Internal gravity waves convectively forced in the atmospheric residual layer during the morning transition



Generation of internal gravity waves in the boundary layer is investigated from observations. Simultaneous measurements from a 2 µm Doppler lidar and a 0.5 µm backscatter lidar are combined to analyse the occurrence, or not, of internal gravity waves in the residual layer during the morning transition on two days, 10 and 14 June 2005. Three cases are studied for illustrating three different flow configurations in the residual layer: no wave, evanescent wave and propagating wave. Comparison of the three cases suggests two necessary conditions for the generation of gravity waves: a stably stratified residual layer and a convective boundary layer with mechanical forcing frequencies less than the Brunt–Vaïsala frequency. The horizontal wind shear probably plays a role in the dynamics of the waves, but, in the cases analysed, it is not sufficient alone to generate the observed waves through the obstacle effect. In the case of wave propagation, the waves tilt upstream and against the wind shear, with a typical horizontal wavelength and a line phase direction with respect to the vertical of 2.4 km and 32°, respectively. Unexpectedly, we found that measurements of the wave-associated vertical velocity and the displacement of tracers (0.5 µm depolarization ratio or 2 µm backcatter, both indicative of relative humidity fluctuations) are in phase. Possible explanations include: (i) aerosol particles are not passive with respect to temperature or water vapour fluctuations; or (ii) a nonlinear wave-turbulence interaction is at work and needs further investigation. Copyright © 2011 Royal Meteorological Society

1. Introduction

The residual layer (RL) is usually defined as a neutrally stratified layer resulting from the decay of turbulence in the formerly convective boundary layer (CBL) (Stull, 1988). Almost isolated from the ground by the nocturnal stable layer (NSL), the RL is usually not affected by turbulent transport of surface-related properties but by subsidence and generally by the shear-related burst of turbulence (Mahrt, 1999; Yi et al., 2001). In this study we investigate RL–CBL interactions during the morning transition when the RL is progressively eroded by the CBL. Exchanges between RL and CBL are of particular interest for air quality studies and for a general understanding of the vertical mixing and the budget of atmospheric boundary layer (ABL) constituents during the morning transition (aerosols, O3, CO2, H2O) (Doran et al., 2003; Gibert et al., 2008; Kolev et al., 2008).

It is generally thought that the RL remains isolated from the ground and its top is thus expected to remain constant overnight (excluding effects of subsidence and radiative cooling), until fully eroded by the developing CBL (Stull, 1988), yet coupling between the RL and the developing CBL and entrainment at the top of the RL has been observed (Fochesatto et al., 2001). Fochesatto etal., suggested that internal gravity waves in the RL played a role in this interaction and in the turbulence leading to entrainment at the top of the RL, however, they did not show evidence of the wave phenomena. In addition, Sun et al. (2004) showed that the passage of an internal gravity wave in a stable layer can be associated with a flux of momentum, heat, H2O and CO2.

Among the different sources of atmospheric gravity waves (see Nappo (2002) or Fritts and Alexander (2003) for a general view), convection is recognized as a major source, others being topography and jet or front systems. Three simplified mechanisms have been proposed to describe convective generation of gravity waves: (i) pure thermal forcing, which is linked to a latent heat flux source (Salby and Garcia, 1987); (ii) an ‘obstacle’ or ‘transient mountain’ effect in strong wind-shear conditions (Clark et al., 1986); and (iii) a ‘mechanical oscillator’ effect linked to the periodic intrusion of CBL overshoots in a stable stratified layer (Fovell et al., 1992). In the present article we consider only cloud-free conditions that exclude thermal forcing. However, the relative importance of the other two generation mechanisms in wave-generation, i.e. obstacle effect versus mechanical oscillator effect, is still an open issue that has been mainly addressed by mesoscale model simulations (Clark et al., 1986; Carruthers and Moeng, 1987; Alexander et al., 1995; Kershaw, 1995; Lane and Clark, 2002) but needs new dedicated observations. One other important motivation of this article is to study the contribution of these waves to convective initiation and organization.

The first observations of internal gravity waves in the ABL were made with an array of acoustic sounders (Hooke et al., 1972). Since then observations of convectively generated waves have rather focused on the free troposphere using in situ airborne measurements (Kuettner et al., 1987; Hauf, 1993) and radars (Böhme et al., 2004, 2007; Chagnon and Gray, 2008). Fot this article, the study of convective wave generation during the morning transition required wind field measurements at higher space–time resolutions than those provided by wind profilers, due to the short wavelength and limited area of propagation of these waves in the RL. Observations of an elastic backscatter lidar (EBL) and a heterodyne Doppler lidar (HDL) are used hereafter in combination with surface meteorological sensors, soundings and also the Pennsylvania State University/National Center for Atmospheric Research (PSU/NCAR) mesoscale model MM5 (hereafter MM5) reanalysis. Three cases are presented, analysed and compared: I, showing no wave; II, in which an evanescent wave occurs (no propagation upward); III, with a wave propagating in the entire RL. The three cases were selected because they present a clear contrast in the dynamical behaviour of the RL while the background conditions remained comparable, making it possible to attribute the occurrence of the waves to those specific features that differ between the three cases.

Section 2 presents the experimental set-up and section 3 the observations and their analysis. Section 4 is devoted to the application of the linear theory of wave excitation by turbulence in the ABL for the different cases. In particular we provide an experimental estimate of wave forcing frequency, details of wave characteristics and a discussion of the wind-shear effects. Section 5 extends the work of Fochesatto et al. (2001) by analysing coupling effects between the CBL, the RL and the free troposphere during the morning transition. A conclusion and discussion are presented in Section 6.

