Quarterly Journal of the Royal Meteorological Society Q. J. R. Meteorol. Soc. (2011)

Internal gravity waves convectively forced in the atmosphericresidual layer during the morning transition

Fabien Gibert,a* Nicolas Arnault,a Juan Cuesta,a Riwal Plougonvenb and Pierre H. Flamanta

aLaboratoire de Meteorologie Dynamique (LMD)/Institut Pierre Simon Laplace, Ecole Polytechnique, Palaiseau, FrancebLaboratoire de Meteorologie Dynamique (LMD)/Institut Pierre Simon Laplace, Ecole Normale Superieure, Paris, France

*Correspondence to: F. Gibert, Laboratoire de Meteorologie Dynamique (LMD)/Institut Pierre Simon Laplace, Ecole

Polytechnique, Palaiseau, France. E-mail: gibert@lmd.polytechnique.fr

Generation of internal gravity waves in the boundary layer is investigated fromobservations. Simultaneous measurements from a 2 µm Doppler lidar and a 0.5 µmbackscatter lidar are combined to analyse the occurrence, or not, of internalgravity 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 flowconfigurations in the residual layer: no wave, evanescent wave and propagatingwave. Comparison of the three cases suggests two necessary conditions for thegeneration of gravity waves: a stably stratified residual layer and a convectiveboundary layer with mechanical forcing frequencies less than the Brunt–Va ısalafrequency. The horizontal wind shear probably plays a role in the dynamics of thewaves, but, in the cases analysed, it is not sufficient alone to generate the observedwaves through the obstacle effect. In the case of wave propagation, the waves tiltupstream and against the wind shear, with a typical horizontal wavelength and aline phase direction with respect to the vertical of 2.4 km and 32◦, respectively.Unexpectedly, we found that measurements of the wave-associated vertical velocityand the displacement of tracers (0.5 µm depolarization ratio or 2 µm backcatter,both indicative of relative humidity fluctuations) are in phase. Possible explanationsinclude: (i) aerosol particles are not passive with respect to temperature or watervapour fluctuations; or (ii) a nonlinear wave-turbulence interaction is at work andneeds further investigation. Copyright c© 2011 Royal Meteorological Society

Key Words: convective boundary layer; depolarization ratio; gravity wave; lidar; residual layer; thermal

forcing

Received 15 December 2010; Revised 30 March 2011; Accepted 5 April 2011; Published online in Wiley Online

Library

Citation: Gibert F, Arnault N, Cuesta J, Plougonven R, Flamant PH. 2011. Internal gravity waves

convectively forced in the atmospheric residual layer during the morning transition. Q. J. R. Meteorol.

Soc. DOI:10.1002/qj.836

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

Copyright c© 2011 Royal Meteorological Society

F. Gibert et al.

of atmospheric boundary layer (ABL) constituents duringthe morning transition (aerosols, O3, CO2, H2O) (Doranet al., 2003; Gibert et al., 2008; Kolev et al., 2008).

It is generally thought that the RL remains isolated fromthe ground and its top is thus expected to remain constantovernight (excluding effects of subsidence and radiativecooling), until fully eroded by the developing CBL (Stull,1988), yet coupling between the RL and the developing CBLand entrainment at the top of the RL has been observed(Fochesatto et al., 2001). Fochesatto et al., suggested thatinternal gravity waves in the RL played a role in thisinteraction and in the turbulence leading to entrainmentat the top of the RL, however, they did not show evidence ofthe wave phenomena. In addition, Sun et al. (2004) showedthat the passage of an internal gravity wave in a stable layercan be associated with a flux of momentum, heat, H2O andCO2.

Among the different sources of atmospheric gravity waves(see Nappo (2002) or Fritts and Alexander (2003) for ageneral view), convection is recognized as a major source,others being topography and jet or front systems. Threesimplified mechanisms have been proposed to describeconvective generation of gravity waves: (i) pure thermalforcing, which is linked to a latent heat flux source (Salbyand 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 periodicintrusion of CBL overshoots in a stable stratified layer (Fovellet al., 1992). In the present article we consider only cloud-free conditions that exclude thermal forcing. However, therelative importance of the other two generation mechanismsin wave-generation, i.e. obstacle effect versus mechanicaloscillator effect, is still an open issue that has been mainlyaddressed by mesoscale model simulations (Clark et al.,1986; Carruthers and Moeng, 1987; Alexander et al., 1995;Kershaw, 1995; Lane and Clark, 2002) but needs newdedicated observations. One other important motivationof this article is to study the contribution of these waves toconvective initiation and organization.

The first observations of internal gravity waves in the ABLwere made with an array of acoustic sounders (Hooke et al.,1972). Since then observations of convectively generatedwaves have rather focused on the free troposphere usingin situ airborne measurements (Kuettner et al., 1987; Hauf,1993) and radars (Bohme et al., 2004, 2007; Chagnon andGray, 2008). Fot this article, the study of convective wavegeneration during the morning transition required windfield measurements at higher space–time resolutions thanthose provided by wind profilers, due to the short wavelengthand limited area of propagation of these waves in the RL.Observations of an elastic backscatter lidar (EBL) and aheterodyne Doppler lidar (HDL) are used hereafter incombination with surface meteorological sensors, soundingsand also the Pennsylvania State University/National Centerfor Atmospheric Research (PSU/NCAR) mesoscale modelMM5 (hereafter MM5) reanalysis. Three cases are presented,analysed and compared: I, showing no wave; II, in whichan evanescent wave occurs (no propagation upward); III,with a wave propagating in the entire RL. The three caseswere selected because they present a clear contrast in thedynamical behaviour of the RL while the backgroundconditions remained comparable, making it possible toattribute the occurrence of the waves to those specificfeatures that differ between the three cases.

Figure 1. Experimental site at IPSL-LMD, Ecole Polytechnique cam-pus, France.

Section 2 presents the experimental set-up and section 3the observations and their analysis. Section 4 is devoted tothe application of the linear theory of wave excitation byturbulence in the ABL for the different cases. In particular weprovide 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 Fochesattoet al. (2001) by analysing coupling effects between theCBL, the RL and the free troposphere during the morningtransition. A conclusion and discussion are presented inSection 6.

