Wavelet methods: application to the study of the stable atmospheric boundary layer under non-stationary conditions

Transcription

1 Dynamics of Atmospheres and Oceans 34 (2001) Wavelet methods: application to the study of the stable atmospheric boundary layer under non-stationary conditions E. Terradellas a,, G. Morales b,1, J. Cuxart b, C. Yagüe b a Instituto Nacional de Meteorología, Barcelona, Spain b Instituto Nacional de Meteorología, Madrid, Spain Received 25 September 2000; received in revised form 27 February 2001; accepted 27 February 2001 Abstract In this work, some wavelet methods are introduced to study the atmospheric boundary layer under stable conditions, where intermittent events and non-stationary turbulence take place. Such behavior makes classical methods, based on Fourier transform, difficult to use or even of no application. The wavelet transform is used to detect and characterize some structures in the stable atmospheric boundary layer. First, a wave-like event with a 16 min period is detected and analyzed in a wind record. The sum of some Morlet wavelets is proposed as a model for the oscillations. Afterwards, the wavelet transform is introduced to the study of non-stationary small scale turbulence. It provides the time evolution of the energy and a good location in time of the spots of turbulence. Finally, some wavelet tools are used to characterize a traveling structure, provided that it is simultaneously detected at different locations. The phase differences in the wavelet transform give the wavelength and the phase speed of the oscillations, whereas a double transform method is introduced to estimate the group velocity of the structure Elsevier Science B.V. All rights reserved. Keywords: Gravity wave; Intermittency; Stable boundary layer; Turbulence; Wavelet transform 1. Introduction The planetary boundary layer (PBL) is the lowest region of the atmosphere, directly modified by the physical conditions of the underlying surface. These conditions together Corresponding author. Present address: Centre Meteorològic Territorial a Catalunya, Arquitecte Sert 1, Barcelona, Spain. Fax: addresses: enric.bar@inm.es (E. Terradellas), pny@inm.es (G. Morales), j.cuxart@inm.es (J. Cuxart), c.yague@inm.es (C. Yagüe). 1 Fax: /01/$ see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S (01)

2 226 E. Terradellas et al. / Dynamics of Atmospheres and Oceans 34 (2001) with the dynamics and thermodynamics of the lower atmosphere determine its features. The stable boundary layer (SBL) is a special regime of the PBL that develops over the surface usually at night. Its structure and the physical processes taking place in it are less known than those of the unstable or neutral boundary layers are. The SBL hardly ever reaches the equilibrium state of the unstable boundary layer (Kaimal and Finnigan, 1994): wind, temperature and humidity profiles evolve continuously; internal gravity waves grow and propagate and turbulence appears intermittent and sporadically (Mahrt, 1989; Yagüe and Cano, 1994a; Mahrt, 1999). Knowledge of the SBL is mainly empirical and comes from the analyses of time series of meteorological variables wind speed, temperature, humidity and pressure (Hunt et al., 1985; Mahrt, 1985; Smedman, 1988; Yagüe and Cano, 1994b; Cuxart et al., 2000). Classical methods based on statistical approaches and Fourier analysis are appropriate for homogeneous and stationary states. However, they are not suitable for the study of the SBL, where these characteristics are not present because of the intermittency and constant evolution of the phenomena taking place in it. Autocorrelation and cross-correlation estimates in time space, or spectra and cospectra in frequency space, are not revealing, as they are tools that treat the time series as a whole. For this reason, peculiarities in a signal such as bursts of turbulence or coherent structures of small duration are almost unnoticeable because they are smoothed into the whole series. Based on the same mathematical principles, classical methods for studying the movement of these propagating structures, such as cross-correlation or beam-steering (Rees and Mobbs, 1988; Rees et al., 2000) have the same problems. When using them, it is necessary to delimit in time the extension of the event, in order to perform the analysis in such a period. Usually, analysis of turbulence is based on the averaging process where mean values are separated from the turbulent part of the magnitude (Hussain and Reynolds, 1970). In this way, the differential equations of fluid dynamics are highly simplified. Nevertheless, choosing the averaging interval is not a trivial issue. We need it to be larger than the characteristic periods of the turbulent part, but smaller than the periods of the averaged values (Monin and Yaglom, 1971). In other words, we need a gap between large (mean part) and small (fluctuating part) scales of motion. During daytime, under convective conditions, to assume the gap existence is a good approximation. However, the gap is not always present in the SBL, where the scales of motion are not clearly defined. Classical flux calculations might become questionable since it will not be possible to establish which scales are making the most significant contributions to heat, energy and momentum transport. In this paper, we introduce wavelet methods in order to resolve some of the problems mentioned above (Farge, 1992; Mallat, 1998). The wavelet transform is able to describe a time series in time frequency terms, which provides an opportunity to determine the dominant modes of the signal as well as their time evolution. It also provides easier detection of short duration events in large series. Before the wavelet analysis was developed, attempts to isolate structures had been made using the windowed Fourier transform (Cartwright, 1990), but problems arose in the choice of the window width. Once the window is fixed, the resolution in time is unique for all the modes analyzed. By contrast, the wavelet transform uses a fully scaleable window whose width changes as the transform is computed at different scales, optimizing the time resolution.