2. Experimental set-up

The experimental site is shown in Figure 1. The EBL and HDL lidars are located at Laboratoire de Météorologie Dynamique, Institut Pierre Simon Laplace (IPSL-LMD, Palaiseau, France), 20 km southwest of Paris in a nearby suburb on a 165 m high plateau that extends west and north for several kilometres. Table I summarizes the main characteristics of the two lidars, and more details can be found in Gibert et al. (2008) and Loth et al. (2004). The EBL signal is backscattered by molecules and particles whereas the HDL signal is due to particles only (heterodyne detection). The EBL also implements a polarization diversity capability, which indirectly provides information on the relative humidity. The relevant quantity is the depolarization ratio Δ = P/P//, where P and P// are the perpendicular and parallel signal, respectively. For a linear polarization of the incident laser beam, the backscatter signal depolarization usually varies with the type and the shape of particles and equals 1.4% for molecules. This value is taken as a reference in the free troposphere. The depolarization ratio is expected to change with the relative humidity for RH > 60% as both size and shape of particles are then modified significantly (Fitzgerald, 1989). Using both backscatter and depolarization ratio signals, the EBL is used to retrieve the 15 m space and 10 s time resolved vertical structure and layering in aerosol content and sphericity (proxy for relative humidity), the heights of the CBL, zi, and the RL, zr. Located 200 m northeast, the HDL provides 75 m space and 25 s time averaged vertical velocity measurements. This distance between the lidars is smaller than usual CBL heights so the two lidars are assumed to be embedded in the same eddy and to probe the same wave phenomenon. Some representativity error may occur depending on the wind direction, strength and shear. Lidar measurements provide cross-sections of the atmosphere for the different variables listed in Table I, but only for z > 200 m. Both an incomplete overlap between the field of view of the telescope and the laser beam for the EBL and parasitic optical reflections for the HDL preclude measurements below this altitude.

Figure 1.

Experimental site at IPSL-LMD, Ecole Polytechnique campus, France.

Table I. Instrumentation at IPSL-LMD, Ecole Polytechnique, Palaiseau, France used for the dataset in this article.
Instrument/dataMain characteristicsVariables used in this articleTime resolutionVertical resolution
Elastic backscatterLaser Nd: YAG0.5 µm bakscatter signalP0.5 μm.z210 s15 m
lidar (EBL)     
 Wavelength 532 nm Energy 30 mJDepolarization ratioΔ  
 PRF 20 HzABL heightszi, zr  
Heterodyne Doppler lidar (EDL)Laser Tm,Ho: YLF2 µm bakscatter signalP2 μm.z225 s75 m
 Wavelength 2064 nmVertical velocityuz  
 Energy 10 mJ    
 PRF 10 Hz    
Sonic anemometer at 10 m Surface sensible heat fluxequation image30 min 
Soundings TemperatureT2 day−1<10 m
  Specific and relative humidityq, RH  
  Wind speed and directionux, uy, uz  
MM5 reanalysis TemperatureT1 h50–200 m
  Specific and relative humidityq, RH  
  Wind speed and directionux, uy, uz  

A sonic anemometer at 10 m is also used to infer 30 min rolling-average surface heat flux measurements. In addition vertical profiles of humidity, temperature and horizontal wind speed and direction were obtained from daily soundings at a nearby meteorological station located 10 km to the west of IPSL-LMD site at approximately 0000 and 1200 UTC.

To complete the dataset of meteorological variables during all measurement days, we use hourly mesoscale MM5 model reanalysis interpolated at the IPSL-LMD. These reanalyses are produced for real-time air quality modelling in the framework of the CHIMERE model (IPSL-LMD), with a time resolution of 1 h, 32 levels between 0 and 13 km (15 levels between 0 and 2 km), and a domain resolution of 5 × 5 km2. The MM5 profiles provide useful guidance about the evolution of the flow during the morning transition for the time intervals in between two soundings.

3. Observations and analysis

The observations were made on 10 and 14 June 2005. These days are characterized by clear sky conditions and similar surface heat flux during the morning transition. In addition to measurements listed in Table I, we also used the soundings and the MM5 reanalyses to calculate:

  • the Brunt–Vaïsala pulsation in rad s−1, commonly called the ‘Brunt–Vaïsala frequency’, as the criterion for static stability:

    equation image(1)

    where θ is the virtual potential temperature in K and g = 9.8 m s−2 is the strength of the gravitational field (in this article the Brunt–Vaïsala frequency will be written as fN = N/2π); and

  • the gradient Richardson number as the criterion for the dynamic stability of the flow:

    equation image(2)

    where ux and uy are the two components of the horizontal wind. A laminar stratified shear flow becomes unstable and may break down into clear-air turbulence for Ri < 0.25 and a turbulent flow becomes laminar for Ri > 1 (Stull, 1988). These results are derived from theory for parallel flows, which may not be the fully justified in real atmospheric conditions but provides a guideline.

3.1. Observations on June 10: nocturnal, residual and convective layers

Figure 2 displays a general view of the morning transition during 10 June (called J10 hereafter). Using the lidar backscatter signal at 0.5 µm we retrieved the height zi and zr for the CBL–RL and RL–free-troposphere interfaces respectively (see Menut et al. (1999) for the algorithms). Figure 3 completes the data set with night (2314 UTC) and day (1122 UTC) soundings and MM5 reanalysis profiles.

Figure 2.

Lidar measurements on 10 June: (a) and (b) 2.064 and 0.532 µm backscatter signal (P.z2) in logarithmic scale and arbitrary units; (c) depolarization ratio Δ; and (d) vertical velocities uz as functions of the altitude. Thin solid lines indicate zi and zr. Local time is UTC time +2 h.

Figure 3.

Soundings for 10 June (black solid lines) and MM5 reanalysis profiles (markers) for the (a) night (sounding is at 2314 UTC) and (b) morning transition (0800, 0900 and 1000 UTC) (c) convective boundary layer at 1100 UTC (sounding at 1122 UTC). The variables displayed are the temperature (T), the virtual potential temperature (θ), the relative humidity (RH), the wind speed (V) and direction (dirV) and the gradient Richardson number (Ri).

The night potential temperature profile shows that the former nocturnal layer is characterized by a rather small inversion close to the ground (z < 150 m). The maximum speed of a northeastward nocturnal jet reaches 8.2 m s−1 at z = 200 m. This height coincides with the top of the nocturnal layer. The MM5 reanalysis does not provide the observed nocturnal layer properties (low inversion and nocturnal jet).

The RL starts above the nocturnal layer with a quasi-constant virtual potential temperature profile showing near neutral conditions. The RH profiles show that the nocturnal layer is covered by a low-RH layer from 0.2 to 0.6 km where RH < 50%. Then, above 600 m, the relative humidity increases to reach 82% at the top of the RL. Both MM5 reanalysis and lidar measurements show that this layer keeps the same properties during the morning. The depolarization ratio profiles show large values (Δ > 0.1) below 0.7 km and small values above (Δ < 0.05), following the RH profile according to particle hygroscopicity (Figure 2). This RL is characterized by a negative wind shear dV/dz of −510−3 s−1 up to z > 0.9 km which, together with the weak stability, explains the low values of the gradient Richardson number: Ri < 1 for 0.4 < z < 0.9 km (Figure 3a). The flow is expected to be turbulent at z = 0.7 km where Ri = 0.1. The MM5 reanalysis provides a good potential temperature profile in the RL but generally fails to represent the wind shear, the layering of RH and the local low gradient Richardson values.

