Kelvin–Helmholtz Billows in the Rising Turbulent Layer During Morning Evolution of the ABL at Dome C, Antarctica

Kelvin–Helmholtz billows (KHBs) within a rising turbulent layer during the transition period from stable to unstable stratification occurring in the morning hours in summertime at the interior of Antarctica (Dome C, Concordia station) are examined in this study. The wave pattern captured by high-resolution sodar echograms from November 2014–February 2015 exhibits regular braid-like structures, associated with Kelvin–Helmholtz shear instabilities. This phenomenon is observed in more than 70% of days in the selected period. Two main regimes of the morning evolution with KHBs are identified roughly, distinguished by the presence or absence of turbulence in the preceding night-time. The weather and turbulent conditions favouring the occurrence of these regimes are analyzed. Also, two distinct patterns of KHBs are identified: (i) quasi-periodical (with periods ≈ 8–15 min) trains containing 5–10 braids, (ii) about continuous series lasting 20–90 min containing 20–80 braids. A composite shape of KHBs is determined. The periodicity of these waves is estimated to be between 20 and 70 s, and their wavelength is estimated roughly to be 100–400 m. The vertical thickness of individual braids at the wave crests ranges between 5 and 25 m. The total depth of a rising turbulent layer containing these waves varies between 15 and 120 m, and the ratio of the wavelength to the depth of the wave layer varies from 3 to 12 with a mean value ≈ 8.2. The morphology of the turbulence structure in the ABL is studied as a function of both temperature and wind field characteristics retrieved from an instrumented 45-m tower and an ultrasonic anemometer-thermometer at 3.5 m. The observational results highlight the necessity of considering the interaction between convective and wave processes when occurring simultaneously.


Introduction
In the present paper we study two phenomena which were rarely considered together, apart from a few recent papers (e.g., Finnigan 2000;Lapworth 2015;Petenko et al. 2016;Kallistratova et al. 2019). Namely, these are (1) the occurrence of Kelvin-Helmholtz billows (KHBs) during the morning transition from stable to unstable stratification in the lower shear atmospheric layer, and (2) the morning rise of the inversion turbulent layer.
The atmospheric boundary layer (ABL) over land often has a pronounced diurnal cycle induced by solar heating (e.g., Stull 1988). The diurnal cycle usually consists of convective and neutral phases in daytime and a period of stable stratification at night. In the early morning hours (after the surface sensible heat flux becomes upward), a shallow convective mixed layer grows into the surface-based inversion layer which begins to rise as a whole, expanding in the vertical direction. This morning evolution from the nocturnal stable boundary layer (SBL) to the daytime convective boundary layer (CBL) is of interest both for basic understanding and for correct initializing prognostic models. A comprehensive set of morning transition observations was described by Angevine et al. (2001Angevine et al. ( , 2020, and they concluded that wind shear is important in controlling the morning evolution processes. Moreover, they emphasized that despite general outlines of the morning behaviour of the ABL being well known (e.g., Hooper and Eloranta 1986;Stull 1988;Lapworth 2006 among others), some features and details of this process are not fully described (Petenko 1996;Holtslag et al. 2013). Some prognostic and diagnostics models were developed and verified experimentally (e.g., Van Ulden and Holtslag 1985;Gryning and Batchvarova 1990;García et al. 2002;Argentini et al. 2005;Lapworth 2006;Steeneveld et al. 2007;Sorbjan 2007;Holtslag et al. 2013;Casasanta et al. 2014;Liu et al. 2018, among others). Beare (2008) modelled the full evolution from the stable boundary layer to the convective one and showed that in the early stages of the morning transition the boundary layer is a combination of a shallow mixed layer capped by a significant shear driven stable boundary layer (the so-called mixed CBL-SBL state). Though much study has already been carried out on the morning behaviour of the ABL, a commonly accepted theory does not exist, and there is still no full agreement between models and observations. The presence of KHBs occurring in a rising inversion layer above the CBL has not been examined comprehensively, to our knowledge. Theoretical studies were mainly devoted to convectively generated gravity waves at heights of several kilometres with wavelengths of several tens of kilometres (e.g., Beres et al. 2004). Understanding the transition between stably stratified phase by night and convective phase by day remains an ongoing area of research (Gibert et al. 2011). It is important for improving the numerical weather forecast, climate modelling, and air pollution predictions. Unfortunately, large-eddy simulation (LES) studies of the morning transition (e.g., Beare 2008;Carneiro et al. 2021) revealed no signs of the presence of wave activity. Another recent comprehensive LES study by Jiang (2021) showed that KHBs in an elevated shear layer (ESL) influence the underlying atmospheric layer enhancing turbulence due to the secondary instability, increasing its depth and weakening stratification; a possible impact appears to be dependent on both ESL and lower ABL characteristics. However, the phase of the morning transition was out of consideration. So, further development of modelling algorithms is needed. LES models with adaptive mesh refinement (Van Hooft et al. 2018 seems to be well suited to provide particularly high resolution to capture the wave/turbulence interactions in the ESL. Wavelike motions of varying forms, amplitudes, and periods are ubiquitous and often observed in stably stratified flows with shear being one of the main mechanisms responsible for production of turbulent mixing in such conditions both in the atmosphere and in the underwater environment (Woods 1969;Hooke et al. 1973;Gossard and Hooke 1975;Corcos and Sherman 1976;Einaudi et al. 1978;De Silva et al. 1996;Geyer et al. 2010 among others). When talking about atmospheric waves, we should distinguish two main types of wavy undulations: (i) Kelvin-Helmholtz (KH) waves (also referred to as vorticity waves or internal shear-gravity waves) and (ii) propagating buoyancy waves (Chimonas and Fua 1984;Einaudi et al. 1978;Sun et al. 2015;Jiang 2021) having different generation mechanisms and phase speeds. In this study, we are interested in the former developing in a shear layer and propagating along the mean wind direction approximately at the mean wind speed in the shear layer. There are several definitions of KHBs. Smyth and Carpenter (2019) applied this term to finite-amplitude manifestations of the instability having a form of trains of corotating vortices resembling either cat's eyes or braids. It is possible to observe them visually at cloud layers or in the ocean surface near coasts. However, in the clear-air atmosphere and below the water surface, they are practically invisible and cannot be observed directly and detected with only in situ measurements. Only remote-sensing tools such as radars, lidars and sodars can provide effective means for studying these phenomena in the atmosphere, and sonars in underwater environments. They reveal clearly the internal wave structure in echosounder images. Kelvin-Helmholtz instability is usually recognized by its alternating braid-core structure in echograms (Gossard et al. 1970;Emmanuel et al. 1972;Hooke et al. 1973;Gossard and Hooke 1975;Kallistratova and Petenko 1993;Geyer et al. 2010;Nappo 2012).
