1 Introduction

Experimental investigation of internal shear-induced waves due to a Kelvin-Helmholtz (KH) instability is an important problem in the study and parametrization of the stably-stratified atmospheric boundary layer (ABL). It is now generally recognized that peculiarities of the stable ABL, including wave activity, affect not only regional weather, but also the general circulation of the atmosphere (see, e.g. Zilitinkevich and Esau 2003; Esau and Zilitinkevich 2010; Holtslag et al. 2013; McGrath-Spangler et al. 2015). A poor understanding of the exchange processes in the stable ABL and difficulties in its parametrization inhibit the development of numerical modelling of regional and local weather (Mahrt 1998; Cohen et al. 2015). The breakdown of KH billows is considered to be the main mechanism of turbulence excitation in the stable ABL (Patterson et al. 2006; Venkatesh et al. 2014). Interest in KH waves in Antarctica is linked, first of all, to their role in the exchange processes over the ice and snow surface, under conditions of long-lived inversions (Neff et al. 2008; Argentini et al. 2014).

Sodars often provide a clear representation of wavelike structures appearing as layers of enhanced turbulence oscillating in the vertical plane (e.g., Brown and Hall 1978). The duration of the wavy patterns varies from a few minutes to several hours, appearing as billows, braids or cat’s eye patterns in radar, sodar and lidar echograms, and cloud photos. Eymard and Weill (1979) observed oscillations resembling regular herringbones [similar to braids in the classification by Gossard and Hooke (1975)] at nighttime with a wave period of about 2 min. A similar herringbone pattern attributed to the KH instability was observed by Neff et al. (2008) at the South Pole. A few episodes of larger-scale undulations of several minutes in Antarctica were observed by Kouznetsov (2009) and Kouznetsov and Lyulyukin (2014), and examples of small-scale wavy structures with periods of <1 min were presented by Petenko et al. (2013) and Argentini et al. (2014). During the sodar measurements at Dome C, observations of the smoke flow from the chimney of the Concordia power station sometimes showed a wavelike pattern (Fig. 1) similar to that shown by the sodar.

Fig. 1
figure 1

A smoke flow from the Concordia power station, which shows the wavy structure

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The internal structure of an inversion layer capping a convective ABL was observed by Browning (1971), Browning et al. (1973a) and Readings et al. (1973) using pulsed Doppler radar, tethered balloon and radiosonde ascents. They showed how convective circulations in the ABL perturb the height of an inversion, thereby producing undulations of the entire inversion layer. Readings et al. (1973) also hypothesised the possibility of the presence of trapped gravity waves within an inversion. Air motions within KH billows at a height of \(\approx \)7 km were investigated by Browning et al. (1973b) using radar, aircraft and radiosonde measurements. Undulations of an inversion layer overlying convection during the morning hours were also observed by means of a sodar by Taconet and Weill (1983), with periods of 5–7 min.

The composite shape and structure of KH billows observed with a sodar was analyzed by Lyulyukin et al. (2013), and in Lyulyukin et al. (2015) a climatology of wavelike structures in the stable ABL and a comprehensive bibliography were presented. However, the climatological data on the occurrence of wavelike structures observed with conventional sodars are not very accurate, since these measurement systems are unable to detect the fine short-lived wavelike turbulent patterns with periods of a few tens of seconds. This leads to the underestimation of the role of the wave processes in the energy transfer in the lower ABL.

A deeper insight into atmospheric processes in the ABL is possible with the use of advanced high-resolution sodars (Neff et al. 2008; Argentini et al. 2012). New features of the surface-based turbulent layer were shown during the one-year experiment conducted at the Concordia station (Dome C, Antarctica) in 2012 (Petenko et al. 2013, 2014a; Argentini et al. 2014). A second long-term experiment using this sodar was made in 2014 at the same site investigating the diurnal behaviour of thermal turbulence in the ABL. In previous studies, it was shown that the diurnal cycle at Dome C during the summer season includes a “night” part, with the stably stratified shallow ABL, and a “day” part. During the “day” part, convective activity develops under the capping inversion layer that reaches heights of 100–300 m (Argentini et al. 2005; Casasanta et al. 2014). “Night” and “day” are quoted because the sun is always above the horizon during the polar summer (from the end of October to the end of February). The presence of a distinct diurnal cycle differentiates the Dome C site from other well-investigated sites in Antarctica, such as the South Pole and Halley, where the daily cycle during the summer is absent (King et al. 2006; Neff et al. 2008).

