Study of the overturning length scales at the Spanish planetary boundary layer

The focus of this paper is to analyse the behaviour of the maximum Thorpe displacement (dT)max and the Thorpe scale LT at the atmospheric boundary layer (ABL), extending previous research with new data and improving our studies related to the novel use of the Thorpe method applied to ABL. The maximum Thorpe displacements vary between −900 and 950 m for the different field campaigns. The Thorpe scale LT ranges between 0.2 and 680 m for the different data sets which cover different stratified mixing conditions (turbulence shear-driven and convective regions). We analyse the relationship between (dT)max and the Thorpe scale LT and we deduce that they verify a power law. We also deduce that there is a difference in exponents of the power laws for convective conditions and sheardriven conditions. These different power laws could identify overturns created under different mechanisms.


Introduction
Atmospheric boundary layer (or ABL) is almost always turbulent. In the absence of turbulence, atmospheric temperature profiles become increasingly monotonic, due to 15 the smoothing effect of molecular diffusion. Turbulence gives rise to an effective eddy diffusivity and as well as other causes (as fluid instabilities or internal wave breaking) makes vertical overturns appear as inversions in measured temperature profiles. These overturns produce small-scale turbulent mixing which is of great relevance for many processes ranging from medium to a local scale. Unfortunately, measuring at small 20 scales is very difficult. To overcome this disadvantage it is interesting to use theories and parameterizations which are based on larger scales. For example, the theories of turbulent stirring which often depend on hypotheses about the length scales of turbulent eddies. Vertical overturns, produced by turbulence in density stratified fluids as lakes or the ABL, can often be quantified by the Thorpe  Next we present the atmospheric data used for the analysis. In Sect. 3 we present the Thorpe method and the definitions of the scale descriptors used. In Sect. 4, the results of Thorpe displacements, the maximum Thorpe displacement and the Thorpe scale L T at ABL are presented and discussed.
2 Atmospheric data sets and meteorological instrumentation 5 The results presented in this paper are based on three ABL field campaings made at Spain and called Almaraz94-95, Sables98 and Sables2006. ABL data from 98 zeppelin-shaped tethered balloon soundings ranging from 150 to 1000 m were carried out in Almaraz94-95 field campaign made in Almaraz (Cáceres, Spain). The ABL profiles were obtained from 25 to 29 September 1995 in the time intervals 06:00-12:00 and 10 15:00-00:00 GMT. And from 5 to 10 June 1994 in the time intervals 05:00-12:00 and 17:00-00:00 GMT. Almaraz94-95 experiment collects data over a whole day and, therefore, covers different stratified conditions and mixing conditions -from shear-driven turbulence to convective regions. For more details see López et al. (2008). Sables98 (Stable Atmospheric Boundary Layer Experiment in Spain) took place over the north- 15 ern Spanish plateau in the period 10-28 September 1998. The campaign site was the CIBA (Research Centre for the Lower Atmosphere). Two meteorological masts (10 and 100 m) were available at CIBA with high precision meteorological instruments (Cuxart et al., 2000). Additionally, a triangular array of cup anemometers was installed for the purpose of detecting wave events and a tethered balloon was operated at nighttime. 20 A detailed description can be consulted in Cuxart et al. (2000). Sables98 field campaign only collects data over the night and, therefore, under neutral to stable conditions. Sables2006 field campaign took place from 19 June to 5 July 2006 at the CIBA. As in Sables98, different instrumentation was available on a tower of 100 m, a surface triangular array of microbarometers was also deployed and a tethered balloon was used 25 to get vertical profiles of the atmosphere up to 1000 m. As in Sables98, Sables2006 field campaign also collects data over the night. Therefore, Sables98 and Sables2006 Introduction experiments let us to analyze the behaviour of overturns under stable conditions while Almaraz94-95 under unstable conditions (and also stable ones). These three sets of data were selected for this analysis because they cover different mixing conditions (turbulence shear-driven and convective regions).
3 Thorpe method and overturn length scales 5 Thorpe devised an objective technique for evaluating a vertical length scale associated with overturns in a stratified flow (Thorpe, 1977;Itsweire, 1984;Gavrilov et al., 2005). Thorpe's technique consists of rearranging a density profile (which contains gravitationally unstable inversions) so that each fluid particle is statically stable. If the sample at depth z n must be moved to depth z m to generate the stable profile, the Thorpe Thorpe, 1977;López et al., 2008López et al., , 2015. The Thorpe displacement d T is not necessarily the real space actually traveled by the fluid sample, is an estimate of the vertical distance from the given vertical profile to the statically stable one that each fluid particle has to move up-or downward to its position in the stable monotonic profile (Thorpe, 1977;Dillon, 1982). Over most of a typical profile, the lo-15 cal stratification will be stable and the Thorpe displacement zero. A turbulent event is, therefore, defined as a region of continuously nonzero d T , i.e, overturns are defined as a profile section for which i d Ti = 0 while d Ti = 0 for most i (Dillon, 1982;Peters et al., 1995). The maximum of the Thorpe displacements scale (d T ) max = max [d T (z)] represents 20 the larger overturns which might have ocurred at earlier time when buoyancy effects were negligible (Thorpe, 1977;Dillon, 1982;Itsweire, 1984) and it could be considered as an appropiate measure of the overturning scale. The Thorpe scale L T is the root mean square (rms) of the Thorpe displacements . Therefore, it is a statistical measure of the vertical size of 25 overturning eddies (Thorpe, 1977;Dillon, 1982;Itsweire, 1984;Fer et al., 2004)  proportional to the mean eddy size as long as the mean horizontal potential temperature gradient is much smaller than the vertical gradient. For our field ABL measurements, we can consider that the ABL is horizontally homogenous because the average horizontal temperature gradient is 4 × 10 −4 (K m −1 ) and is smaller than the average vertical temperature gradient, which is 2 × 10 −2 (K m −1 ) (López et al., 2015).

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Because of the expensive nature of collecting data at microscale resolution, there is a great interest to use parameterizations for small-scale dynamics which are based on larger scales -as L T or (d T ) max . Therefore, it is very important to analyze the relation between L T and (d T ) max for selecting the most aproppiate overturning scale.

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Our methodology is based on reordering 111 measured potential temperature profiles, which may contain inversions, to the corresponding stable monotonic profiles. Then, the vertical profiles of the displacement length scales d T (z) or Thorpe displacements profiles can be calculated by using a bubble sort algorithm with ordering beginning at the shallowest depth (Thorpe, 1977;Dillon, 1982;Itsweire, 1984;López et al., 2008López et al., , 15 2015. This simple sorting algorithm works by repeatedly stepping through the data list to be sorted, comparing each pair of adjacent items and swapping them if they are in the wrong order (López et al., 2015).

Thorpe displacement profiles at ABL
Usually, the signature that might be expected for a large overturning eddy is: sharp 20 upper and lower boundaries with intense mixing inside -displacement fluctuations of a size comparable to the size of the disturbance itself are found in the interior. While common in surface layers strongly forced by the wind, these large features are not always found as in our ABL case (López et al., 2008(López et al., , 2015. For our ABL studies, