Gravity waves above a convective boundary layer: A comparison between wind-profiler observations and numerical simulations

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1 QUARTERLY JOURNAL OF THE ROYAL METEOROLOGICAL SOCIETY Q. J. R. Meteorol. Soc. 133: (7) Published online in Wiley InterScience ( DOI: 1.1/qj.7 Gravity waves above a convective boundary layer: A comparison between wind-profiler observations and numerical simulations T. Böhme, a T. P. Lane, b *W.D.Hall c and T. Hauf a a Universität Hannover, Germany b The University of Melbourne, Australia c National Center for Atmospheric Research, USA ABSTRACT: Wind-profiler observations and three-dimensional numerical modelling are used to investigate gravity waves generated over convectively active boundary layers on 1 and 13 May 1 at Lindenberg, Germany. The results illustrate the importance of changes in the background wind and stability profile in influencing the vertical propagation of the gravity waves. The resultant changes in vertical propagation cause the dominance of relatively long-wavelength, long-period waves at high levels and the dominance of relatively short-wavelength, short-period waves near the top of the convective boundary layer. There are important differences between the two days in the wave-propagation characteristics, due to subtle changes in the background flow conditions. Copyright 7 Royal Meteorological Society KEY WORDS troposphere; wave generation; wave propagation; wavelet analysis Received 9 May 6; Revised 14 December 6; Accepted 19 December 6 1. Introduction Convective boundary layers generate gravity waves via the interaction between boundary-layer thermals and the overlying stable troposphere. These waves, often called convection waves, define patterns of ascent and descent that can extend vertically throughout the troposphere and possibly into the stratosphere. They can be generated in dry or moist conditions, and have been shown to influence the organization of convection on a variety of scales. Convection waves are also of great interest to glider pilots, who achieve higher altitudes of flight by exploiting the sustained ascent within the upward phases of the waves. Although glider pilots have been aware of the existence of convection waves for some time (e.g. Kuettner, 197), it was not until the early 198s that substantial progress was made in understanding the processes controlling the generation and propagation of these waves. Through theory (Mason and Sykes, 198; Sang, 1993), observations (Kuettner et al., 1987), and modelling studies (Clark et al., 1986; Hauf and Clark, 1989), a better picture of the mechanisms underlying convection waves was obtained. These studies all described the dominance of deep gravity-wave modes that generally propagated in the opposite direction to the background flow and had a well-defined horizontal scale. * Correspondence to: T. P. Lane, School of Earth Sciences, The University of Melbourne, Melbourne, Victoria 31, Australia. tplane@unimelb.edu.au The importance of convection waves for the initiation and organization of moist convection was highlighted by Balaji and Clark (1988) and Balaji et al. (1993). They showed that the horizontal scale of the waves could influence the horizontal scale of the thermals in the boundary layer, defining the dominant convective mode. This interaction plays an important role in the development of cloud streets (e.g. Schultz et al., 4) and horizontal convective rolls (e.g. Weckwerth et al., 1997). Balaji and Clark (1988) and Balaji et al. (1993) also showed that the upward phases of convection waves provide preferential locations for the deepening of convective cells into the troposphere aloft. Understanding the processes that select the horizontal scale of the dominant gravity waves is therefore of fundamental importance to understanding the contribution of these waves to convective initiation and organization. Clark et al. (1986) attributed the scale selection to a combination of the depth of the convective layer, the transient nature of the source, and possibly partial reflection at the tropopause. Both Balaji et al. (1993) and Sang (1993) showed that the wave field generally consists of a combination of propagating wave modes and evanescent wave modes, with a strong response from those propagating modes that are close to being evanescent. Evanescence can lead to trapping of low-level gravity waves, and is a result of changes in the wind shear and stability with height (e.g. Crook, 1988). In a recent modelling study, Lane and Clark () examined the role of changes in the wind profile for wave generation and propagation above a two-dimensional Copyright 7 Royal Meteorological Society

2 14 T. BÖHME ET AL. convective boundary layer (CBL). Their primary focus was on the role of the background flow conditions in selecting the scale of the gravity waves. They showed that in conditions without vertical wind shear, a broad spectrum of gravity waves is generated, and virtually the entire spectrum of waves is able to propagate vertically. In their cases with wind shear, the changes in wind speed strongly filtered the generated spectrum of waves, through critical-level dissipation and wave trapping. Even in cases with relatively weak wind shear, the spectrum was filtered so strongly that only a narrow part of the wave spectrum was able to propagate vertically. Lane and Clark also identified what they considered a special case, where the gravity waves interacted strongly with the thermals in the CBL and induced a resonant effect. In the resonant case, the waves and the thermals were phase-locked and the spacing of the thermals matched the dominant wavelength of the gravity waves. Böhme et al. (4) used wind-profiler observations to examine the development and generation of gravity waves above a CBL. These observations had almost uninterrupted coverage from the surface to 6 km altitude, documenting the growth and decay of the CBL and the generation of the waves aloft. By using wavelet analysis of the data, they were able to show that the observed waves were strongly influenced by