From visual inspection of the echograms obtained with conventional sodars, the morning evolution of the ABL presents a gradual ascent of the inversion layer with enhanced turbulence, under which convective plumes expand and rise. A quite surprising internal structure appears in the high-resolution sodar echograms with an extended time scale. Clear wavelike braid patterns with periods of several tens of seconds are observed within the elevated turbulent layer above the convective plumes rising from the surface.

Here, we analyze the characteristics of wavelike structures within an overlying turbulent inversion layer observed by sodar during the morning development of convection near the surface at Dome C in summer January-February 2014. A description of the experimental set-up and the meteorological conditions are given in Sect. 2. In Sect. 3, the characteristics of the diurnal variation of the spatial and temporal structure of turbulence in the ABL, and clear examples of undulation processes are presented. Periods, wavelengths, and vertical extension of the wavy patterns were estimated by visual and spectral analysis. A summary is presented in Sect. 4.

2 Experimental Set-up and Meteorological Conditions

2.1 Site Location and Characterisation

Concordia station is located at Dome Charlie (Dome C), Antarctic plateau, 900 km inland from the nearest coast (75\(^{\circ }\)06\(^{\prime }\)S, 123\(^{\circ }\)21\(^{\prime }\)E, 3233 m a.s.l.) with a surface slope \(\approx \) 0.1%. Its position is shown on the geographical map in Fig. 2. The Sun culminates at \(38^{\circ }\) on 21 December, with the weather in summer dominated by the alternation of a surface-based temperature inversion layer and a weak convective ABL capped by an elevated inversion layer. Episodic synoptic perturbations, due to maritime air intrusions produce wind speeds of up to 10 m s\(^{-1}\), significantly higher temperatures (up to \(-20\,^{\circ }\hbox {C}\)) and dense cloudiness (e.g. Argentini et al. 2001; Genthon et al. 2013).

Fig. 2
figure 2

Geographical map of Antarctica, with the position of Dome C, where the French–Italian station of Concordia is located

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2.2 Sodar Measurements

An advanced version of a high-resolution sodar (Argentini et al. 2012) developed by the Institute of Atmospheric Sciences and Climate of the National Research Council of Italy (ISAC-CNR) was used for observations of the turbulence structure in the ABL at altitudes from 2 m up to 200 m above the surface (with a vertical resolution of \(\approx \)2 m). The sodar has collected data continuously since 20 January 2014, and for this study, we used data collected up to the end of February 2014. The acoustic pulse with a carrier frequency of \(\approx \)4800 Hz had a duration of 10 ms and was emitted every 2 s. The four vertically-pointed sodar antennae (three transmitting horns and one receiving parabolic dish with a noise-protection shield) (Fig. 3a) were installed 400 m south-west of the main buildings of the Concordia station. This position was chosen considering the prevailing atmospheric flow to minimize the influence of the station buildings.

Fig. 3
figure 3

a Sodar antennae. b The sonic anemometer USA-1 and the net radiometer CNR1

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Acoustic remote sensing, based on the scattering of acoustic waves by small-scale turbulent temperature and wind fluctuations, provides a clear pattern of the structure of the ABL (Brown and Hall 1978). Sodar allows continuous monitoring of vertical profiles of the temperature structure parameter \(C_{T}^{2}\), since the intensity of the backscattered acoustic signal is proportional to \(C_{T}^{2}\) (Kallistratova 1962; Tatarskii 1971). The parameter \(C_{T}^{2}\) is a proportionality factor in the 2/3-law for the structure function \(D_{T},\) valid within the inertial subrange of locally isotropic turbulence (Obukhov 1949),

$$\begin{aligned} D_T \left( r \right) \equiv \overline{\left[ {T^{{\prime }}\left( {\mathbf{r}_1 } \right) -T^{{\prime }}\left( {\mathbf{r}_2 } \right) } \right] ^{2}} =C_T^2 r^{2/3}, \end{aligned}$$
(1)

where \(T'(\mathbf{r})\) is the temperature fluctuation around its mean at the point r, and \(r=\left| {\mathbf{r}_1 -\mathbf{r}_2 } \right| \) is the distance between points \(\mathbf{r}_{1}\) and r \(_{2}\). In acoustic remote sensing, \( C_{T}^{2}\) is determined from measurements of the intensity of the backscattered acoustic signal (Tatarskii 1971),

