3.1.

Doppler Wind Lidar Images and Spectral Analysis

Nocturnal LLJs were very common in the clear, undisturbed nights and early mornings during JU2003. LLJs appeared in 9 out of 10 IOPs, the lone exception being IOP1 which had convective disturbances due to a passing front.^9–12 From wind profiler/rawinsonde observations¹² and inspection of numerous lidar images,¹⁴ it appeared that the nocturnal LLJs start to form 1 to 2 h after sunset when the ground has cooled, causing a stable layer to develop just above the surface. The LLJs were fully formed around 10 PM local time and were generally strongest just before the sunrise, after which, they started to dissipate from the ground up during the transition from stable to convective boundary layer. Before the LLJs were totally destroyed by the underlying convective boundary layer growth, there was a period during which the atmospheric stability conditions are favorable in the cap inversion layer for shear-generated gravity waves. The gravity waves were fairly common during the transition period associated with the growing of the convective boundary layer. We have seen evidence of wavelike motion in many mornings during JU2003 from Doppler wind lidar images, but only two IOPs’ (IOP2 7/2/2003 and IOP4 7/9/2004) data were chosen for current analysis, since the complete dataset is available for these two cases. Figures 2 and 3 show the RHI (range-height-indicator) scan images of IOPs 2 and 4 at different times in which the wave motions appeared. The lidar scanning directions in these images were approximately parallel to the mean wind directions (see Fig. 1) at the LLJ levels with less than 10 deg of deviation according to mean wind profiles from the radiosondes. The elevation angle of the scan ranged from 0 to 45 deg. Horizontal wind speed was derived from the radial wind signal by dividing the cosine of the elevation angle, while assuming a two-dimensional flow at the jet level.¹⁴ A negative sign represents the wind direction that flows into the lidar in the RHI scanning plane. The gravity waves appeared to be nonlinear, because their wavelength depended on their distance from the lidar. The waves in the IOP2 were shorter in wavelength than that of IOP4, and the LLJ height in IOP4 was much higher than the heights in IOP2. In both cases, the layer above the wave motion, usually characterized as the residual layer from the previous day’s convective turbulence decay, had little wave signatures. Although the wavelengths and heights of the LLJ in those cases were different, they showed some similarity in distinct turbulent air motion near the ground surface below the wave. The gravity waves and turbulent motions appeared to have a strong interaction. A slightly elevated LLJ and wave motion over the urban area was observed in IOP2 cases. This is probably due to the lower altitude of the LLJ in the IOP2 and the urban heat island effect,¹⁴ which was able to reach this height and not the higher altitude of the LLJ in IOP4. As opposed to a nocturnal gravity wave observed when the entire boundary layer is stable,^21,23 the wave did not display overturn or billows in IOP2 or IOP4. This is probably because the lower part of the boundary layer below the cap inversion layer was already in a neutral condition in these IOPs, which had a dissipative effect on the wave motion.

Fig. 2

Doppler wind lidar horizontal wind at different times during IOP2. The azimuth direction for the RHI scan was approximately parallel to wind direction (238 deg). In the figure, the lidar is located at (0, 0), and positive horizontal wind speeds indicate wind flow away from the lidar.

Fig. 3

Doppler wind lidar horizontal wind at different times during IOP4. The azimuth of the RHI scan was approximately parallel to wind direction (230 deg).

Figure 4 shows the spectral analysis that was performed on the lidar horizontal wind speed for IOP2 and IOP4. These are the averaged power spectrum values of the five frames shown in Figs. 2 and 3 and normalized with the highest power spectra values from each scanning frame. The spectrum shows distinct peaks for the dominant average lengths for 1.1 and 1.8 km for IOP2 and IOP4, respectively. The spectral analysis indicated that the power spectrum is different at different heights for both of these cases, and the strongest spectra were at the heights of the LLJ maxima for both cases. The decay of wave motion above the LLJ maximum occurred more rapidly than below the LLJ maximum. By tracking the wave peaks in the lidar image, the wave phase speeds were estimated to be $4.42\mathrm{m}/\mathrm{s}$ for IOP2 and $3.41\mathrm{m}/\mathrm{s}$ for IOP4.

Fig. 4

Power spectral analysis [(a) IOP2; (b) IOP4] of the lidar-observed horizontal winds. The spectra were normalized by the maximum value at the wave altitude locations. The spectra were the average taken from the five frames of the lidar spatial images shown in Figs. 2 and 3.

3.2.