3 E. Terradellas et al. / Dynamics of Atmospheres and Oceans 34 (2001) First, the basis of the wavelet theory is reviewed. Then, wavelet methods are applied to detect and analyze some of the structures appearing in the SBL: small scale turbulence and coherent structures of lower frequency. Some wavelet tools are then applied to characterize the movement of propagating structures. Once an event is detected at different spatial locations, the internal motion of the structure and its global displacement can be studied. Techniques based on phase differences and a double transform method are applied. 2. Data collection Several time series are used to illustrate the methods. They were recorded during two field campaigns on SBL held recently. The stable atmospheric boundary layer experiment in Spain (SABLES-98) took place during September 1998 at the research center for the lower atmosphere (CIBA) in the northern Spanish plateau (Cuxart et al., 2000). The purpose of the experiment was to study the characteristics of the stable atmospheric boundary layer in mid-latitudes. Details on instrument deployment can be seen in the mentioned paper. Data used in this work were gathered by several sonic anemometers mounted on a 100 m mast and operating at 20 Hz. These instruments were provided and calibrated by the Risö National Laboratory. On the night of September, the synoptic meteorological situation is dominated by high pressure at surface, with a weak north-easterly flow near the ground. The temperature inversion in the surface layer exceeds 5 C and a low level jet is observed at 150 m, with peak winds of 12 m s 1. Some important fluctuations with periods around 16 min were observed in several meteorological variables. Fig. 1 shows the wind speed at 13.5 m above ground level recorded that night by one of the sonic anemometers. The cooperative atmosphere surface exchange study (CASES-99) field campaign was carried out during October 1999 in Kansas, USA. It was designed to study events in the night Fig. 1. Wind speed at 13.5 m, recorded at night on September 1998 during SABLES-98 field campaign (notice that 26 in the plot corresponds to 2.00 UTC, 28 to 4.00 UTC and so on).

4 228 E. Terradellas et al. / Dynamics of Atmospheres and Oceans 34 (2001) time boundary layer and to investigate the physical processes associated with the evening and morning boundary layer transition regimes. Data gathered by an array of microbarographs operating at 0.1 Hz have been supplied by the Atmospheric Turbulence and Diffusion Division (NOAA/ATDD). On 6 October, high pressure and relatively calm conditions are observed at the surface. There is a temperature inversion with an 8 C increase over the lowest m, and a low level jet with peak winds of 10 m s 1 develops. Some turbulence was recorded during the night, with an especially relevant event detected in the pressure records at 6.00 UTC (0.00 local standard time). This event has been used to illustrate the horizontal motion of coherent structures. 3. Wavelet transform In this section we review some basic concepts of wavelet analysis. A more detailed description has been presented in many works (Daubechies, 1991; Farge, 1992; Meyers et al., 1993; Lau and Weng, 1995; Torrence and Compo, 1998). The wavelet transform is a mathematical tool that enables the analysis of non-stationary time series. It gives a time frequency representation of them providing time and frequency information simultaneously. In this way, the wavelet transform gives the time evolution of the different frequencies that take part in the series. By contrast, the Fourier transform, the classical method to analyze time series, gives the spectral contents of the whole series. A wavelet is a function generated from a mother wavelet, translating and scaling it according to the following expression: Ψ st (t) = 1 ( ) (t t) s 1/2 Ψ 0 (1) s where s is the scale and t the central point. The mother wavelet has to verify the admissibility condition, which can be expressed in the following way (Daubechies, 1991): C Ψ = 2Π dξ ξ 1 Ψ (ξ) 2 < (2) where Ψ (ξ) is the Fourier transform of Ψ (t), defined as Ψ (ξ) = (2Π) 1/2 Ψ(t)exp( iξt)dt (3) This condition is satisfied if the mother wavelet has zero mean, that is Ψ 0 (t) dt = 0 (4) In addition, a good location in time and frequency is required. The ideal wavelet should have a narrow time and frequency support. However, there is no mathematical function with compact support whose Fourier transform also has a compact support (Mallat, 1998). This is the reason why, in practice, a rapid decay in the function and in its Fourier transform is required to the wavelet.