The J10 day is characterized by a rapid growth of the CBL height from 0.4 to 1.7 km in 2 h (0800–1000 UTC) until the complete erosion of the RL. During all the morning transition, the RL top keeps the same altitude (1.72 ± 0.02 km). Throughout the morning transition, strong updrafts produce large overshoots at the top of the CBL, reaching δzi = 400 m at 0850 and 0920 UTC (Figure 2). These observations correspond well to the near neutral condition observed in the RL. The MM5 reanalysis potential temperature profiles at 0800, 0900 and 1000 UTC show that the statically neutral condition remains during the rise of the CBL (Figure 3b). No stratification and temperature inversion slow down the rise of the CBL. In summary, J10 illustrates a case with a neutral RL that is eroded quickly by the vigorous convection in the developing CBL.

3.2. Observations on 14 June: nocturnal, residual and convective layers

The morning transition on 14 June (called J14 hereafter) contrasts with that of J10 mainly due to the stratification of the RL, and its slower erosion. Lidar observations are displayed in Figure 4 with backscatter signals at 2 and 0.5 µm, depolarization ratio and vertical velocity. Figures 4a and 4b show no contrast in backscatter signal owing to the same CBL and RL properties. Therefore, zi is determined using the depolarization ratio contrast. Figure 5 presents the soundings at 2319 and 1117 UTC and MM5 reanalysis profiles.

Figure 4.

Same as Figure 2 but for 14 June.

Figure 5.

Same as Figure 3 but for 14 June 14. The night- and daytime soundings were at 2319 and 1117 UTC respectively.

The J14 nocturnal layer is characterized by large relative humidity and a near-ground nocturnal jet from the northeast, which reaches 6 m s−1 at z = 50 m. The night-time potential temperature profile shows that this relatively thick nocturnal layer (∼350 m) is capped by a strong inversion of 6 K (to be compared with 2.5 K for J10) located between 0.3 and 0.6 km (Figure 5a). The MM5 reanalysis profiles are in relatively good agreement with the sounding profiles identifying the nocturnal jet from the north, temperature inversion and the large value of RH close to the ground.

During the night and above the inversion there is a dry and dynamically stable layer (see the large value of Ri) with a change of wind direction from northward to southwestward (Figure 5a). The RL on J14 is characterized by a stable stratification, with a temperature gradient of −5 K km−1 close to the free troposphere condition of −6 K km−1 (to be compared with the J10 decrease of −9 K km−1) and by a stratification of the atmosphere (see the RH profile in Figure 5a and 0.5 µm backscatter and depolarization in Figure 4c). A strong positive wind shear of dV/dz of 1.510−2 s−1 up to z > 1 km and a weak stratification explain the low values of the gradient Richardson number Ri < 1 for 0.6 < z < 1.1 km, as for the J10 night. The flow is expected to be turbulent at z = 0.7 km where Ri = 0.1. The 2319 UTC and 1117 UTC soundings show that this wind shear is reduced to 5.710−3 s−1 during the morning transition due to a decrease of the wind speed from 11 to 6 m s−1 for z > 1.5 km (Figures 5a and 5c). The MM5 reanalyses show a smaller difference between night and day RL wind shear but suggest that the stable stratification remains during the morning transition (see the 0800, 0900 UTC potential temperature profiles in Figure 5b). Lidar measurements provide additional evidence that the stratification of the atmosphere remains the same as for the night. The dry stable layer between 0.6 and 0.8 km is identified by a relatively large depolarization ratio and low backscatter signal (Figures 4b and 4c).

In contrast to J10, the morning transition is characterized by a slow rise of the CBL before 1020 UTC, with a rate of increase of CBL height dzi/dt = 0.15 km h−1 for zi < 0.7 km (to be compared with the rate of dzi/dt = 0.75 km h−1 during the same period of time for J10) corresponding to a slow erosion of the strong temperature inversion (Figure 2c and 4c). Updraft vertical velocities and the overshoot amplitudes (δzi = 150 m) are globally smaller than for J10 (Figures 2d and 4d). After 1020 UTC and for zi > 0.7 km, the increase of CBL height is rather faster b–reaching a speed of dzi/dt = 0.85 km h−1.

3.3. Occurrence of waves in the residual layer: comparison between cases I, II and III

During case I (J10), from 0815 to 0900 UTC, the vertical velocities in the RL reached significant values. Figure 2d shows positive vertical velocity of +0.5 m s−1 around 0850 UTC. However, no wave phenomenon is suggested by lidar backscatter, the depolarization ratio or the vertical velocity time series.

In case II (J14—from 0830 to 0915 UTC), 0.5 µm backscatter and depolarization ratio signals show that the increase of CBL height entails some oscillations in the dry layer above (Figures 4b and 4c). It is worth noticing that these oscillations are not seen above z > 0.9 km and that the fluctuations of zr are negligible during this period of time. Velocity fluctuations can be seen but they remain unorganized and very close to the noise level of the signal, i.e. 0.2 m s−1 (Figure 4d). To conclude, the observations during case II show a wave phenomenon that seems to be linked to the rise of the CBL but does not propagate upward.

Later in the morning of 14 June, from 0945 to 1030 UTC, which corresponds to the case III period of time, the depolarization ratios show remarkable coherent vertical structures in the RL, which seem to be well correlated with the positive vertical velocities measured by the HDL. The wave phenomenon is observed in the entire RL up to the free troposphere interface.

4. Wave excitation by convection

We now investigate under which conditions the thermals of the developing CBL may force gravity waves in the RL. Essential elements and notations of the linear theory of gravity waves are recalled in section 4.1. The frequencies of the thermals and of the gravity waves are then estimated in several ways and compared to show under what conditions gravity waves are excited (section 4.2).

4.1. Linear theory in the atmospheric boundary layer

We report here a linear theory in the ground-based lidar reference frame. We represent the thermodynamic variables as a sum of the base-state variables plus small deviations (Stull, 1988):

equation image(3)

The deviations are assumed to be small so we can linearize about the base state. The base state variables are indicated with the bar over the letters and deviations with the prime in Eq. (3). For all of this study we neglect the effect of diffusivity, viscous dissipation and heat transfer, so the entropy and potential temperature are conserved, Ds/Dt = 0, Dθ/Dt = 0, which after linearization leads to:

equation image(4)

where we used Eq. (1), writing equation image, and uz is the vertical wind component.