$$\begin{aligned} C_T^2 =0.25\times 10^{3}T^{2}k^{-1/3}\sigma _{180}, \end{aligned}$$
(2)

where \(\sigma _{180}\) is the effective backscattering cross-section per unit of scattering volume per unit solid angle, T is the absolute temperature, and k is the wavenumber. In the short form, the relationship between \(C_{T}^{2}~(z)\) and the measured intensity of the sodar electric return signal I is

$$\begin{aligned} C_{T}^{2} (z) = B(P_{t}, k, { T, c, \tau , S}) I(z) z^{2}\hbox {exp}(4{az}), \end{aligned}$$
(3)

where B is constant that can be calculated from a complex calibration procedure (Danilov et al. 1994) or determined from comparison with in situ \(C_{T}^{2}\) measurements (Petenko et al. 2014b). Here, \(P_{t}\) is the transmitted power, c is the velocity of sound, \(\tau \) is the pulse duration, z is the distance of the scattering volume from the transmitter, S is the effective area of the sodar antenna, and a is the absorption coefficient of sound.

Note that only those small-scale turbulent temperature inhomogeneities whose vertical dimensions are equal to one half of the wavelength of the interrogating sound wave produce scattering at an angle of \(180^{\circ }\) (Tatarskii 1971). For our sodar these dimensions are 30 mm.

The turbulence structure of the atmosphere is depicted by sodar echograms that show a time-height distribution of \(C_{T}^{2}\) (e.g. Brown and Hall 1978), and the possibility of reliable quantitative measurements of \(C_{T}^{2}\) by using a sodar was shown in many studies (e.g. Coulter and Wesely 1980; Weill et al. 1980; Gur’yanov et al. 1987; Asimakopoulos et al. 1983; Danilov et al. 1994). The applicability of the concept of locally isotropic turbulence, and consequently the structure parameter \(C_{T}^{2}\) and Eqs. 1–3, to the turbulent field within wavelike layers will be discussed in Sect. 3, following a detailed description of the observed characteristics of quasi-periodical undulations.

2.3 Other Measurements

Air temperature, wind speed and direction, humidity, pressure measurements were provided by an automatic weather station (AWS) Milos 520 (Vaisala) that was located 500 m from the sodar with an acquisition rate of 1 point min\(^{-1}\) at heights of 1.4 and 3.6 m above the surface for temperature and wind, respectively. A net radiometer type CNR1 (Kipp & Zonen) was used for radiation measurements, and an ultrasonic anemometer USA-1 (Metek) was used for measurements of parameters characterizing small-scale turbulence. A mast equipped with the net radiometer at a height of 1.2 m and the sonic at a height of 3.5 m (Fig. 3b) was installed about 15 m from the sodar antennae. All data were quality controlled, and to remove outliers from the observations, a median absolute deviation filter (Barnett and Lewis 1984; Petenko et al. 2014a) was applied to the measured time series.

To determine \(C_{T}^{2}\) from air temperature and wind velocity sonic data (Kohsiek 1982), the line between \(\mathbf{r}_{1}\) and \(\mathbf{r}_{2}\) in Eq. 1 is chosen in the direction of the mean horizontal flow. With the assumption of validity of Taylor’s frozen turbulence hypothesis, \(r= V~\delta t\), where V is the mean wind speed, \(\delta t\) is the timestep between samples. Sonic data were used to calibrate the sodar measurements and to convert the sodar return signal intensity I(z) to values of \(C_{T}^{2}~(z)\). The calibration procedure was the same as described in Petenko et al. (2014b): based on the validity of the Obukhov–Wyngaard “\(z^{-4/3}\)” height dependence of \(C_{T}^{2}\) for convective conditions, we calculated the corresponding values for each range gate of the sodar profile from the sonic \(C_{T}^{2}\) measured at 3.5 m. The converting coefficients \(C = C_{T}^{2} (z)/(I z^{2}\hbox {exp}(4{az}))\) at several levels were then averaged providing the value \(C = (2.67 \pm 0.58)\times ~10^{-9} \hbox {K m}^{-2/3}~ \Delta ^{-2 }\) to give \(C_{T}^{2}\) values expressed in units of \(\hbox {K m}^{-2/3}\), where \(\Delta \) is the analog-to-digital converter count. In addition, temperature and wind-velocity profiles from a 45-m tower located at a distance of approximately 1 km were available; see a description of the tower equipment in Genthon et al. (2013).