5 E. Terradellas et al. / Dynamics of Atmospheres and Oceans 34 (2001) The wavelet transform of a time series at a scale s and a time t, represented F st,isthe convolution of the series with the dilated and translated wavelet. That is F st = f(t)ψ (t) dt (5) where ( ) denotes complex conjugation. st 3.1. Continuous and discrete wavelet transform The continuous wavelet transform (CWT) is calculated by continuously shifting the scale and time in Eq. (5). The mother wavelet with all its dilations and translations form an over-complete basis. Thus, there is redundant information in the coefficients of the wavelet transform as they are not independent. The discrete wavelet transform (DWT) is worked out using the same procedure expressed in Eq. (5) but just in a discrete amount of scales {s i } and positions {t i }. This transform turns out to be especially relevant when the resulting group of dilations and translations constitute an orthonormal basis and there is no redundant information in the coefficients. Nevertheless, orthogonality is not always the best option. In the analysis of time series, a CWT is preferred because the redundancy itself may enhance some features in the series, such as shape and singularities (Farge, 1992). By contrast, a DWT is advisable when we want to reduce the information needed to rebuild a signal from its wavelet coefficients. Our goal is the analysis of events occurring in the SBL, to find out information about their shape and motion. Therefore, a continuous wavelet transform is used The Morlet wavelet The Morlet wavelet (Fig. 2) is a plane wave modulated by a Gaussian, and it is expressed by Ψ 0 (t) = exp( 1 2 t2 ) exp(iω 0 t) (6) Fig. 2. Morlet wavelet with a base frequency ω 0 = 5.

6 230 E. Terradellas et al. / Dynamics of Atmospheres and Oceans 34 (2001) where ω 0 is usually known as the base frequency, although it is a non-dimensional quantity. This wavelet has been widely used (Farge, 1992; Meyers et al., 1993; Wang and Wang, 1996; Torrence and Compo, 1998). Before starting with the analysis, the most suitable wavelet function must be chosen. Wavelet and structures we are looking for in the series should have similar shape. The sinusoidal form of the Morlet wavelet makes this function an appropriate one to study wave-like phenomena. Not all events in the SBL show that behavior, but there is no way to represent them with just one wavelet. Nevertheless, the choice of the wavelet is not a crucial decision when the interest is focused on wavelet power spectra (Farge, 1992; Torrence and Compo, 1998). Since the Morlet wavelet is a complex function, we can obtain modulus and phase of the transform. The modulus refers to the proximity of the series to the wavelet at a particular scale and time. The more similar they are, the higher the wavelet coefficient is. The modulus is also related to the energy of the time series at one particular time scale step. The phase will be useful when analyzing the spatial displacement of disturbances. An objection to the Morlet wavelet is that it does not exactly fulfill the admissibility condition (4). The drawback could be solved adding corrective terms to the mother wavelet, so that it has zero average. In practice, the error terms turn out to be negligible just by choosing a base frequency equal or higher than 5 (Farge, 1992) Reconstruction of a series from its wavelet transform Sometimes the reconstruction of a signal from its wavelet transform coefficients is needed. Recovering the time series is straightforward when using an orthonormal basis. By contrast, the reconstruction from a continuous wavelet transform is more difficult because of the redundant information supplied by the coefficients. However, the continuous wavelet transform is reversible provided the admissibility condition is satisfied (Daubechies, 1991) f(t)= C 1 Ψ ds dt F st Ψ st (t) s 2 (7) where C Ψ is defined in Eq. (2). If the series is real and the wavelet is an analytic function, that is, its Fourier transform is zero for negative frequencies (the Morlet wavelet satisfies approximately this condition for ω 0 5), Eq. (7) can be rewritten as f(t)= 2C 1 Ψ 0 ds dt Re(F st Ψ st (t)) s 2 (8) where Re(x) means real part of x. We can therefore reconstruct the series with only the wavelet transform coefficients corresponding to positive scales. The redundant information generated by the continuous wavelet transform allows several synthesizing formulas. For instance, we could use a reconstructing wavelet function different from the analyzing one. The easiest procedure is the introduction of a Dirac function to work out the inverse transform (Farge, 1992). In this way, the value of the series in a time location t is obtained just by considering the wavelet transform coefficients at t. With this property,

7 E. Terradellas et al. / Dynamics of Atmospheres and Oceans 34 (2001) we keep from extending the calculations beyond the time for which the reconstruction is desired. For real series where f(t) = 2C 1 Ψδ 0 Re(F st ) ds s 3/2 (9) C Ψδ = (2Π) 1/2 dξ ξ 1 Ψ (ξ) (10) 3.4. Time and frequency distribution of the energy From the reconstruction formula (7), it can be proved that the wavelet transform fulfills the energy conservation of the whole series (Farge, 1992; Mallat, 1998) E = f(t) 2 dt = CΨ 1 2 ds dt F st s 2 (11) This expression leads to the definition of an energy density (energy per time and scale unit), e st = CΨ 1 F st 2 s 2 (12) The energy density depends on the analyzing wavelet. When the wavelet has a good time resolution there is a high frequency uncertainty and the local energy spectra will be highly smeared. Alternatively, if the wavelet has a good frequency resolution there is a high time uncertainty and spectra will smoothly evolve with time. Energy density per scale and time unit may be substituted by energy density per period and time unit. In order to do so, the scale must be replaced by an equivalent Fourier period. The relationship between the equivalent Fourier period and the scale is derived analytically by substituting a wave of a known frequency into Eq. (5) and calculating the scale at which the wavelet transform modulus reaches its maximum (Meyers et al., 1993; Torrence and Compo, 1998). For the Morlet wavelet 4Πs T = (ω 0 + (2 + ω0 2 (13) )1/2 ) For real series, only positive periods need to be computed and the energy density is ( ) 8Π e Tt = ω 0 + (2 + ω0 2 C 1 F st 2 )1/2 Ψ T 2 (14) where a2factor has been introduced to include energy associated to negative scales. 4. Detection and analysis of coherent structures Gravity waves and other coherent structures, such as Kelvin Helmholtz waves and drainage fronts, are usually present during stable conditions. Knowledge of their dynamics