Making the Boussinesq approximation, the equation for mass conservation simplifies to:

equation image(5)

and the equation of momentum is:

equation image(6)

where ∇H = ∂/∂xex + ∂/∂yey.

Using Eqs (4)–(6), we find the following wave equation for uz (Holton, 2004):

equation image(7)

In the same way, we may find a wave equation for pressure p′, temperature θ′ and density of all components in the atmosphere ρ′.

Assuming that N is constant with height, we look for a monochromatic plane wave solution, equation image, with equation image the vertical velocity amplitude of the wave, k the wavenumber and ω the pulsation in the lidar reference frame, i.e. fixed to the ground. We find the dispersion relation:

equation image(8)

where kH = kxex + kyey is the horizontal wave number vector and

equation image(9)

is the intrinsic pulsation of the wave. The corresponding intrinsic frequency will be noted fΩ = Ω/2π.

For Ω < N or Fr < 1 (with Fr, the Froude number), kz is real and the solution of Eq. (7) is a propagating wave. In this case, we usually write:

equation image(10)

where β is the angle of the wave vector with respect to the horizontal or the angle of the phase lines with respect to the vertical (e.g. Gill, 1982).

For Ω > N or Fr > 1, kz is imaginary and the solution of Eq. (7) is an evanescent wave that propagates only in the horizontal direction. The wave amplitude decreases exponentially in the vertical direction with a characteristic length scale of equation image where ΛH = 2π/|kH| is the horizontal intrinsic wavelength (Gill, 1982).

4.2. Application

After recalling the meaning of the intrinsic pulsation of the wave (section 4.2.1), we will estimate the frequencies observed in the waves and in the convection in the CBL (section 4.2.2), in order to explain the forcing mechanism of the observed waves (section 4.2.3).

4.2.1. Intrinsic frequency of the wave

For any fluid motion the vertical parcel velocity w is related to the vertical parcel displacement ξ by w = Dξ/Dt, where D/Dt denotes the time derivative following the parcel motion. Thus, assuming horizontal homogeneity, we expect a π/2 phase shift between vertical wind velocity and vertical scalar displacements or fluctuations such as temperature and density. Indeed, for a monochromatic gravity wave, we can write:

equation image(11)

where the tilde denotes the amplitude of the wave and Ω is the Lagrangian or intrinsic wave pulsation as used in Eq. (8). Equation (11) will be used to obtain direct experimental measurement of intrinsic frequency of the wave.

4.2.2. Forcing frequency due to convection

Two different kinds of forcing mechanisms of wave in the RL can be associated with the convection during the morning transition (in free-cloud conditions): (i) the ‘mechanic oscillator’ mechanism, in which the forcing frequencies due to convection result from the vertical motion of up and downdrafts (Fovell et al., 1992); and (ii) the ‘obstacle effect’ in which the wave generation results from advection by the horizontal wind over the pattern of overshoots at the top of the CBL (Clark et al., 1986). Of course, these two forcing mechanisms are simplified models trying to capture essential mechanisms for wave generation. Reality lies between the two extremes, with some time-dependence of the forcing and some shear always present.

The following paragraphs present the experimental estimates of a dominant forcing frequency through the mechanic oscillator or the obstacle effect. Given a forcing frequency, it is expected that the dominant response will be at that frequency. Other forcing frequencies and other harmonics of the response may be present but weaker and will hence just modulate the main forcing and wave frequency. This assumption will be justified in the analysis for case II in section 4.2.3.

(1) Mechanic oscillator forcing frequency. The forcing frequency ft is estimated looking at the vertical amplitude of the overshoots and using Doppler lidar vertical velocities at the top of the ABL. From Eq. (11) we can write the mechanic oscillator forcing frequency as:

equation image(12)

where the variance of the vertical velocity is corrected from the noise variance.

We assume that Taylor's frozen turbulence hypothesis is valid at the top of the CBL as the vertical velocity standard deviation (<0.5 m s−1) is weak compared with the horizontal wind speed (>5 m s−1). Therefore, backscatter lidar measurements provide a non-biased estimate of overshoot amplitude at the top of the CBL. A 30 min rolling time gate is chosen to calculate each variance, as a good compromise to having a sufficient number of samples (i.e. overshoots) in the time gate and to resolve the different wave regimes during the morning transition.

Carruthers and Hunt (1987) also suggested another way to estimate the main mechanic oscillator vertical forcing frequency of the energy-containing eddies that does not require Doppler lidar measurements:

equation image(13)

where Ut and Lt are the velocity and time scales of the eddies.

Using backscatter lidar and surface heat flux measurements, ft,scale can be calculated as a function of the turbulent velocity scale and the height of the CBL (Appendix A):

equation image(14)

where w is the convective velocity scale and the 0.9 coefficient is determined in Appendix A. Variables w and zi are calculated over a rolling time gate of 30 min.

(2) Obstacle forcing frequency. The obstacle horizontal wavelength of thermal overshoots and the vertical gradient of the horizontal wind above the CBL are the critical parameters for waves generated through the obstacle effect. We assume that the mean wind is uniform in the CBL and that there is a vertical gradient of the horizontal wind above the CBL (equation image over the mean height of the overshoots). In the reference frame of the CBL wind, the overshoots are then fixed (i.e. in a mountain-like situation). In this case, there are solutions of Eq. (7) in the form of waves that are stationary relative to the CBL (i.e. ω = 0). For such stationary waves, uz depends only on x, and Eq. (7) simplifies to:

equation image(15)

The dispersion relation is then:

equation image(16)

A Froude number is then usually written as:

equation image(17)

where equation image is the increase of horizontal wind speed at the top of the CBL (with the notation equation image) and L is the obstacle horizontal wavelength of excitation, i.e. the horizontal wavelength of thermal overshoots. This Froude number can be defined at any time t at a mean CBL height zi(t) using values for equation image and L that are obtained with a time gate of 30 min. We will express Fr as a function of zi: Fr(zi).

Assuming the CBL top is moving at a mean horizontal wind speed of equation image and that the turbulence horizontal pattern is homogeneous along the wind direction, the overshoots obstacle forcing frequency can be estimated using a lidar Eulerian cross-section of the CBL:

equation image(18)

equation image is calculated over the mean height of the overshoots (i.e. equation image), i.e. the mean height over which the mean flow is disturbed by the obstacle. For this purpose, we use the horizontal wind velocity profile of the sounding at 1117 UTC, which is in agreement with the MM5 profile at 1000 UTC.