The study of the stability of shear flows was initiated by Helmholtz (1868), who determined the stability criterion for an unbounded atmosphere with step discontinuities in wind speed and density. Kelvin (1871) then showed that a braided, or cat's-eye, pattern arises as a result of nonlinear disturbance. Continuous wind speed and density profiles (Drazin 1958;Hazel 1972) and more realistic boundary conditions (Jones 1968) were analysed much later. The properties of stratified KHBs have been widely studied in the laboratory (e.g., Thorpe 1987;Fernando 1991;Patterson et al. 2006) and numerically (e.g., Peltier and Caulfield 2003;Mashayek and Peltier 2012). A summary of these studies in the field of fluid mechanics has been presented by some authors (e.g., Readings et al. 1973;Thorpe 1987;Smyth and Carpenter 2019). Certain features of KH waves are accepted as common in all studies: a phase speed equal to the speed of the background wind at the centre of the shear layer, a horizontal wavelength proportional to the depth of the shear layer, and an amplitude which decayed rapidly away from the region of shear.
Experimental studies in the real atmosphere and in the underwater environment are much more scarce. There is a principal difference between laboratory and outdoor conditions due to the great difference in Reynolds numbers Re = Ud/ν (here, ν is the fluid's kinematic viscosity, and U and d are the characteristic velocity and length scales of the shear layer, respectively). For flows with sufficiently high Re, KHBs trigger turbulent mixing due to secondary instability (Corcos and Sherman 1976;Smyth 2003). Davis and Peltier (1976) applied linear stability analysis to consider a realistic atmospheric boundary layer as a compressible, inviscid, stratified parallel shear flow, bounded below by a rigid wall. Comparison of their theoretical predictions concerning disturbances with sodar observations of KH waves by Hooke et al. (1973) showed a good agreement.
It is interesting to note that the appearance of KHBs captured by remote sensing techniques is identical both in the atmosphere and in the underwater environment. Figure 1 shows two examples of KHB trains recorded by sodar (in the atmosphere) and by sonar (in an estuary). The development of high-resolution underwater acoustic echosounders allowed to capture KH waves below the water surface with high temporal and vertical resolution both at moderate depths of 100-200 m, and at several metres. High-resolution fast-response sensors Fig. 1 Examples of patterns of KHBs captured by acoustic echosounders: a in the atmosphere in Antarctica (current study), and b in the underwater environment in the Connecticut River estuary (adapted after Geyer et al. 2010). Braids show the classic S-shape with strong signal within them. Details of individual billows: braids, cores, upper eyelids, lower eyelids are indicated for measurements of salinity and temperature installed on moving ships allowed to determine fine-scale mean and turbulent characteristics of flow within braided wave structures (Geyer and Smith 1987;Moum et al. 2003;van Haren and Gostiaux 2010;Geyer et al. 2010Geyer et al. , 2017Chang et al. 2016;Tu et al. 2020). Fukao et al. (2011) reviewed atmospheric studies of KHBs using radars, lidars, and sodars in the height range 0.2-20 km. A comprehensive review of different types of wave processes in the atmosphere was made by Sun et al. (2015). Results of a comprehensive study of statistics of the occurrence of KHBs and their features such as altitudes of location, amplitudes, durations, periods, and wavelengths from multi-year sodar observations in the Moscow region were presented by Lyulyukin et al. (2015).
There is some confusion and uncertainty concerning existing terminology in KH wave studies. Some authors (e.g., Davis and Peltier 1976;Einaudi et al. 1978) noted that the term "Kelvin-Helmholtz instability (waves)" originally was applied for discontinuous (both in density and velocity) flows, however, then it began to be used widely for all fluid processes resembling in its pattern the "braid" structures occurring in flows with continuous density and velocity profiles. Sun et al. (2015) considered different terminology classifying such kind of wave patterns as vorticity-generated or simply vorticity waves. Lyulyukin et al. (2015) suggested a more appropriate term for such phenomena, calling them "gravity-shear waves". We agree with the authors and believe that this term is more adequate and less confusing. Nevertheless, to avoid some possible misunderstanding, we follow the widely accepted terminology and use the term "Kelvin-Helmholtz waves" in describing alike wavy processes occurring in stratified shear flows.