2.4 Meteorological Conditions

The histograms shown in Figs. 4 and 5 give an indication of the statistical distributions of the relevant meteorological variables measured by the AWS and the radiometers near the surface. The temperature varies in the range \(-60\) to \(-25\,^{\circ }\hbox {C}\) (Fig. 4a), with the average daily temperature variation (the difference between maximum and minimum values) of about \(15\,^{\circ }\hbox {C}\). The wind regime (observed at 3.6 m) was characterized mainly by low (<4 \(\hbox {m s}^{-1})\) and moderate (4–6 m s\(^{-1})\) wind speeds and wind directions from the south and south-west sectors as shown by the frequency distribution wind rose in Fig. 4b. Low wind speeds were observed >80 % of the time, moderate wind speeds were observed for \(\approx \)15 % of the time, with wind speeds >7 m s\(^{-1}\) usually occurring during synoptic perturbations due to intrusions of maritime airmasses from coastal zones accompanied by cloudiness and strong turbulence extending up to a few hundred metres. The values of the downwelling longwave radiative flux \({LW}{\downarrow }\) (Fig.  5a) and shortwave radiation \({SW}{\downarrow }\) (Fig.  5b) were used as an objective criterion for the presence of clouds or mist.

Fig. 4
figure 4

Statistics of the meteorological variables for selected days with fair weather conditions in the period 20 January to 25 February 2014. Histograms of: a temperature, c pressure, d relative humidity. b wind rose diagram showing the joint probability distribution of wind speed and wind direction (colour intensity is proportional to the value of the joint probability density function)

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Fig. 5
figure 5

Histograms of downward (red) and upward (blue) radiation components for selected days with fair weather conditions in the period 20 January to 25 February 2014: a shortwave, b longwave

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3 Results

3.1 Diurnal Behaviour of the Meteorological and Turbulent Parameters

First, we consider the diurnal variations of the relevant mean and turbulent parameters measured by the sonic thermo-anemometer at a height of 3.5 m. Parameters relevant to the study of thermal and mechanical turbulence are considered: (1) temperature T (Fig. 6a), sensible heat flux \(H_{0}\) (Fig. 6c), and temperature structure parameter \(C_{T}^{2}\) (Fig. 6e); (2) wind speed V (Fig. 6b), friction velocity \(u_*\) (Fig. 6d), and turbulent kinetic energy TKE (Fig. 6f). Definitions of these turbulent parameters can be found in Tatarskii (1971). All plots show the 60-min average measurements presenting the available dataset for 15 days with fair weather conditions. Almost all of these parameters show a typical diurnal behaviour that is consistent with the turbulent ABL structure shown by the sodar (Fig. 7a) and does not differ from the results of previous studies at Dome C.

The “daytime” behaviour of \(C_{T}^{2}\) near the surface shows a remarkable repeatability with relative variations of 13–25 %, with two local minima around 0800 and 1600 LST (local standard time = UTC + 8) similar to that observed at mid latitudes (e.g., Beyrich et al. 2005; Wood et al. 2013). The diurnal maximum observed around 1300 LST is markedly lower than nocturnal \(C_{T}^{2}\) values. The other turbulence parameters indicate a much larger variability: for \(u_* -27{-}42\,\%\), \(H_{0} - 28\)–51 %, \(\textit{TKE} - 44\)–56%. We emphasize that the data presented are for fair weather conditions.

Fig. 6
figure 6

Diurnal behaviour of selected parameters obtained using a sonic thermometer-anemometer at a height of 3.5 m over 15 selected days in the months of January and February 2014. a temperature, b wind speed, c sensible heat flux \(H_{0}\), d friction velocity \(u_*\), e temperature structure parameter \(C_{T}^{2}\), f turbulent kinetic energy TKE. Circles are the 60-min averaged values for individual days. Solid lines show the values averaged over the all days; red and green lines show arithmetic and median averages, respectively

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3.2 24-h Behaviour of the ABL Spatial and Temporal Structure

Here we consider the diurnal behaviour of the spatial and temporal distribution of turbulence together with variations of selected atmospheric parameters. As a representative example, we chose the observations made on 3 February 2014. The sodar echogram in Fig. 7a shows the 24-h behaviour of a time-height cross-section of the turbulence intensity in the ABL during this day. During the “nighttime”, the turbulent ABL is shallow with height <10 m, with \(C_{T}^{2}\) values within it reaching 5 \(\times 10^{-2}\) \(\hbox {K}^{2}\hbox { m}^{-2/3}\). After 0700 LST, the turbulent layer deepens reaching heights of 150–200 m and capping the convective boundary layer growing from the surface. After 1600 LST, the elevated turbulent layer disappears. After 2100 LST, the surface-based turbulent layer returns to a shallow depth and the thermal turbulence within it intensifies having values of \(C_{T}^{2}\) higher than those during the “daytime”. A more detailed analysis of the echograms shown in Fig. 7 is made in Sect. 3.2.

