch scheme, the triggering is based on a check of the temperature of a boundary layer parcel that has a temperature perturbation added to it upon being lifted to its LCL (lifting condensation level) (Kain and Fritsch 1993). The temperature perturbation is based on the gridpoint vertical velocity of the parcel, and convection commences if the parcel’s modified temperature exceeds the environmental temperature.

b. Experiments performed

The experiments, listed in Table 2, vary model configuration and physics in simulations of the 15 December 1987 event. Unless otherwise noted, the runs are nonhydrostatic, use the explicit/Grell moist physics package and the radiative upper boundary condition, and begin at 0000 UTC 15 December 1987 (following the 12-h preliminary forecast).

Simulation 1 (30KCTRL) is a full-physics, 30-km, 24 σ-level run that serves as a control for the other 30-km, single-domain experiments (experiments 2 and 3). Experiment 2 (REFL) is identical to experiment 1 except that it substitutes a reflective upper boundary condition for the radiative one: it tests the impact of a reflective model top on the development and maintenance of model mesoscale gravity waves. The MM4 simulations in PR used a reflective upper boundary condition and raised the issue of a reflective top’s assistance in model wave growth and maintenance.

Experiment 3 (EVR) increases the vertical resolution in the single 30-km grid setup from 24 to 49 σ levels. Its purposes are to determine the effect on wave simulation of strict consistency between the horizontal and vertical resolution, and to determine the effects of enhanced vertical resolution on model wave development and structure. Regarding the former issue, Persson and Warner (1991) have shown that resolution disparity can yield spurious gravity waves in frontal zones. In this case the warm front aloft is implicated. To avoid the problem, Persson and Warner have advocated a grid aspect/frontal slope (AS) ratio of 1, where
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Here, Δx = 30 km, the estimated frontal slope of 1:172 yields s ≈ 0.630 mb km−1, and the Δp between vertical levels is approximately 19 mb. Therefore, the use of 49 σ levels gives AS = 1.0.

Experiment 4 (10KCTRL) is the control simulation for the 10-km nested runs; it will alternately be referred to as the “control run.” Its purposes are to reveal the effects of finer resolution on the simulation of the event and to serve as a touchstone for the other 10-km experiments. Experiment 5 (HYD) is identical to experiment 4 except for its use of hydrostatic physics. It is designed to reveal the impact of hydrostatic model physics at 10-km resolution.

Experiments 6–11 examine the effects of moist physics on wave simulation. Experiment 6 (RD) is designed to reveal the effects of the removal of latent heating on developed model waves. The latent heat of condensation is set to zero following 4 h of full-physics simulation; this type of run is referred to as “fake dry.” In contrast, PR performed an experiment that barred latent heating from the outset of a run. An extension of RD, experiment 7 (RM) reintroduces latent heating after 4 h of fake dry simulation in RD. This run thus consists of an initial 4 h of full-physics simulation, followed by 4 h of fake dry simulation, followed by full-physics simulation. The latent heating effects recommence at hour 8. The purposes of RM are to uncover whether model mesoscale gravity waves can be generated after the environmental deprivation of latent heating and, if so, how readily this can occur.

Experiment 8 (NLC) aims to reveal the impacts of latent cooling on wave development. The latent cooling from evaporation/sublimation and melting is set to zero from the outset, yet latent heating from condensation/deposition and freezing remains.

Experiments 9 (KS) and 10 (KFS) substitute the Kuo and Kain–Fritsch cumulus schemes for the Grell as the cumulus parameterization. They probe the impact on model wave production of changing the moist physics package.

Experiment 11 (3.3K) is a 3.3-km simulation exploring the effects of very fine resolution on wave development and wave scale. Otherwise identical to 10KCTRL, it nests a 3.3-km domain within the 10-km grid. The 3.3-km grid is turned on at hour 4 and operates for 8 h (0400–1200 UTC 15 December 1987). High cost limited the length of this simulation to the given period.

4. Experiment results and analyses

In the results to be presented, it is the mesoscale gravity wave simulations that are of primary concern. Thus, the synoptic element verifications, performed elsewhere for this study (Powers 1994), are not presented here for all of the experiments.

a. 30-km control

This run (30KCTRL) is the control for the 30-km experiments. Figure 3a shows the sea level pressure field for simulation hour 3 (0300 UTC 15 December 1987). A strong mesoscale gravity wave (labeled “1”) appears in eastern Missouri as a trough–ridge couplet.1 It has a wavelength and phase speed of 174 km and 27 m s−1, respectively. As in PR, wavelengths are derived from average trough–ridge axis separations. Wave speeds are determined by averaging results from the “center-tracking” and “axis-tracking” methods (Koch and Golus 1988). The center-tracking method equates wave speed with the speed of the wave’s surface mesolow or mesohigh. The axis-tracking method estimates phase speed from the movement of the wave front as a whole, done through averaging the movements of three points along the wave front. The axis-tracking method is used alone when wave pressure centers are absent or indistinct.

By hour 6 (0600 UTC; Fig. 3b) wave 1 has traveled to north-central Illinois, while two new waves are marked “2” and “3.” The length of these waves is approximately 140 km. The mesoscale gravity waves here are similar in scale, synoptic placement, and amplitude to those in PR’s 30-km MM4 control simulation (PR, Fig. 16b).

By hour 12 (Fig. 3c) a wave marked “4” has traveled to eastern Michigan, while a weak trough (unmarked) extends through northern Illinois. In the actual event, observed wave C was still strong over southern Wisconsin/northern Illinois, while observed wave B was weak and decaying over northern Wisconsin and Michigan. Table 3 summarizes the characteristics of 30KCTRL’s significant waves.

b. Reflective upper boundary

In this experiment (REFL) the upper boundary condition permits the reflection of gravity wave energy; 30KCTRL, in contrast, used a radiative condition. As the hour 6 output illustrates (Fig. 4), the REFL results are virtually identical to those of 30KCTRL. At this time REFL has produced waves 1, 2, and 3 with the same strength and positioning seen in 30KCTRL (Fig. 3b). The weak, decaying wave over northern Iowa and central Wisconsin in 30KCTRL also appears in REFL with the same intensity. Wave comparisons at all other times are similar, including those times featuring decaying waves in REFL, waves whose amplitudes are not greater than their counterparts in 30KCTRL. Cross-sectional analyses of wave structure in 30KCTRL and REFL (not shown; Powers 1994) have revealed that such structure is the same in both runs.

c. Enhanced vertical resolution

This run (EVR) is configured identically to 30KCTRL except that it employs 49 σ levels instead of 24, yielding an AS ratio of 1.0. Throughout EVR the wave simulation parallels that in 30KCTRL. Figure 5 provides an example from hour 3 (cf. Fig. 3a), and wave comparisons at all other times are similar.

The waves in 30KCTRL and EVR have the same structure, which Figs. 6a and 6b illustrate for wave 1 from both runs. Here the cross-sectional views of θ and vertical velocity w reveal the same structure seen in PR: an elevated convective updraft over a low-level, stable-layer wave ridge. While the strength of the updraft is slightly greater in the EVR wave, neither the EVR updraft core’s height nor its position are significantly different from those of the 30KCTRL updraft. The vertical motion and isentropic patterns are also essentially the same.

d. Enhanced horizontal resolution

1) 10KCTRL

This simulation serves as the control run for all of the 10-km experiments. At hour 3 (0300 UTC; Fig. 7a) the model is simulating strong waves in central Missouri (e.g., wave 1), with weaker ones in northwestern Illinois (S0, S1). These wave groups are moving at 25–32 m s−1 to the northeast, which is in the direction of the mid- to upper-level flow (shown in PR, Fig. 2a) and which agrees with observed wave motion and speed. The single amplitude of wave 1 is about 6 mb, and this wave and those behind it have horizontal lengths of 60–70 km.

By hour 6 (0600 UTC; Fig. 7b) the wave field is complex, but, as observed, the dominant wave activity remains confined to the region northwest to northeast of the synoptic low. In this region, the waves2 are due to grid-resolved elevated convection breaking out above a regional stable layer lying below a warm-frontal surface. In contrast are the pressure perturbations in the nonducted environment (i.e., one without the regional, low-level stable layer) of the warm sector. Examples of the latter appear in extreme western Tennessee, along the cold front, and along the northern border of Tennessee, in the warm sector. Reflecting mostly parameterized convective activity, these pressure perturbations do not represent ducted waves, in contrast to the perturbations in the region from northwest of the low to north of the warm front. Figures 8a and 8b reveal the different profiles of the ducted and nonducted environments. The sounding in Fig. 8a (location marked “D” in Fig. 7b) presents the typical duct profile of a low-level isothermal/inversion layer (here, to 700 mb) overlain by a deep conditionally unstable layer (see Lindzen and Tung 1976). In contrast, the sounding in Fig. 8b (location marked “N” in Fig. 7b) shows no duct and indicates, apart from the stable nocturnal surface layer, generally moist neutral to conditionally unstable/conditionally moist neutral conditions through the troposphere.