8 232 E. Terradellas et al. / Dynamics of Atmospheres and Oceans 34 (2001) and their interaction with turbulence is essential to understand the mechanisms that govern the SBL. In this section, we show the usefulness of the wavelet transform to detect these structures and analyze some of their characteristics. During the SABLES-98 field campaign, there was an interesting period of observations from 14 to 21 September. The synoptic meteorological situation was almost stationary, with predominance of local effects during daytime and a weak synoptic flow during night time. Stably stratified temperature profiles and vertical wind shear developed at night. The September night is chosen to illustrate the present study because a significant stable layer became established close to the ground. Fig. 3a and b show the wavelet transform modulus and the energy density per period and time unit of the wind speed record at 13.5 m AGL from 1.00 to 5.00 UTC (the same local standard time). The energy maximum appears at a slightly lower period than the modulus maximum, as could be expected looking at Eq. (14). Maxima in modulus and energy plots would appear at the same period if an energy density per time and frequency unit was used. Looking at Fig. 3a, a defined structure of a 16 min period with a maximum around 2.00 UTC can be observed. A weaker peak appears around 4.00 UTC. Its period is slightly higher. These periods are not unusual for gravity waves (Rees and Mobbs, 1988). However, some other processes might cause this wave-like behavior, so further analysis is needed to ensure that a real gravity wave has been detected. Fig. 3. Coherent structure in the wavelet transform (using a Morlet wavelet with a base frequency ω 0 = 5) of the wind speed recorded on 15 September (a) Wavelet transform modulus; (b) energy density per period and time unit (m 2 s 3 ). The solid line encloses the area where the values are 99% statistically significant; (c) energy density distribution for a model with five wave trains.

9 E. Terradellas et al. / Dynamics of Atmospheres and Oceans 34 (2001) Fig. 4. Mean spectrum of the wind speed recorded on September from to 5.00 UTC (solid line) and 99% confidence level for a Chi-square distribution with 2 degrees of freedom (dots). The significance of the peaks is tested following the method described by Torrence and Compo (1998). Let us assume that the energy density can be expressed as the sum of the square of two variables, both normally distributed for a given period, with zero mean and σ 2 variance. It can be deduced that e/σ 2 will be Chi-square distributed with 2 degrees of freedom, denoted by χ2 2. Since the mean value of a χ 2 2 distributed variable is 2, 2e/e m will also follow a χ2 2 distribution, being e m the average energy density for the given period. A mean energy spectrum has been calculated integrating the wavelet energy density from to 5.00 UTC at different periods. This spectrum and the 99% confidence levels can be seen in Fig. 4. The solid line in Fig. 3b encloses the area where the energy density exceeds this 99% confidence level. Some authors have applied the same significance test in a slightly different form. They have built a red-noise background spectrum using autoregressive models (Kestin et al., 1998; Torrence and Compo, 1998) to substitute the mean spectrum. However, the use of the averaged values seems to be a more robust procedure, because no additional hypotheses are required other than the Chi-square distribution of the energy. When a significance test is applied, it is essential to keep in mind the non-stationary behavior of the SBL. The use of a stationary test such as that given above has some advantages as argued by Torrence and Compo (1998), but there is no way to define a standard background spectrum for the SBL. With regards to the event amplitude, it can be related with the wavelet transform modulus. In order to quantify it, we must state some additional hypotheses about the structure shape. We may assume that this event can be represented by a wave train looking like a Morlet wavelet ( ) ( ω0 (t t) (t t) 2 ) f(t)= A cos + θ exp (15) s max 2s 2 max where s max is obtained by means of Eq. (13) from the period at which the wavelet transform modulus reaches its maximum value.