wave trapping, which caused distinct wave layers to form. The results of their study were consistent with the mechanisms suggested by Lane and Clark (); however, because of the limitations of only having vertical profiles, the propagation characteristics of the waves could not be unambiguously determined. The primary aim of this study is to revisit the cases studied in Böhme et al. (4) using numerical modelling, in an attempt to learn more about the details of the waves that were observed: in particular, the details of the wave propagation and the influence of wave evanescence and trapping on the wave field. To address this aim, we configure a numerical model to simulate two specific days that were analysed in Böhme et al. (4): 1 and 13 May 1. Results from the model simulations and the wind-profiler observations are analysed and compared in detail. The observations from 1 and 13 May 1 are described in Section ; a description of the numerical model simulations follows in Section 3. In Section 4, the simulation results and wind-profiler observations are compared, by examining the vertical velocity amplitudes and by using wavelet analysis to examine the amplitudes of the component periods. In Section 5, the propagation characteristics of the waves and the influence of wave trapping are determined from the simulations. Finally, we summarize our conclusions in Section 6.. Observations from 1 and 13 May 1 On 1 and 13 May 1 a campaign with the 48 MHz tropospheric wind profiler (TWP) was undertaken at the meteorological observatory of the German Weather Service at Lindenberg, approximately 5 km southeast of Berlin. The synoptic weather situation was marked by high surface pressure (around 13 hpa), maximum surface temperatures around 5 C, and relatively dry air (a surface relative humidity of around 4%). Complete details of the campaign, the synoptic situation and the observational platforms may be found in (Böhme et al., 4). Figure 1 shows the 1 UTC radiosonde soundings taken on 1 and 13 May 1. The profiles indicate that the CBL top (horizontal dashed lines) rises to around km at noon on both days. Both days show relatively weak surface winds. The 1 May wind profile features stronger wind speeds and wind shear in the lower troposphere, compared to the 13 May profile. There was no precipitation on either day. The TWP measured the vertical velocity w at heights between 59 m and approximately 7 m, with a vertical spacing of 15 m and a temporal resolution of 14 s. The measurements from 1 and 13 May 1 both show wave fluctuations that begin in the late morning just above the CBL and propagate vertically into the troposphere. Wave activity persists after the strength of the convection subsides in the late afternoon. Examples of the observed vertical velocity are shown in Figures and 3 at heights of 1. km,.8 km and 4.6 km. These three heights correspond to regions within the CBL, just above the CBL, and in the free troposphere, respectively. On 1 May, the vertical velocity amplitudes are between.3 ms 1 and 1. ms 1 ; these are stronger than the amplitudes on 13 May, which are between.1 ms 1 and.5 ms 1. On both days the vertical velocity amplitudes in the CBL are at least twice as large as the amplitudes above the CBL. Above the CBL, the vertical velocity fluctuations continue to decrease in amplitude with height: the amplitudes at 4.6 km are generally smaller than at.8 km. On both days, the signals at.8 km exhibit higher-frequency fluctuations than at 4.6 km. The vertical velocity fluctuations at.8 km feature a strong contribution from a signal with a period of about 1 minutes, whereas at 4.6 km the strongest signal has a period of about 15 minutes. Because of weak radar-signal reflection, there are irregular wind-profiler signatures at some heights on 1 May between 9 and 18 UTC (not shown). The wind-profiler measurements generally show an average downward vertical velocity. Angevine (1997) identifies a similar downward bias from daytime windprofiler measurements, and attributes it to small particles with low fall velocities; it is likely that the bias exhibited in the Lindenberg measurements is caused by the same phenomenon. Using similar wind-profiler observations taken on 11 May 1, Böhme et al. (4) showed that the vertical velocity fluctuations above the CBL were consistent with gravity waves, the vertical velocity being in quadrature with the potential temperature. They also showed that the observed waves just above the CBL had phase structures consistent with trapped waves. This is a possible reason Copyright 7 Royal Meteorological Society Q. J. R. Meteorol. Soc. 133: (7) DOI: 1.1/qj

3 CONVECTION WAVES Height in m (a) Potential temperature in K (b) Wind u component in m s 1 (c) Wind v component in m s Height in m (a) Potential temperature in K (b) Wind u component in m s 1 (c) Wind v component in m s 1 Figure 1. Profiles of (a) potential temperature, (b) zonal wind, and (c) meridional wind, at 1 UTC on 1 May 1 (upper panels) and 13 May 1 (lower panels). The horizontal dashed lines show the CBL top at 1 UTC. W in m s 1 (a) W in m s 1 (b) 1. Height 46 m Height 8 m for the decrease in wave frequency with height. However, because these observations are limited to vertical profiles only, horizontal structures, and therefore phase velocities, cannot be determined. This limitation means that there is some ambiguity in determining the wave characteristics. In the following section, 1 and 13 May 1 are simulated with an idealized numerical model in order to gain a more complete picture of the wave field by determining the horizontal scales of the waves, their propagation characteristics, and the extent of wave trapping. 3. Numerical model simulations In this section we describe the numerical model and its configurations, and present the results from the simulations of 1 and 13 May 1. W in m s 1 (c) Height 1 m Figure. Time series of vertical velocity from the wind profiler at heights of (a) 4.6 km, (b).8 km, and (c) 1. km, on 1 May Model outline The numerical model used in this study was originally introduced by Clark (1977), and received subsequent updates and improvements. The model is a finite-difference approximation to the three-dimensional, non-hydrostatic, anelastic equations of motion. It features advection schemes that are second-order accurate in time and space, and a first-order Smagorinsky (1963) subgrid-scale closure. In its current configuration, moist processes and the Earth s rotation are neglected. Copyright 7 Royal Meteorological Society Q. J. R. Meteorol. Soc. 133: (7) DOI: 1.1/qj