It is known that echo-signals of sodars and radars is produced by small-scale turbulent fluctuations of the sound or radio wave refractive indexes. So, the fact that we see in echosounder images braid-like structure means that within zones of braids, turbulent fluctuations with scales of half of the wavelength of the sounding signal occur (in the case of sodars, they are temperature and humidity fluctuations). Observations by Geyer et al. (2010) using a combination of acoustic imaging and in situ measurements supported the conjecture that secondary instabilities should lead to mixing within the braids at high Re (e.g., Corcos and Sherman 1976;Mashayek and Peltier 2012).
A knowledge of the wind field within KHBs is important for estimation of energy transfer. Blumen et al. (2001) and Newsom and Banta (2003) observed KHB structures in the wind field using a scanning high-resolution Doppler lidar in combination with an equipped 60-m tower. In this study, the spatial (both vertical and horizontal) structure of the instability and its evolution were captured on lidar images that allowed direct determination of the wavelength (distance between braids) of KH waves. Their estimates provided the values of 200-300 m and inverse aspect ratios of 5-8. Browning et al. (1971Browning et al. ( , 1973 studied air motion within KHBs from the simultaneous Doppler radar and aircraft measurements. Later, Lyulyukin et al. (2015) and Lyulyukin (2018) analyzed wind field features within KHBs using sodar measurements. The composite analysis applied for wind field data showed circular flow patterns for different forms of wind speed profiles. Luce et al. (2018) analyzed time-height cross sections of the wind field determined by radar at heights of 4-9 km.
From the first observations in the late 1960s, waves, both buoyancy (Browning 1971;Gossard et al. 1970;Weill et al. 1987;Kouznetsov 2009;Gibert et al. 2011;Petenko et al. 2011 among others) and shear-induced (KH) (Emmanuel et al. 1972;Eymard and Weill 1979;Neff et al. 2008;Kouznetsov and Lyulyukin 2014 among others), mainly with periods > 200 s and amplitudes > 50 m were observed occasionally. However, recent studies showed the common occurrence of KH waves with smaller scales in different atmospheric contexts. So, the study by Petenko et al. (2020) with a sodar in a coastal zone showed a high occurrence of KHBs within the turbulent layers induced by nocturnal low-level jets (LLJs) in the height range between 100 and 600 m. Periods of these waves were between 70 and 120 s, and their braids had opposite slopes below and above the LLJ nose in correspondence with the sign of the wind speed shears which varied between 0.01 and 0.05 s -1 . They occurred in 20% of the total time through the year, more frequently in spring and autumn.
Highly frequent occurrence of the KH instability events in the lowest atmosphere over the interior of Antarctica was revealed recently thanks to 3-year observations conducted at Dome C using a high-resolution sodar Petenko et al. 2016Petenko et al. , 2019. Some new morphological features of turbulent flows in the lowest ABL (< 150 m) were captured. In the wintertime (from April to October), under very stable near-surface stratification, trains of KHBs with braid periods of 20-50 s and amplitudes of 10-50 m in the surface-based turbulent layer with shear  were observed in 40% of the time.
An amazing feature found recently is the clear and frequent presence of long-lived KHBs, which occur regularly within the rising turbulent layer above the layer of developing convection (Petenko et al. 2016;Kallistratova et al. 2019). Here we report the results of analyzing the characteristics of KH billows within a rising turbulent layer captured by sodar in the morning hours at Dome C in the austral summer season. The experimental site, instrumentation and the meteorological conditions are described in Sect. 2. In Sect. 3, the characteristics of the diurnal variation of the spatial and temporal structure of turbulence in the ABL for different regimes, and clear examples of wave structures are presented. Periods, wavelengths, and vertical extension of the KHBs are determined. The influence of weather conditions on their occurrence in the morning hours was analyzed in correspondence to the morphology of the wind and temperature fields. Conclusions are presented in Sect. 4.

Site, Instrumentation and Meteorological Conditions
The experiment was carried out at the French-Italian station of Concordia (75°06 S, 123°21 E, 3233 m above sea level) located at Dome Charlie (Dome C), East Antarctic plateau (Fig. 2a). The station is situated at a distance of 900 km inland from the nearest coast; the surface at this site has a slope of ≈ 0.1% in the absence of orography. The Sun culminates at 38°on 21 December, being permanently below the horizon from 6 May to 12 August. The measurements carried out from 1 November 2014 to 28 February 2015 were analyzed. The austral summer weather is dominated by long-lasting high-pressure systems favouring fair weather. The environmental conditions are ideal for observations because the influence of external sources of perturbations (both spatial and temporal) is very low due to the absence of orographic forcing and low occurrence of synoptic changes (stationary fair-weather conditions lasted for a few weeks). Thanks to the near ideal, similar to laboratory site conditions, the main features and principal intrinsic causes influencing the turbulent structure of the ABL can be determined.

High-Resolution Sodar System
We observe, record and describe atmospheric turbulence in terms of the intensity of temperature fluctuations quantified by the value of the temperature structure parameter C 2 T (in arbitrary units) measured with a sodar (Kallistratova 1962;Tatarskii 1971;Brown and Hall 1978;Emeis 2021). For more details, see the Appendix. An advanced version of a highresolution sodar Petenko et al. 2016Petenko et al. , 2019 developed by the Institute of Atmospheric Sciences and Climate of the National Research Council of Italy (ISAC-CNR) Fig. 2 a Schematic geographical map of Antarctica, with the position of Dome C (red star). b Sodar antennae was used to observe temperature turbulence in the height range of 5-200 m. The four vertically pointed sodar antennae (three transmitting and one receiving) were installed 400 m south-west of the main buildings of Concordia station (Fig. 2b). This position minimized the impact of the station, as the south-west wind prevailed. Acoustic pulses with duration of 10 ms were emitted every 2 s; this results in a corresponding vertical extent of the scattering volume of ≈ 1.7 m. A step of vertical profiles of instantaneous values of the echo intensity recorded was < 1 m. A data processing methodology was described by Argentini et al. (2012Argentini et al. ( , 2014. A noise-subtraction procedure was developed and applied to the echo-signal intensity profiles ) to increase the data quality.