By hour 9 (0900 UTC; Fig. 7c) the area of ducted wave activity has shifted northward but still occupies the sector from northwest to northeast of the synoptic low. As at hour 6, the pressure perturbation in north-central Tennessee is associated with intense, mostly parameterized convection in the warm sector. The lack of waves in the postfrontal southwest quadrant of the domain reflects the absence of a pair of conditions associated with ducted wave activity elsewhere in the model: (i) a two-layer ducting profile (Fig. 8a) and (ii) elevated convection.

One wave that develops late in the simulation is wave 9. It organizes west and north of the synoptic low and achieves peak amplitude between hours 11 and 12 (1100–1200 UTC), a period during which the strongest observed wave C was exhibiting its maximum amplitude (in the Chicago area; Table 1). To compare observed and model conditions during this time, Figs. 9a and 9b present observational and model surface analyses for 1100 UTC. Regarding the synoptic low depth, the observed minimum is approximately 980 mb, while the model minimum is 984 mb. The difference reflects the delay in the onset of rapid deepening in the model and the resultant temporal lag of the model low’s minimum pressure. As for the low’s placement, the model center is about 70 km south of observation. Regarding the mesoscale gravity waves, Fig. 9a shows the troughs of observed waves B and C.3 In the model (Fig. 9b), the main mesoscale waves occupy the same position relative to the synoptic low as, and are similar in scale to, the observed waves. Despite the similarities, however, the model and observed waves are not in 1:1 correspondence; the model has not forecast observed waves B and C but rather has produced its own waves in the same region at the same time.

At hour 12 (1200 UTC; Fig. 7d) the strongest model wave is 9, extending northwest from the surface low and propagating northeastward. Other waves of lower amplitude are wave 10 over central Wisconsin and those over Lake Michigan, traveling mostly northward and eastward, respectively. Note that wave 9 is one of the simulation’s strongest and arises over 9 h into the simulation. Furthermore, in MM5 simulations of other mesoscale gravity wave cases, strong waves develop over 20 h after initialization and model startup (Powers 1994). Thus, neither wave generation nor amplitude hinge on initialization procedure or initial model adjustment. Regarding initialization procedure, note that PR found that adiabatic nonlinear normal mode initialization (NNMI) had no significant effect on the wave simulation. The potential influence of diabatic NNMI (Fox-Rabinovitz and Gross 1993) on the simulation, however, remains unknown because this technique is not available in the MM5.

To view and to check the accuracy of synoptic upper-level conditions in the control run, Fig. 10 presents the observed and model 500-mb analyses for 1200 UTC (hour 12). The model has simulated the strong trough over the eastern Plains, with the forecast 500-mb closed low minimum of 5256 m over western Illinois (Fig. 10b) verified by the observed closed low minimum of approximately 5250 m in that location (Fig. 10a). The observed jet maximum lies over western Tennessee/Kentucky to southeastern Missouri and northeastern Arkansas (Fig. 10a), and wind speeds of greater than 123 kt (63.1 m s−1) are indicated. In comparison, the model jet maximum is 127–137 kt (65–70 m s−1) (Fig. 10b). The model 60 m s−1 contour covers the area of the 60 m s−1 observed maximum but also extends to the southwest over northeastern Arkansas. The mid- and upper-level flow over the wave core region during the event, as seen in the model output and as implied by the 0000 UTC and 1200 UTC upper-air data, has been from southerly to west-southwesterly (Fig. 8a; see also Schneider 1990; PR). At 1200 UTC, the model output (Fig. 10b) shows similar flow directions over the area of observed wave activity at that time: central and southern Michigan and Wisconsin.

Table 3 catalogs the specifications of the strongest waves of 10KCTRL. Although the 10-km control waves have shorter average wavelengths than observed waves A–C, it deserves noting that the picture of the observed wave activity is biased toward those longer wavelengths that could be tracked with the resolution of the spacing of the barograph stations. Thus, the limitations of the observational network inhibit a complete analysis and comparison of actual cousins of all of the 10KCTRL waves.

The average phase speeds of the observed and 10KCTRL waves are similar at approximately 30 m s−1. As for wave amplitude comparisons, these are clouded because the estimates listed for the observed waves are from filtered data, while the model estimates are not. Nonetheless, looking at unfiltered data, Schneider (1990) found a single amplitude of 4.5 mb for the largest observed wave (C). The maximum amplitudes (subjectively determined) of model waves 8 and 9 compare well with this, but the 6.5-mb maximum amplitude of model wave 1 is 2 mb higher.

Figure 11 reveals the model wave structure. This cross section cuts through wave 1 at hour 3 along line AB shown in Fig. 7a. The prominent, low-level isentropic ridge (location marked “R” in Fig. 11) overlies the ridge in SLP (sea level pressure) seen in Fig. 7a. Consistent with trapped gravity wave dynamics, the maximum low-level upward and downward motions are in quadrature with the isentropic ridge, which is moving to the right in this view. The vertical velocity maximum aloft over the ridge is an elevated convective updraft core that is grid resolved and has arisen in conjunction with the explicit scheme. The convection is coupled to the wave, moves with it, and both strengthen together (accord PR). Precipitation (not shown) from the updraft both falls through the low-level ridge (see, e.g., PR, Fig. 21a) and, in the form of snow, is advected forward and ahead of the disturbance at upper levels. The bold, long-dashed contour marks the wave’s steering level. In this azimuthal cross section, the steering level is the level at which the along-section wind speed matches the wave phase speed (28 m s−1).

2) HYD

Through the 8 h of this hydrostatic experiment, the appearances, amplitude, length, speed, and positioning of its waves are virtually identical to the appearances, amplitude, etc., of 10KCTRL’s waves (Powers 1994). As no appreciable differences appear between HYD and 10KCTRL (Fig. 7), HYD’s output is not reproduced here. Wave simulation in the MM5 is insensitive to hydrostatic physics at 10-km resolution.

3) 3.3K

This full-physics simulation features a third domain of 3.3-km resolution placed within the 10-km grid (location shown in Fig. 7a). At hour 6 (0600 UTC; Fig. 12a) the waves numbered 1 and 2.5 are counterparts to those in 10KCTRL shown in Fig. 7b. These wave pairs in 3.3K and 10KCTRL have the same scale: about 80 km. Their calculated phase speeds over hours 5–6 are also essentially the same in both experiments: approximately 31 m s−1 for wave 1 and 26 m s−1 for wave 2.5. However, 3.3K also simulates other, shorter disturbances; 3K2 and 3K3 in the southern part of wave 2.5 are examples. Their wavelengths are 20–23 km (6–7Δx), and their phase speeds are 21 m s−1.

At hour 9 (Fig. 12b) 3.3K has reproduced waves 5, 8, and 8.5 from 10KCTRL (Fig. 7c), although their organization differs somewhat between the two simulations. Consider wave 8.5, for example. Whereas in 10KCTRL wave 8.5 is a single trough and ridge (Fig. 7c), in 3.3K separate troughs and ridges appear. The latter are identified 8.5T1, 8.5T2 and 8.5R1, 8.5R2, with 8.5T1 and 8.5R2 corresponding to the trough and ridge of wave 8.5 in 10KCTRL. The wavelength from 8.5T1 to 8.5T2 is approximately 9Δx, or 30 km. The components intersecting the southern extensions of 8.5T2 and 8.5R2, identified 8.5T2a, 8.5R2a, and 8.5R2b, are phase shifted from 8.5R2. The shortest waves in the 3.3-km run have approximately 6Δx wavelengths.

To assess the realism of the 3.3K short-wave production, observed raw (unfiltered) and high-pass-filtered barograms from the region have been examined. Figure 13 presents an example in the high-passed barogram for Dubuque, Iowa (DBQ; location in Fig. 1), for 0020–1220 UTC 15 December 1987. This record reveals the presence of low-amplitude waves of less than 40-min period such as those seen in 3.3K. Thus, both the observational and model data exhibit an embroidery of short-period waves on longer-period waves.

Lastly, at hour 12 (1200 UTC; Fig. 12c) wave 9 from 10KCTRL (Fig. 7d) has been reproduced. This trough and ridge manifest larger-scale organization than that reflected in other waves analyzed in the 3.3K output, and the wave has a 80–90-km length, like 10KCTRL wave 9. Thus, while 3.3K is capable of simulating shorter scales than 10KCTRL, 3.3K produces the longer wavelengths as well.

4) Horizontal resolution sensitivities

Table 4 presents the characteristics of the waves of the three grid size experiments: 30KCTRL, 10KCTRL, and 3.3K. The first half of the table lists the spectrum of all of the wavelengths and speeds appearing in the output. The second half of the table describes a subset of the waves, those of approximately the “smallest robust scale.” This is the smallest scale seen to persist with appreciable amplitude: approximately 6Δx. The smallest robust waves are considered separately because there is overlap for waves on the high end of the scale ranges. For example, the 3.3K larger-scale organization typically corresponds to waves also captured on the 10-km grid in 10KCTRL and listed under that heading.