4 144 T. BÖHME ET AL. W in m s 1 (a) W in m s 1 (b) W in m s 1 (c) Height 46 m Height 8 m Height 1 m Figure 3. As Figure, but for 13 May 1. The model is configured with 15 m horizontal grid spacing and 1 m vertical grid spacing. The model boundaries are 48.3 km long in both horizontal directions, and km in the vertical direction. Therefore, there are 3 3 grid points. The uppermost 1 km of the grid features a Rayleigh friction absorber to reduce the reflection of disturbances off the model s upper boundary. The lateral boundaries are cyclic. The time step is 3 seconds. Surface friction is parametrized using a roughness length of.1 m. The relatively large computational domain is made possible by the recent upgrade of the model s parallel-processing directives to the Message Passing Interface standard by T. Clark of the University of British Columbia. Sensitivity tests were conducted with varied grid spacing and domain size. To approximate the conditions on the observed days and locations, the model is initialized with the soundings observed at 6 UTC on 1 and 13 May 1. These soundings define the potential-temperature field and the wind field, which are horizontally uniform at the initial time. A time-dependent surface sensible heat flux a sinusoidal function with a 4-hour period is applied in the surface layer. In line with the observations, the phase of the sinusoid is chosen so that the sensible heat flux is zero at 6 UTC and 18 UTC, and is maximal at 1 UTC. The maximum value of the sensible heat flux was observed over flat grassland as approximately 175 Wm on 1 May and 15 Wm on 13 May. (These observations were made by the German Weather Service at.4 m above ground level using a METEK GmbH ultrasonic anemometer (USA-1).) These two maximum values of the observed sensible heat flux are used for the corresponding simulations. A small-amplitude random perturbation is applied to the sensible heat flux at every surface grid point. Each simulation is initialized at 6 UTC and continues until 18 UTC. From the above model configurations it is clear that this is not a model simulation tailored to reproduce exactly the conditions observed on 1 and 13 May 1. Rather, we incorporate those features of the flow that have been shown in previous studies to be important in defining the wave field: namely, the background wind and stability profiles and the strength of the surface heating. These simulations do not include synoptic-scale forcing, nor do they include features unique to the Lindenberg region, such as changes in land use and topography. These omissions will be one source of the differences between the observations and the simulations. 3.. Simulation results Horizontal cross-sections of vertical velocity for 13 UTC on 1 May are shown in Figure 4. This time was chosen because it is close to the period of strongest wave activity, and also to the time of best agreement with the observations (see below). By 13 UTC the mixed layer is about km deep and contains convective structures that resemble open cell convection (Figure 4(c)). Updraughts have a horizontal scale of 1 km, and vertical velocities of about 4 ms 1 in magnitude. The thermals do not show any tendency to be organized into streets or rolls. Just above the top of the mixed layer (Figure 4(b)), the vertical velocity perturbations represent gravity waves in the stable troposphere. These waves have an amplitude of about 1 ms 1, and wavelengths of about 4 km, but do not show clear evidence of a preferred direction of propagation; there is little phase coherence, as would be expected from a rectilinear wave propagating in a single direction. In the middle troposphere (4.6 km, Figure 4(a)), the horizontal scale of the gravitywave perturbations has increased to about 8 km, with amplitudes of approximately.5 ms 1. At this height the waves show a preferential alignment of wave fronts in the northwest-to-southeast plane, implying wave propagation towards the southwest or the northeast. The vertical structure of the thermals and gravity waves is illustrated in a southwest-to-northeast crosssection of the vertical velocity (Figure 5). The horizontal scale of the waves just above the mixed layer approximately matches the scale of the thermals below; the dominant wavelength is about twice the width of individual thermals. However, these small-scale waves are evanescent i.e. their phase lines are vertical and their amplitudes decrease with height and by 4 km these evanescent waves have mostly been attenuated. Above 4 km, the remaining waves have a horizontal scale that is larger than the thermals, yet they also appear evanescent, Copyright 7 Royal Meteorological Society Q. J. R. Meteorol. Soc. 133: (7) DOI: 1.1/qj

5 145 CONVECTION WAVES (a) Vertical velocity at Z = 46 m, 13 UTC 1 May 4 Y (km) X (km) (b) Vertical velocity at Z = 8 m, 13 UTC 1 May and just above the mixed layer (Figure 6(c) and (b)) are similar to those shown for the 1 May case. However, the wave field at 4.6 km (Figure 6(a)) appears to have two dominant modes of propagation, one in the southwestto-northeast plane and one in the northwest-to-southeast plane. The 1 May case only had one dominant mode of propagation. The presence of two dominant modes of propagation results in complicated interference patterns. The horizontal scale of the waves at 4.6 km is smaller than before only about 6 km and the amplitudes are also slightly smaller. The southwest-to-northeast crosssection (Figure 7) shows similar patterns to those of the 1 May case, with strong filtering of the short-scale waves below 4 km. Above 4 km, there are fewer complete wavelengths in comparison to the 1 May case, but this is probably a result of the interference between the two dominant modes of propagation highlighted in Figure 6(a). 