10 234 E. Terradellas et al. / Dynamics of Atmospheres and Oceans 34 (2001) Introducing Eq. (15) into Eq. (5), the following expression can be derived: A = 2 max( F st ) (Πs max ) 1/2 (16) A1.0ms 1 amplitude is obtained in the current event The wavelet transform as a band-pass filter The reconstruction formula (9) can be used as a time series band-pass filter by performing the integration over a specific range of scales. Once a structure is identified, the display of the filtered series allows to see its features more clearly in the physical (time) space. A band-pass filter for periods between 10 and 20 min has been applied to the wind record. Fig. 5 shows the oscillations that are present in the coherent structure. The filtered series will depend on the properties of the chosen wavelet, especially on its time and frequency resolutions. When a wavelet with a good time resolution is used, the oscillations are well represented in time but the range of the selected frequencies becomes blurred. On the other hand, when the wavelet has a good frequency resolution, the passing frequencies are much better delimited but their amplitudes are smoothed in time. In the example mentioned above, the band-pass filter is acting over a range of frequencies in which a maximum in the wavelet transform was detected. Because of the smoothing, the amplitudes in the filtered series are underestimated. The worse the frequency resolution in the wavelet, the higher the loss in the filtered series is Wavelet modeling The sum of several functions similar to the Morlet wavelet can sometimes be used to model the real time series. That is Fig. 5. Wind speed at 13.5 m, recorded on 15 September 1998 during SABLES-98 field campaign. Original series (up) and filtered series using a Morlet wavelet with a base frequency ω 0 = 5 as a band-pass filter (down).

11 E. Terradellas et al. / Dynamics of Atmospheres and Oceans 34 (2001) f(t)= N (M i (t)) = 1 [ N ( ω0 (t t i A i cos ) ) ( (t t i + θ i exp )2 1 s i 2s 2 i )] (17) We want the model to be as close as possible to the original series in a selected time and period window, Ω = [(t 1,T 1 ), (t 2,T 2 )]. This substitution would be trivial in the case of an orthonormal basis, but much more difficult for the continuous wavelet transform. In this case, it can only be accomplished by successive iterations. Let W 1 be the maximum of the wavelet transform modulus in the analyzing window and θ 1 its phase. The time and period where it occurs are t 1 and T 1. We assume that a function M 1 (t) has produced the maximum. This function must be characterized by the same parameters θ 1, t 1. Its scale can be calculated by Eq. (13) and its amplitude by Eq. (16). They are s 1 = (ω 0 + (2 + ω 2 0 )1/2 )T 1 4Π (18) A 1 = 2W 1 (Πs 1 ) 1/2 (19) M 1 (t) is the first model of the series in the window. In order to quantify its goodness, the fraction of energy that the model explains is estimated, (E f E r )/E f, where E f is the energy of the original series in the window, and E r is the energy of the residual series f(t) M 1 (t). Next, the wavelet transform of the residual series is computed. It is likely to have a new maximum in the window. Let W 2 be the modulus of this second maximum, with θ 2, t 2 and T 2 having the same meaning as before. A new function M 2 (t) can be found. The new model will be M 1 (t) + M 2 (t) and, provided the residual energy is lower, it will be considered an improved representation of the series. M 1 (t) and M 2 (t) may be refined using an iterative method. The procedure is performed by working out M 1 (t) again from f(t) M 2 (t) and then M 2 (t) from f(t) M 1 (t). After every iteration, the residual energy must be checked. Only if it decreases, will the new model be accepted. Afterwards, new functions M i (t) can be added to the model and the previous ones (M 1 (t), M 2 (t),...) refined. The model has been applied to the wind series recorded during SABLES-98. Table 1 shows the characteristics of the five wave trains used to represent the series for periods Table 1 Wave trains used to model the wind speed at 13.5 m AGL recorded on 15 September 1998 during SABLES-98 field campaign Amplitude (m s 1 ) Central time (UTC) Scale (min) Equation Fourier period (min) Phase ( ) :14: :37: :02: :50: :31:

12 236 E. Terradellas et al. / Dynamics of Atmospheres and Oceans 34 (2001) between 10 and 20 min. This model explains 97.2% of the energy in the window. Fig. 3b and c show the energy distribution in the original series and in the model. A very good agreement between the plots is observed, as expected in view of the high energy percentage explained by the model. The linear correlation coefficient between the energy densities in the series and the model is 0.99 and the Spearman rank-order correlation coefficient, Characterization of intermittent small scale turbulence The wavelet transform allows to study the whole range of scales present in the flow, provided that enough resolution in time is available in the measurements as well as the record length is large enough. In the previous section, we have focused on the detection and characterization of phenomena with time scales between 10 and 20 min. Discussion in this section is mainly centered on smaller scale phenomena, which we arbitrarily define as small scale turbulence. Large scale phenomena may also hold some embedded turbulence, but we have chosen in this work to categorize them as coherent structures. Fig. 6 shows the wavelet transform modulus and the energy density per period and time unit of the series analyzed in the previous section, but for smaller periods (from a few seconds to 5 min). It is assumed that the peaks in the energy density plot correspond to spots of turbulence. These peaks can also be seen in the wavelet transform plot, but they do not appear so clearly. For small periods, the energy density per period and time unit is more sensitive than the wavelet transform modulus, and, consequently, more suitable for the detection of small scale turbulence. The low amount of energy associated to periods between 3 and 8 min (see the mean spectrum in Fig. 7a) may be considered as a spectral gap. Therefore, the classical averaging method may be used with some confidence, as in Cuxart et al. (2000). Nevertheless, this analysis assumes stationarity, and, as it usually occurs under stable conditions, the turbulence in the series is not stationary, and it may be suspected that it is not isotropic either. When the energy density is integrated along the time axis, a mean energy spectrum is obtained for the whole window, similar to what is done using a Fourier transform, except for a smoothing factor (Farge, 1992). Fig. 7a shows this power spectrum. In Fig. 7b, the Fig. 6. Bursts of turbulence detected in the wavelet transform of the wind speed recorded on 15 September (a) Wavelet transform modulus; (b) energy density per period and time unit (m 2 s 3 ).