The main eddies horizontal ‘spatial frequency’, fzi, is estimated with the maximum in the power spectra of lidar CBL-height time series (Appendix B) and is used to estimate L(zi), the main horizontal wavelength of CBL overshoots as indicated in Eq. (18). equation image and fzi are calculated over a rolling time gate of 30 min.

4.2.3. Conditions to force gravity waves with convection in the residual layer

We now explain a simple criterion to identify favourable conditions for the forcing of gravity waves in the RL. Figure 6 summarizes our main findings, and requires some care in interpretation: in the left panels of Figure 6 we display the vertical profile of the Brunt–Vaïsala frequency (fN = N/2π) for 10 June (Figure 6a) and 14 June (Figure 6c) as a function of altitude, from the night-time sounding and MM5 reanalysis for 0000, 0800, 0900 and 1000 UTC. We can see that the night-time fN profile from MM5 does not provide the fine resolution of the stratification in the RL. However, lidar reflectivity and depolarization measurements (Figures 4b and 4c) make it clear that the fine stratification in the RL seen on the midnight sounding is still in place during the morning transition. We choose then to compare the thermals' forcing frequencies during CBL rise with fN values in the RL as calculated with the night-time sounding. Hence this sounding is also reproduced in the right panels of Figures 6b and 6d, superposed on mechanic oscillator and obstacle forcing frequencies (section 4.2.2).

Figure 6.

Brunt–Vaïsala frequency profiles for (a) 10 and (c) 14 June calculated from the night-time sounding (black solid line) and from MM5 reanalysis: night, solid line; 0800 UTC, dashed line; 0900 UTC, dashed and dotted line; 1000 UTC, dotted line). (b) and (d) Mechanic oscillator forcing frequency (Eq. (12)) using Doppler lidar measurements, ft,mech (blue solid line), and scales in the CBL, ft,scale (Appendix B; green solid line) and obstacle forcing frequency, ft,obs (pink solid line) are displayed as a function of zi(t).ft,mech, ft,scale and ft,obs are the rolling mean and main forcing frequencies over 30 min. The variations of zi(t) during cases I, II and III are also indicated with a thick black solid line. Experimental calculation of the intrinsic wave frequency for case II is indicated with the red asterisk.zr, the residual layer height is indicated by a grey shaded area.

The key points to understanding Figure 6 are:

  • the vertical axis for the curves is zi(t) and therefore can be read as a time axis, which makes it possible to make a correspondence between the time intervals defining cases I, II and III, and height intervals, as shown in Figure 6;

  • the forcing frequency estimates at zi(t) have to be compared with Brunt–Vaïsala frequencies at heights above CBL height zzi(t), i.e. in the RL, to predict a possible forcing of RL gravity waves.

(1) No wave in the residual layer ft(zi) > fN(z > zi) (case I). On 10 June, Figure 6b shows that at any time during the rise of the CBL, the forcing frequencies are larger than the Brunt–Vaïsala frequencies, which take smaller values than 10−3 s−1. Following the theoretical calculations in section 4.1, the CBL cannot generate gravity waves in these conditions (assuming the intrinsic frequency fΩ will coincide with forcing frequency values). The thermals' forcing frequencies calculated with time- and length-scale estimates of eddies, ft,scale, follow a linear decrease while the CBL height increases, as expected from Eq. (14). ft,scale is in rather good agreement (except for zi ∼ 0.8 km) with ft,mech, which was calculated independently. These values are always larger than the Brunt–Vaïsala frequencies, yielding the same conclusion. No estimates of obstacle forcing frequencies are provided as no evidence of significant vertical gradient of the horizontal wind is observed in the 1100 UTC sounding and MM5 reanalysis profiles for 0.6 < zi < 1.1 km.

(2) Evanescent wave convectively forced in the residual layer ft(zi) ≥ fN(z > zi) (case II). For 0.6 < z < 0.8 km, we retrieve similar conditions as for case I: both ft,mech or ft,scale are larger than Brunt–Vaïsala frequencies at heights above the CBL (i.e. above the delimited area or block displayed in Figure 6). The main difference between cases I and II appears inside the blocks that indicate the range of zi(t) variations during the two cases. Inside the block for case I we have the relationship ft(zi) > fN(z > zi), which prevents wave generation from convection. Inside the block for case II we have ft(zi) < fN(z > zi), which means that for the lowest values of zi(t) inside the block there is a possible wave propagation in the layer just above zi(t). This explains why we can see some oscillations at z = 625 m in Figure 4b.

Following section 4.2.1 we use lidar depolarization contours and Doppler vertical velocity fluctuations to estimate the typical wave amplitude equation image = 15.5 ± 2.0 m (calculated as the standard deviation of depolarization anomaly height variations) and vertical velocity amplitude equation image m s−1 (standard deviation of vertical velocity fluctuations) at a mean altitude of z = 625 m (Figure 7). The intrinsic wave frequency is then fΩ = (2.2 ± 0.3) 10−3 s−1. This frequency equals vertical forcing frequency ft,mech at lower altitude (see ft,mech values for case II in Figure 6d), which allows the occurrence of vertical convectively forced gravity waves (no vertical gradient of horizontal wind is observed for z < 0.65 km, which rules out wave generation from the obstacle effect). However, for 0.6 < z < 0.8 km (i.e. above the delimited area of case II), both fΩ and ft,mech or ft,scale are larger than Brunt–Vaïsala frequencies at heights above the CBL and we have fΩ/fN3. This indicates that the observed wave is evanescent. The wave vector is then expected to be horizontal.

Figure 7.

(a) Lidar depolarization ratio during case II. Wave displacement is estimated using depolarization contours and indicated with a thin solid line. (b) Lidar vertical velocity and depolarization ratio time series at z = 0.625 m. Wave amplitude displacement ξ and vertical velocity w are indicated.

Auto-covariance calculations using displacement and depolarization ratio oscillations provide an accurate estimate of the apparent horizontal pulsation of the wave: ω = 0.0144 ± 0.0006 rad s−1 (see Appendix C). Given the intrinsic pulsation of the wave Ω = 2πfΩ = 0.014 ± 0.002 rad s−1 and using Eq. (9) we find kH.u0. The uncertainties prevent us from knowing if the along-wind component of the horizontal vector is positive or negative. We estimate an upper bound for the intrinsic wavelength (wave vector being in the direction of the wind): ΛH ≤ 1.09 ± 0.04 km. From section 4.1 we can estimate an upper bound for the vertical length scale of absorption of the wave in the RL equation image km (with fΩ/fN3). Figure 7a confirms this result as we do not see any oscillations in the lidar depolarization ratio above 850 m (i.e. 200 m above the location of the wave measurements). To complete the case II analysis, a cross-covariance between vertical velocity and depolarization ratio time series (or backscatter lidar signal at 2 µm) shows no phase shift (Appendix C).