Other Measurements
Measurements of air temperature, humidity, wind speed and direction were provided by an automatic weather station (AWS) Milos 520 (Vaisala, Helsinki, Finland) with an acquisition rate of 1 min at heights of 1.4 and 3.6 m above the surface for the temperature and wind speed, respectively. A mast equipped with an ultrasonic anemometer-thermometer (hereafter, "sonic") USA-1 (Metek, Elmshorn, Germany) at a height of 3.5 m and a net radiometer CNR1 (Kipp & Zonen, Delft, Netherlands) was installed ≈ 15 m from the sodar antennae. The sonic was used to determine turbulence characteristics in the surface layer. The value of the downward longwave radiation > 120 W m -2 was used as an objective criterion to detect the presence of clouds or mist. Temperature and wind-speed profiles from a 45-m tower located at a distance of ≈ 1 km were available (see Genthon et al. 2013;Vignon et al. 2017 for a description of the tower equipment), yielding measurements at six heights of about 3, 10, 17, 25, 32 and 41 m. Radiosoundings were performed daily at 1930 LST (local standard time = UTC + 8) with a radiosonde RS92-GSL (Vaisala, Finland) providing temperature, humidity and wind-speed profiles. The same instrumentation set was used as that described by Petenko et al. (2019).
Instability of the flow is often assessed basing on the value of the speed shear S = dV /dz, where V is the mean wind speed and z is the height, and the buoyancy (or, Brunt-Väisälä) frequency N 2 = (g/T ) (dϑ/dz). In fluid mechanics, the Richardson number Ri = N 2 /S 2 is used as a valid measure of instability. In practice, the Brunt-Väisälä frequency N characterizing temperature stratification is calculated as , with the period corresponding to the Brunt-Väisälä frequency termed the "buoyancy period", and calculated as T b = 2π N −1 . To characterize the wind speed shear, we calculate the value To assess the instability quantitatively, in practice, we take the bulk Richardson number Ri B , and follow Mahrt and Vickers (2006) by computing the Ri B between the levels z 1 and z 2 as: where g is the acceleration due to gravity, z is the height above the surface, V is the wind speed, θ is the potential temperature in K calculated as θ (z) = T [1000/p(z)] 0.286 , where T is the absolute temperature in K, p is the air pressure in hPa, and θ 0 is the average value of θ between two levels. As V and T can vary nonlinearly and non-monotonously with height, there is no unique Richardson number applicable to the entire layer. Some authors believe the Ri at the inflection point height provides a more adequate value (e.g., Nappo 2012). As was proven by Miles (1961), and Miles and Howard (1964), a necessary (although not always sufficient) condition for a flow to become unstable is that the value of Ri must be < Ri cr = 0.25, which means that the shear is strong enough to overcome buoyancy and to break down the stability. Further experience has shown that this criterion Ri < 0.25 is generally a useful indicator of shear instability under different conditions (Nappo 2012;Smyth and Carpenter 2019). An important factor that influences measurements of Ri is the spatial and temporal scales at which the shear and stratification are measured. In general, not enough both spatial and temporal resolution leads to overestimation of Ri influencing its statistics; observations of Ri < 0.25 become more frequent with finer resolution and less frequent with coarser resolution (e.g., Smyth and Moum 2012). Hence, we believe that shear instabilities may occur even though the measured estimate of bulk Ri B > 0.25. As Einaudi et al. (1978) and Jiang et al. (2010) pointed out, the local values of the Richardson number are influenced by the gravity wave flow field, as expected, since gravity waves both generate shear and tilt the surfaces of constant density (e.g., Phillips 1966). The local Richardson number can become less than 0.25, so secondary instabilities may develop. Later, Zilitinkevich (2002), Galperin et al. (2007), and Grachev et al. (2013) among others considered the existence of turbulence at Ri > Ri cr = 0.25 and discussed the problem of the determination of the critical Ri cr .

Characteristics of Meteorological Conditions
Analysis of relevant meteorological variables measured during the considered period showed that they are in reasonable agreement with the results of previous studies (e.g., Argentini et al. 2001;Genthon et al. 2013;Pietroni et al. , 2014Petenko et al. 2016), and no significant anomalies were revealed; such weather conditions are typical for this site in austral summer. Examples of time series of meteorological parameters from the AWS and radiometric measurements are shown in Fig. 3, turbulent parameters from sonic in Fig. 4. The weather was characterized by predominance of clear-sky conditions. Synoptic perturbations accompanied with the increase in the temperature, wind speed and cloud cover occurred with periodicity 7-10 days and lasted 2-3 days. The temperatures during the 4 months varied between − 55 and − 20°C with a clear daily cycle with an amplitude of ≈ 10 K; a regular daily cycle is a specific feature of this site (King et al. 2006). During passages of warming events, temperature reached − 20°C. Wind speed at 3 m also varies with a diurnal cycle from 2-4 m s -1 at night and to 4-6 m s -1 by day.