2. Experimental set-up

The experimental site is shown in Figure 1. The EBL andHDL lidars are located at Laboratoire de MeteorologieDynamique, Institut Pierre Simon Laplace (IPSL-LMD,Palaiseau, France), 20 km southwest of Paris in a nearbysuburb on a 165 m high plateau that extends west andnorth for several kilometres. Table I summarizes the maincharacteristics of the two lidars, and more details can befound in Gibert et al. (2008) and Loth et al. (2004). TheEBL signal is backscattered by molecules and particleswhereas the HDL signal is due to particles only (heterodynedetection). The EBL also implements a polarization diversitycapability, which indirectly provides information on therelative humidity. The relevant quantity is the depolarizationratio 1 = P⊥/P//, where P⊥ and P// are the perpendicularand parallel signal, respectively. For a linear polarization ofthe incident laser beam, the backscatter signal depolarizationusually varies with the type and the shape of particlesand equals 1.4% for molecules. This value is taken as areference in the free troposphere. The depolarization ratiois expected to change with the relative humidity for RH >

60% as both size and shape of particles are then modifiedsignificantly (Fitzgerald, 1989). Using both backscatter anddepolarization ratio signals, the EBL is used to retrieve the15 m space and 10 s time resolved vertical structure andlayering in aerosol content and sphericity (proxy for relativehumidity), the heights of the CBL, zi, and the RL, zr. Located200 m northeast, the HDL provides 75 m space and 25 stime averaged vertical velocity measurements. This distancebetween the lidars is smaller than usual CBL heights so thetwo lidars are assumed to be embedded in the same eddy andto probe the same wave phenomenon. Some representativityerror may occur depending on the wind direction, strength

Copyright c© 2011 Royal Meteorological Society Q. J. R. Meteorol. Soc. (2011)

Convectively Forced Internal Gravity Waves

Table I. Instrumentation at IPSL-LMD, Ecole Polytechnique, Palaiseau, France used for the dataset in this article.

Instrument/data Main characteristics Variables used in Time Verticalthis article resolution resolution

Elastic backscatter Laser Nd: YAG 0.5 µm bakscatter signal P0.5 µm.z2 10 s 15 mlidar (EBL)

Wavelength 532 nmEnergy 30 mJ

Depolarization ratio 1

PRF 20 Hz ABL heights zi, zr

Heterodyne Dopplerlidar (EDL)

Laser Tm,Ho: YLF 2 µm bakscatter signal P2 µm.z2 25 s 75 m

Wavelength

2064 nm

Vertical velocity uz

Energy 10 mJPRF 10 Hz

Sonic anemometer at

10 m

Surface sensible heat

flux

uz′θ ′ 30 min

Soundings Temperature T 2 day−1 <10 mSpecific and relativehumidity

q, RH

Wind speed and direc-tion

ux, uy, uz

MM5 reanalysis Temperature T 1 h 50–200 mSpecific and relative

humidity

q, RH

Wind speed and direc-tion

ux, uy, uz

and shear. Lidar measurements provide cross-sections of theatmosphere for the different variables listed in Table I, butonly for z > 200 m. Both an incomplete overlap betweenthe field of view of the telescope and the laser beam for theEBL and parasitic optical reflections for the HDL precludemeasurements below this altitude.

A sonic anemometer at 10 m is also used to infer 30 minrolling-average surface heat flux measurements. In additionvertical profiles of humidity, temperature and horizontalwind speed and direction were obtained from daily sound-ings at a nearby meteorological station located 10 km to thewest of IPSL-LMD site at approximately 0000 and 1200 UTC.

To complete the dataset of meteorological variablesduring all measurement days, we use hourly mesoscaleMM5 model reanalysis interpolated at the IPSL-LMD. Thesereanalyses are produced for real-time air quality modellingin 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 of5 × 5 km2. The MM5 profiles provide useful guidance aboutthe evolution of the flow during the morning transition forthe time intervals in between two soundings.

3. Observations and analysis

The observations were made on 10 and 14 June 2005. Thesedays are characterized by clear sky conditions and similarsurface heat flux during the morning transition. In additionto measurements listed in Table I, we also used the soundingsand the MM5 reanalyses to calculate:

• the Brunt–Vaısala pulsation in rad s−1, commonlycalled the ‘Brunt–Vaısala frequency’, as the criterion

for static stability:

N =(

g

θ

dθ

dz

)1/2

, (1)

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

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

Ri =N2

[

(∂ux/∂z)2 + (∂uy/∂z)2] , (2)

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

3.1. Observations on June 10: nocturnal, residual andconvective layers

Figure 2 displays a general view of the morning transitionduring 10 June (called J10 hereafter). Using the lidarbackscatter signal at 0.5 µm we retrieved the height zi andzr for the CBL–RL and RL–free-troposphere interfacesrespectively (see Menut et al. (1999) for the algorithms).

Copyright c© 2011 Royal Meteorological Society Q. J. R. Meteorol. Soc. (2011)

F. Gibert et al.

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 1; 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 completes the data set with night (2314 UTC) andday (1122 UTC) soundings and MM5 reanalysis profiles.

The night potential temperature profile shows that theformer nocturnal layer is characterized by a rather smallinversion close to the ground (z < 150 m). The maximumspeed of a northeastward nocturnal jet reaches 8.2 m s−1

at z = 200 m. This height coincides with the top of thenocturnal layer. The MM5 reanalysis does not providethe observed nocturnal layer properties (low inversion andnocturnal jet).

The RL starts above the nocturnal layer with a quasi-constant virtual potential temperature profile showing nearneutral conditions. The RH profiles show that the nocturnallayer is covered by a low-RH layer from 0.2 to 0.6 kmwhere RH < 50%. Then, above 600 m, the relative humidityincreases to reach 82% at the top of the RL. Both MM5reanalysis and lidar measurements show that this layer keepsthe same properties during the morning. The depolarizationratio profiles show large values (1 > 0.1) below 0.7 kmand small values above (1 < 0.05), following the RHprofile according to particle hygroscopicity (Figure 2). ThisRL is characterized by a negative wind shear dV/dz of−5 10−3 s−1 up to z > 0.9 km which, together with the weakstability, explains the low values of the gradient Richardsonnumber: Ri < 1 for 0.4 < z < 0.9 km (Figure 3a). Theflow is expected to be turbulent at z = 0.7 km whereRi = 0.1. The MM5 reanalysis provides a good potentialtemperature profile in the RL but generally fails to representthe wind shear, the layering of RH and the local low gradientRichardson values.