4. Comparison of wind-profiler observations and model simulations Vertical velocity amplitudes Y (km) X (km) (c) Vertical velocity at Z = 1 m, 13 UTC 1 May 4 Y (km) X (km) Figure 4. Contours of vertical velocity at (a) 4.6 km, (b).8 km, and (c) 1. km, at 13 UTC for the 1 May simulation. The contours intervals are.1 ms 1,.5 ms 1 and 1. ms 1, respectively, with negative values shown by dashed lines. with phase lines that are close to vertical and amplitudes that decay with height. The results of the 13 May simulation are shown in Figures 6 and 7. The patterns of vertical velocity within Copyright 7 Royal Meteorological Society In this section we examine the temporal development of the convective activity in the CBL and the gravity-wave activity aloft from the model simulations and the windprofiler measurements. These activities are estimated by taking a 6-minute moving average of the vertical velocity squared: w, where angle brackets represent the time-averaging. This is calculated directly from the windprofiler data at the heights considered earlier 1. km,.8 km and 4.6 km and it is calculated from the model simulations by taking a vertical profile in the horizontal centre of the model domain, and then following the same procedure as for the wind-profiler data. To the extent that the average vertical velocity over 6 minutes is zero, w approximates the time-dependent vertical-velocity variance, and provides an indication of convective activity in the CBL and wave activity aloft. The modelled and observed time series of w from 6 UTC to 18 UTC are shown in Figures 8 and 9, for 1 and 13 May respectively. On 1 May, the observed and simulated gravity waves above the CBL (Figure 8(a) and (b)) show very good agreement. They show similar temporal development, and similar amplitudes. The model underestimates the wave activity by about a factor of at 4.6 km, and overestimates the maximum wave activity by about a factor of at.8 km at about 14 UTC. Note that a factor-of- difference in w corresponds to only a factor of 1.4 in amplitude. The convective activity in the CBL (Figure 8(c)) does not show such good agreement. The model and the observations show correspondence before 1 UTC, with similar intensification rates and amplitudes. The value of w is overestimated by a factor of.5 at about 133 UTC by the model, yet both graphs show qualitatively similar local maxima around this time. After 14 UTC, the two time series diverge: Q. J. R. Meteorol. Soc. 133: (7) DOI: 1.1/qj

6 146 T. BÖHME ET AL UTC 1 May 6 Z (km) Distance (km) from SW corner Figure 5. Cross-section of vertical velocity from the southwest corner of the model domain to the northeast corner, at 13 UTC for the 1 May simulation. The contour levels are ±.15 ms 1, ±.5 ms 1, ±.5 ms 1, ±1 ms 1, ± ms 1 and ±4 ms 1, with negative values shown by dashed lines. the model features an absolute maximum at around 143 UTC, whereas the convective activity in the observations is decaying at this time. This disagreement between the observations and the model simulations in the CBL in the late afternoon could be due to a number of factors. First, and probably most important, this model does not include any synoptic-scale forcing, and therefore the large-scale subsidence associated with the high-pressure system over Lindenberg is not present. These large-scale effects would act to suppress convective activity in the late afternoon once the surface heating decays. The simple subgrid mixing scheme may also be ineffective at dissipating the turbulent eddies in a realistic fashion. The disagreement may also be due to differences between the idealized model configuration and the real scenario. Nonetheless, it is clear that the differences in the CBL in the afternoon do not lead to substantial differences in the wave fields aloft, and the simulation prior to 14 UTC shows very good agreement with the observations. The comparison between the model simulations and the wind-profiler observations on 13 May (Figure 9) also shows good correspondence. The estimates of wave activity at.8 km agree very well in both amplitude and temporal development. The strength of the convective activity in the CBL (Figure 9(a)) shows very good agreement until 14 UTC; after 14 UTC the values from the model and the observations diverge similarly to the 1 May case. At 4.6 km the model and observations agree well in terms of peak amplitude; however, the observations show the existence of background wave activity early in the morning that continues to intensify in the afternoon and evening. Despite these differences, this simulation shows good general agreement with the observations, especially before 14 UTC. Both cases (1 and 13 May) show wave activities in the upper levels that lag behind the activity in the CBL by about 3 6 minutes. This result is to be expected because of the finite vertical group velocity of the waves. For example, a gravity wave with a horizontal and vertical wavelength of 5 km (similar to the waves here) in an environment with a Brunt Väisälä frequency of.1 s 1 has a vertical group velocity of approximately.8 ms 1. Moreover, both days show the strongest wave activity at.8 km at around 13 UTC, i.e. shortly after the boundary layer has reached its maximum depth. This result highlights the importance of the boundary-layer growth in generating the gravity waves aloft. 4.. Wavelet analysis To compare the measured and simulated spectral composition of the gravity waves in the free troposphere, we conduct a wavelet analysis of the vertical velocity. This technique allows a localization of waves in time and frequency, and is therefore a convenient tool for analysing highly transient phenomena like convection and gravity waves. Thermals within the CBL generate gravity waves in packets, resulting in a wave field that is not spatially uniform in terms of amplitude. Wavelet analysis is very well suited to analysing such a wave field. Using the wave-like Morlet wavelet as the mother function, the vertical velocity amplitudes are calculated for signals with periods in the ranges 4 11 minutes, 1 minutes, and 1 3 minutes. Details of this technique can be found in (Böhme et al., 4) and (Hauf et al., 1996). Copyright 7 Royal Meteorological Society Q. J. R. Meteorol. Soc. 133: (7) DOI: 1.1/qj