The experiments show the following to be insensitive to grid size: 1) wave synoptic placement and timing, 2) wave structure, and 3) wave forcing. First, the model waves are consistently simulated in the observed wave activity area during the observed wave activity period. Figures 3b, 7b, and 12a, for example, show similar placements of wave activity at hour 6 of 30KCTRL, 10KCTRL, and 3.3K. Note, however, that the model does not simulate waves in a deterministic sense: specific observed waves are not forecast. As for wave structure, this is consistently that of a stable layer trapped mode coupled to an elevated convective entity (Fig. 11). Such elevated convection forces the waves. The process is one in which the waves initially arise in response to concentrated ascent and convergence under the developing updrafts and at the top of the stable layer (Powers 1994; see also Fig. 11 and section 4e below). That the modes are trapped is evidenced by the lack of phase tilt with height in the duct (Fig. 11) and has been confirmed by analyses of the relations of perturbation winds, temperatures, and pressures in the wave (not shown). The wave/convection structure is consistent with that implied by the weather attending the observed waves of 15 December 1987 (Schneider 1990; PR).

Model wave characteristics that, to varying degrees, are sensitive to grid size are: 1) speed, 2) amplitude, and 3) scale. Wave speed exhibits little difference between 30–10-km resolution but decreases for the shortest waves simulated at 3.3-km resolution (Table 4). Regarding the similar 30KCTRL and 10KCTRL average speeds (∼30.8 m s−1), these are close to the average phase speed for observed waves A–C (29 m s−1) and, on average, are consistent with the ducted mode speeds associated with the observed environmental conditions. Such ducted mode speeds would translate to ground-relative phase speeds of up to 35 m s−1 (PR). Consistency with ducted mode speeds associated with the model environment has been checked by extracting duct characteristics from the 30KCTRL and 10KCTRL data and applying them to the expression for such speeds:
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(Lindzen and Tung 1976). Here, N is the average Brunt–Väisälä frequency in the duct, H is the duct height (Lindzen and Tung 1976), and the N and H data are for points in the wave activity region for hours 6 and 9 of 30KCTRL and 10KCTRL. Equation (5) yields phase speeds for the 30KCTRL and 10KCTRL runs of 25.4 and 24.8 m s−1, respectively. Taking a 5 m s−1 mean azimuthal flow in the duct (Ū) (accord PR), these ducted mode speeds translate to 30.4 and 29.8 m s−1 ground-relative (i.e., CD + Ū) phase speeds, in agreement with the 30.8 m s−1 average ground-relative speed of the waves analyzed in these simulations (Table 3).

As for 3.3K, its waves of greater than 60-km length move at speeds like those of 10KCTRL’s waves. However, 3.3K’s smallest robust waves (20–22 km) average 21 m s−1 (Table 4). This speed decrease for the shortest waves is addressed below.

It is noted that in PR the average phase speeds of three waves in their 30-km control run were estimated at 42.8, 37.7, and 56.7 m s−1, a range generally higher than the range for the 30KCTRL waves (Table 3).4 It is believed that the main factor in the generally lower phase speed estimates for the 30KCTRL waves is an alternate method of phase speed determination and that the method applied in PR’s control simulation resulted in overestimates. Powers and Reed (1993) estimated phase speeds by the center-tracking method (Koch and Golus 1988; see also section 4a above) but used the estimated centroid of the wave trough when there was no clear surface mesohigh or mesolow (pressure center). However, estimating the centroid of a wave trough or ridge is subjective, determinations of wave movements are sensitive to the point taken as the centroid, and movement of the estimated centroid has been found to be greater in a given period than the movement represented by the successive wave front positions considered in axis tracking. In this study, axis tracking is used when pressure centers are absent or indistinct, which is usually the case in the 30-km runs. While axis tracking may introduce error where there is spread of the phase velocity vectors (see Koch and Golus 1988), it has been found in this work to be less subjective and procedurally less sensitive than the centroid-tracking method.

Using the axis-tracking method, the speeds of two of the PR 30-km control run waves are estimated at 36.7 and 30.1 m s−1, consistent with the range of the 30KCTRL waves (Table 3). For the third wave, the axis-tracking estimate of 53.6 m s−1 is not much different from the PR listing of 56.7 m s−1; this reflects the fact that this wave was more consistently amenable to center tracking. Thus, although this one wave is outside of the 30KCTRL range, the wave speeds in 30KCTRL and PR’s 30-km simulations are not systematically different.

Returning to the 30KCTRL and 10KCTRL results, the lack of significant variation in wave speeds between these runs despite the reduction in minimum horizontal wavelengths is consistent with hydrostatic gravity wave dynamics, within measurement error (Powers 1994). The difference between the average speeds of the 10KCTRL waves and the shortest 3.3K waves (i.e., those shown in the second half of Table 4) is consistent with shorter vertical wavelengths of the latter. The effects of differing wavenumbers may be examined with the nonhydrostatic gravity wave dispersion relation, from which phase speed c is given by
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Here, k and m are the vertical and horizontal wavenumbers, and N is the Brunt–Väisälä frequency. While in 10KCTRL vertical wavelengths are o(10 km), in 3.3K cross sections indicate vertical wavelengths of o(8 km) for the waves of the smallest horizontal scales. Thus, constructing the ratio of c10KCTRL/c3.3K, considering the 6Δx scales in both runs (see Table 4), and assuming the same average N in both runs, Eq. (6) yields c10KCTRL/c3.3K = 1.33. For a 30 m s−1 10KCTRL wave, this implies a 3.3K wave phase speed of 22.6 m s−1, which is within the range of speeds for the smallest 3.3K waves and which is in line with the 21 m s−1 average for such waves (Table 4). Nonlinear effects also influence the speeds of the smallest 3.3K waves, but such effects have not been quantified.

Wave amplitude also displays a limited sensitivity to grid size. The 10KCTRL waves exhibit greater maximum and average amplitudes than the 30KCTRL waves. This is primarily due to stronger resolved convection, as assessed by updraft intensities, forcing the 10KCTRL waves (cf. Figs. 6a and 11). As for the 10-km to 3-km comparison, the longer waves in 3.3K have amplitudes comparable to or somewhat greater than those seen on the 10KCTRL grid. Where 3.3K shows higher amplitudes, there are stronger convective updrafts than in the same area on the 10KCTRL grid. The shortest 3.3K waves have weak to moderate amplitudes (Table 4). Regarding such waves, their elevated convection is less deep than that of the 10KCTRL, and longer-wavelength 3.3K, waves. Thus, the vertical scale of the forcing of the shortest 3.3K waves is reduced. The reduction in vertical wavelengths of these waves, mentioned above, is consistent with the reduction in the forcing scale.

Regarding wave horizontal scale, the results first show that the scale range, in terms of Δx, broadens with decreasing grid size. This range increases from 4–6Δx for 30KCTRL to 6–30Δx for 3.3K. Second, the results reveal that minimum wavelength decreases as grid size decreases (Table 4). The decrease in the minimum mirrors the reduction in the absolute length of the minimum resolvable scale, 4Δx,5 with decreasing grid size. It is stressed that, as resolution is increased, the model waves do not all merely collapse to the minimum resolvable (4Δx) or smallest robust (6Δx) scales. For example, 10KCTRL at hour 3 (Fig. 7a) features 6Δx waves, such as wave 1 in Missouri, as well as 12Δx (∼120 km) waves, such as wave S1 in Illinois. Similarly, 3.3K at hour 6 (Fig. 12a) has produced both the short 20–23-km waves (∼6–7Δx; e.g., wave 3K2) and longer 80-km waves (24Δx; e.g., wave 2). Thus, as grid size decreases, the Δx wavelength spectrum expands: the model resolves finer components but retains organization on larger scales.

In the model, the initial horizontal scale of a disturbance is usually associated with the initial scale of the convective forcing (Powers 1994). With respect to the scales of actual waves, the question arises as to how convection that might originally be on a meso-γ scale might lead to a meso-β-scale wave. It is hypothesized that wave interference may be a mechanism for the emergence of longer waves from waves forced on shorter scales. Note first that observational (Bosart and Sanders 1986; Koch et al. 1988) and model (see figures herein and Powers 1994) data show that the scale of a given mesoscale wave in an event is not fixed. Second, MM5 simulations of wave events have shown that through interference much longer wave scales can evolve from shorter ones. One of the clearest examples comes from a 10-km simulation, configured like 10KCTRL, of the mesoscale gravity wave event of 9 May 1979 in the central United States (Stobie et al. 1983). While a full analysis of this simulation is beyond the scope of this paper, Fig. 14 shows a series of sea level pressure analyses at 20-min intervals between simulation hours 17.33 and 18.33 (0533–0633 UTC 9 May 1979). The wave region shown is ducted, and the mid- to upper-level flow is predominately southwesterly (see Stobie et al. 1983). In Fig. 14a, three trough–ridge pairs, labeled T1/R1, T2/R2, and T3/R3, have an average wavelength of 66 km or of 6.6Δx. Over the next hour (Figs. 14b–d) trough T2 becomes indistinct and is superseded by a larger-scale ridge from the interference of R1 and R2. The new, unified ridge becomes R1.5 in Fig. 14d. Thus, from the interference and merger of individual 60–70-km waves, a 140-km wave evolves (cf. Figs. 14a,d).