(3) Wave convectively forced in the residual layer ft(zi) < fN(z > zi) (case III). On 14 June, during the case III period of time, 0945–1100 UTC, the forcing frequencies at the top of the CBL are lower than the Brunt–Vaïsala frequency above in Figure 6d (0.75 < z < 1.4 km). zi(t) keeps nearly the same height around 0.75 km. During case III, ft,mech = (1.5 ± 0.1) 10−3 s−1 and is slightly lower than the Brunt–Vaïsala frequency in the RL above, fN = 1.810−3 s−1 between 0.8 and 1.2 km. Of interesting is the fact that fN takes a typical value that is usually found in the free troposphere.

According to the theoretical considerations of section 4.1, the propagation of convectively forced gravity waves is possible in the RL: ft,mech/fN = 0.83 ± 0.05. It is worth noticing that vertical forcing frequencies inferred from CBL length scales,ft,scale, take larger values than fN in the RL, which leads to a different conclusion for case III (the wave would be evanescent).

Assuming that the wave frequency belongs to the thermal forcing main frequencies as shown for case II, i.e. fΩ = ft,mech, and using Eq. (10), we obtain β= 33° ± 5°. Using depolarization maxima and minima in the RL (Figure 8a), which represent the line phases in a Eulerian view, and using the MM5 wind speed profile at 1000 UTC (which, as we can see in Figures 5b and 5c, is in good agreement with the sounding at 1117 UTC), we retrieved the phase lines orientation in a Lagrangian view. Figure 9c shows that: (i) the phase lines are tilted upstream and against the shear; (ii) the mean tilted angle is βexp = 32° ± 3° for 0.8 < z < 1.2 km. βexp is the projected line phase angle in the horizontal wind direction and so we expect ββexp. Given that we obtain βexp≅β for 0.8 < z < 1.2 km, the direction of the horizontal component of the wave vector is therefore that of the horizontal wind and shear directions (Figures 9a–c). This result is consistent with numerical simulations (Hauf and Clark, 1989; Fovell et al., 1992; Kershaw, 1995; Lott, 1997). This experimental result, however, must be viewed with caution, given the uncertainties of β and βexp. Further investigations which will involve three-dimensional scanning Doppler lidar will be necessary to fully address the question of wave vector direction with respect to the horizontal wind. We assume that we can write equation image (x being the mean horizontal wind direction). The group and the phase velocities can then be written:

equation image(19)
equation image(20)
Figure 8.

(a) Lidar depolarization ratio and (b) Doppler vertical velocity colour plot during case III. (c) Time series of the vertical velocity, uz (black solid line) and depolarization ratio, Δ (grey solid line) at z = 1.05 km. Thin black solid lines indicate zi and zr are indicated. Depolarization ratio and vertical velocity anomalies indicate the phase lines of the wave in the residual layer.

Figure 9.

(a) Horizontal wind speed (line with dots) and shear (line with squares) from 1000 UTC MM5 reanalysis and horizontal wind speed from 1117 UTC sounding (thick solid lines) as a function of altitude. (b) Same as (a) but for the horizontal wind direction taken as the x direction. (c) Maxima and minima of the depolarization ratio as a function of z in a Lagrangian view using Figure 8a and the wind profile at 1000 UTC from MM5 reanalysis (thin grey solid lines). The black squares are for the average profile. We estimated the mean phase line direction using a linear fit of the average profile. βexp is estimated as the angle of the mean direction of the phase lines with respect to the vertical. βexp is also the angle of the wave vector with respect to the horizontal as indicated at the bottom of panel (c). Phase (cp) and group (cg) wave velocity directions are also indicated.

Using Eqs (8)—(10), one can write in the (x, z) plane:

equation image(21)
equation image(22)

We can also infer the useful relationship: kz = kx tanβ. Using a cross-covariance of Doppler lidar velocity or depolarization ratio time series (Figure 8c) and the wind speed at 1.05 km from MM5 reanalysis (Figure 9a), ux = 4.5 m s−1, we can estimate a horizontal wavelength ΛH = Λx = 2.42 ± 0.06 km (see Appendix C). With the typical values at z = 1.05 km, β = βexp = 32° ± 3° and fΩ = ft,mech = 1.510−3 s−1, we obtain cgz = 1.38 ± 0.03 m s−1. This vertical group velocity is indeed sufficient for the wave to penetrate the whole RL in a few minutes.

The relevant characteristics of cases I, II and III are summarized in Table II.

Table II. Experimental parameters for cases I, II and III: mean Brunt–Vaïsala (fN) and forcing frequencies due to convection (ft), Froude number calculated with mechanical forcing frequency (Fr), tilted angle of the line phases (β), wavelength (Λ) and group (cg) and phase (cp) velocities.fN is calculated using night-time soundings and ft is calculated from lidar measurements (section 4.2.2).fΩ is the intrinsic frequency of the wave, which is calculated using lidar depolarization and vertical velocity time series for case II and which is assumed to be ft for case III. The uncertainties for each parameter can be found in the text and are generally less than 10%.
 equation imageequation image (s−1)Fr =fΩ 10−3Wind shearβΛ (km)cg (m s−1)cp (m s−1)
 (s−1)equation imageequation imageequation image(s−1)10−3 (s−1)(°)ΛHΛzcgHcgzcpHcpz
Case I0.         
No wave             
Case II0.72.12.432.2 0<1.1    <2.6
Evanescent wave             
Case III1.81.520.831.55.5322.43.80.851.43.14.9
Propagating wave             

(4) Discussion on the role of wind shear in convectively forced gravity waves. Figure 6d shows that the overshoots, acting as obstacles in the vertical gradient of the horizontal wind, can also generate forcing frequency lower than the Brunt–Vaisala frequency in the RL. Unlike mechanical oscillator forcing, the obstacle forcing suggests a large variation in the ratio: ft,obs/fN from 0.27 for zi = 0.7 km or zi = 0.9 km to 0.85 for zi = 0.8 km, i.e. a value similar to that for mechanical oscillator forcing (fN = 1.810−3 s−1 for 0.8 < z < 1.2 km). This suggests a large variation in β from 74° to 31° when the CBL is rising, which is not observed in βexp = 32° ± 3° during case III. Assuming that we have fΩ = ft,obs, the relationship ββexp makes us understand that the obstacle forcing is generally not possible except during a short period of CBL rising and is consequently not the source of the gravity waves observed during case III.