13 E. Terradellas et al. / Dynamics of Atmospheres and Oceans 34 (2001) Fig. 7. Mean spectrum of the wind speed recorded on September 1998 from to 5.00 UTC. (a) Energy density per period and time unit (m 2 s 3 ); (b) classical representation of the energy density per time and frequency unit. Frequency times the spectrum (m 2 s 2 ) vs. frequency. same spectrum is represented in a more common way, in terms of the frequency instead of the period. Both have been extended to time scales up to 20 min. Some important scales of the flow show up clearly in the plot. The peak in Fig. 7b corresponds to the sustained event with a 16 min period described in the previous section. The intermittent events have a temporal scale below 2.5 min (frequencies above s 1 ). The Brunt Väisälä frequency (N) computed from the series is 0.05 s 1, the last local minimum in the plot. Some portions of the spectrum between the energy maximum and N seem to have a 2 slope, indicating the possible presence of a buoyancy subrange where quasi-2d turbulence may take place. However, the presence of this slope is not clear. It might be explained by the fact that the turbulence is intermittent and sporadic, and therefore the averaged spectrum is not significant (see Szilagyi et al., 1996 and references therein). For frequencies higher than N, the inertial subrange is expected, but the ( 2/3) slope does not appear in the plot. The spectrum of the vertical velocity (not shown) indicates that the inertial subrange starts at 0.5 s 1, a much Fig. 8. Bursts of turbulence in the wind speed recorded on 15 September The energy density has been integrated for periods between 30 s and 3 min.

14 238 E. Terradellas et al. / Dynamics of Atmospheres and Oceans 34 (2001) higher frequency. Therefore, turbulence between N and 0.5 s 1 is not isotropic and does not follow the Kolmogorov theory. For comparison purposes, see Fig in Stull (1988). Up to this point, it is seen that the same mean spectrum and similar conclusions can be reached using Fourier or wavelet transforms. However, knowledge of the time distribution of the events allows to make a more refined interpretation of the information. An example is given in Fig. 8, where the energy density integrated for periods from 30 s to 3 min is shown. This representation allows to have a clear picture of the time evolution of the small scale turbulent activity. The identification of individual spots of turbulence is particularly interesting in the SBL, because it will make possible to search the connection between turbulence and the development or the breaking of structures with a higher time and spatial magnitude. More turbulence diagnostics can be derived from the wavelet transform, and this might be the subject of future work. 6. Application to the analysis of structure movements The wavelet transform phase can be used to describe the motion of coherent structures in the boundary layer, provided these structures are simultaneously detected at different locations. The movement speed and direction are calculated from the differences between the wavelet transform phases of different time series. Classical methods used to characterize the motion (cross-correlation and beam-steering) are based on the Fourier transform. They just provide one speed and direction value for the whole series, and therefore the outcome is sensitive to the extension of the analysis. When the structure is only present in a small fraction of the series, the results can be distorted. Sometimes, the proper length of the analyzed series is difficult to determine. The series should be long enough to avoid aliasing problems but short enough to isolate the structure we want to analyze. If applied to series of different magnitudes, the phase difference method can provide additional information about the event characteristics, by looking at the concordance between the oscillation phases of horizontal and vertical speed, potential temperature, pressure, and so on Vertical displacement of coherent structures Let f(t) be a time series recorded at height z and f (t) a time series of the same magnitude recorded at a different height z. If a coherent structure is detected in both series at the same scale, phase differences at a certain time permit calculation of the time the oscillation takes to travel from z to z. This time is δt = (θ θ )T (20) 2Π where T is the period corresponding to the scale s, defined in Eq. (13). Hence, the vertical velocity is ( 2Π V = T ) z z θ θ (21)