7 CONVECTION WAVES (a) Vertical velocity at Z = 46 m, 13 UTC 13 May 4 Y (km) X (km) (b) Vertical velocity at Z = 8 m, 13 UTC 13 May 4 Y (km) X (km) (c) Vertical velocity at Z = 1 m, 13 UTC 13 May 4 Y (km) X (km) Figure 6. As Figure 4, but for 13 May. The wavelet analyses of the wind-profiler measurements and the model simulations for 1 and 13 May are shown in Figures 1 and 11 respectively. In the last section it was shown that the model performs well until about 14 UTC; in this section we only consider the results of the wavelet analysis prior to this time. These two figures show that three general layers can be Copyright 7 Royal Meteorological Society 147 identified from the wind-profiler observations. The first layer is the CBL: the top of the CBL rises from around 1 km at 9 UTC to around km in the afternoon. The second layer is the wave layer, which extends to about 5 m above the CBL top at 9 UTC, and to more than 5 m above the CBL top in the afternoon. The lowest 1 15 m of the wave layer is dominated by waves with shorter periods; the top of the wave layer is dominated by longer-period waves. The third layer is above about 6 km, where only a few gravity waves occur (not shown). Comparison of the measured and simulated amplitudes on 1 May shows that the amplitudes, as well as the temporal and vertical distributions of the wave patterns, do not differ much. Most of the convection waves start between 9 UTC and 1 UTC. The top of the wave layer rises from 15 m at 9 UTC to above 5 m at 14 UTC. Both the model simulation and the windprofiler measurements show maximum amplitudes of approximately.6 ms 1. The wave layer itself shows a division into two regions at approximately 3 m. While shorter waves with periods T between 4 min and 11 min dominate the region between CBL top and approximately 3 m, the medium- to longer-period waves (1 min T 3 min) dominate the wave field between 3 m and 5 m. The comparison suggests that the numerical simulation tends to underestimate the amplitude and vertical extent of the shortest-period waves, but provides reasonable estimates for the amplitude of the longerperiod waves. Although the synoptic pattern did not change substantially overnight, the local situation on 13 May is quite different. The wind speed in the lower-to-middle troposphere has decreased from about 15 ms 1 to about ms 1. Another difference is the change in w from 1 to 13 May. This quantity is similar in the CBL on both days, yet on 13 May the value of w above the CBL is substantially smaller than on 1 May, in both the observations and the simulations. The wave-layer region is shallower, and rises to a maximum height of 45 m at 15 UTC. The numerical model data on 13 May are qualitatively in accordance with the wind-profiler data. However, there are larger quantitative differences between both records, in comparison to 1 May. In some regions, the modelled wave amplitudes are about twice as large as the wind-profiler amplitudes for example, the longer-period waves at heights above 6 m. In general, however, the simulation underestimates the contribution of shorter-period waves (4 min T 11 min) and overestimates the contribution of longer-period waves (1 min T 3 min). Both days show one important similarity: longer-period waves dominate the upper levels, while shorter-period waves dominate the low levels. As a test of the model sensitivity to resolution, the 1 May and 13 May simulations were run again with 5 m horizontal and m vertical grid spacings. On 1 May, the two simulations performed similarly, with the higherresolution simulation having slightly larger amplitudes in comparison to the observations (not shown). The largest Q. J. R. Meteorol. Soc. 133: (7) DOI: 1.1/qj

8 148 T. BÖHME ET AL UTC 13 May 6 Z (km) Distance (km) from SW corner Figure 7. As Figure 5, but for 13 May. w in m s w in m s w in m s (a) (b) (c) Height 46 m Wind profiler measurement 3 D simulation Height 8 m Wind profiler measurement 3 D simulation Height 1 m Wind profiler measurement 3 D simulation Figure 8. Time series of w on 1 May 1 at heights of (a) 4.6 km, (b).8 km, and (c) 1. km, for the wind-profiler measurements (dashed lines) and model simulation (solid lines). differences were for the waves with periods greater than or equal to 1 minutes. In contrast to the 15 m simulation, the 5 m simulation underestimates the amplitudes of waves with periods greater than minutes (especially above 3 m), and overestimates the amplitudes of localized events with periods less than or equal to minutes. On 13 May, the coarser-resolution model run shows the same general differences in comparison to the windprofiler measurements. The contribution of the shorter waves is underestimated while the contribution of the longer waves is overestimated. The coarser-resolution results show even weaker amplitudes for waves with periods less than 8 minutes, in comparison to those present in the 15 m simulation. 5. Further analysis of the simulations In this section the model simulations are analysed to determine the processes controlling the horizontal scale of the waves, and to determine their propagation direction. This information can not be unambiguously determined from the wind-profiler data May The 1 May simulations showed gravity waves with a vertical structure that suggested vertical trapping (Figure 5). The horizontal wavelength of these waves was about 4 km just above the top of the mixed layer, and about 8 km aloft. A horizontal cross-section suggested that these waves were propagating either towards the southwest or towards the northeast; however, the precise direction of propagation could not be determined from these figures. In this section the 1 May simulation is analysed further to determine more information about the propagation characteristics of the gravity waves. The vertical velocity from the simulation was analysed in both the southwest northeast plane and the Copyright 7 Royal Meteorological Society Q. J. R. Meteorol. Soc. 133: (7) DOI: 1.1/qj