Fig. 7
figure 7

a Typical sodar echogram showing the diurnal variation of the ABL structure in summer (3 February 2014). The grey-scale intensity is proportional to the thermal turbulence intensity. Double arrows show the time location of the selected close-ups. b 4-h close-up of the echogram for the same day (0700-1100 LST). c, d Two 20-min close-ups of the echogram with undulation pattern during morning development of the convective ABL

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Temperature, wind speed, shortwave and longwave downward radiation components can be assumed as relevant variables characterizing the state of the ABL (Argentini et al. 2014; Pietroni et al. 2012, 2014). The time series of these parameters for 3 February 2014 are shown in Fig. 8. The 24-h variation of temperature and wind speed on 3 February 2014 shown in Fig. 8a, b are rather close to the mean diurnal behaviour of these parameters given in Fig. 6a, b, confirming that this example is quite representative.

Fig. 8
figure 8

24-h behaviour of a temperature at a height of 1.4 m, b wind speed and direction at a height of 3.6 m, c relative humidity and pressure, d shortwave radiation components, and e longwave radiation components on 3 February 2014

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In Fig. 9, the 24-h behaviour of the vertical structure of temperature and wind speed measured at six levels of the 45-m meteorological tower on 3 February 2014 is shown. The temperature field shows clear alternation between the strong surface-based inversion at nighttime and the weak convective layer capped with an elevated inversion during the daytime. The wind field is characterized by strong wind shear at night that disappears at noon.

Fig. 9
figure 9

24-h behaviour of the vertical structure of a temperature and b wind speed measured at the 45-m tower on 3 February 2014

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Figure 10 shows the time series of \(C_{T}^{2}\) measured by the sonic at 3.5 m and by the sodar at 7 and 14 m. Both the sonic-measured \(C_{T}^{2}\) (Fig. 6e) and that derived from the sodar return power (Fig. 10) show that the “nocturnal” values within the surface-based turbulent layer are in many cases larger the “diurnal” ones. This means that small-scale temperature fluctuations under stable stratification are significant and can be much larger than those observed in convection. This behaviour was not observed in previous experiments, due to the lack of reliable sodar measurements below 30 m at this site. Although \(C_{T}^{2}\) itself is not considered in any ABL model, this characteristic, and especially its variation with height, could be used to estimate more conventional parameters such as the turbulent fluxes or the mixing-layer height. Such attempts were made by Coulter and Wesely (1980), Coulter (1990), Wood et al. (2013).

Fig. 10
figure 10

24-h behaviour of \(C_{T}^{2}\) observed by the sonic at a height of 3.5 m (black) and by the sodar at two heights 7 m (red) and 14 m (blue) on 3 February 2014

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Fig. 11
figure 11

Profiles of temperature (top panel) and wind speed (bottom panel) averaged over 30 min for 3 February 2014, 0800-1000 LST

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Fig. 12
figure 12

Schematic diagram of characteristics of the wavelike pattern with indication of the symbols used in Fig. 13 and in Table 1. P quasi-period of the wave, \(H_{t}\) height of the top of the wavy layer, \(H_{b}\) height of the bottom of the wavy layer, dH vertical extension of the whole layer of wave activity, \(\Delta h\) the vertical thickness of the high \(C_{T}^{2}\) layer measured at the wave crests, \(\Delta t\) time duration of the turbulent contour of the wavelike pattern

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3.3 Wavy Structures During the Morning Evolution of the ABL