These graphs were used to identify an association between the morning evolution events and the behaviour of meteorological and turbulent characteristics. (i) In the presence of cloudiness, morning evolution processes were mainly absent or not evident. Nevertheless, a few of such events have occurred also under cloudy conditions. The increased values of downward longwave radiation LW ↓ were used as an indicator of the presence of clouds (Fig. 3e). (ii) Strong winds near the surface at night often preclude the morning evolution, although sometimes it occurs despite strong nocturnal wind. (iii) The values of turbulent parameters at night, apparently, do not have regular effect on the occurrence of morning evolution. Although, the higher values of TKE accompany them. Corresponding colour (rose, light green and grey) shaded areas (vertically extended) highlight (indicate) the presence of mentioned events to facilitate searching the common features in the behaviour of different considered parameters in correspondence to the phenomenon of interest (i) to determine atmospheric conditions favouring the occurrence of the morning evolution accompanied with KHBs, (ii) to determine their characteristics and features within a rising turbulent layer in the morning hours.

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 at a height of 3.5 m. Parameters relevant to the study of thermal and mechanical turbulence are considered: (1) temperature T (Fig. 5a), sensible heat flux H 0 (Fig. 5c), and temperature structure parameter C 2 T (Fig. 5e); (2) wind speed V (Fig. 5b), friction velocity u* (Fig. 5d), and the stability parameter z/L (Fig. 5f) (L is the Obukhov length) (Fig. 5f). Conventional definitions of these turbulent parameters (e.g., Tatarskii 1971;Wyngard 2010;Foken 2017) are given in Appendix. All plots show the 60-min average measurements presenting the available dataset for the whole observational period. Almost all of these parameters show a typical diurnal behaviour that is consistent with the daily cycle of the turbulent ABL structure shown by the sodar and does not differ significantly from the results of previous studies at Dome C Petenko et al. 2016). The "daytime" behaviour of C 2 T (Fig. 5e) near the surface is quite similar to that found by Petenko et al. (2016), being a quite regular day by day with two local minima similar to those observed at mid latitudes (e.g., Beyrich et al. 2005;Wood et al. 2013). The other turbulence parameters (H 0 , u*, z/L) indicate a much larger variability day to day. The minima in C 2 T occur closely to the transitions from stable to unstable stratification (after 0700 LST) and vice versa (around 1600 LST) indicated by zero-crossing of the stability parameter z/L (L is the Obukhov length) (Fig. 5f). The diurnal maximum observed around 1100-1200 LST is markedly lower than "nocturnal" values. The increase of the intensity of small-scale turbulence at night, especially under lower temperatures, evidences indirectly the influence of the wave activity on mixing processes. We emphasize that the data shown represent all the observed weather conditions.
To provide a holistic view of the diurnal behaviour of some relevant parameters over the entire observational period, we use so called Hovmöller diagrams (Hovmöller 1949;Persson 2016), which sometimes are called "fingerprint plots" (e.g., Pirk et al. 2017). The visual examination of these plots seems to be a useful first stage when analyzing processes with pronounced daily cycle. This kind of visual analysis allowed us to get an idea of the connection between the analysed phenomenon and the diurnal behaviour of the relevant atmospheric characteristics.  Figure 6b shows time variations of the daily behaviour of the difference of wind speed between the 3rd and 1st levels of the 45-m tower. We considered this kind of plot also to analyse the behaviour of temperature and wind speed, along with their gradients, and turbulent parameters H 0 , u*, and z/L (not shown for brevity).
In addition to the colour representation of the values of considered parameters, we placed some symbols indicating the presence of some phenomena of interest under consideration. Analyzing these diagrams, we got an idea about long-term variations of the daily cycles of different meteorological and turbulent parameters and their correspondence to morning transition and KHB events through the entire observational periods. So, from Fig. 6b we can conclude that the morning evolution is absent often, when the wind shear is weak in the night.

Regimes of the Boundary-Layer Morning Evolution in Summer 2014-2015
The visual examination of the sodar echograms revealed the existence of several predominant types of the diurnal behaviour of the ABL. They can be classified into several groups, according to the pattern of the spatial (over height) and temporal distribution of turbulence during night and morning hours. Here, we present examples of two principally different regimes of morning evolution with growing KHBs: (i) first regime (Regime 1) is characterized by the presence of the intense turbulent layer with KHBs in the night and its following rise in the morning hours (Fig. 7a-c); (ii) the other regime (Regime 2) is characterized by the absence of significant turbulence in the night and the following appearance and developing of the rising turbulent layer with KHBs ( Fig. 7d-f). We should note that the presence of KH instability events at night occurred 52% of the days in the selected period, so the Regime 1 was prevailing. We should mention that some sub-regimes can be identified in each of these main regimes. But we leave such a detailed classification into more regimes and their characterisation for further study. Figure 8a and b, and i and j show the 24-h behaviour of the vertical structure of temperature and wind speed measured at six levels of the 45-m meteorological tower on 23 December 2014 (Regime 1) and 24 November 2014 (Regime 2), respectively. The temperature field shows clear alternation between the strong surface-based inversion at night-time and the weak convective layer near the surface during the daytime. The wind field is characterized by strong wind shear at night which disappears at noon. We put on the same graph both squared shear S 2 and 4N 2 ; the factor 4 before N 2 makes it easy to see when Ri varied around 0.25.
The temperature and wind fields in the lowest 45 m for Regimes 1 and 2 show significant difference in the night-time hours preceding the beginning of the morning evolution. In comparison with Regime 1, Regime 2 is characterized by stronger gradients of temperature and wind speed. At the beginning hours of morning transition, for Regime1, a zone with Ri < 0.25 is elevated between 10 and 30 m. As for Regime2, a zone of instability locates near the surface below 15 m in earlier morning.