As for observational support of the possibility of interference, the bandpass-filtered barograms in Fig. 15 reveal an evolution of waves A and B in which they merge into a combined, broader wave as they propagate northward from Decatur (DEC) to Marseilles (MMO) to Chicago, Illinois (NBU; locations shown in Fig. 1). Similarly, from the genesis region, Fig. 16 presents raw barograms that suggest scale growth from a meso-γ wave field. First, the Joplin, Missouri (JLN; located in Fig. 1), barogram reveals the high-frequency wave activity in the convective period before the arrival of the polar air in this area. The wave periods generally are less than 10–30 min. At Springfield, Missouri (SGF; located in Fig. 1), 100 km downstream from JLN, what has emerged as the signature of the longer-period wave C appears as a pressure fall (of 4.78 mb) and rebound over about 70 min (Fig. 16; arrows over SGF trace mark approximate wave period, hours 8.16–9.33 on plot). Preceding this wave appears the higher-frequency activity seen at JLN. Such observational data suggest scale growth, and satellite evidence of wave interference in this case has been described by Seimon (1994). In summary, observed mesoscale gravity waves and their scales may at times represent a synthesis of shorter, separately generated disturbances, and it is possible that the scale of a gravity wave that is ultimately detected may be significantly larger than the scale of the original forcing mechanism.6

e. Removal and reintroduction of latent heating

Experiment RD is a 10-km run that sets the latent heat of condensation to zero after 4 h of full-physics simulation. Upon a dry restart of 10KCTRL at hour 4 (Fig. 17a), wave 1 is strong and straddles the Missouri–Illinois border, while wave S2 is moving across northeast Illinois (cf. hour 3 in Fig. 7a). Two hours later, however, the wave activity is dramatically diminished. Figures 17b and 7b compare RD and the control run at hour 6. In RD, waves 3–8 have not developed. Some waves that were present upon dry restart, however, persist in RD: waves 1, 2, and S1 in northern Wisconsin, northern Illinois, and northern Indiana, respectively.

By hour 8 (Fig. 17c) the wave activity in RD has ceased. No old waves remain of significance in the fine domain and no new ones have been generated. After hour 8, RD remains quiescent with respect to wave activity.

Experiment RM reintroduces the latent heat of condensation at hour 8 of RD. Figure 17c shows RM’s initial state—RD at hour 8. Two hours later (Fig. 18a) new mesoscale waves have developed over northern Illinois (wave 1RM) and southwestern Michigan (wave 2RM). The first signs of these waves appeared as early as hour 9 (not shown), and the wave activity is being forced by new elevated convection (e.g., Fig. 11). New wave generation and amplification continue in RM through hour 12 (Fig. 18b).

These results demonstrate that latent heating is necessary for model mesoscale gravity wave development. The wave response is due to the enervation and redevelopment of the convection forcing these waves (see Figs. 6 and 11).

f. No latent cooling

In experiment NLC the latent heats of cooling from evaporation/sublimation and melting are set to zero from the outset. At hour 3 the wave development in NLC (Fig. 19a) is similar to that in the control run (Fig. 7a), although wave 1’s ridge is a bit weaker in NLC. Comparisons of NLC and control 850-mb temperatures through this wave (not shown) reveal that the cold anomaly at the wave ridge at this level is weaker (warmer) by 0.4°C in NLC. Latent cooling from the melting and evaporation of precipitate produced in the updrafts and falling through the low-level wave ridges does enhance the ridges’ cold anomalies.

The NLC and control simulations diverge after hour 3. In the bulk of the NLC simulation, the absence of latent cooling leads to stronger and more coherent waves, distinguished by deeper, more uniform surface troughs. NLC hour 6 (Fig. 19b) provides an example. In NLC the troughs waves 1 and NLC2 are stronger and more coherent than their control counterparts, 10KCTRL waves 1 and 2 (Fig. 7b). Cross sections (not presented) indicate that the main factor is stronger forcing of the NLC waves in the form of stronger updraft cores and compensating, upper-level subsidence ahead of the ridge (see also Fritsch and Chappell 1980; PR Figs. 20a and 21b). Note that some of the precipitate produced in the updrafts is advected forward into subsaturated regions over the wave troughs and enhances the subsidence aloft through precipitate loading. In NLC, however, the column warming attending the enhanced subsidence is not offset to any degree by cooling from melting or evaporation/sublimation. Thus, subsidence warming is greater in NLC than in the full-physics simulation and the associated surface troughs are enhanced in NLC.

In summary, for most of NLC waves of generally higher amplitude and coherence than in the control are seen due to the unmitigated warming in the convective columns and compensating subsidence regions. However, in contrast to latent heating, latent cooling is not determinative in wave generation.

g. Altered moist process schemes

1) Kuo scheme

In experiment KS the Kuo parameterization substitutes for the Grell in the moist physics package. At hour 3 (0300 UTC; Fig. 20a) waves 1 in Missouri and S1K in Illinois are weaker than their control run counterparts (Fig. 7a). This relative response persists throughout the run. While, overall, KS generates both fewer and lower-amplitude waves, the propagation directions, lengths, and phase speeds of such waves are comparable to those of the control waves. At hour 12 (1200 UTC; Fig. 20b) the KS waves remain of relatively low amplitude.

The lower wave amplitudes in KS reflect weaker convection than in the control. For example, whereas the maximum updrafts at 500 mb associated with control run waves 1 and S0 at hour 3 are 502 and 174 cm s−1, respectively, those of their KS closest counterparts, 4K and S1K, are only 57 and 20 cm s−1, respectively (vertical velocities not shown; waves shown in Figs. 20a and 7a).

Before comparing the precipitation in the KS and control experiments, consider first the observed precipitation for 0000–1200 UTC 15 December 1987 in Fig. 21. Precipitation totals over the wave core area (central to northern Illinois) are 2–4 cm, with a maximum of greater than 4 cm occurring in eastern Illinois and into Indiana. The 12-h (0000–1200 UTC) total (i.e., explicit and parameterized) precipitation for the control and KS runs appears in Figs. 22a and 22b. The 2–3-cm totals in the wave core region in both experiments compare favorably with observation, although neither experiment’s total matches the greater than 4 cm observed maximum in eastern Illinois. The control maximum of over 3 cm in northern Missouri reflects, in part, the accumulation associated with the passage of a number of simulated mesoscale waves. Experiment KS’s smaller totals of 2–2.4 cm in this area are better verified, as the observed maximum in north-central Missouri (outlined by the 2-cm contour in Fig. 21) was 2–2.5 cm. Comparing the control and KS, one sees that the precipitation distribution is similar in both experiments. Except for the control’s enhanced maximum in Missouri, over most of the wave activity area the control and KS amounts are comparable.

Figures 23a and 23b show the control and KS parameterized precipitation only. In the control run, nearly all of the precipitation in the main Missouri–Illinois wave area is explicit, while in KS approximately half of the precipitation in the wave activity area is parameterized (except for the maximum on the Missouri–Iowa border) and approximately half is explicit.

2) Kain–Fritsch scheme

In experiment KFS the Kain–Fritsch parameterization substitutes for the Grell in the moist physics package. At hour 3 (0300 UTC; Fig. 24a) the model is producing mesoscale wave activity over Missouri and Illinois similar to that in the control (Fig. 7a). Counterparts to control waves S0 and S1 appear in KFS over northern Illinois. Absent from KFS, however, is a clear counterpart to wave 1 in central Missouri.

During the KFS simulation the wave activity is, as in the control, most prominent in central to northern Illinois, and the scales, phase speeds, and average amplitudes of the KFS waves are like those of the control waves. By hour 12 (1200 UTC; Fig. 24b) the KFS wave simulation remains similar to that of the control (Fig. 7d). Both runs yield wave 9 in northwest Illinois, as well as waves over Wisconsin and Lake Michigan, which are propagating to the northeast/north-northeast. As also seen at hour 3, at this time the KFS waves are stronger than those in KS (Fig. 20b).

Figure 22c shows the 12-h (0000–1200 UTC) total precipitation for KFS. Like the control and KS, KFS produces 2–3 cm across the wave core area. While it too underpredicts the greater than 4.0 cm observed maximum in eastern Illinois, its totals of 1–3 cm in the northwestern half of Illinois and 1–2.4 cm in north-central Missouri are verified (cf. Figs. 21 and 22c). Figures 22a, 22c and 23a, 23c compare the control and KFS 12-h total and parameterized precipitation. The pattern of the total precipitation is similar in both simulations (Figs. 22a and 22c), but quantitatively the control has produced about 13 mm more than KFS in the maximum in northern Missouri. In this area the KFS total is closer to the KS total (Fig. 22b). Although in the Missouri–Illinois wave area the majority of precipitation in KFS is explicit, KFS’s Kain–Fritsch scheme is indeed active there, as revealed in Fig. 23c. In contrast, in the control run, the Grell scheme is inactive there (Fig. 23a).