9 CONVECTION WAVES Height 46 m Wind profiler measurement 3 D simulation (a) w' in m s w' in m s (b) w' in m s (c) Height 8 m Wind profiler measurement 3 D simulation Height 1 m Wind profiler measurement 3 D simulation Figure 9. As Figure 8, but for 13 May. northwest southeast plane, using two-dimensional spectral analysis to determine the spectral power as a function of frequency ω and horizontal wave number k (not shown). In the southwest northeast plane, the spectral power of the vertical velocity perturbations was located almost exactly along the line in (ω k)-space that corresponds to a horizontal phase speed (ω/k) of 7 ms 1 (i.e. towards the southwest). At lower altitudes there were stronger contributions from waves of high wave number (short wavelength) than at higher altitudes. In the northwest southeast plane, the majority of the spectral power was aligned along k =, consistent with waves whose fronts are aligned parallel to this direction. The vertical wave number m of a two-dimensional nonhydrostatic gravity wave is defined by m = l k, where k is the horizontal wave number and l is defined here in simplified form as l = N (U c) 1 U U c z. Here l is the Scorer parameter, U is the background wind, c is the horizontal phase speed of the gravity wave, and N is the Brunt Väisälä frequency. In situations where k >l, the vertical wave number is complex and the amplitude of the wave decays exponentially with height (i.e. the wave is evanescent). The wavelength λ = π/k at which the wave becomes evanescent is called the critical wavelength, and satisfies π/λ = 1. The square root of the Scorer parameter for the 1 May case is shown in Figure 1(b). l(z) is calculated using the southwest northeast component of the domain-averaged 13 UTC simulated wind (Figure 1(a)), assuming that the phase velocity is 7 ms 1. Also shown in Figure 1(b) are lines that correspond to critical wavelengths of 1 km, km, 4 km and 8 km; when l(z) is to the left of one of these lines, waves with the corresponding wavelength will be evanescent. The profile of l shows that the height to which gravity waves with horizontal wavelengths between km and 4 km (and a horizontal phase speed of 7 ms 1 ) can propagate lies between.5 km and 4 km. Above 4 km altitude, waves with horizontal wavelengths less than 4 km are evanescent. Gravity waves with wavelengths of 8 km are able to propagate vertically up to about 6 km, where they also become evanescent. This result is consistent with Figure 5, which shows that the shortest waves have their largest amplitudes between the top of the mixed layer and 3.5 km. Longer waves exist above this height, but their amplitudes decay with height further aloft. The above discussion is consistent with the windprofiler observations, which show a general reduction in wave frequency (increase in period) with altitude. The 1 May model simulation shows a dominance of a single wave phase speed ( 7 ms 1 ). The phase speed is equal to ω/k, so if the phase speed is constant, waves with larger wave numbers (shorter wavelengths) must have higher frequencies (lower periods). The profile of l shows that the waves with horizontal wavelengths of 4 km or less are evanescent above 4 km altitude. When the shorterwavelength waves are evanescent, at upper levels the only waves present are the longer-wavelength, lowerfrequency waves May Like the 1 May simulation, the 13 May case showed a dominance of short-wavelength waves just above the top of the CBL that appeared to be evanescent aloft (Figure 7). The waves able to propagate vertically had a horizontal wavelength of approximately 6 km, which is shorter than that of the waves at the same height on 1 May. At 4.6 km (Figure 6(a)), there was no single dominant direction of propagation; rather, waves appeared to propagate in both the southwest northeast and the northwest southeast planes, causing interference patterns. Spectral analysis in both of these directions shows a predominance of waves with phase speeds close to zero (not shown). There was little spectral power at non-zero phase speeds in the southwest northeast plane. The southwest northeast component of the modelled wind at 13 UTC (Figure 13(a)) is used to calculate the vertical profile of the square root of the Scorer parameter Copyright 7 Royal Meteorological Society Q. J. R. Meteorol. Soc. 133: (7) DOI: 1.1/qj

10 15 T. BÖHME ET AL. Height in metres Height in metres Height in metres Height in metres Height in metres Height in metres (a) Wind profiler (a) 3 D simulation (b) Wind profiler (b) 3 D simulation (c) Wind profiler (c) 3 D simulation Amplitudes of the w signal for each period class Figure 1. Wavelet analysis of the vertical velocity from the wind-profiler measurements and the numerical model, for 1 May 1. Shading represents the amplitude of w, inms 1, for signals with periods in the range (a) 4 11 minutes, (b) 1 minutes, and (c) 1 3 minutes. (Figure 13(b)). For this calculation, the phase speed is assumed to be zero. Figure 13(b) shows that while kmwavelength waves are evanescent above about 3.5 km, gravity waves with wavelengths greater than 4 km can propagate freely, and will not become evanescent. The middle-tropospheric values of l are generally larger on 13 May than on 1 May, reducing the influence of wave evanescence. Sang (1993) and Lane and Clark () explain that the strongest wave response is from those waves that are almost evanescent. Therefore, larger values of l in the middle troposphere mean that shorterwavelength gravity waves will dominate the wave field. This is consistent with the differences between the 1 May and 13 May simulations. Copyright 7 Royal Meteorological Society Q. J. R. Meteorol. Soc. 133: (7) DOI: 1.1/qj