Previous studies have emphasized the role of wind shear in the generation of waves and the selection of their characteristics, i.e. filtering in the spectrum of mechanical forcing frequencies (Lane and Clark, 2002; Böhme et al., 2007). Figures 4, 5a and 9a show that the generation of the propagating gravity wave occurs in case III when the height of the boundary layer reaches the height of maximum wind shear and minimum gradient Richardson number (Ri < 0.2). The gravity wave is then probably enhanced by the wind shear, as previously demonstrated in numerical simulations (Clark et al., 1986; Kershaw, 1995; Lott, 1997), but it is not initially produced by this wind shear because the gravity wave, in its evanescent form, is already observed in case II at z = 0.6 km, where the wind shear is negligible. In addition, as the shear is positive and the waves tilt upstream, simple WKB predictions suggest that the phase lines will become more and more vertical with height. There is some indication of this behaviour in the observations (Figure 4c). To conclude, the dynamics of the convection in the CBL appear to be the determining factor for the generation of the 14 June gravity waves, with wind shear possibly enhancing and modifying the waves' propagation.

From cross-covariance calculations between vertical velocity and depolarization ratio time series, we found that velocity and depolarization fluctuations are in phase, which is unexpected for wave dynamics (Appendix C). This result is confirmed if we look at the phase shift between the HDL backscatter signal and vertical velocity fluctuations. We discuss this particular point in section 5.

5. Consequences for the pseudo-neutrality of the residual layer

5.1. Gravity wave in a stable stratified atmosphere: vertical flux of relative humidity?

For 10 and 14 June, non-zero vertical velocities are measured in the RL, implying significant dynamical activity. Vertical motion in the RL can occur at different scales, from meso-scale with subsidence to smaller scales with wind shear and convective forcing, which is addressed in this paper.

Figure 10e displays a 30 min rolling cross-covariance of vertical velocities and the depolarization ratio, the so-called ‘depolarization flux’, equation image, as a function of height. Some organized vertical structures are seen in the CBL following the structures of turbulent eddies, as the depolarization ratio is usually a good tracer of dynamics in the ABL (Gibert et al., 2007). The entrainment layer can also be well identified with specific variations of Δ. Until 0950 UTC no vertical flux of depolarization is observed in the RL. During case III, Figure 10e shows a positive flux in the RL that vanishes close to the free troposphere interface.

Figure 10.

(a) and (d) Relative (blue thick solid line) and specific (blue thin solid line) humidity profiles from the 2319 and 1117 UTC soundings. (b) and (c) Lidar depolarization ratio Δ (black symbols) as a function of altitude for cases II and III. The green and yellow solid lines are for the averaged profiles. (e) The 30 min rolling cross-covariance of vertical velocities and depolarization ratio time series as a function of altitude. Black solid lines indicate zi and zr.

The positive sign of this flux is due to (i) upward motion bringing some high Δ values from the specific layer at z = 0.7 km (and later by the CBL) and (ii) downward motion bringing some air from the free troposphere with low Δ (Figure 10c).

Δ is a complex tracer changing with the type of particles (biogenic, pollution, dust, etc.) and temperature and moisture (from RH). The 0.5 µm backscatter signal is mainly sensitive to the concentration load or type of aerosol particles rather than relative humidity (see the small variations of the 0.5 µm reflectivity with RH in figure 13 of Gibert et al. (2007) for J14). Given the nearly constant value of 0.5 µm reflectivity in the RL (Figure 4b), we conclude that the fluctuations in depolarization are probably due to RH variations. This is also confirmed by negative anticorrelations that are observed between HDL vertical velocity and backscatter signal time series at z = 1.05 km during case III (see CCV( uz,Pz2) calculations in Figure C1b, Appendix C). This negative anticorrelation is directly linked to the effect of RH fluctuations on the 2 µm backscatter signal (Gibert et al., 2007).

The vertical velocity fluctuations due to the wave cause an increase of Δ and a decrease of RH in the RL. However, in classic linear wave theory, the fluctuations of RH (expected to be within 30% given the RH profile displayed in Figure 10a) are not expected to generate a flux due to a π/2 phase shift between vertical velocity and scalar fluctuations (see e.g. wind profiler observations in Böhme et al., 2004). A possible explanation to the unusual phase relationship calculated by cross-covariance functions (Appendix C) may be that particles are not a passive tracer of the wave. A latent desorption of the particles passing from 75% to 50% of surrounding air RH can cause a delay in the RH effect on lidar reflectivity, in agreement with the RH hysteresis effect reported in Randriamiarisoa etal. (2005).

A different explanation may be a nonlinear wave interaction with turbulence. In that case, it is observed that the phase shift between velocity and density fluctuations is different than π/2. A vertical positive flux of temperature can be observed locally (Einaudi and Finnigan, 1993). This wave–turbulence interaction is supported by the fact that the CBL rise decreases during case III and that low frequencies of CBL height fluctuations are suddenly observed during that period.

5.2. Coupling between convective layer, residual layer and free troposphere

In case III (Figure 8a) some oscillations of zr can be seen that suggest a coupling of CBL, RL and free troposphere. Actually, we do not expect a change in zr either due to advection, as the terrain is homogeneously flat in the west direction, or due to wind shear, as it is negligible at the top of the RL (Figure 10a). However, RL height fluctuations are poorly correlated with CBL height fluctuations or with wave oscillations (see vertical velocities or depolarization fluctuations), which suggests at best a partial transmission of wave momentum through the RL-free troposphere interface. The Brunt–Vaïsala frequency in Figure 6d indicates that fN becomes smaller at 1.3 km, i.e. at the free troposphere interface, and thus the propagating gravity wave may become trapped in the RL; these waves can be partially reflected downward from their trapping height (Lott, 1997; Sutherland, 2000). This may explain why the RL height (zr) fluctuations (see Figures 4c and 4d) are poorly correlated with RL velocity and depolarization ratio fluctuations.

6. Conclusion

The generation of internal gravity waves in the RL during the morning transition has been investigated from observations. Simultaneous measurements by two lidars have provided a description of vertical velocities and of the heights of the RL and the developing CBL for two mornings (10 and 14 June 2005). Three cases have been identified and studied, showing either no waves in the RL (case I), evanescent waves (case II) or propagating gravity waves (case III). Analysis of the different conditions suggests that: (i) a necessary condition to generate convectively forced gravity waves in the RL is to have a statically stable and stratified RL (thus the 14 June conditions are similar to those in the free troposphere with a Brunt–Vaïsala frequency of 1.810−3 s−1); (ii) the ‘mechanic oscillator’ (vs. ‘obstacle’) thermal forcing is the source of gravity waves in the RL; and (iii) the horizontal wind shear is not a sufficient condition for the occurrence of an internal gravity wave through the ‘obstacle effect’, but can still have a significant impact on their propagation.