The J10 day is characterized by a rapid growth of theCBL height from 0.4 to 1.7 km in 2 h (0800–1000 UTC)until the complete erosion of the RL. During all themorning 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 ofthe CBL, reaching δzi = 400 m at 0850 and 0920 UTC

(Figure 2). These observations correspond well to the nearneutral condition observed in the RL. The MM5 reanalysispotential temperature profiles at 0800, 0900 and 1000 UTCshow that the statically neutral condition remains during therise of the CBL (Figure 3b). No stratification and temperatureinversion slow down the rise of the CBL. In summary, J10illustrates a case with a neutral RL that is eroded quickly bythe vigorous convection in the developing CBL.

3.2. Observations on 14 June: nocturnal, residual andconvective layers

The morning transition on 14 June (called J14 hereafter)contrasts with that of J10 mainly due to the stratificationof the RL, and its slower erosion. Lidar observations aredisplayed in Figure 4 with backscatter signals at 2 and0.5 µm, depolarization ratio and vertical velocity. Figures 4aand 4b show no contrast in backscatter signal owing to thesame CBL and RL properties. Therefore, zi is determinedusing the depolarization ratio contrast. Figure 5 presentsthe soundings at 2319 and 1117 UTC and MM5 reanalysisprofiles.

The J14 nocturnal layer is characterized by large relativehumidity and a near-ground nocturnal jet from thenortheast, which reaches 6 m s−1 at z = 50 m. The night-timepotential temperature profile shows that this relatively thicknocturnal layer (∼350 m) is capped by a strong inversionof 6 K (to be compared with 2.5 K for J10) located between0.3 and 0.6 km (Figure 5a). The MM5 reanalysis profilesare in relatively good agreement with the sounding profilesidentifying the nocturnal jet from the north, temperatureinversion and the large value of RH close to the ground.

During the night and above the inversion there is a dryand dynamically stable layer (see the large value of Ri) with achange of wind direction from northward to southwestward(Figure 5a). The RL on J14 is characterized by a stablestratification, with a temperature gradient of −5 K km−1

Copyright c© 2011 Royal Meteorological Society Q. J. R. Meteorol. Soc. (2011)

Convectively Forced Internal Gravity Waves

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) morningtransition (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).

Figure 4. Same as Figure 2 but for 14 June.

Copyright c© 2011 Royal Meteorological Society Q. J. R. Meteorol. Soc. (2011)

F. Gibert et al.

close to the free troposphere condition of −6 K km−1 (tobe compared with the J10 decrease of −9 K km−1) andby a stratification of the atmosphere (see the RH profilein Figure 5a and 0.5 µm backscatter and depolarizationin Figure 4c). A strong positive wind shear of dV/dz of1.5 10−2 s−1 up to z > 1 km and a weak stratification explainthe low values of the gradient Richardson number Ri < 1 for0.6 < z < 1.1 km, as for the J10 night. The flow is expectedto be turbulent at z = 0.7 km where Ri = 0.1. The 2319UTC and 1117 UTC soundings show that this wind shearis reduced to 5.7 10−3 s−1 during the morning transitiondue 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 reanalysesshow a smaller difference between night and day RL windshear but suggest that the stable stratification remains duringthe morning transition (see the 0800, 0900 UTC potentialtemperature profiles in Figure 5b). Lidar measurementsprovide additional evidence that the stratification of theatmosphere remains the same as for the night. The dry stablelayer between 0.6 and 0.8 km is identified by a relatively largedepolarization ratio and low backscatter signal (Figures 4band 4c).

In contrast to J10, the morning transition is characterizedby a slow rise of the CBL before 1020 UTC, with a rate ofincrease 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) correspondingto a slow erosion of the strong temperature inversion(Figure 2c and 4c). Updraft vertical velocities and theovershoot amplitudes (δzi = 150 m) are globally smallerthan for J10 (Figures 2d and 4d). After 1020 UTC andfor zi > 0.7 km, the increase of CBL height is rather fasterb–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 verticalvelocities in the RL reached significant values. Figure 2dshows positive vertical velocity of +0.5 m s−1 around 0850UTC. However, no wave phenomenon is suggested by lidarbackscatter, the depolarization ratio or the vertical velocitytime series.

In case II (J14 – from 0830 to 0915 UTC), 0.5 µmbackscatter and depolarization ratio signals show that theincrease of CBL height entails some oscillations in the drylayer above (Figures 4b and 4c). It is worth noticing thatthese oscillations are not seen above z > 0.9 km and thatthe fluctuations of zr are negligible during this period oftime. Velocity fluctuations can be seen but they remainunorganized and very close to the noise level of the signal,i.e. 0.2 m s−1 (Figure 4d). To conclude, the observationsduring case II show a wave phenomenon that seems to belinked to the rise of the CBL but does not propagate upward.

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

4. Wave excitation by convection

We now investigate under which conditions the thermalsof the developing CBL may force gravity waves in the RL.Essential elements and notations of the linear theory ofgravity waves are recalled in section 4.1. The frequencies ofthe thermals and of the gravity waves are then estimated inseveral ways and compared to show under what conditionsgravity 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 lidarreference frame. We represent the thermodynamic variablesas a sum of the base-state variables plus small deviations(Stull, 1988):

p = p(z) + p′(x, t);ρ = ρ(z) + ρ′(x, t);T = T(z) + T′(x, t);θ = θ(z) + θ ′(x, t);u = u + u′(x, t). (3)

The deviations are assumed to be small so we canlinearize about the base state. The base state variables areindicated with the bar over the letters and deviations withthe prime in Eq. (3). For all of this study we neglect theeffect 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:

∂θ ′

∂t= −uz

θN2

g, (4)

where we used Eq. (1), writing dθ/dz = θN2/g, and uz isthe vertical wind component.