Radar Wind Profiler, Radiosonde, and Linear Atmospheric Stability Analysis

The wave generation is determined by the mean atmospheric wind shear and temperature stratification. Figure 5 shows vertical profiles of horizontal wind and potential temperature before and after the wave motions for both IOP2 and IOP4. The potential temperature was measured by the ANL rawinsonde, while the horizontal winds were measured by the radar wind profiler since the radiosonde’s wind profiles did not have enough data points at altitudes below 200 m. Before the wave activity, the data shows that the strongest shear of the jet is located just below the jet nose or maxima, and the potential temperature profile indicated that a near-neutral stratification was present near the ground at the time. There was a strong stable stratification layer right below the height of the jet speed maximum, and this is where the gravity waves were produced. After the wave motion, the lower boundary layer temperature warmed and the wind profiles generally showed reduced wind maxima at the altitude of the LLJ. In addition, the stratification at the LLJ maxima reduced, and the LLJs moved higher than before the wave motion. The convective boundary layer grew significantly during the wave motion periods which also resulted in a well-mixed equilibrium state, as indicated by the wind and temperature profiles.

Fig. 5

Average wind and temperature profiles before (thick lines) and after (thin lines) gravity wave events for IOP2 and IOP4 measured by ANL radar wind profiler and radiosonde.

A linear stability analysis was carried out for these two wave events using the radiosonde and radar wind profiler data. Figure 6 shows the horizontal wind profiles from the radar wind profiler and temperature profiles from both the PNNL and the ANL sites. As similar temperature stratification and wind shear were observed in previous studies that have used standard linear stability analysis,^29,27 we have used the basic theoretical conclusions resulting from these studies for the analysis that we perform. The PNNL and ANL data agreed reasonably well before the wave episodes, even though the two sites were about 6.3 km apart. Since the ANL data were more complete at the lower levels, the Brunt–Väisälä (or buoyancy) frequency ($N$) and gradient Richardson number ($R_i$) were computed from the ANL data as follows:

Fig. 6

Wind and temperature profiles at the PNNL and ANL sites and the computed Richardson number ($R_i$) and Brunt–Väisälä frequency ($N$) using the ANL data before the gravity episodes for IOP2 (a–c) and IOP4 (d–f).

The wavelength is very much related to the depth of the stratified shear layer. The typical wavelengths range from $2\pi h$ to $7.5h$, where $h$ is the depth of the stratified shear layer.²⁸ Using the estimate of the stable cap layer depth from Fig. 6 from both IOP2 ($h=180\mathrm{m}$) and IOP4 ($h=240\mathrm{m}$) and the dominant wavelengths (Fig. 4, $L=1.1$ and 1.8 km for IOP2 and IOP4, respectively) from spectral analysis results, the relationship between the shear layer depth and the wavelength was established, $L=6.1h$ to $7.5h$. This relation from the observational data agreed with the theoretical analysis.

As the Doppler lidar wind data show (Figs. 2 and 3), the gravity waves are not quite linear, because they have slightly different wavelengths and amplitudes. This nonlinearity is expected in these two cases of ABL not only for the nonuniform atmospheric shear stratification, but also for the different ground surface conditions along the wave-propagation direction. The linear stability analysis gives a reasonably good prediction of the wave characteristics over a relatively uniform and flat terrain. Nonlinear variation and the intermittency of the wave characteristics in space and time cannot be described by the linear theory.

3.3.

Separation of Wave and Turbulence Wind Signals

Although the gravity waves originated from the stable cap layer several hundred meters above the ground, the question to ask is how do they affect both the atmospheric momentum transport in the ABL below the cap and the overall ABL transition from stable to unstable conditions? To answer these questions, we will use data from eight sonic anemometers on the 83-m LLNL crane-tower.^9,31

Sonic anemometer-observed wind signal time series are analyzed using a wavelet technique to characterize the wave impact on the vertical momentum fluxes. We chose to use the wavelet transform technique, because it is more appropriate than the Fourier transform for the nonstationary intermittent turbulence time series.^33,34 In contrast to the Fourier transform, the wavelet analysis is performed locally in time domain by dilating or contracting a wavelet basis function. While both transforms conserve the energy in the physical space, the advantage of wavelet analysis over the Fourier transform is that it preserves the local information of a signal. The local wavelet transform is not affected by the behavior of the signal far away in the physical space. The temporal variation of a turbulent wind signal is especially important for understanding the interactions between nonstationary, spatially coherent wave structures and incoherent turbulence. Furthermore, use of orthogonal wavelet bases allows one to decompose and reconstruct a signal which is useful to separate the gravity waves and turbulence signals. For detailed general descriptions of the wavelet transform and its application in turbulence research, the readers are referred to monographs³⁵ and general reviews.^33,34 We only describe the basic wavelet analysis methods used in this research.