Boundary-Layer Characteristics Favouring Different Regimes of Wavy Morning Evolution
One of the main aims of our study is to find, understand and describe the weather conditions and identify the main variables favouring the occurrence of the growing elevating turbulent layer during morning hours with KH waves developed within it during morning hours. We considered the behaviour of several relevant mean and turbulent characteristics that could be responsible for being the source of this phenomenon. In the previous section, we roughly described the two main regimes distinguished by the turbulence behaviour in the nocturnal boundary layer (Fig. 7). Here, the behaviour of some relevant meteorological and turbulent parameters that could influence this phenomenon is analysed. The statistics and behaviour of the mean meteorological variables, wind and temperature gradients, turbulent fluxes favouring each morning evolution regimes were determined. The obtained results, in the form of histograms associated with different regimes, are summarized below for the two main regimes defined above. Histograms of the selected variables measured late in the night (between 0300 and 0500 LST) preceding the beginning of the morning evolution are shown in in Fig. 9 (red and green coloured bars for Regimes 1 and 2, respectively). Histograms of the considered variables measured when morning evolution with KH waves was absent (blue coloured bars) are also shown in the same figures for comparison. When we classified the conditions in the ABL as the absence of the morning evolution with KH waves, we meant that there was no significant rising or growing of a turbulent layer. We attributed some rare cases, when KH waves were not detected within rising turbulent layers (usually, such layers are weak and short-living) to the same class of conditions. So, when we say "no morning evolution", we mean that no patterns corresponding to either Regime 1 or Regime 2 (as in Fig. 7) occur.
From analysis of the presented histograms we can conclude that the most important parameters associated with the occurrence of the morning evolution with KHBs are cloudiness, wind speed, wind speed shear, gradient of temperature, and TKE. More intense winds favour the occurrence of Regime 1, while Regime 2 occurs mainly under weaker winds. Higher values of wind speed shear favour the occurrence of both morning evolution regimes. Regime 1 occurs when the temperature gradients are weak, while stronger gradients favour Regime 2. Regime 1 appears when higher TKE occurs; while lower TKE accompanies Regime 2. Weak or no cloudiness accompanies both morning evolution regimes; under increased cloudiness, KH events in the morning hours are not observed. Fig. 9 Histograms of meteorological and turbulent characteristics preceding to the occurrence of morning evolution events classified as Regime 1 ('ME-R1', red coloured), Regime 2 ('ME-R2', green coloured) and the absence of any morning evolution with a rising turbulent layer ('NoME', blue coloured). a wind speed shear; b temperature gradient; c wind speed; d turbulence kinetic energy TKE; e downwelling long-wave radiation; f sensible heat flux. Average values of correspondent parameters for different regimes are presented on plots The probability distribution of wind shears measured at a 45-m tower in regions where KH instabilities occurred during morning evolution events is shown in Fig. 10. The values measured in our observations are between 0.05 and 0.3 s −1 with a mean equal to 0.16 s −1 , and differ significantly from the values reported in the literature from the earlier radar observations of KH waves (Fukao et al. 2011;Luce et al. 2018). The mean values presented by them are ≈ 0.023 s −1 . The results summarized by Lyulyukin et al. (2015) from sodar, radar and lidar  . The values of wind shear at lower heights (< 45 m) measured by us are markedly higher than those occurring at heights above 100 m. This is probably due to the fact that static stability in the lowest layer is quite strong on average at Dome C, and, therefore, stronger shear is needed to overcome stratification when creating instabilities.

Characteristics of Kelvin-Helmholtz Waves in the Morning ABL
Our interest in this phenomenon was attracted when the first time we saw on high-resolution sodar echograms clearly visible wave structures resembling typical Kelvin-Helmholtz billows well-known from cloud observations and from hydrodynamics laboratory and numerical experiments (Petenko et al. 2016).

Appearance of KH Waves in the Rising Turbulent Layer During Morning Evolution
The braid-core structure could be discerned from high-resolution sodar echograms. Important difference of KH braids occurring during morning evolution (Fig. 11) from the classic S-shape braids (Fig. 1) is the absence of lower eyelids.
Two principal patterns of KHBs during morning evolution were identified: (i) quasiperiodical (every 4-15 min) trains containing 5-10 breaking braids that is similar to night conditions (Fig. 12a), and (ii) long-living series of KH billows lasting 30-90 min and consisting of 20-40 braids with monotonously growing amplitudes (Fig. 12b), that, to our knowledge, were never described in the literature.

Composite Shape of Kelvin-Helmholtz Billows
To determine the representative features of the KHB shape and structure, the method of composite averaging of the sodar echo-signal (Lyulyukin et al. 2013) was used. In that work, the method was successfully applied to several KHB episodes and made it possible to reveal the eddy structure of the wind velocity field inside individual billows (Lyulyukin et al. 2013;Lyulyukin 2018). The scheme of composite averaging is shown in Fig. 13. Here, the averaging is carried out in three steps. First, samples containing individual billows (about two dozen samples for each episode) are manually selected in the part of the echogram containing the KHB episodes. Figure 13a shows an echogram with KHB episodes, in which selected averaging windows are indicated by shaded rectangles. Second, the averaging window is manually positioned for each sample in such a way that the location of the billow crests in the averaging window was the same for all samples (phase synchronization). Also, samples with billows poorly distinguishable or having very different shapes are rejected at this step. All individual samples selected for composite averaging are shown in Fig. 13b. Third, all  Fig. 13c. The composite shape shows a quasi-continuous structure of billows, in which the braids about adjoin each other. The lower part of braids on the composite has a less clear structure, because the braid's inclination varies, and braids are not always traced to the lowest level. Figure 14 shows a fragment of a typical KHB train with the designation of parameters of braids which were revealed in the sodar echograms and used in the following text. Different characteristics of KHBs derived from our measurements are summarized in Table 1.