Making the Boussinesq approximation, the equation formass conservation simplifies to:

∇.u′ = 0, (5)

and the equation of momentum is:

(

∂

∂t+ u.∇H

)

u′ = − 1

ρ∇p′ + g

θ ′

θez, (6)

where ∇H = ∂/∂x ex + ∂/∂yey.Using Eqs (4)–(6), we find the following wave equation

for u′z (Holton, 2004):

(

∂

∂t+ u.∇H

)2

∇2u′z + N2∇2

Hu′z = 0. (7)

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

Assuming that N is constant with height, we look for amonochromatic plane wave solution, w. exp[i(k.x − ωt)],with w the vertical velocity amplitude of the wave, k thewavenumber and ω the pulsation in the lidar referenceframe, i.e. fixed to the ground. We find the dispersionrelation:

Ä2k2 − N2k2H = 0, (8)

Copyright c© 2011 Royal Meteorological Society Q. J. R. Meteorol. Soc. (2011)

Convectively Forced Internal Gravity Waves

Figure 5. Same as Figure 3 but for 14 June 14. The night- and daytime soundings were at 2319 and 1117 UTC respectively.

where kH = kxex + kyey is the horizontal wave numbervector and

Ä = ω − u.kH (9)

is the intrinsic pulsation of the wave. The correspondingintrinsic frequency will be noted fÄ = Ä/2π .

For Ä < N or Fr < 1 (with Fr, the Froude number), kz

is real and the solution of Eq. (7) is a propagating wave. Inthis case, we usually write:

Ä/N = cos β = Fr, (10)

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

For Ä > N or Fr > 1, kz is imaginary and thesolution of Eq. (7) is an evanescent wave that propagatesonly in the horizontal direction. The wave amplitudedecreases exponentially in the vertical direction with a

characteristic length scale of 3H/(

2π√

1 − N2/Ä2)

where

3H = 2π/|kH| is the horizontal intrinsic wavelength (Gill,1982).

4.2. Application

After recalling the meaning of the intrinsic pulsation ofthe 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 mechanismof the observed waves (section 4.2.3).

4.2.1. Intrinsic frequency of the wave

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

w = iÄξ , (11)

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

4.2.2. Forcing frequency due to convection

Two different kinds of forcing mechanisms of wave in the RLcan be associated with the convection during the morningtransition (in free-cloud conditions): (i) the ‘mechanicoscillator’ mechanism, in which the forcing frequencies

Copyright c© 2011 Royal Meteorological Society Q. J. R. Meteorol. Soc. (2011)

F. Gibert et al.

due to convection result from the vertical motion of up anddowndrafts (Fovell et al., 1992); and (ii) the ‘obstacle effect’in which the wave generation results from advection by thehorizontal wind over the pattern of overshoots at the topof the CBL (Clark et al., 1986). Of course, these two forcingmechanisms are simplified models trying to capture essentialmechanisms for wave generation. Reality lies between thetwo extremes, with some time-dependence of the forcingand some shear always present.

The following paragraphs present the experimentalestimates of a dominant forcing frequency through themechanic oscillator or the obstacle effect. Given a forcingfrequency, it is expected that the dominant response willbe at that frequency. Other forcing frequencies and otherharmonics of the response may be present but weakerand will hence just modulate the main forcing and wavefrequency. This assumption will be justified in the analysisfor case II in section 4.2.3.

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

ft,mech =1

2π

(

u′2z (zi)

z′2i

)1/2

, (12)

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

We assume that Taylor’s frozen turbulence hypothesis isvalid at the top of the CBL as the vertical velocity standarddeviation (<0.5 m s−1) is weak compared with the horizontalwind speed (>5 m s−1). Therefore, backscatter lidarmeasurements provide a non-biased estimate of overshootamplitude at the top of the CBL. A 30 min rolling time gateis chosen to calculate each variance, as a good compromiseto having a sufficient number of samples (i.e. overshoots)in the time gate and to resolve the different wave regimesduring the morning transition.

Carruthers and Hunt (1987) also suggested another wayto estimate the main mechanic oscillator vertical forcingfrequency of the energy-containing eddies that does notrequire Doppler lidar measurements:

ft,scale = Ut/Lt, (13)

where Ut and Lt are the velocity and time scales of the eddies.Using backscatter lidar and surface heat flux measure-

ments, ft,scale can be calculated as a function of the turbulentvelocity scale and the height of the CBL (Appendix A):

ft,scale∼= 0.9w∗/zi, (14)

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

(2) Obstacle forcing frequency. The obstacle horizontalwavelength of thermal overshoots and the vertical gradient ofthe horizontal wind above the CBL are the critical parametersfor waves generated through the obstacle effect. We assumethat the mean wind is uniform in the CBL and that there is avertical gradient of the horizontal wind above the CBL (δuover the mean height of the overshoots). In the referenceframe of the CBL wind, the overshoots are then fixed (i.e. in

a mountain-like situation). In this case, there are solutionsof Eq. (7) in the form of waves that are stationary relative tothe CBL (i.e. ω = 0). For such stationary waves, u′

z dependsonly on x, and Eq. (7) simplifies to:

∇2u′z + N2/(δu)2u′

z = 0. (15)

The dispersion relation is then:

k2z = N2/(δu)2 − kH. (16)

A Froude number is then usually written as:

Fr = δu/fN L, (17)

where δu is the increase of horizontal wind speed at thetop of the CBL (with the notation u = ||u||) and L isthe obstacle horizontal wavelength of excitation, i.e. thehorizontal wavelength of thermal overshoots. This Froudenumber can be defined at any time t at a mean CBL heightzi(t) using values for δu and L that are obtained with a timegate of 30 min. We will express Fr as a function of zi: Fr(zi).

Assuming the CBL top is moving at a mean horizontalwind speed of |u(zi)| and that the turbulence horizontalpattern is homogeneous along the wind direction, theovershoots obstacle forcing frequency can be estimatedusing a lidar Eulerian cross-section of the CBL:

ft,obs = |δu(zi)|L(zi)

= |δu(zi)|fzi

|u(zi)|(18)

δu(zi) is calculated over the mean height of the overshoots

(i.e. 2[

(zi − zi)2]1/2

), i.e. the mean height over which the

mean flow is disturbed by the obstacle. For this purpose, weuse the horizontal wind velocity profile of the sounding at1117 UTC, which is in agreement with the MM5 profile at1000 UTC.

The main eddies horizontal ‘spatial frequency’, fzi, isestimated with the maximum in the power spectra of lidarCBL-height time series (Appendix B) and is used to estimateL(zi), the main horizontal wavelength of CBL overshootsas indicated in Eq. (18). δu(zi) and fzi are calculated over arolling time gate of 30 min.