3) Moist physics sensitivities

The control, KS, and KFS suite of experiments demonstrate that, at the resolution used, mesoscale wave production and amplitude are maximized when the explicit scheme operates alone in simulating convection rather than jointly with these parameterizations in doing so. This is because the jointly produced vertical motion on the grid is weaker. The weaker vertical motions result because the triggered parameterizations expedite relief of the buildup of instability. Specifically, in environments permitting elevated convection over deep stable layers, the Kuo and Kain–Fritsch parameterizations may be triggered while the version of the Grell scheme used might not be. When the explicit scheme operates alone (i.e., in the company of an untriggered parameterization), the necessary convective adjustment prompted by developing elevated instabilities is effected less rapidly because the scheme must do so with vertical velocities characteristic of grid scales (e.g., 10 km). In contrast, with the explicit/Kuo and explicit/Kain–Fritsch packages, the triggered parameterizations help to relieve the buildup of such instabilities faster through parameterized inclusion of the effects of cloud-scale vertical motions (see also Kuo et al. 1996).

Wave response in KFS is closer to that in the control run than that in KS because the Kain–Fritsch scheme is less active (in terms of precipitation produced) than the Kuo scheme in the wave region (cf. Figs. 23b,c). Thus, the convection driving the waves is primarily the product of the explicit scheme. The KS simulation produces the weakest waves, in part due to weaker convective updrafts associated with them. Although recognized is the issue of the validity of the Kuo scheme’s assumptions, which originally contemplated scales greater than those of the 10-km grid, the KS experiment is nonetheless valuable in illuminating wave simulation sensitivity.

Note in closing that the waves in the control run are not simply a product of the explicit scheme operating on overly large grid boxes. The continued production of waves in connection with the explicit scheme in 3.3K indicates this, as 3.3 km is a scale at which a fully explicit approach is more appropriate than a parameterized one (Molinari and Dudek 1992).

5. Spectral analysis

Power spectral analyses have been performed to (i) uncover significant frequencies for the actual event, (ii) analyze objectively the nature and distribution of mesoscale gravity wave energy in a numerical simulation, (iii) compare the power spectra of the actual and simulated events, and (iv) investigate changes in model power spectra with varying grid size. The data analyzed are the observed and model pressure records, with the model barograms constructed from 10KCTRL and 30KCTRL simulations. 3.3K has not been spectrally analyzed because its high cost prohibited the production of a sufficiently long record.

a. Methodology

The stations for which observed and model barograms were spectrally analyzed are SPI, DEC, PIA, MLI, MMO, ORD, RFD, MSN, MKE, and GRB (Fig. 1). For the observed barograms, digitization was the first step in processing the data, accomplished by sampling the analog records at intervals of 5 and 10 min. The corresponding model data were obtained by saving the surface pressure output for grid points corresponding to the NWS barogram sites.

In preparation for the spectral computations, the digitized records were high-pass filtered to eliminate trends (see Jenkins and Watts 1968; Stobie et al. 1983; Koch and Golus 1988). The high-pass filters removed periods of greater than 6 h. The subrecords spectrally analyzed were 8 h long, with such 8-h periods encompassing the observed wave event at each of the stations chosen.

The power spectra were obtained through both the autocorrelation/autocovariance function and maximum entropy methods (described in the appendix). The products of the former will be referred to as the autocorrelation function (ACF) and the autocovariance function (ACVF) spectra and those of the latter as the maximum entropy (ME) spectra. Windowing and smoothing (see appendix) have both been used in computing the spectra. In the spectral plots, the “bandwidth” shown offers a measure of the resolution of the window for the number of lags used. To obtain good estimates of a spectral peak, the window bandwidth must be of the same order as the width of the peak (Jenkins and Watts 1969).

Confidence intervals reveal the statistical significance of the ACF/ACVF spectral peaks. Red noise spectra serve as the null hypothesis curves on which the confidence intervals are based (see appendix), and a 90% confidence level is used. The appendix describes the calculation of the confidence limits.

The ME spectra are offered as corroboration of the ACF and ACVF spectra and to reveal spectral peaks more sharply. No confidence limits have been calculated for the ME spectra.

b. Results

Consider first the high-pass-filtered 8-h subrecords for SPI for 0400–1200 UTC (Fig. 25). The observed and 10KCTRL records (Figs. 25a,b) are similar in terms of (i) having wave activity of approximately the same period and double amplitude (∼1–4 mb) and (ii) the cessation of wave activity after about subrecord hour 5.5, or 0930 UTC. The two records differ in the specific waves that are seen (model does not forecast specific observed waves) and in the model’s lack of the higher-frequency, low-amplitude signals seen in observation (unresolved by the model grid). In the 30KCTRL record (Fig. 25c), longer-period, higher-amplitude wave activity is seen. It is apparent that the 10-km simulation better resembles observation than the 30-km; the raw barograms (not shown) also reveal this.

Figure 26 presents plots of the smoothed power spectral density for the SPI data. Both the ACF (solid) and ME (dashed) spectra are shown, with the 90% confidence intervals (dotted) applying to the ACF curves only. The period over which the spectra are analyzed is 0400–1200 UTC 15 December. Consider first the observed SPI ACF spectrum in Fig. 26a. The broad, low-frequency peak is not clearly significant at the 90% level (upper dotted line), and thus this spectrum is essentially that of red noise. Of the model spectra (Figs. 26b,c) only the 30-km record has a significant peak at low frequency: at 0.36 cph (cph—cycles per hour, 2.8 h). At higher frequencies, however, both model spectra have significant secondary peaks: at 0.88 cph (68 min) in the 30-km data and at 1.98 cph (30 min) in the 10-km data.

For comparisons of total power in the model and observed datasets, Fig. 27 presents the ACVF spectra for SPI. The power is proportional to the area under each curve, and the plots show that the 30-km power [variance (σ2) = 1.93] dominates, with total power in the observed (σ2 = 0.73) and 10-km records (σ2 = 0.70) much lower and almost equal. As examples from other sites, Figs. 28a and 28b offer ME spectrum comparisons for Rockford (RFD) and Chicago, Illinois (ORD). For both stations the observed power dominates, and, in fact, both of these sites lay in the area of maximum observed wave amplitudes. Table 5 lists for these sites the significant spectral frequencies, which are indicated with arrows in Fig. 28.

Tables 5 and 6 summarize the results of the spectral analyses in terms of significant peaks, secondary peaks, and variance statistics. Consider first the significant peak results in Table 5. The spectra from all sources typically have one dominant peak at low frequency. These are referred to as “dominant” based on their appearances as the largest-amplitude peaks in the spectra, not on whether they consistently exceed the confidence limits by greater relative margins than other peaks. Secondary, higher-frequency peaks also emerge, however. These are more common in the model data.

The observed dominant period of 138 min (2.3 h) differs from the range of periods of the main mesoscale gravity waves (A, B, and C). The latter was estimated first by Schneider (1990) at 50–90 min and later by PR at 63–85 min (Table 1), estimates supported by analyses of surface observations, satellite imagery, and filtered barograms. Thus, the frequency of the dominant observed spectral peak is not that of the main waves; instead, a secondary peak, where it occurs, reflects them. For ORD, for example, a secondary peak at 64 min (Fig. 28b and Table 5) is within the period range of the main observed waves (see also MKE in Table 5). A visual examination of the observed high-passed records reveals correlations on the longer timescales of the dominant peaks, and such periods seem to be reflecting the ensemble propagation of two or three of the strongest waves (see Fig. 15). The model spectral results confirm that individual waves may have shorter periods than that of the dominant peak in a spectrum. For example, in the 10-km run the individual wave periods average 40 min (Table 3), whereas the dominant low-frequency peaks average 1.8 h (Table 5).

The results in Table 5 address what the model secondary peaks represent. The secondary spectral periods of 38 and 75 min for the 10- and 30-km runs, respectively, closely approximate the average periods of 40 and 81 min for the cataloged 10- and 30-km waves. Thus, as in the observed spectra, the model secondary peaks reflect the individual mesocale gravity waves.

In the model spectra there is a shift to a higher frequency of both the dominant and secondary peaks as resolution increases from 30 to 10 km (Table 5). The low-end peak shifts from 0.38 cph (2.6 h) to 0.56 cph (1.8 h); the secondary peak shifts from 0.77 cph (78 min) to 1.59 cph (38 min). The shift in the secondary peak reflects the shorter periods of individual waves, whose average wavelengths have decreased; the shift in the low-frequency peak represents the impact on the longer-lag correlations from groups of the individual, shorter-period waves.

The total power of the spectra (i.e., the record variances) further illuminate the observed and modeled events. Table 6 presents a variance analysis. First, the variance of the observed records dominates. The ranking is σ2obs > σ230 km > σ210 km. Average variance ratios show that the 30- and 10-km runs have 76% and 62%, respectively, of the observed power. Second, variance maxima for each data source all occur in the wave core area—observed: RFD; 10-km: MMO; 30-km: MLI (locations in Fig. 1). Third, the results in Table 6 confirm that the wave activity in the 30-km run was more spatially variable than in the 10-km run. For example, in the 30-km run, the variance at DEC is 0.66, while at nearby SPI it is 1.93, nearly three times as great. In contrast, in the 10-km run the variance at DEC is 0.95, while at SPI it is 0.70. The variability is reduced in the 10-km run because of (i) greater wave production and (ii) more uniform wave distribution. The model surface plots confirm the former, while variance statistics confirm the latter. For example, for each record type the variance of the total power at each station is
i1520-0493-125-8-1838-e7
where N is the number of stations spectrally analyzed and σ¯2 is the average power for all of the records of the given type. Illustrating (ii) above, σ2power for the 10-km run is 0.10, while σ2power for the 30-km run is 0.29.