15 E. Terradellas et al. / Dynamics of Atmospheres and Oceans 34 (2001) Fig. 9. Coherent structure with a 17 min period detected in the wind speed recorded on 15 September (a) Wavelet transform modulus; (b) vertical velocity (m s 1 ). As the information is available at every time and scale, the vertical speed can be depicted as a function of both time and period. The wind speed series recorded on 15 September at 5.8 and 13.5 m AGL present a similar behavior between 1.00 and 5.00 UTC. Their wavelet transform modulus show a main peak at 2.00 and a secondary one at 4.00, as may be noted in Fig. 3a. In spite of the fact that only the first one has proved to be statistically significant, we will use the last one to illustrate this section, because the results seem to be more revealing. In fact, we have to be aware that some structures in the SBL will only appear at some levels, and others will have independent evolutions at different heights, without a clear vertical displacement. The absence of an area around the peak with a slow variation in the speed with time and period appears to be a symptom of such an independent evolution. This absence of a clear pattern has been observed in the main peak. We zoom in on the secondary peak in Fig. 9a, where 17 min period oscillations are highlighted. The vertical velocity is calculated from Eq. (21) and depicted in Fig. 9b. The value in the range of scales and times of the event occurrence is about 0.3 m s 1 and is directed upward. Notice that the speed does not follow a regular behavior on the right part of the picture, once the coherent structure has disappeared. When a Fourier method is used to calculate the velocity, it is very difficult to build a window, which does not contain other oscillations besides the structure that is being analyzed. Sometimes, these spurious oscillations move in a completely different way that the coherent structure. Occasionally, they are present only in one of the levels, resulting in a random phase difference. These facts represent an important source of error when the average motion in the window is obtained using Fourier methods Horizontal displacement of coherent structures Following an analogous procedure, the horizontal displacements can be analyzed. Let f A, f B and f C be three series of the same magnitude recorded at locations A, B, C. If a coherent structure is simultaneously detected in them, the wavelet transform phases θ A, θ B, and θ C at a given time and scale permit calculation of the oscillation wave number, by using the

16 240 E. Terradellas et al. / Dynamics of Atmospheres and Oceans 34 (2001) following expressions: θ B θ A = K x (x B x A ) + K y (y B y A ), θ C θ A = K x (x C x A ) + K y (y C y A ) (22) where (x i, y i ) are A, B, C coordinates and K = (K x,k y ) is the oscillation wave number. Wavelength, phase speed and direction of the structure can be obtained from this wave number. In order to illustrate the method, we present the analysis of three pressure time series recorded during CASES-99 field campaign on 6 October Data were gathered by an array of three microbarographs. The distances between the sensors were 350.5, and 506 m. A 6 min period pressure disturbance is detected around 6.00 UTC (0.00 local standard time). The peak appears in the wavelet transform of the three series and, following the method described in Section 5, it has proved to be 99% significant. The wavelet transform of the three series has been computed between 6.00 and 6.10 UTC for periods ranging from 4.5 to 7.5 min. The wavelet transform modulus of one of the series can be seen in Fig. 10a. The wavelet transforms of the other series look very similar. Their mutual Spearman rank-order correlation coefficients are 0.95, 0.85 and A perfect correlation is not possible provided the structure is moving. Fig. 10. Pressure disturbance recorded on 6 October 1999 during CASES-99 field campaign. (a) Wavelet transform modulus; (b) wavelength (m); (c) phase speed (m s 1 ); (d) propagating direction ( ).

17 E. Terradellas et al. / Dynamics of Atmospheres and Oceans 34 (2001) Using Eq. (22), the wavelength, phase speed and direction are obtained as a function of period and time. They can be seen in Fig. 10b d. Looking at the periods and time at which the wavelet transform modulus reaches its maximum values, we can conclude that the disturbance has a 3700 m wavelength, an 11 m s 1 phase speed and a propagating direction of Group velocity: the double transform method In the previous section, the phase speed of the oscillations has been calculated. As a rule, this velocity does not match the speed at which the overall structure is traveling. To calculate the group velocity, we initially work out the wavelet transform of the original series, referred later as first transform. Once the scale of the analyzed structure is identified, we keep the time series of the modulus at this scale. This new series can be viewed as the time evolution of the amplitude at the frequency corresponding to the considered scale. Next, a wavelet transform is applied to the new time series (second transform) and their phase differences provide the group velocity of the structure. Calculation of group velocities from the wavelet transform modulus was made by Meyers et al. (1993). Their method is based on the time the maximum in the transform takes to travel between different points. The double transform method is able to determine the traveling time at any point of the amplitude series, not only at the maximum, giving a more complete image of the motion. As discussed in Section 6.1, a strong variation in the results with time and period can label the results as suspicious. This diagnostic is not possible when the method described by Meyers et al. is used. The quantification of the acceptable range of variations is the object of future work. The pressure event described in the previous section presents maxima in the wavelet transform modulus for 349, 335 and 335 s periods at A, B and C locations, respectively. We keep the modulus time series at an intermediate 340 s period. One of the modulus time evolutions can be seen in Fig. 11. In this plot, the peak at 6.00 UTC represents the analyzed structure. Fig. 11. Time evolution of the wavelet transform modulus (for a 340 s period) of the pressure recorded on 6 October 1999.