4.2.3. Conditions to force gravity waves with convection inthe residual layer

We now explain a simple criterion to identify favourableconditions for the forcing of gravity waves in the RL.Figure 6 summarizes our main findings, and requires somecare in interpretation: in the left panels of Figure 6 wedisplay the vertical profile of the Brunt–Vaısala frequency(fN = N/2π) for 10 June (Figure 6a) and 14 June (Figure 6c)as a function of altitude, from the night-time sounding andMM5 reanalysis for 0000, 0800, 0900 and 1000 UTC. Wecan see that the night-time fN profile from MM5 does notprovide 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 stratificationin the RL seen on the midnight sounding is still in placeduring the morning transition. We choose then to comparethe thermals’ forcing frequencies during CBL rise with fNvalues in the RL as calculated with the night-time sounding.Hence this sounding is also reproduced in the right panels ofFigures 6b and 6d, superposed on mechanic oscillator andobstacle forcing frequencies (section 4.2.2).

The key points to understanding Figure 6 are:

Copyright c© 2011 Royal Meteorological Society Q. J. R. Meteorol. Soc. (2011)

Convectively Forced Internal Gravity Waves

Figure 6. Brunt–Vaısala frequency profiles for (a) 10 and (c) 14 Junecalculated from the night-time sounding (black solid line) and from MM5reanalysis: night, solid line; 0800 UTC, dashed line; 0900 UTC, dashedand dotted line; 1000 UTC, dotted line). (b) and (d) Mechanic oscillatorforcing frequency (Eq. (12)) using Doppler lidar measurements, ft,mech

(blue solid line), and scales in the CBL, ft,scale (Appendix B; green solidline) and obstacle forcing frequency, ft,obs (pink solid line) are displayedas a function of zi(t).ft,mech, ft,scale and ft,obs are the rolling mean and mainforcing frequencies over 30 min. The variations of zi(t) during cases I,II and III are also indicated with a thick black solid line. Experimentalcalculation of the intrinsic wave frequency for case II is indicated with thered asterisk.zr , the residual layer height is indicated by a grey shaded area.

• the vertical axis for the curves is zi(t) and thereforecan be read as a time axis, which makes it possibleto make a correspondence between the time intervalsdefining cases I, II and III, and height intervals, asshown in Figure 6;

• the forcing frequency estimates at zi(t) have to becompared with Brunt–Vaısala frequencies at heightsabove CBL height z > zi(t), i.e. in the RL, to predict apossible 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 timeduring the rise of the CBL, the forcing frequencies are largerthan the Brunt–Vaısala frequencies, which take smallervalues than 10−3 s−1. Following the theoretical calculationsin section 4.1, the CBL cannot generate gravity waves inthese conditions (assuming the intrinsic frequency fÄ willcoincide with forcing frequency values). The thermals’forcing frequencies calculated with time- and length-scaleestimates of eddies, ft,scale, follow a linear decrease while theCBL height increases, as expected from Eq. (14). ft,scale is inrather good agreement (except for zi ∼ 0.8 km) with ft,mech,which was calculated independently. These values are alwayslarger than the Brunt–Vaısala frequencies, yielding the sameconclusion. No estimates of obstacle forcing frequencies areprovided as no evidence of significant vertical gradient ofthe horizontal wind is observed in the 1100 UTC soundingand MM5 reanalysis profiles for 0.6 < zi < 1.1 km.

(2) Evanescent wave convectively forced in the residuallayer ft(zi) > fN (z > zi) (case II). For 0.6 < z < 0.8 km, weretrieve similar conditions as for case I: both ft,mech or ft,scale

are larger than Brunt–Vaısala frequencies at heights abovethe CBL (i.e. above the delimited area or block displayedin Figure 6). The main difference between cases I and IIappears inside the blocks that indicate the range of zi(t)variations during the two cases. Inside the block for case Iwe have the relationship ft(zi) > fN (z > zi), which prevents

Figure 7. (a) Lidar depolarization ratio during case II. Wave displacementis estimated using depolarization contours and indicated with a thin solidline. (b) Lidar vertical velocity and depolarization ratio time series atz = 0.625 m. Wave amplitude displacement ξ and vertical velocity w areindicated.

wave generation from convection. Inside the block for caseII we have ft(zi) < fN (z > zi), which means that for thelowest values of zi(t) inside the block there is a possible wavepropagation in the layer just above zi(t). This explains whywe can see some oscillations at z = 625 m in Figure 4b.

Following section 4.2.1 we use lidar depolarizationcontours and Doppler vertical velocity fluctuations to

estimate the typical wave amplitude ξ = 15.5 ± 2.0 m(calculated as the standard deviation of depolarizationanomaly height variations) and vertical velocity amplitudew = 0.21 ± 0.01 m s−1 (standard deviation of verticalvelocity fluctuations) at a mean altitude of z = 625 m(Figure 7). The intrinsic wave frequency is then fÄ =(2.2 ± 0.3) 10−3 s−1. This frequency equals vertical forcingfrequency ft,mech at lower altitude (see ft,mech values for caseII in Figure 6d), which allows the occurrence of verticalconvectively forced gravity waves (no vertical gradient ofhorizontal wind is observed for z < 0.65 km, which rulesout wave generation from the obstacle effect). However, for0.6 < z < 0.8 km (i.e. above the delimited area of case II),both fÄ and ft,mech or ft,scale are larger than Brunt–Vaısalafrequencies at heights above the CBL and we have fÄ/fN ∼= 3.This indicates that the observed wave is evanescent. The wavevector is then expected to be horizontal.