Spatial and Temporal Characteristics of KHBs During Morning Evolution
The mean periods of KH braids for every event are determined visually from sodar echograms. The averaged periods P KH of KH braids (time intervals between consecutive braids) were determined by measuring the time between Mb consecutive KH braids and by dividing by (Mb − 1). More complicated methods were examined (e.g., calculating spectra of time series of signal intensity), and rather similar results were obtained. Hence, for simplicity and rapidity, the method described above was further used. The histogram of the estimated periods is shown in Fig. 15a. Periods are mainly distributed between 20 and 70 s, with the mean and median values of P KH estimated as 45 s and 47 s, respectively.   The duration of a train of KHBs was simply defined as the time interval during which KH braids were visible on sodar echograms. Their periodicity was estimated by the same way. The mean values of train duration and periodicity were estimated as 9 min and 13 min, respectively.
In studying KH waves, the wavelength is usually calculated by multiplying the braid period estimated from the echosounder images by the phase velocity. As is well known, the phase velocity of a KH wave equals the wind speed at the altitude of maximum shear (e.g., Gossard et al. 1970;Gossard and Hooke 1975;Fukao et al. 2011). So, horizontal wavelengths can be roughly estimated from the braid periods by multiplying by the wind speed. Of course, we are aware of the large uncertainty in wavelength estimates due to the difficulty of determining phase velocity from wind velocity profiles. The probability distribution of wavelength values is shown in Fig. 15b. Note, estimated wavelengths are available only for the lowest 45-m layer where wind speed at the tower was measured.
The depth of the KHBs H was taken as the vertical distance between the trough and peak of the largest braid in a KHB train (Fig. 14). It is found that 80% of the KH braids had a maximum depth between 20 and 70 m when observed. It is very likely that smaller amplitude KH braids (say, less than 5 m) also occur, but they are difficult to recognise and might be missed due to insufficient height and time resolution. The relationship λ = A H between the wavelength λ and the depth of the layer occupied by waves H was considered in earlier theoretical works (e.g., Drazin 1958;Hazel 1972;Davis and Peltier 1976). The numerical coefficient A estimated by Drazin (1958) based on the linear theory is ≈ 7.5. In some experimental both atmospheric and hydrographic studies (e.g., Scorer 1969;Eymard and Weill 1979;Fukao et al. 2011;Lyulyukin et al. 2015;Petenko et al. 2016;Geyer et al. 2017;Luce et al. 2018), values of A from 3 to 20 were obtained. Estimated values of A ≈ 4-12 in Fig. 15c are consistent with those obtained earlier. Note, in some of cited studies, they used a so-called aspect ratio defined as a ratio H/λ; in fact, this is an inverse value of A calculated as 1/A. Note, changes in the shape and parameters of the KHBs were determined visually from echograms presented in height-current time coordinates (the latter being equivalent to the horizontal direction across the axes of billows). Our single-component vertical sodar does not provide information on changes in KHBs in the third direction, i.e., along the axis of billows, and our results were obtained assuming that the KHBs were two-dimensional.

Conclusions
In this paper, we presented a further field-observation evidence of the occurrence of Kelvin-Helmholtz billows in the shear-driven inversion layer rising during the morning transition from stable to convective conditions. Analysis of turbulence observations at Concordia station (Dome C, Antarctica) collected during an entire austral summer (November 2014-December 2015) using a high-resolution sodar and in situ instrumentation allowed us to detail the features and characteristics of this phenomenon described earlier by Petenko et al. (2016). We seeked to show its important role in the developing of mixing processes in the lower ABL.
The wave patterns captured in sodar echograms exhibited a regular braiding structure, attributed to KH shear instabilities. The frequent occurrence of this phenomenon (> 70% of days in the selected period) indicates that it, apparently, is not a casual event, but regularly takes place in the morning ABL under fair weather conditions in austral summers at this site. A similar phenomenon occurs also at mid-latitudes as shown by Kallistratova et al. (2019). We believe the real frequency of occurrence of KHBs, especially with small and moderate amplitudes (less than 100 m) and periods (less than 100 s), is much higher than previously thought: KHBs are invisible by bare eyes, and often are not detected by in-situ sensors and conventional sodars due to insufficient temporal and spatial resolution. Nevertheless, they are regularly observed by a high-resolution sodar.
From visual inspection of sodar echograms, we identified roughly two main regimes of the morning evolution distinguished by the presence of turbulence in the preceding night-time. It was found that KHBs within a rising turbulent layer occur not only when they were already being formed during night in the shear layer, but even though they (and the turbulence generated by them) were quasi-absent in the night-time. From this we can conclude that formation of KH waves in the initial phase of the development of the near-surface CBL is an intrinsic feature of the coexistence of the elevated shear layer and the CBL during morning evolution. The statistics and behaviour of the relevant characteristics (mean temperature and wind speed along with their gradients, turbulent and radiation parameters) favouring each type of the morning-evolution regime were determined. The most important parameters influencing the occurrence of different morning evolution regimes are cloudiness, wind speed, wind speed shear, and gradient of temperature.
Two specific patterns of KH billows during the morning evolution were identified: (i) quasi-periodical (every 8-15 min) trains containing 5-10 breaking KH billows that is more typical for night conditions; (ii) long-living series of KH billows lasting 30-90 min and consisting of 20-40 braids with monotonously growing amplitudes, that, to our knowledge, has never been described in the literature so far.