The statistics in Table 6 have broad implications for wave event simulations. First, the model spectra are less energetic overall than the observed spectra. The lower power (average variance) in the model thus implies that the model is not, in general, overproducing energy in the mesoscale gravity wave spectral range. In light of the individual spectra and the results in Table 5, however, it is apparent that in certain bands at certain sites the model can display more energy than the actual atmosphere. Second, the distribution of mesoscale gravity wave power is more variable in the real atmosphere than in the model. The variances of the record variances (i.e., the power variances) for each station [Eq. (3)] rank as follows: σ2power, obs > σ2power, 30 km > σ2power, 10 km. For the model, barograms (Figs. 25b,c) and surface analyses (Figs. 3a,b and 7a,b) show that as grid size is decreased, wave production is increased and wave distribution is more uniform. Because wave energy is distributed more evenly at higher model resolution, the spatial variability of the power is lower.

6. Summary and conclusions

This paper has examined simulation sensitivities in the application of a mesoscale model to a mesoscale gravity wave event. Using the Pennsylvania State University–NCAR MM5, the purposes of the investigation have been to settle a number of mesoscale gravity wave modeling issues raised by the work of Powers and Reed (1993; PR), to probe new ones, and to explore spectrally numerical simulations of a wave event. The large-amplitude case of 15 December 1987 in the central United States has served as the vehicle.

The model sensitivity experiments first show that upper boundary reflectivity has no significant effect on the development, structure, strength, or maintenance of model mesoscale gravity waves at 30-km resolution. Thus, the model waves analyzed in PR were not artifacts of a reflective upper boundary condition. The experiments, furthermore, show that (i) the model waves are not spurious byproducts of horizontal/vertical resolution inconsistency, (ii) model wave characteristics are not highly sensitive to vertical resolution, and (iii) nonhydrostatic effects are negligible in wave simulation for grid sizes down to 10 km.

It is recognized that the experiments are initialized relatively close to the wave activity period, and this may aid in the general prediction of this event. The initialization time was kept at 0000 UTC 15 December to match the procedure of PR. As for the impact of initial model adjustment on wave simulation, however, the experiments show that the simulation of strong waves throughout an event period does not hinge on initialization procedure or initial adjustment. Waves can develop in the MM5 at over 9 and at over 20 h (Powers 1994) into a simulation of a mesoscale gravity wave event. A reviewer has suggested that it is possible that weak gravity waves arising during model startup could trigger convection, which then could force a gravity wave, which, in turn, could trigger new convection, which could force a new wave, etc. While this is within the realm of possibility, any early convection, whether or not triggered by a numerically generated gravity wave, would not be the proximate cause of waves seen 12, 20, etc., hours into a run. In considering early convection in the model runs, note too that natural, strong mechanisms for its triggering (e.g., synoptic ascent and frontal lifting) are fully present.

Grid size experiments have found the following wave and event simulation characteristics to be insensitive to grid size: wave synoptic placement and timing, wave structure, and wave forcing. Regardless of grid size, waves are consistently simulated in the region and during the period observed, their structures are primarily those of trapped modes at low levels coupled to convective cells aloft, and elevated convection is the primary wave forcing mechanism.

The following wave characteristics have been found to be sensitive, to varying degrees, to grid size: speed, amplitude, and scale. Wave speed sensitivity is minimal for the 30- to 10-km grid reduction. For the 10- to 3.3-km reduction, however, while the 3.3-km grid’s long waves have speeds comparable to those of the waves on the coarser grids, lower average wave speeds emerge for the shortest waves that the 3.3-km grid can resolve (less than 30-km wavelength). Regarding wave amplitude, larger amplitudes are seen as the resolution is increased from 30- to 10-km. At 3.3-km resolution the larger-scale waves have amplitudes comparable to or somewhat greater than the waves on the 10-km grid, while the shortest waves at 3.3-km resolution have weak to moderate amplitudes.

Horizontal wavelength also exhibits some sensitivity to horizontal resolution. As grid size decreases, the spectrum of wavelengths broadens; that is, as resolution increases, the model simulates finer components while retaining organization on larger scales. With respect to the shortest scales, it is found that the minimum wavelength decreases with decreasing grid size. This reflects the translation of the minimum resolvable scale, 4Δx, to shorter absolute wavelengths as Δx decreases. Despite the sensitivity of the minimum wavelength to grid size, the model waves do not all merely collapse to the smallest resolvable or smallest robust (∼6Δx) scales.

Moist physics experiments demonstrate that latent heating is necessary for model mesoscale gravity wave development, confirming a result of PR. When latent heating is removed, model waves decay; when it is reintroduced, they redevelop. This is because latent heat is the energy source for convection, which is primarily forcing the model waves. Latent cooling, from evaporation and melting, is secondary to latent heating in wave forcing. With respect to the observed spectra of scales of convection and of meso-β gravity waves, it is hypothesized that wave interference may be a means for the ultimate emergence of longer wave scales from forcing on smaller scales. Note that while the model waves are forced by convection in the case simulated, and while evidence supports it as a possible source mechanism here (Schneider 1990; Marwitz and Toth 1993; PR), it is not maintained that this is the forcing mechanism in all cases.7

Wave simulation is sensitive to the moist process package used. It is found that at the 10-km grid size examined, waves are stronger and generally more numerous when the explicit scheme operates alone in producing their driving convection, rather than when it operates jointly with triggered cumulus parameterizations. This is because at 10-km resolution the convection is more vigorous in the fully explicit case. The above sensitivity notwithstanding, the model wave production process is qualitatively the same in the case simulated regardless of the moist process package used.

Regarding the statistical analyses, filtered observed and model barograms indicate that the simulation of wave activity becomes more realistic with increasing model resolution. From spectral analyses, it is found that the observed spectra display a low-frequency dominant peak consistently and a higher-frequency secondary peak less often. The secondary, higher-frequency peak represents the individual observed mesoscale gravity waves, while the dominant peak appears to reflect the signal of the two or three largest waves traveling as an ensemble. The model also produces bimodal spectra. As in the observed spectra, the model secondary peaks reflect individual mesoscale gravity waves.

The model spectra have lower average power than the observed spectra. Thus, in general, the model does not overproduce energy in the frequency range of mesoscale gravity waves. As seen from spectrum and significant peak comparisons, however, the model can put relatively more energy into frequency bands differing from those preferred by the actual atmosphere. In addition, as horizontal resolution increases, there is an upward frequency shift in both the dominant and secondary model peaks.

From variance analyses it is found that the distribution of mesoscale gravity wave energy (σ2power) is more variable in observation than in the model, with the results showing that σ2power, obs > σ2power, 30 km > σ2power, 10 km. In the model, wave energy is distributed more evenly with decreasing grid size because more waves are produced and because they are distributed more uniformly.

The limitations on the mesoscale gravity wave simulations presented herein have been noted above where appropriate. Apart from these concerns, in future work simulations could be improved and observational verification could be enhanced through (i) a better representation of initial conditions via four-dimensional variational data assimilation, (ii) smaller grid sizes over larger areas, and (iii) collection of an enhanced observational dataset from a high-amplitude wave event. Better observations and simulations can only help to improve our understanding of mesoscale gravity waves, truly “wrinkles in the weather’s face.”8

Acknowledgments

Sincere gratitude is extended to Dr. Richard Reed for assistance during the course of this research. Thanks are also extended to Drs. Ying-Hwa Kuo, Richard Rotunno, Dale Durran, Clifford Mass, and K. K. Tung for valuable discussions and helpful comments. This research was supported by the National Science Foundation under Grant ATM9213851 and by the NCAR Advanced Study Program.

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APPENDIX

Spectral Methods

For the spectral calculations, both the autocorrelation/autocovariance function (ACF/ACVF) and maximum entropy (ME) methods have been used. In calculating the ACF or ACVF spectra, first determined is the sample autocorrelation function [acf, ρXX(u)] or the autocovariance function [acvf, γXX (u)] at lag u of the process represented by the time series of station pressure. Sample acfs calculated for the observed and model barograms have revealed high initial correlations, which indicate dependence on prior values and argue against a white noise model for the underlying process (Jenkins and Watts 1969).