11 CONVECTION WAVES 151 Height in metres Height in metres Height in metres Height in metres Height in metres Height in metres (a) Wind profiler (a) 3 D simulation (b) Wind profiler (b) 3 D simulation (c) Wind profiler (c) 3 D simulation Amplitudes of the w signal for each period class Figure 11. As Figure 1, but for 13 May. The above discussion identifies a dominant phase speed of the waves in the southwest northeast plane. This phase speed can be used to determine the frequency of the signal for a given wavelength. However, such a frequency should not be directly compared to the estimates of frequency (or period) made using wavelet analyses of the wind-profiler observations. These observed frequencies represent the frequency of the entire signal, not just of the dominant wave propagating in a certain direction. To further examine the influence of changes in the wind profile on the vertical wave propagation, we present a number of other simulations that are similar to the 13 May simulation but with idealized wind profiles and lower model resolution (5 m horizontal and m vertical grid spacing). These simulations use the same Copyright 7 Royal Meteorological Society Q. J. R. Meteorol. Soc. 133: (7) DOI: 1.1/qj

12 15 T. BÖHME ET AL. 8 (a) Wind 1 May 8 (b) I(Z) 1 May 4 km 1 km 6 6 Z (km) 4 Z (km) (m s 1 ) (rad m 1 ) Figure 1. (a) The southwest northeast wind (solid line) and the northwest southeast wind (dashed line), calculated from the domain-averaged wind profile at 13 UTC from the 1 May simulation. (b) The square root of the Scorer parameter for the 1 May case. l(z) is calculated using the southwest northeast component of the wind, assuming a horizontal phase speed of 7 ms 1. The vertical dashed lines indicate the critical wavelengths of 1 km, km, 4 km and 8 km. 8 (a) Wind 13 May 8 (b) I(Z) 13 May 4 km 1 km 6 6 Z (km) 4 Z (km) (m s 1 ) (rad m 1 ) Figure 13. As Figure 1, but for the 13 May case. In this case the Scorer parameter is calculated assuming a phase speed of zero. thermodynamic profile and the same surface sensible heat flux as the 13 May case. The wind profile is replaced by a unidirectional (zonal) wind that is zero below km altitude, whose speed increases linearly between km and 4 km and is constant above 4 km. The three simulations to be considered have free-tropospheric (above 4 km) zonal winds of 5 ms 1, 1 ms 1 and ms 1. The background meridional wind is zero in all cases, so the shear vector (between km and 4 km) points towards the west. The general shape of Copyright 7 Royal Meteorological Society Q. J. R. Meteorol. Soc. 133: (7) DOI: 1.1/qj

13 CONVECTION WAVES m s (a) X (km) 1. (b) Y (km) Figure 14. Vertical velocity at z = 4.6 km at 13 UTC, at (a) y = 5 km and (b) x = 5 km, for simulations using idealized wind profiles with free-tropospheric zonal winds of ms 1 (upper lines), 1 ms 1 (middle lines), and 5 ms 1 (lower lines). Constant values of.6 ms 1, and.6 ms 1 respectively have been added to the vertical velocities for the three simulations; the line representing zero vertical velocity is shown for each simulation. this zonal wind profile is similar to that observed on 13 May (Figure 13(a)). For waves with near-zero phase speeds, this increase in the magnitude of the zonal wind results in a systematic decrease in the (zonal) Scorer parameter. There is no change in the Scorer parameter in the meridional direction. Figure 14 shows the instantaneous (13 UTC) vertical velocity at 4.6 km altitude along the zonal direction and along the meridional direction for each of the three idealized simulations. In the zonal direction (Figure 14(a)), there is a systematic reduction in gravity-wave scale for those simulations with lower wind speed. The larger Scorer parameter in the case with 5 ms 1 wind allows the short-scale waves to propagate vertically, i.e. they do not become evanescent below 4.6 km. The simulation with ms 1 wind has a much smaller Scorer parameter, so the short-scale waves are evanescent below 4.6 km; the longer-scale waves are able to propagate vertically and dominate the wave field aloft. This is consistent with the aforementioned differences between the 1 May and 13 May cases. The vertical velocity in the meridional direction (Figure 14(b)) shows no systematic change in wave scale, because the Scorer parameter in the meridional direction is unaffected by the change in the zonal wind. Therefore there is no mechanism that filters the small-scale signals in the meridional direction, and they are able to propagate vertically to this altitude. This result illustrates that while unidirectional shear does filter the waves in one direction, it does not filter waves propagating in a direction perpendicular to the shear vector. Directional shear represents a more effective mechanism for wave filtering in a three-dimensional wave field, and plays a key role in defining the wave spectrum aloft Propagation speed and direction The above discussion using the Scorer parameter explains one of the main differences between the 1 May and 13 May cases: the dominance of longer-wavelength waves at upper levels on 1 May. This analysis does not, however, explain the other main difference between the simulations: on 1 May there is one dominant direction of propagation, whereas on 13 May there is not. The wind profiles on 1 May (Figure 1(a)) and 13 May (Figure 13(a)) are not substantially different. The northwesterly wind profiles are very similar on both days. The differences are in the southwesterly wind profiles. The first main difference is that on 1 May the magnitude of the southwesterly component of the wind is stronger at the surface, in comparison to 13 May. Secondly, and possibly more importantly, there is a strong, shallow shear layer capping the CBL on 1 May. The magnitude of the wind shear on 1 May is about twice as large as on 13 May (not shown). This stronger wind shear may play a role in defining the dominant direction of propagation on 1 May. Conversely, the lack of strong shear near the top of the CBL on 13 May means that there is no dominant direction of propagation. On both days, the direction of the shear vector changes considerably with height. On 1 May, the shear vector points towards the south below 1 km altitude, towards the south southwest between 1 km and km (the strong shear layer capping the CBL), and towards the south further aloft. Therefore, the dominant direction of wave Copyright 7 Royal Meteorological Society Q. J. R. Meteorol. Soc. 133: (7) DOI: 1.1/qj