The wavelet transformation coefficient $C(a,b)$ of a function $f(x)$ is an integral transformation defined as

A turbulent wind signal $S(t)$ can be decomposed into orthogonal wavelet series with the wavelet basis ${\psi }_{j,k}(t)$:

Using the wavelet multiresolution analysis technique,³⁶ the turbulent wind signals can be decomposed into the fine-scale turbulent and large-scale coherent parts. The wavelet decomposition produces a family of hierarchically organized signals. The choice of a suitable level for the hierarchy to separate the turbulence from the signal is largely dependent on the signal being analyzed. Howell and Mahrt³⁷ have used the ratio of turbulence variance to total variance in the low-pass filtered signal to determine the level of the wavelet decomposition. In our analysis, we choose to use the recursive method by Farge et al.³⁸ to separate the turbulent and wave parts from the signal. The wavelet basis function is the Symlets,³⁵ which is an orthogonal function. In this algorithm, an optimal threshold is iteratively determined without any adjustable parameters. It has been proven to be very robust in the separation of the coherent large-scale and turbulence signals.³⁸ Given a time series $S(t)$ with $M$ discrete samples, we initially set $M^I=M$, where $I$ represents the number of data points in the incoherent turbulent signals. A wavelet transform is performed to obtain the wavelet coefficient $C(j,k)$. The threshold (${\epsilon }_0$), a dependent variable of the wavelet coefficient $C(j,k)$, is estimated by

A recursive loop ($n=0$) is entered after initialization. For the $n$’th iterative step, we count the number of wavelet coefficients less than ${\epsilon }_n$, which gives a new $M^I$, and then set $M_{\text{old}}^I=M^I$ and compute the new variance using the wavelet coefficient less than the ${\epsilon }_n$, ${\sigma }_{n+1}^2=1/M^I\sum _{j,k}{|C^I(j,k)|}^2$, and recomputed a new threshold for the next iteration, ${\epsilon }_{n+1}={\sigma }_{n+1}{(2\ln M^I)}^{1/2}$. Next, in an iterative fashion, we repeatedly set $n=n+1$ until $M^I=M_{\mathrm{old}}^I$. The final step is reconstructing the coherent signal using all wavelet coefficients greater than the threshold, ${\epsilon }_n$, using Eq. (4). The difference between the original signal and the coherent signal is the turbulence signal.

Before the wavelet analysis, the sonic $u$, $v$, $w$ signal was rotated in such a way that the mean wind is oriented along the $u$ direction. Since the wave observed during IOP2 and IOP4 are quasi two-dimensional, only $u$ and $w$ components are analyzed via the wavelet decomposition method. Furthermore, the mean wind speed was subtracted from the signal before the analysis. Figure 7 shows an example of wavelet decomposition of IOP4 sonic anemometer signals into three parts—the mean, the wave (∼), and the turbulent fluctuation (´) at $z=82.3\mathrm{m}$.

Fig. 7

An example of wavelet decomposition of $u$ and $w$ wind components of sonic anemometer wind data at 83.2-m height during IOP4. It is decomposed into mean, wave (∼), and turbulent fluctuation (´) parts.

Figures 8 and 9 show the LLNL tower sonic anemometer-observed horizontal and vertical winds during IOP2 and IOP4 after a signal decomposition using the wavelet technique. The gravity waves generated by the LLJ at the inversion cap layer are measured in situ by the sonic anemometer and shown in the time series plots of Figs. 8 and 9. The wave motion appeared in both the horizontal and vertical wind signals which were in coherent, near-quadrature fluctuations with $u$ ahead of $w$, which is a typical signature of wave motion. The wave motions were nonlinear with some variations of amplitude and period. Further inspection of the wave signals from the tower sonic anemometers indicated that the wave originated at higher levels with the lower level signals lagging behind. The wave signals at different levels also show a damping effect with smaller amplitudes in lower levels.

Fig. 8

Horizontal (a) and vertical (b) wind time series from LLNL tower sonic anemometers during IOP2. The signals are low-passed wave part of the wind using the wavelet separation technique. The heights of anemometers are labeled on the color-coded curves, applying for both panels. The signals are also transformed and plotted by adding a number, so the curves are readable.

Fig. 9

Horizontal (a) and vertical (b) wind time series from LLNL tower sonic anemometers during IOP4. The signals are the low-pass-filtered part of the wind signal using the wavelet separation technique. The heights of anemometers are labeled on the color-coded curves. The signals are transformed and plotted by adding a number, so the curves are readable.

Gravity waves not only feed energy to small-scale turbulence through wave breaking and instabilities, but also directly contribute to the transport of momentum. Figure 10 shows that the wave momentum flux is 1.5 to 3 times greater than the turbulent momentum flux at higher levels and approaches it in magnitude near the ground surface. The negative sign indicates that wave momentum is transported to the ground and is absorbed by the ground surface. The higher momentum transport was transported downward by the wave momentum flux, which indicates the importance of the role that gravity waves play in the ABL transition from stable to unstable condition; namely, it accelerates the ABL transition toward the well-mixed equilibrium condition.

Fig. 10

Vertical wave momentum flux and turbulent momentum flux using the decomposed signals.