The ACVF power spectrum ΓXX(f) is the Fourier transform of the acvf:
i1520-0493-125-8-1838-ea1
where f is frequency. The ACF power spectrum is the ACVF power spectrum normalized by the record variance σ2X, or the Fourier transform of the acf.
The smoothing employed in some of the spectral computations reduces the variance of the spectral estimator, a desired effect (Jenkins and Watts 1969), and a spectrum is considered smoothed when the maximum lag used in windowing is less than the number of points in the record. Windowing involves multiplying the acf or acvf by a series of weights, the lag window. The windowed ACF spectrum has the form
i1520-0493-125-8-1838-ea2
where w(u) is the window function. The Parzen window is used in this study and has the form
i1520-0493-125-8-1838-ea3
where u is the current lag and M is the maximum lag. The Parzen window has been chosen over others because it (i) leads to fewer and smaller spurious peaks in the spectrum and (ii) always yields positive spectral estimates. The bandwidth for the Parzen window is represented by the relation b = 1.86/M, where b is the bandwidth in units of frequency [cycles per Δ(sampling interval)] and M is the maximum lag used.
The ACF/ACVF confidence intervals are based on red noise spectra. Red noise spectra have been used to base the confidence limits because of the consistently high one- and two-lag autocorrelations seen in the barogram acfs and because of the conformity of such spectra to atmospheric time series (Gilman et al. 1963). The red noise spectrum Lh is formulated as
i1520-0493-125-8-1838-ea4
(Gilman et al. 1963). Here, ρ is the one-lag autocorrelation [ρXX(1)], M is the maximum lag, and h is the frequency. In practice, the parameter ρ may be obtained as an average of the one- and two-lag autocorrelations; namely, ρ = [ρXX(1) + ρXX(2)]/2 (Hartmann 1990).
The 100(1 − α)% confidence interval about the red noise null hypothesis spectrum Lh is given by
i1520-0493-125-8-1838-ea5
where χν refers to a value in the χ2 distribution and ν is the number of degrees of freedom. For the Parzen window used, ν is given by
i1520-0493-125-8-1838-ea6
where T is the record length and M is the maximum lag.
The maximum entropy method has also been used to calculate power spectra. The power spectrum estimate [P(f)] from the maximum entropy method is given by
i1520-0493-125-8-1838-ea7
(Press et al. 1987). In this formula the ak’s are the prediction filter coefficients, ze{2πifΔ} (where f is the frequency and Δ is the sampling interval), and M is known as the number of poles or prediction filter points. Limiting the number of poles smooths the spectrum. In this study M is taken as either approximately 0.15T or 0.30T (T is the number of points in record), resulting in more- and less-smoothed spectra, respectively. For the theory of ME spectral estimation, the reader may consult Childers (1978), Press et al. (1987), and Percival and Walden (1993).
Fig. 1.
Fig. 1.

Wave isochrones and wave activity zones for 15 December 1987 event. Isochrones (times in UTC 15 December) are of troughs of observed waves B (dashed) and C (solid). Times in parentheses indicate that wave is ill defined. For earliest times, waves are weak and positions more approximate. The “L” indicates position of the surface low at the given hour UTC (after Powers and Reed 1993).

Citation: Monthly Weather Review 125, 8; 10.1175/1520-0493(1997)125<1838:NMSOAM>2.0.CO;2

Fig. 2.
Fig. 2.

Model domains for 30/10 experiments. Outer frame is 30-km domain and inner frame is 10-km domain.

Citation: Monthly Weather Review 125, 8; 10.1175/1520-0493(1997)125<1838:NMSOAM>2.0.CO;2

Fig. 3.
Fig. 3.

Sea level pressure fields for hours 3, 6, and 12 of experiment 30KCTRL (0300, 0600, and 1200 UTC 15 December 1987). Contour interval is 1 mb, and frontal positions are indicated. Waves labeled, and wave trough axes analyzed with solid lines. (a) Hour 3, Line AB marks position of cross section in Fig. 6a; (b) hour 6; (c) hour 12.

Citation: Monthly Weather Review 125, 8; 10.1175/1520-0493(1997)125<1838:NMSOAM>2.0.CO;2

Fig. 4.
Fig. 4.

Sea level pressure field for hour 6 of experiment REFL (0600 UTC). Contour interval is 1 mb. Waves labeled and wave trough axes analyzed with solid lines.

Citation: Monthly Weather Review 125, 8; 10.1175/1520-0493(1997)125<1838:NMSOAM>2.0.CO;2

Fig. 5.
Fig. 5.

Sea level pressure field for hour 3 (0300 UTC) of experiment EVR. Contour interval is 1 mb. Waves labeled and wave trough axes analyzed with solid lines. Line AB marks position of cross section in Fig. 6b.

Citation: Monthly Weather Review 125, 8; 10.1175/1520-0493(1997)125<1838:NMSOAM>2.0.CO;2

Fig. 6.
Fig. 6.

Cross section of θ and w through wave 1 in experiments 30KCTRL and EVR at hour 3 (0300 UTC). Cross-sectional lengths are 732 km, and locations are marked as line AB in Figs. 3a and 5; θ is solid line and the interval is 4 K; w is short-dashed line upward; zero contour is the solid boldface line; regions of downward motion bounded by zero contour and negative minima marked; interval is 10 cm s−1. (a) 30KCTRL and (b) EVR.

Citation: Monthly Weather Review 125, 8; 10.1175/1520-0493(1997)125<1838:NMSOAM>2.0.CO;2

Fig. 7.
Fig. 7.

Sea level pressure field for hours 3, 6, 9, and 12 of experiment 10KCTRL (0300, 0600, 0900, and 1200 UTC). Contour interval is 1 mb. Waves labeled and wave trough axes analyzed with solid lines. (a) Hour 3. Area in dashed outline in northern Illinois is that of the 3.3-km grid of experiment 3.3K. Line AB marks position of cross section in Fig. 11. (b) Hour 6. Surface frontal positions marked; “D” and “N” mark locations of soundings in Figs. 8a and 8b, respectively. (c) Hour 9. Surface frontal positions marked. (d) Hour 12.

Citation: Monthly Weather Review 125, 8; 10.1175/1520-0493(1997)125<1838:NMSOAM>2.0.CO;2

Fig. 8.
Fig. 8.

Model soundings from hour 6 of experiment 10KCTRL (0600 UTC). Full barb is 10 kt; pennant is 50 kt. (a) Ducted region sounding. Sounding location marked “D” in Fig. 7b. (b) Nonducted region sounding. Sounding location marked “N” in Fig. 7b.

Citation: Monthly Weather Review 125, 8; 10.1175/1520-0493(1997)125<1838:NMSOAM>2.0.CO;2

Fig. 9.
Fig. 9.

Observational and 10KCTRL surface analyses for 1100 UTC 15 December 1987. Sea level pressure solid; contour interval is 2 mb. (a) Observational analysis. Approximate positions of troughs of waves B and C analyzed with dotted lines. Temperatures and dewpoints are in degrees Celsius. Winds: full barb is 10 kt. (b) 10KCTRL analysis. Wave troughs analyzed with dotted lines.

Citation: Monthly Weather Review 125, 8; 10.1175/1520-0493(1997)125<1838:NMSOAM>2.0.CO;2

Fig. 10.
Fig. 10.

Observed and 10KCTRL 500-mb analyses for 1200 UTC 15 December 1987. Height contours solid; interval is 60 m. Isotachs dashed; contour interval is 10 m s−1. Wind vectors plotted in (b). (a) Observed and (b) 10KCTRL. Coarse grid shown.

Citation: Monthly Weather Review 125, 8; 10.1175/1520-0493(1997)125<1838:NMSOAM>2.0.CO;2

Fig. 11.
Fig. 11.

Cross section of θ, w, and steering-level isotach through wave 1 in experiment 10KCTRL at hour 3 (0300 UTC). Cross-sectional length is 270 km, and location is marked as line AB in Fig. 7a; θ thin solid, interval is 4 K; w dashed upward, bold solid downward; interval is 65 cm s−1; zero contour bold solid and labeled; updraft core maximum is 6.12 m s−1. Heavy long-dashed line represents the steering level wind isotach (28 m s−1), “R” marks the location of the wave ridge, and arrows indicate the sense of vertical motion.

Citation: Monthly Weather Review 125, 8; 10.1175/1520-0493(1997)125<1838:NMSOAM>2.0.CO;2

Fig. 12.
Fig. 12.

Sea level pressure fields for hours 6, 9, and 12 of experiment 3.3K (0600, 0900, and 1200 UTC). Contour interval is 1 mb. Wave trough and ridge axes analyzed with solid and dashed lines, respectively. Gridpoint numbers shown on plot border. Grid position relative to 10-km grid shown in 7a. (a) Hour 6; (b) hour 9; (c) hour 12.

Citation: Monthly Weather Review 125, 8; 10.1175/1520-0493(1997)125<1838:NMSOAM>2.0.CO;2

Fig. 13.
Fig. 13.

High-pass-filtered observed barogram for DBQ for period 0020–1220 UTC 15 December 1987. Ordinate in millibars; abscissa lists time in hours from 2120 UTC 14 December 1987. Periods of less than 3 h retained. Location of DBQ shown in Fig. 1.

Citation: Monthly Weather Review 125, 8; 10.1175/1520-0493(1997)125<1838:NMSOAM>2.0.CO;2

Fig. 14.
Fig. 14.

Sea level pressure fields for hours 17.33–18.33 (1720–1820 UTC 9 May 1979) of simulation of 9 May 1979 event. Model grid size (10 km) and physics as in 10KCTRL. Contour interval is 1 mb. Gravity wave troughs (T) and ridges (R) labeled and marked by solid and dashed lines, respectively. (a) Hour 17.33; (b) hour 17.67; (c) hour 18; (d) hour 18.33.