14 154 T. BÖHME ET AL. propagation (towards the southwest) is at a small angle to the direction of the shear vector in the (stable) shear layer just above the CBL top. This result agrees with most previous work, which shows gravity-wave fronts aligned perpendicularly to the shear vector. On 13 May, the direction of the shear vector also rotates, pointing towards the east near the surface, towards the west at the CBL top, and towards the south above about 3 km. Both 1 May and 13 May show dominant gravity-wave phase speeds that are approximately equal to the wind speed in the CBL: 7 ms 1 on 1 May and approximately zero on 13 May. Lane and Clark () suggest that such a scenario provides the opportunity for maximum resonance between the waves and the thermals in the CBL. But in their two-dimensional simulations this was considered a special case, only occurring in a few of the simulations. It is difficult to determine whether resonance is occurring here, and whether this is indeed a special case. This is a topic of continuing research. 6. Conclusions On 1 and 13 May 1, a 48 MHz tropospheric wind profiler based at Lindenberg in Germany made detailed observations of boundary-layer growth and gravity waves in clear-sky conditions. The observations showed evidence of strong wave activity that varied with the growth of the boundary layer, and was maximal in the early afternoon. These waves were shown to be consistent with low-level wave trapping, with the frequency of the signal steadily decreasing with altitude. The observations are described in detail by Böhme et al. (4). The wind-profiler observations lack any information about the horizontal structure of the waves, and are therefore unable to determine their wavelength, propagating speed, or propagation direction. In an attempt to determine these wave properties, a numerical model was configured to simulate the growth of the observed boundary layer and subsequent wave generation aloft. The model s configuration was highly idealized and did not include all of the features surrounding the events, such as large-scale forcing or details of topography and land use. Despite the limitations of the model configuration, the simulations were shown to perform very well until the early afternoon. The simulated convective activity and wave amplitudes showed very good agreement with the observations prior to 14 UTC on both days. Detailed wavelet analyses were also performed; these showed that the modelled and simulated waves contained similar spectral quantities in the temporal domain. The main general difference was that the simulations slightly overestimated the amplitude and extent of lower-frequency signals, and underestimated the amplitude of the highest-frequency signals. Analysis of the model simulations showed that on each day there was a strong wave response, just above the mixed layer, from short-wavelength gravity waves. These waves were evanescent and only extended a few kilometres into the troposphere. On 1 May, the wind shear was strong and the waves in the upper troposphere had a single preferential direction of propagation. These waves had a southwestward propagation direction and wavelengths of about 8 km, and they were evanescent above about 6 km. On 13 May, the wind shear was weaker, and the waves did not have a single preferential direction of propagation. On this day, the Scorer parameter was larger, leading to a dominance of gravity waves with shorter wavelengths (around 6 km) in the upper troposphere. These results were consistent with the wind-profiler observations, which indicated that the higher-frequency waves were dominant in the lower troposphere. The observations and simulations highlight the important role of wave filtering in controlling the characteristics of gravity waves above convectively active boundary layers. The analyses presented here show consistency with the conclusions of Böhme et al. (4) and the mechanisms presented by Lane and Clark (). However, they do not show good evidence of a resonant interaction. The dominant direction of propagation of the waves was shown to be related to the low-level shear vector, but this relationship is less well defined in complicated windshear conditions. Therefore, further research is required, under more idealized conditions than those considered here, to understand the processes in three dimensions when the shear vector is not unidirectional. The good performance of the model in the cases presented here will provide further confidence in future studies. Acknowledgements The authors would like to thank Volker Lehmann for providing the wind-profiler measurements, Terry Clark for providing the latest version of his numerical model, and three anonymous reviewers for their comments on an earlier version of the paper. The National Center for Atmospheric Research is sponsored by the National Science Foundation. References Angevine WM Errors in mean vertical velocities measured by boundary layer wind profilers. J. Atmos. Oceanic Technol. 14: Balaji V, Clark TL Scale selection in locally forced convective fields and the initiation of deep cumulus. J. Atmos. Sci. 45: Balaji V, Redelsperger J-L, Klaassen GP Mechanisms for the mesoscale organization of tropical cloud clusters in GATE Phase III. Part I: Shallow cloud bands. J. Atmos. Sci. 5: Böhme T, Hauf T, Lehmann V. 4. Investigation of short-period gravity waves with the Lindenberg 48 MHz tropospheric wind profiler. Q. J. R. Meteorol. Soc. 13: Clark TL A small-scale dynamic model using a terrain following coordinate transformation. J. Comput. Phys. 4: Clark TL, Hauf T, Kuettner JP Convectively forced internal gravity waves: Results from two-dimensional numerical experiments. Q. J. R. Meteorol. Soc. 11: Crook NA Trapping of low-level internal gravity waves J. Atmos. Sci. 45: Hauf T, Clark TL Three-dimensional numerical experiments on convectively forced internal gravity waves. Q. J. R. Meteorol. Soc. 115: Copyright 7 Royal Meteorological Society Q. J. R. Meteorol. Soc. 133: (7) DOI: 1.1/qj