Fifteen days with fair weather conditions (from 20 January to 25 February 2014) were selected for analysis. The diurnal evolution of the turbulence structure in the ABL on these days was similar to that presented in Fig. 7a. We focus on the specific transition period in the morning hours, when convection begins to develop and the turbulent layer (normally associated with a layer of temperature inversion) begins to rise and ascend, capping the layer of convection. The evolution of the profiles of temperature and wind speed in this period are shown in Fig. 11, where the rising inversion layer and the substantial wind shear are seen. Sodar echograms always showed the presence of undulations within an elevated turbulent layer. We refer to the “wavy layer” when we consider the whole layer of wave activity visible on the echograms in Fig. 7 and extending for heights of several tens of metres. The wavy layer elevates as a whole and contains a wavelike pattern, or braids that have a noticeable turbulent contour as in Fig. 7c, d. As shown by the echograms in Fig. 7c, d, the braid-type quasiperiodic structures occupy (and, in fact, form) the whole turbulence layer. They behave as waves induced by the KH instability within an elevated inversion layer. A schematic diagram of characteristics of the wavelike pattern in the braid region (its vertical dimensions and time durations) is shown in Fig. 12. Periods of these structures range mainly between 40 and 50 s.

The vertical dimension of these turbulent contours increases during the morning hours as the entire elevated turbulent layer rises and spreads As an estimate of the vertical extension of the whole wavy layer we take the difference \({ \mathrm{d}H} = H_{t} -H_{b}\) between two characteristic heights: (1) the height \(H_{t}\) of the upper boundary of the braid contour at the crests of the wavy layer (see Fig. 12), and (2) the height \(H_{b}\) of the lowest boundary of the braid contour at the hollows. In Fig. 13, the time behaviour of \(H_{t}\), \(H_{b}\), and dH is shown for the period of the morning evolution 0700-1200 LST averaged over 15 days with fair weather conditions. While the top and bottom heights continue to grow, the thickness dH of the layer seems to reach an equilibrium value of \(45 \pm 10\) m after 0930 LST.

Fig. 13
figure 13

Time behaviour of \(H_{t}\) (red), \(H_{b}\) (blue) and dH (green) for the morning period 0700-1200 LST averaged over selected fifteen days with fair weather. Vertical bars show the standard deviation of the presented characteristics

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Note that spatial dimensions of the braid contours are not small. Rough visual estimates of the vertical thickness of the contours at the wave crests, \(\Delta h\), in Fig. 7c, d are about 15-20 m. The horizontal thickness \(\Delta x = V \Delta t\) estimated using Taylor’s hypothesis is about 50-100 m. These values that might be considered as the outer scale of turbulence \(L_{0}\) provide a sufficiently large inertial range of turbulence within the contours. Note that the scale of turbulent inhomogeneities which provides backscattering is equal to one half of an interrogating sound wavelength, which for the sodar used here is about 0.03 m. Such a scale, apparently, lays within the inertial range. Thus, we have sufficient reasons to apply Eq. 1–3 to the case of the wave activity.

A spectral analysis of sodar \(C_{T}^{2}\) time series was used to accurately estimate the periods of the wavy structures shown in Fig. 7c, d. Before the spectral processing, some filtering procedures were applied to reduce the noise impact and to extract the regular part of the undulation processes. A bi-dimensional median filter, median absolute deviation filter and a low-pass filter were used. The power spectra were calculated using Welch’s averaged modified periodogram method of spectral estimation (see Welch 1967). All the spectra were then normalized by the variances of the time series. Spectra of \(C_{T}^{2}\) at three heights are shown as a function of period in Fig. 14 separately for both the considered close-ups. The heights were chosen to be within the layer with wavy structures: 19, 24 and 32 m for the first close-up, and 43, 49, and 54 m for the second one. The spectra show peaks at the similar periods to those estimated earlier visually from the echograms and time series. In both the cases, the spectra for the selected heights are quite similar, and the peak positions are close to each other. In the first close-up, the main peaks are located at periods of about 42–43 s; in the second close-up, the positions of the peaks are shifted to \(\approx \)50 s. These periods are less than the buoyancy periods corresponding to the Brunt-Väisälä frequency, which are estimated to be about 75 and 150 s for the considered time intervals. The histogram in Fig. 15 shows the distribution of periods observed in all cases of wave activity.