Citation: Monthly Weather Review 125, 8; 10.1175/1520-0493(1997)125<1838:NMSOAM>2.0.CO;2

Fig. 15.
Fig. 15.

Bandpass-filtered observed barograms for DEC, MMO, and NBU for period 2120–2320 UTC 14–15 December 1987. Ordinate in millibars; abscissa in hours from 2120 UTC 14 December. Peaks of observed waves A, B, and C labeled. Filter retains periods of about 30 min–3 h. Locations of DEC, MMO, and NBU shown in Fig. 1.

Citation: Monthly Weather Review 125, 8; 10.1175/1520-0493(1997)125<1838:NMSOAM>2.0.CO;2

Fig. 16.
Fig. 16.

Observed barograms for JLN and SGF for period 2000–0600 UTC 14–15 December 1987. Locations of JLN and SGF shown in Fig. 1. Ordinate in millibars, with 8 mb added to all SGF values for plotting purposes. Abscissa in hours from 2000 UTC 14 December. Arrows over SGF trace indicate approximate period of wave C.

Citation: Monthly Weather Review 125, 8; 10.1175/1520-0493(1997)125<1838:NMSOAM>2.0.CO;2

Fig. 17.
Fig. 17.

Sea level pressure field for hours 4, 6, and 8 of experiment RD (0400, 0600, and 0800 UTC). Contour interval is 1 mb. Waves labeled and trough axes analyzed with solid lines. (a) Hour 4; (b) hour 6; (c) hour 8.

Citation: Monthly Weather Review 125, 8; 10.1175/1520-0493(1997)125<1838:NMSOAM>2.0.CO;2

Fig. 18.
Fig. 18.

Sea level pressure fields for hours 10 and 12 of experiment RM (1000 and 1200 UTC). Contour interval is 1 mb. Waves labeled and wave trough axes analyzed with solid lines. (a) Hour 10; (b) hour 12.

Citation: Monthly Weather Review 125, 8; 10.1175/1520-0493(1997)125<1838:NMSOAM>2.0.CO;2

Fig. 19.
Fig. 19.

Sea level pressure field for hours 3 and 6 (0300 and 0600 UTC) of experiment NLC. Contour interval is 1 mb. Waves labeled and wave trough axes analyzed with solid lines. (a) Hour 3; (b) hour 6.

Citation: Monthly Weather Review 125, 8; 10.1175/1520-0493(1997)125<1838:NMSOAM>2.0.CO;2

Fig. 20.
Fig. 20.

Sea level pressure field for hours 3 and 12 of experiment KS (0300 and 1200 UTC). Contour interval is 1 mb. Waves labeled and wave trough axes analyzed with solid lines in (a). (a) Hour 3; (b) hour 12.

Citation: Monthly Weather Review 125, 8; 10.1175/1520-0493(1997)125<1838:NMSOAM>2.0.CO;2

Fig. 21.
Fig. 21.

Observed precipitation (cm) for 0000–1200 UTC 15 December 1987. Contour interval is 1 cm except for first plotted contour, which is 0.5 cm. Circles locate precipitation recording stations.

Citation: Monthly Weather Review 125, 8; 10.1175/1520-0493(1997)125<1838:NMSOAM>2.0.CO;2

Fig. 22.
Fig. 22.

Total precipitation (mm) on fine grid for experiments 10KCTRL, KS, and KFS for hours 0–12. Contour interval is 10 mm except for first plotted contour, which is 5 mm. Gridpoint numbers shown on plot borders. (a) 10KCTRL; (b) KS; (c) KFS.

Citation: Monthly Weather Review 125, 8; 10.1175/1520-0493(1997)125<1838:NMSOAM>2.0.CO;2

Fig. 23.
Fig. 23.

Parameterized precipitation (mm) on fine grid for experiments 10KCTRL, KS, and KFS for hours 0–12. Contour interval is 5 mm except for the first and second plotted contours, which are 0.1 and 0.5 mm, respectively. Gridpoint numbers shown on plot borders. (a) 10KCTRL; (b) KS; (c) KFS.

Citation: Monthly Weather Review 125, 8; 10.1175/1520-0493(1997)125<1838:NMSOAM>2.0.CO;2

Fig. 24.
Fig. 24.

Sea level pressure field for hours 3 and 12 of experiment KFS (0300 and 1200 UTC). Contour interval is 1 mb. Selected waves labeled and wave trough axes analyzed with solid lines. (a) Hour 3; (b) hour 12.

Citation: Monthly Weather Review 125, 8; 10.1175/1520-0493(1997)125<1838:NMSOAM>2.0.CO;2

Fig. 25.
Fig. 25.

Observed, 10KCTRL, and 30KCTRL high-pass-filtered barograms for SPI. Abscissa in hours from 0400 UTC 15 December 1987; total period shown corresponds to 0400–1200 UTC 15 December. Ordinate in millibars. (a) Observed; (b) 10KCTRL; (c) 30KCTRL.

Citation: Monthly Weather Review 125, 8; 10.1175/1520-0493(1997)125<1838:NMSOAM>2.0.CO;2

Fig. 26.
Fig. 26.

Observed, 10KCTRL, and 30KCTRL smoothed ACF and ME spectra for SPI. ACF spectra solid; ME spectra dashed. Lag (M) and pole (P) ratios shown, and numbers of points in the records are 97 (observed), 99 (10KCTRL), and 91 (30KCTRL); 90% confidence intervals marked with dotted lines. (a) Observed spectra; M = 48, P = 15. (b) 10KCTRL spectra; M = 49, P = 15. (c) 30KCTRL spectra; M = 45, P = 14.

Citation: Monthly Weather Review 125, 8; 10.1175/1520-0493(1997)125<1838:NMSOAM>2.0.CO;2

Fig. 27.
Fig. 27.

Observed, 10KCTRL, and 30KCTRL smoothed ACVF spectra for SPI. Observed spectrum solid, 10KCTRL spectrum dashed, and 30KCTRL spectrum dotted. Lag ratios are as in Figs. 26a–c for each record source.

Citation: Monthly Weather Review 125, 8; 10.1175/1520-0493(1997)125<1838:NMSOAM>2.0.CO;2

Fig. 28.
Fig. 28.

Observed, 10KCTRL, and 30KCTRL smoothed ME spectra for RFD and ORD. Arrows indicate significant peaks. Observed spectrum solid, 10KCTRL spectrum dashed, and 30KCTRL spectrum dotted. Pole ratios are as in Figs. 26a–c for each record source. (a) RFD and (b) ORD.

Citation: Monthly Weather Review 125, 8; 10.1175/1520-0493(1997)125<1838:NMSOAM>2.0.CO;2

Table 1.

Characteristics of main observed waves A, B, and C. The maximum double amplitudes (from filtered data) and the stations exhibiting them are also listed (from Powers and Reed 1993).

Table 1.
Table 2.

Summary of MM5 experiments.

Table 2.
Table 3.

Characteristics of waves of 30KCTRL and 10KCTRL experiments. Only model waves that were sufficiently distinct and long-lived are analyzed. The maximum (single) amplitudes of only the strongest model and observed waves are listed.

Table 3.
Table 4.

Collective characteristics of waves of 30KCTRL, 10KCTRL, and 3.3K experiments.

Table 4.
Table 5.

Significant spectral peaks in observed, 10KCTRL, and 30KCTRL spectra. Significance is at the 90% level. Frequencies given in cycles per hour; periods in minutes or hours shown in parentheses for significant peaks. Blanks indicate no significant peak in the spectrum of the given record. Average frequencies and periods for 10KCTRL and 30KCTRL waves also listed. Sampling intervals for SPI, RFD, ORD, MSN, GRB, and MKE records are 5 min. Sampling intervals for PIA, MLI, and MMO records are 10 min.

Table 5.
Table 6.

Variance statistics for observed, 10KCTRL, and 30KCTRL spectra.

Table 6.
1

Powers and Reed (1993) confirmed the gravity wave structure of such features.

2

Not all of the mesoscale waves are numbered in the plots.

3

The positions of these waves were determined from analyses of filtered barograms, barogram cross correlations, and observations (see PR).

4

In a different PR 30-km experiment (1200 UTC 14 December 1987 initialization), the average model phase speed (per axis tracking) was 29 m s−1, similar to the 31 m s−1 average in 30KCTRL.

5

This is the smallest possible wavelength given the model’s fourth-order horizontal diffusion.

6

Demonstrations of wave scale expansion and upscale scattering, in the context of shear-generated gravity waves, appear in Fritts (1984) and Chimonas and Grant (1984).

7

Considering the modeling result more broadly, perhaps the wave-yielding process occurring from simulated convection [i.e. the displacement (narrow lifting/depression) of the stable layer], may result from other conditions as well. For example, a positive vorticity advection tail aloft (described by Schneider 1990) that moved over the duct may have had associated vertical motions [evidenced by a commalike cloud signature (Schneider 1990)] sufficiently focused to perturb the underlying stable fluid.

8

The phrase is respectfully borrowed from Jewell (1994) in a discussion of V. Bjerknes’s concern with subsynoptic weather features.

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