18 242 E. Terradellas et al. / Dynamics of Atmospheres and Oceans 34 (2001) Fig. 12. (a) Double wavelet transform of the pressure recorded on 6 October 1999; (b) group velocity of the pressure disturbance (m s 1 ); (c) direction ( ). Once the second transforms are calculated, the coherent structure can be identified in the modulus plots. One of them is shown in Fig. 12a. Obviously, the periods of the maxima in the first and second transforms are not related each other at all. The periods of the second transform maxima are indeed related to the coherent structure time length, whereas the period in the first transform maxima refers to the internal oscillations. Phase differences between these second transforms at the scale and time of the new maxima provide the group velocity and displacement direction of the corresponding structure. The analyzed structure has a 3.2 m s 1 group velocity and moves from 200, as can be seen in Fig. 12b and c. This group velocity approximately matches the wind speed and direction observed at near-surface levels, while the phase velocity calculated in Section 7.2 is considerably higher and veered 50 from it. These results are consistent with other observations (Rees and Mobbs, 1988). 7. Conclusions The wavelet analysis turns out to be a great improvement in the analysis of atmospheric data, especially when we deal with non-stationary time series. In contrast to standard

19 E. Terradellas et al. / Dynamics of Atmospheres and Oceans 34 (2001) methods of analysis, the wavelet transform yields the spectral content of the series as well as its evolution in time. Detecting events using the wavelet transform modulus or the energy density plots is straightforward. Some characteristics of wave-like events, i.e. their period or amplitude, can be estimated. Some other techniques, such as the band-pass filtering or the wavelet modeling can be used to enhance the main features of the detected structures. The energy density plots also provide useful information about turbulent episodes, depicting individual bursts and illustrating the notion of intermittency. The wavelet analysis is also an efficient way to study the displacement characteristics of a moving structure. As the Morlet wavelet is a complex function, the wavelet transform provides a phase for every scale and time. Phase differences between series simultaneously recorded at different locations give the phase speed and wave number of the oscillations. The importance of the wavelet transform to find the speed and direction parameters is that we obtain them as a function of time and scale, whereas classical methods, such as beam-steering or cross-correlation, yield just one value for the whole series length. The double-transform method is an advantageous way to estimate the group velocity. It is based on the displacement of the structure, avoiding the uncertainties of indirect calculations from the phase velocity. References Cartwright, Fourier Methods for Mathematicians, Scientists and Engineers. Ellis Horwood, New York. Cuxart, J., Yagüe, C., Morales, G., Terradellas, E., Orbe, J., Calvo, J., Fernández, A., Soler, M.R., Infante, C., Buenestado, P., Espinalt, A., Joergensen, E., Rees, J.M., Vilà, J., Redondo, J.M., Cantalapiedra, I.R., Conangla, L., Stable boundary layer experiment in Spain (SABLES-98): a report. Bound.-Layer Meteorol. 96, Daubechies, I., Ten Lectures on Wavelets. CBMS Lecture Notes Series. SIAM, Philadelphia. Farge, M., Wavelet transforms and their applications to turbulence. Ann. Rev. Fluid Mech. 24, Hunt, J.C.R., Kaimal, J.C., Gaynor, J.E., Some observations of turbulence structure in stable layers. Q. J. R. Meteorol. Soc. III, Hussain, A.K.M.F., Reynolds, W.C., The mechanics of an organized wave in turbulent shear flow. J. Fluid Mech. 41, Kaimal, J.C., Finnigan, J.J., Atmospheric Boundary Layer Flows. Oxford University Press, New York. Kestin, T.A., Karoly, D.J., Yano, J.I., Rayner, N., Time frequency variability of ENSO and stochastic simulations. J. Climate 11, Lau, K.M., Weng, H.Y., Climate signal detection using wavelet transform. How to make a time series sing. Bull. Am. Med. Soc. 76, Mahrt, L., Vertical structure and turbulence in the very stable boundary layer. J. Atmos. Sci. 42, Mahrt, L., Intermittency of atmospheric turbulence. J. Atmos. Sci. 46, Mahrt, L., Stratified atmospheric boundary layers. Bound.-Layer Meteorol. 90, Mallat, S., A Wavelet Tour of Signal Processing. Academic Press, Cambridge. Meyers, S.D., Kelly, B.G., O Brien, J.J., An introduction to wavelet analysis in oceanography and meteorology: with application to the dispersion of Yanai waves. Mon. Wea. Rev. 121, Monin, A.S., Yaglom, A.M., Statistical Fluid Mechanics Mechanics of Turbulence. MIT Press, Cambridge, MA. Rees, J.M., Mobbs, S.D., Studies of internal gravity waves at halley base, antarctica, using wind observations. Q. J. R. Meteorol. Soc. 114, Rees, J.M., Denholm-Price, J.C.W., King, J.C., Anderson, P.S., A climatological study of internal gravity waves in the atmospheric boundary layer overlying the brunt ice shelf, Antarctica. J. Atmos. Sci. 57,

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