Auto-covariance calculations using displacement anddepolarization ratio oscillations provide an accurateestimate of the apparent horizontal pulsation of thewave: ω = 0.0144 ± 0.0006 rad s−1 (see Appendix C).Given the intrinsic pulsation of the wave Ä = 2π fÄ =0.014 ± 0.002 rad s−1 and using Eq. (9) we find kH .u ∼= 0.The uncertainties prevent us from knowing if the along-wind component of the horizontal vector is positive ornegative. We estimate an upper bound for the intrinsicwavelength (wave vector being in the direction of the wind):3H 6 1.09 ± 0.04 km. From section 4.1 we can estimate anupper bound for the vertical length scale of absorption of

the wave in the RL 3H/(2π

√

1 − f 2N/f 2

Ä) 6 0.18 ± 0.03 km

(with fÄ/fN ∼= 3). Figure 7a confirms this result as we donot see any oscillations in the lidar depolarization ratioabove 850 m (i.e. 200 m above the location of the wavemeasurements). To complete the case II analysis, a cross-covariance between vertical velocity and depolarization ratio

Copyright c© 2011 Royal Meteorological Society Q. J. R. Meteorol. Soc. (2011)

F. Gibert et al.

Table II. Experimental parameters for cases I, II and III: mean Brunt–Vaısala (fN ) and forcing frequencies due toconvection (ft), Froude number calculated with mechanical forcing frequency (Fr), tilted angle of the line phases (β),wavelength (3) 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).fÄ is the intrinsic frequency of the wave, which is calculated using lidar depolarizationand 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%.

fN 10−3 ft 10−3 (s−1) Fr = fÄ 10−3 Wind shear β 3 (km) cg (m s−1) cp (m s−1)

(s−1) ft,mech ft,scale ft,mech/fN (s−1) 10−3 (s−1) (◦) 3H 3z cgH cgz cpH cpz

Case I 0.8 1.4 1.6 1.8No wave

Case II 0.7 2.1 2.4 3 2.2 0 <1.1 <2.6EvanescentwaveCase III 1.8 1.5 2 0.83 1.5 5.5 32 2.4 3.8 0.85 1.4 3.1 4.9

Propagatingwave

time series (or backscatter lidar signal at 2 µm) shows nophase shift (Appendix C).

(3) Wave convectively forced in the residual layer ft(zi) <

fN (z > zi) (case III). On 14 June, during the case IIIperiod of time, 0945–1100 UTC, the forcing frequenciesat the top of the CBL are lower than the Brunt–Vaısalafrequency 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 theBrunt–Vaısala 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 takesa typical value that is usually found in the free troposphere.

According to the theoretical considerations of section4.1, the propagation of convectively forced gravity wavesis possible in the RL: ft,mech/fN = 0.83 ± 0.05. It is worthnoticing that vertical forcing frequencies inferred from CBLlength scales,ft,scale, take larger values than fN in the RL,which leads to a different conclusion for case III (the wavewould be evanescent).

Assuming that the wave frequency belongs to the thermalforcing main frequencies as shown for case II, i.e. fÄ =ft,mech, and using Eq. (10), we obtain β=33◦ ± 5◦. Usingdepolarization maxima and minima in the RL (Figure 8a),which represent the line phases in a Eulerian view, andusing the MM5 wind speed profile at 1000 UTC (which,as we can see in Figures 5b and 5c, is in good agreementwith the sounding at 1117 UTC), we retrieved the phase linesorientation in a Lagrangian view. Figure 9c shows that: (i) thephase lines are tilted upstream and against the shear; (ii) themean tilted angle is βexp = 32◦ ± 3◦ for 0.8 < z < 1.2 km.βexp is the projected line phase angle in the horizontalwind direction and so we expect β 6 βexp. Given that weobtain βexp

∼= β for 0.8 < z < 1.2 km, the direction of thehorizontal component of the wave vector is therefore thatof the horizontal wind and shear directions (Figures 9a–c).This result is consistent with numerical simulations (Haufand Clark, 1989; Fovell et al., 1992; Kershaw, 1995; Lott,1997). This experimental result, however, must be viewedwith caution, given the uncertainties of β and βexp.Further investigations which will involve three-dimensionalscanning Doppler lidar will be necessary to fully addressthe question of wave vector direction with respect to thehorizontal wind. We assume that we can write k2

x + k2y ≈ k2

x

(x being the mean horizontal wind direction). The groupand the phase velocities can then be written:

cpi = (Ä/k2)ki (19)

cgi = ∂Ä/∂ki (20)

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

cg = 3H fÄ(cos β sin2 β , cos2 β sin β) (21)

cp = 3H fÄ(cos β , cos β cot β) (22)

We can also infer the useful relationship: kz = kx tan β .Using a cross-covariance of Doppler lidar velocity ordepolarization ratio time series (Figure 8c) and the windspeed at 1.05 km from MM5 reanalysis (Figure 9a),ux = 4.5 m s−1, we can estimate a horizontal wavelength3H = 3x = 2.42 ± 0.06 km (see Appendix C). With thetypical values at z = 1.05 km,β = βexp = 32◦ ± 3◦ and fÄ =ft,mech = 1.5 10−3 s−1, we obtain cgz = 1.38 ± 0.03 m s−1.This vertical group velocity is indeed sufficient for the waveto penetrate the whole RL in a few minutes.

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

(4) Discussion on the role of wind shear in convectivelyforced gravity waves. Figure 6d shows that the overshoots,acting as obstacles in the vertical gradient of the horizontalwind, can also generate forcing frequency lower than theBrunt–Vaisala frequency in the RL. Unlike mechanicaloscillator forcing, the obstacle forcing suggests a largevariation in the ratio: ft,obs/fN from 0.27 for zi = 0.7 kmor zi = 0.9 km to 0.85 for zi = 0.8 km, i.e. a value similar tothat for mechanical oscillator forcing (fN = 1.8 10−3 s−1 for0.8 < z < 1.2 km). This suggests a large variation in β from74◦ to 31◦ when the CBL is rising, which is not observedin βexp = 32◦ ± 3◦ during case III. Assuming that we havefÄ = ft,obs, the relationship β 6 βexp makes us understandthat the obstacle forcing is generally not possible exceptduring a short period of CBL rising and is consequently notthe source of the gravity waves observed during case III.

Previous studies have emphasized the role of wind shearin the generation of waves and the selection of theircharacteristics, i.e. filtering in the spectrum of mechanical

Copyright c© 2011 Royal Meteorological Society Q. J. R. Meteorol. Soc. (2011)

Convectively Forced Internal Gravity Waves

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 solidline) and depolarization ratio, 1 (grey solid line) at z = 1.05 km. Thin black solid lines indicate zi and zr are indicated. Depolarization ratio and verticalvelocity 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 1117UTC 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 andminima 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 (thingrey 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 thehorizontal as indicated at the bottom of panel (c). Phase (cp) and group (cg) wave velocity directions are also indicated.

forcing frequencies (Lane and Clark, 2002; Bohme et al.,2007). Figures 4, 5a and 9a show that the generation ofthe propagating gravity wave occurs in case III when theheight of the boundary layer reaches the height of maximumwind shear and minimum gradient Richardson number(Ri < 0.2). The gravity wave is then probably enhanced bythe wind shear, as previously demonstrated in numericalsimulations (Clark et al., 1986; Kershaw, 1995; Lott, 1997),but it is not initially produced by this wind shear because thegravity wave, in its evanescent form, is already observed incase II at z = 0.6 km, where the wind shear is negligible. Inaddition, as the shear is positive and the waves tilt upstream,

simple WKB predictions suggest that the phase lines willbecome more and more vertical with height. There is someindication of this behaviour in the observations (Figure 4c).To conclude, the dynamics of the convection in the CBLappear to be the determining factor for the generation of the14 June gravity waves, with wind shear possibly enhancingand modifying the waves’ propagation.