In the case of wave propagation, the results confirm numerical studies that the phase lines are tilted upstream and against the wind shear. Typical horizontal wavelength and line phase direction are measured as 2.4 km and 32°, respectively. Contrary to what is expected from a classic linear wave theory we found that wave-associated vertical velocity and 0.5 µm depolarization ratio or 2 µm backcatter fluctuations are in phase. Possible explanations include: (i) aerosol particles are not passive with respect to temperature or water vapour fluctuations; (ii) a nonlinear wave-turbulence interaction is at work and needs further investigation. Lidar observations of the RL and free troposphere interface suggest that waves are partially trapped in the RL, which is supported by reduced values of the Brunt–Vaisala frequency above the RL.


The authors thank Meteo France and SIRTA for providing Trappes sounding data and sonic anemometer measurements.

Appendix A: Estimates of Convective Boundary Layer Forcing Frequency Using Scales of Turbulence

From Carruthers and Hunt (1987) the main forcing frequency of the energy-containing eddies can be written:

equation image(A1)

The velocity scale Ut is provided by the maximum in the vertical velocity variance. Using, the mixed layer similarity relationship for vertical velocity variance (Caughey and Palmer, 1979), equation image, where equation image is the convective velocity scale and 〈θ〉 is the mean potential temperature in the CBL, the maximum of the velocity variance is equation image and occurs at z/zi = 0.31. Using Doppler lidar measurements we instead found that equation image during J10 and J14 morning transitions (Figure A1). Kaimal et al. (1976) also showed that the most energetic eddies are those produced on a scale that is comparable to the depth of the CBL zi. Following Hunt (1984), the scale Lt of the energy-containing eddies for isotropic turbulence is Lt = zi/2, keeping in mind that some uncertainties exist for such velocity and time scale.

Figure A1.

Atmospheric velocity variance measured by the Doppler lidar normalized by the turbulent velocity scale for different time gates during the morning transitions of (a) 10 and (b) 14 June. The atmospheric velocity variance is determined by the total lidar velocity variance minus the noise variance as calculated in Gibert etal. (2007). The velocity scale variance is calculated using CBL height and surface flux measurements. CBL relative height variations around zi in the time gate are indicated with a grey shaded area.

Hence, assuming equation image and Lt = zi/2, the main forcing frequency of the energy-containing eddies is estimated by:

equation image(A2)

where w and zi are rolling averages over a time gate of 30 min.

Appendix B: Estimates of Overshoot Horizontal Spatial Frequency

The power spectrum (fsp(fsp)) of CBL height time series zi(t) is calculated as follows:

equation image(B1)

where DFT is the discrete Fourier transform, Δt is the time interval between the estimates, i.e. 25 s, Δfsp = 1/(Mpt) is the frequency resolution, i.e. 5.510−4 s−1 and Mp is the number of points in the temporal range gate of 30 min.

Figure B1 shows an example of a CBL height power spectrum at 1000 UTC on 14 June. The power spectrum is normalized by the total variance of zi(t). The area under the curve represents the relative contribution of the spectrum to the total variance. The spectrum shows a typical decrease in fsp−2/3 of the spectral power in the inertial subrange as expected. At 1000 UTC there is a relatively large band of horizontal frequencies fsp between 1 and 410−3 s−1. Here the frequency of spectral power maximum, fsp,max, is 210−3 s−1.

Figure B1.

For 14 June at 1000 UTC, zi = 0.7 km. Power spectrum of CBL height zi(t), normalized by the total variance (dotted line with crosses). The thick solid line is for a two-point rolling averaging of the spectrum. The time gate is 30 min which gives a frequency resolution of 5.510−4 s−1. The main thermal spatial frequency fzi is indicated by the vertical dashed line. The typical decrease of the power spectrum, which is expected to be proportional to equation image in the inertial subrange, is also indicated with the thin black solid line.

Appendix C: Autocovariance and Cross-Covariance Calculations for Accurate Apparent Pulsation and Phase Shift Estimates

In this section we investigate wave characteristics using covariance techniques. We use the autocovariance function to estimate the apparent pulsation of the wave ω. The autocovariance, ACV, for vertical velocity fluctuations can be calculated according to

equation image(C1)

as a function of the time lag X.

A cosine fitting function cos[ωX + φ] is applied to the data to estimate the apparent pulsation of the wave spatially. Figure C1 shows the results of ACV calculations for HDL vertical velocity uz and EBL depolarization ratio Δ for case II at 625 m (Figure C1a) and case III at 1.05 km (Figure C1b). For cases II and III ω = 0.0142 ± 0.0002 rad s−1 (using Δ) and ω = 0.0117 ± 0.0001 rad s−1 (using uz) are, respectively, the most accurate estimates of wave pulsation using the ACV technique.

Figure C1.

Autocovariance functions for the depolarization ratio, ACV(Δ,Δ), and vertical velocity, ACV(uz,uz), and the cross-covariance function between vertical velocity and backscatter signal at 2 µm CCV( uz,Pz2) and depolarization ratio CCV(uz,Δ) time series (grey line and with filled circles). The time series of each variable is taken at 625 m during case II and at 1.05 km for case III. A cosine fitting function cos[ωX + ϕ] (thick solid line) is applied to the data. Autocovariance and cross-covariance functions are used to estimate the apparent pulsation of the wave and the phase shift spatially between uz and Pz2 or Δ, respectively.

A cross-covariance function, CCV, can be calculated between two different quantities, HDL vertical velocity and the EBL depolarization ratio, to estimate the phase shift between velocity and scalar oscillations,

equation image(C2)

as a function of the time lag X.

Given the different location of EBL and HDL lidars, we decided to calculate CCV( uz,Pz2 as a double check. Figure C1 shows the results of the cross-covariance calculations for cases II and III using the time series displayed in Figure 7b, which corresponds to case II. Despite HDL Doppler lidar vertical velocity and backscatter signal oscillations being close to the noise level of the instrument, Figure C1 shows an unambiguous null phase shift between the time series for cases II and III. The same conclusion is obtained for HDL vertical velocity and EBL depolarization ratio fluctuations.