Fig. 14
figure 14

Power spectra of \(C_{T}^{2}\) at three heights for 20-min time series measured on 3 February 2014 a 0805–0825 LST and b 0935–0955 LST. All spectra are normalized by the variance of the respective time series and are presented as a function of period

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Fig. 15
figure 15

Histogram of wave periods for all observation times

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Table 1 Averaged characteristics of the wave processes during the morning development of the convective ABL for the period 21 January–25 February 2014
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The horizontal scale that can be roughly attributed to the wavelength has been calculated as \(\lambda ~=~\textit{VP}\) assuming the validity of Taylor’s frozen hypothesis, where P is the wave period, and V is the wind speed. For \(P \approx 40\)–50 s and V taken from the 45-m tower we estimate \(\lambda \) as approximately 250–350 m. Of course, Taylor’s hypothesis is applicable to small-scale turbulence only, not to waves. In fact, we apply it not to waves, but to the small-scale turbulent fluctuations, which are modulated by standing (relative to the mean flow) KH billows. The assumption that the KH billows are stationary is based also on the experimental results by Gossard et al. (1970) showing that the phase speed of KH waves is close to the velocity of the ambient wind at the height of maximum shear. We are aware that this approach is quite crude, but in this experiment we cannot suggest any more elaborate and reasonable way to estimate a horizontal scale of the wavy structures from single-point measurements. This is a tentative step to obtain information on the spatial characteristics of the wavy pattern.

Two other spatial characteristics which can be determined from the available data are the thicknesses of the contour at the wave crests in the horizontal and vertical directions. A rough estimate can be made by looking at the sodar echograms. The vertical thickness of an inclined fine layers is estimated as 10–15 m; the vertical thickness of the contours at the wave crests is estimated as 15–20 m. Table 1 summarizes the values obtained for geometric parameters of the wave pattern.

Several theoretical works (e.g., Drazin 1958; Davis and Peltier 1976) provide estimates of the relation \(\lambda =A\mathrm{d}H\) between the wavelength \(\lambda \) and the thickness of the layer dH for waves that are due to shear-flow instability in the gravity field. The numerical coefficient A varies in different theoretical works between 4 and 7.6, being dependent on the assumed model of vertical variation of velocity and density, as well as of the Richardson number. In some experimental studies (e.g., Scorer 1969; Eymard and Weill 1979; Lyulyukin et al. 2015), values of A from 3 to 20 were obtained. From the measurements of Table 1, the ratio \(\lambda /{\mathrm{d}H}\) is roughly estimated to be between 5 and 12. As the experimental estimate is close to the theoretical one, we believe that the observed wavy structures are produced by KH disturbances.

4 Summary

We used measurements from a field experiment held during the summer months (January–February) of 2014 at the French–Italian station of Concordia at Dome C in Antarctica to investigate processes occurring in the polar ABL. A distinct and representative diurnal cycle of the ABL turbulence structure and of the relevant meteorological and micrometeorological characteristics was documented using remote-sensing and in situ measurements. The diurnal behaviour shows the alternation of a surface-based temperature inversion and convection capped by the inversion layer. The results are consistent with those from previous studies (e.g., Argentini et al. 2005; Pietroni et al. 2012). The “nighttime” (from 1800 to 0700 LST) \(C_{T}^{2}\) values near the surface are markedly higher than those observed during the “daytime” (from 0800 to 1800 LST). The “daytime” behaviour of \(C_{T}^{2}\) is characterized by a remarkable periodicity having a small variation in comparison with other turbulence parameters.

We mainly focused on the behaviour of the ABL during the development of convection under the capping inversion layer during the morning hours. The use of an advanced high-resolution sodar provided evidence of new features in the spatial and temporal structure of turbulence, which shows clearly the presence of wavelike processes accompanying the development of convection. Within the elevated inversion layer that ascends when convection develops near the surface, high-resolution sodar echograms present regular fine-scale layers with a braid pattern that resembles waves due to the KH instability. Earlier, similar wavelike patterns were observed in Antarctica (Neff et al. 2008; Petenko et al. 2013; Argentini et al. 2014; Kouznetsov and Lyulyukin 2014), but were not associated with the morning development of convection.

The periods of the observed wavy structures range mainly between 40 and 50 s, while rough estimates of the horizontal scales made using Taylor’s hypothesis provides wavelengths \(\lambda \) of \(\approx \)250–350 m. The horizontal width of individual wavy fine-scale layers ranges between 20 and 60 m, and their vertical thickness at the wave crests is 15–20 m. The depth dH of the ascending turbulent layer containing waves varies from 20 to 60 m, with \(\lambda /{\mathrm{d}H}\) between 5 and 12 (close to theoretical expectations for KH waves).

The results call for the development of advanced theoretical approaches that would enable the interaction of convective and wave processes occurring simultaneously to be taken into account.