Some characteristics of KH billows and their association with meteorological and turbulent parameters were determined. The predominant periodicity of KH braids was estimated to be between 20 and 70 s, which corresponds to wavelengths of about 100-400 m. The horizontal width of individual braid-like layers ranges between 15 and 70 m, and their vertical thickness at the wave crests is 5-25 m. The total depth of the rising turbulent layer containing these waves varies from 15 to 120 m. The ratio A = λ/dH was estimated to be distributed between 5 and 12 with a mean value of 8.2. The morphology of the turbulence structure in the ABL was studied as a function of both temperature and wind field characteristics retrieved from a 45-m tower (six levels). Analysis of the location of KHBs and Ri profiles showed that within wave layers, values of bulk Ri vary around 0.25, being in agreement with other studies (Luce et al. 2018;Geyer et al. 2017 among others).
The variety of KH waves both in the atmosphere and in the underwater environment was documented in many previous studies cited above. The spatial and temporal characteristics varied in wide ranges with a ratio between maximum and minimum values reaching 4-5 times. We would like to note that appearance of KH instabilities captured in echosounder images is amazingly similar both in the atmosphere and in the underwater environment (see Fig. 1), despite the strong difference in spatial and temporal scales. There are some principal differences between the characteristics of KH instabilities at the lowest atmosphere (< 45 m) and those observed at higher altitudes (> 1 km) (Fukao et al. 2011). First, the wind speed shears (200-400 m s −1 km −1 ) in lower regions (< 45 m) with KHBs are much larger (about half order of magnitude) than those at high altitudes (20-50 m s −1 km −1 ). Such strong wind speed shear typically measured in our experiment is an essential difference from many previous studies.
We would like to outline one old problem arisen many years ago. What value should be considered as a mixing-layer height (Beyrich 1997;Seibert et al. 2000), especially during the initial phase of the morning transition process: an apparent height of convective plumes, the bottom of the elevated turbulent layer induced by KH waves or the top of this layer? Or, are there some other criteria to define the mixing layer and methods for estimation of its height H ML ? This is still the subject of controversial discussion. The difference between estimates obtained using different criteria can be quite significant. The question of which one of these definitions is more adequate to the "mixing layer height" (considered earlier by Stull 1988;Gryning and Batchvarova 1990;Seibert et al. 2000;Zilitinkevich and Baklanov 2002;Argentini et al. 2005;Casasanta et al. 2014 among others) should be reconsidered taking into account new findings.
The observational results highlight that the daily evolution of the summer boundary layer is unlikely to be explained and parametrized properly without considering the interaction between convective and wave processes when occurring simultaneously; they underscore the need for the development of advanced theoretical approaches that would enable the interaction between convective and wave processes when occurring simultaneously.
So far, understanding processes of wave-turbulence interactions in stably stratified shear flows is a challenging problem both in atmospheric physics and hydrography. Careful consideration of Kelvin-Helmoholtz waves and their interaction with the vortical turbulence is required for an adequate description of the SBL and its parametrization (e.g., L'vov et al. 2009;Jiang 2021). Turbulent wavelike structures especially those KHBs yet cannot be reproduced numerically well enough, although a few attempts have been made already (e.g., van der Linden et al. 2020;Jiang 2021). The precise mechanism of KHBs to elevate in the morning and grow along with the CBL needs to be investigated in the future. High-resolution modelling focusing on this aspect will help to solve this problem. In particular, the aforementioned techniques of adaptive mesh refinement could be attractive, as local grid refinement in the elevated shear zone can be investigate with great detail at limited computational costs (van Hooft et al. 2018(van Hooft et al. , 2019. As the elevated shear layer needs particularly high resolution (higher than in the bulk of the CBL) to capture the wave/turbulence interactions, and the height of this layer is neither fixed nor known in advance, adaptive mesh refinement seems to be very well suited to study this phenomenon.
More observations followed by theoretical studies, both in the atmosphere and in the underwater environment, are required to deepen our understanding of the role of KH instabilities in geophysical processes having importance for forecasting, environmental and climate problems. We hope that our results will help to arouse researchers to pay more attention to wave processes in the dynamics of the lower atmospheric boundary layer. (Brown and Hall 1978;Emeis 2021). Sodar allows continuous monitoring of vertical profiles of the temperature structure parameter C 2 T , since the intensity of the backscattered acoustic signal is proportional to C 2 T (Kallistratova 1962;Tatarskii 1971). The parameter C 2 T 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), where T'(r) is the temperature fluctuation around its mean at the point r, and r = |r 1 − r 2 | is the distance between points r 1 and r 2 . In acoustic remote sensing, C 2 T is determined from measurements of the intensity of the backscattered acoustic signal (Tatarskii 1971;Emeis 2021), where σ 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 2 T (z) and the measured intensity of the sodar electric return signal I is: where B is a constant that can be calculated from a complex calibration procedure (Danilov et al. 1994) or determined from comparison with in situ C 2 T measurements . Here, P t is the transmitted power, c is the velocity of sound, τ 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° (Tatarskii 1971). For our sodar these dimensions are 30 mm. The possibility of reliable quantitative measurements of C 2 T by using a sodar was shown by comparison with in situ measurements (e.g., Gur'yanov et al. 1987;Danilov et al. 1994). Unfortunately, in practice, due to difficulties of the calibration procedure, the C 2 T values are taken in arbitrary units.
The sensible heat flux is calculated as H 0 = c p ρ(w θ ), where ϑ is the potential temperature, ρ is the air density and c p is the specific heat of air at constant pressure. Note, a sonic measures the so-called acoustic temperature. However, at low temperature and humidity, the acoustic temperature is almost identical to the real air temperature, so we will not dwell on their difference from now on.
The friction velocity is calculated as u * 2 = − u w , where u and w are longitudinal and vertical components of wind velocity, respectively.
The stability parameter is calculated as z/L.