From cross-covariance calculations between verticalvelocity and depolarization ratio time series, we foundthat velocity and depolarization fluctuations are in phase,which is unexpected for wave dynamics (Appendix C). Thisresult is confirmed if we look at the phase shift between the

Copyright c© 2011 Royal Meteorological Society Q. J. R. Meteorol. Soc. (2011)

F. Gibert et al.

HDL backscatter signal and vertical velocity fluctuations.We discuss this particular point in section 5.

5. Consequences for the pseudo-neutrality of theresidual layer

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

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

Figure 10e displays a 30 min rolling cross-covariance ofvertical velocities and the depolarization ratio, the so-called

‘depolarization flux’, uz′1′, as a function of height. Some

organized vertical structures are seen in the CBL followingthe structures of turbulent eddies, as the depolarization ratiois usually a good tracer of dynamics in the ABL (Gibert et al.,2007). The entrainment layer can also be well identifiedwith specific variations of 1. Until 0950 UTC no verticalflux of depolarization is observed in the RL. During case III,Figure 10e shows a positive flux in the RL that vanishes closeto the free troposphere interface.

The positive sign of this flux is due to (i) upward motionbringing some high 1 values from the specific layer atz = 0.7 km (and later by the CBL) and (ii) downwardmotion bringing some air from the free troposphere withlow 1 (Figure 10c).

1 is a complex tracer changing with the type ofparticles (biogenic, pollution, dust, etc.) and temperatureand moisture (from RH). The 0.5 µm backscatter signalis mainly sensitive to the concentration load or type ofaerosol particles rather than relative humidity (see the smallvariations of the 0.5 µm reflectivity with RH in figure 13 ofGibert et al. (2007) for J14). Given the nearly constantvalue of 0.5 µm reflectivity in the RL (Figure 4b), weconclude that the fluctuations in depolarization are probablydue to RH variations. This is also confirmed by negativeanticorrelations that are observed between HDL verticalvelocity and backscatter signal time series at z = 1.05 kmduring case III (see CCV

(

uz, Pz2)

calculations in Figure C1b,Appendix C). This negative anticorrelation is directly linkedto the effect of RH fluctuations on the 2 µm backscattersignal (Gibert et al., 2007).

The vertical velocity fluctuations due to the wave cause anincrease of 1 and a decrease of RH in the RL. However, inclassic linear wave theory, the fluctuations of RH (expected tobe within 30% given the RH profile displayed in Figure 10a)are not expected to generate a flux due to a π /2 phase shiftbetween vertical velocity and scalar fluctuations (see e.g.wind profiler observations in Bohme et al., 2004). A possibleexplanation to the unusual phase relationship calculatedby cross-covariance functions (Appendix C) may be thatparticles are not a passive tracer of the wave. A latentdesorption of the particles passing from 75% to 50% ofsurrounding air RH can cause a delay in the RH effect onlidar reflectivity, in agreement with the RH hysteresis effectreported in Randriamiarisoa et al. (2005).

A different explanation may be a nonlinear waveinteraction with turbulence. In that case, it is observed thatthe phase shift between velocity and density fluctuations isdifferent than π /2. A vertical positive flux of temperature

can be observed locally (Einaudi and Finnigan, 1993). Thiswave–turbulence interaction is supported by the fact that theCBL rise decreases during case III and that low frequenciesof CBL height fluctuations are suddenly observed duringthat period.

5.2. Coupling between convective layer, residual layer andfree troposphere

In case III (Figure 8a) some oscillations of zr can be seen thatsuggest a coupling of CBL, RL and free troposphere. Actually,we do not expect a change in zr either due to advection, as theterrain is homogeneously flat in the west direction, or due towind shear, as it is negligible at the top of the RL (Figure 10a).However, RL height fluctuations are poorly correlated withCBL height fluctuations or with wave oscillations (see verticalvelocities or depolarization fluctuations), which suggests atbest a partial transmission of wave momentum through theRL-free troposphere interface. The Brunt–Vaısala frequencyin Figure 6d indicates that fN becomes smaller at 1.3 km, i.e.at the free troposphere interface, and thus the propagatinggravity wave may become trapped in the RL; these wavescan be partially reflected downward from their trappingheight (Lott, 1997; Sutherland, 2000). This may explain whythe RL height (zr) fluctuations (see Figures 4c and 4d) arepoorly correlated with RL velocity and depolarization ratiofluctuations.

6. Conclusion

The generation of internal gravity waves in the RL during themorning transition has been investigated from observations.Simultaneous measurements by two lidars have provided adescription of vertical velocities and of the heights of the RLand the developing CBL for two mornings (10 and 14 June2005). Three cases have been identified and studied, showingeither no waves in the RL (case I), evanescent waves (caseII) or propagating gravity waves (case III). Analysis of thedifferent conditions suggests that: (i) a necessary conditionto generate convectively forced gravity waves in the RL is tohave a statically stable and stratified RL (thus the 14 Juneconditions are similar to those in the free troposphere with aBrunt–Vaısala frequency of 1.8 10−3 s−1); (ii) the ‘mechanicoscillator’ (vs. ‘obstacle’) thermal forcing is the source ofgravity waves in the RL; and (iii) the horizontal wind shearis not a sufficient condition for the occurrence of an internalgravity wave through the ‘obstacle effect’, but can still havea significant impact on their propagation.

In the case of wave propagation, the results confirmnumerical studies that the phase lines are tilted upstreamand against the wind shear. Typical horizontal wavelengthand line phase direction are measured as 2.4 km and32◦, respectively. Contrary to what is expected from aclassic linear wave theory we found that wave-associatedvertical velocity and 0.5 µm depolarization ratio or 2 µmbackcatter fluctuations are in phase. Possible explanationsinclude: (i) aerosol particles are not passive with respect totemperature or water vapour fluctuations; (ii) a nonlinearwave-turbulence interaction is at work and needs furtherinvestigation. Lidar observations of the RL and freetroposphere interface suggest that waves are partially trappedin the RL, which is supported by reduced values of theBrunt–Vaisala frequency above the RL.

Copyright c© 2011 Royal Meteorological Society Q. J. R. Meteorol. Soc. (2011)

Convectively Forced Internal Gravity Waves

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 1 (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 andzr .

Acknowledgements

The authors thank Meteo France and SIRTA for providingTrappes sounding data and sonic anemometer measure-ments.

Appendix A: Estimates of Convective Boundary LayerForcing Frequency Using Scales of Turbulence

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

ft,scale = Ut/Lt (A1)

The velocity scale Ut is provided by the maximum in thevertical velocity variance. Using, the mixed layer similarityrelationship for vertical velocity variance (Caughey and

Palmer, 1979), u′2z/w2

∗ = 1.8(z/zi)2/3(1 − 0.8z/zi)

2, where

w∗ = (gziuz0′θ0

′/〈θ〉)1/3 is the convective velocity scale and〈θ〉 is the mean potential temperature in the CBL, the

maximum of the velocity variance is U2t = 0.47 w2

∗ andoccurs at z/zi = 0.31. Using Doppler lidar measurements

we instead found that U2t

∼= 0.2 w2∗ during J10 and J14

morning transitions (Figure A1). Kaimal et al. (1976) alsoshowed that the most energetic eddies are those producedon a scale that is comparable to the depth of the CBL zi.Following Hunt (1984), the scale Lt of the energy-containingeddies for isotropic turbulence is Lt = zi/2, keeping in mindthat some uncertainties exist for such velocity and time scale.

Hence, assuming Ut∼=

√0.2 w∗ and Lt = zi/2, the

main forcing frequency of the energy-containing eddiesis estimated by:

ft,scale∼= 0.89w∗/zi, (A2)

where w∗ and zi are rolling averages over a time gate of30 min.

Figure A1. Atmospheric velocity variance measured by the Doppler lidarnormalized by the turbulent velocity scale for different time gates duringthe morning transitions of (a) 10 and (b) 14 June. The atmospheric velocityvariance is determined by the total lidar velocity variance minus the noisevariance as calculated in Gibert et al. (2007). The velocity scale variance iscalculated using CBL height and surface flux measurements. CBL relativeheight variations around zi in the time gate are indicated with a grey shadedarea.

Appendix B: Estimates of Overshoot Horizontal SpatialFrequency

The power spectrum (fsp.φ(

fsp

)

) of CBL height time serieszi(t) is calculated as follows:

fsp.φ(

fsp

)

= 21t

Mpfsp

∣

∣DFT(

fsp/1fsp

)∣

∣

2, (B1)

where DFT is the discrete Fourier transform, 1t is the timeinterval between the estimates, i.e. 25 s, 1fsp = 1/

(

Mp.1t)

is the frequency resolution, i.e. 5.510−4 s−1 and Mp is thenumber of points in the temporal range gate of 30 min.

Figure B1 shows an example of a CBL height powerspectrum at 1000 UTC on 14 June. The power spectrum isnormalized by the total variance of zi(t). The area under thecurve represents the relative contribution of the spectrum

Copyright c© 2011 Royal Meteorological Society Q. J. R. Meteorol. Soc. (2011)

F. Gibert et al.

Figure B1. For 14 June at 1000 UTC, zi = 0.7 km. Power spectrum of CBLheight 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 5.5 10−4 s−1.The main thermal spatial frequency fzi is indicated by the vertical dashedline. The typical decrease of the power spectrum, which is expected to be

proportional to f−2/3sp in the inertial subrange, is also indicated with the

thin black solid line.

to the total variance. The spectrum shows a typical decreasein f sp

−2/3 of the spectral power in the inertial subrange asexpected. At 1000 UTC there is a relatively large band ofhorizontal frequencies fsp between 1 and 4 10−3 s−1. Here the

frequency of spectral power maximum, fsp,max, is 2 10−3 s−1.

Appendix C: Autocovariance and Cross-CovarianceCalculations for Accurate Apparent Pulsation and PhaseShift Estimates

In this section we investigate wave characteristics usingcovariance techniques. We use the autocovariance functionto estimate the apparent pulsation of the wave ω. Theautocovariance, ACV, for vertical velocity fluctuations canbe calculated according to

ACV(uz, uz)(X) = 〈uz(x)〉′〈u(x + X)〉′ (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 HDLvertical velocity uz and EBL depolarization ratio 1 for caseII at 625 m (Figure C1a) and case III at 1.05 km (FigureC1b). For cases II and III ω = 0.0142 ± 0.0002 rad s−1

(using 1) and ω = 0.0117 ± 0.0001 rad s−1 (using uz) are,respectively, the most accurate estimates of wave pulsationusing the ACV technique.

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

CCV(uz, 1)(X) = 〈uz(x)〉′〈1(x + X)〉′, (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

Figure C1. Autocovariance functions for the depolarization ratio,ACV(1, 1), and vertical velocity, ACV(uz , uz), and the cross-covariancefunction between vertical velocity and backscatter signal at 2 µmCCV

(

uz , Pz2)

and depolarization ratio CCV(uz , 1) time series (grey lineand with filled circles). The time series of each variable is taken at 625 mduring case II and at 1.05 km for case III. A cosine fitting functioncos[ωX + φ] (thick solid line) is applied to the data. Autocovariance andcross-covariance functions are used to estimate the apparent pulsation ofthe wave and the phase shift spatially between uz and Pz2 or 1, respectively.

C1 shows the results of the cross-covariance calculations forcases II and III using the time series displayed in Figure 7b,which corresponds to case II. Despite HDL Doppler lidarvertical velocity and backscatter signal oscillations beingclose to the noise level of the instrument, Figure C1 showsan unambiguous null phase shift between the time series forcases II and III. The same conclusion is obtained for HDLvertical velocity and EBL depolarization ratio fluctuations.

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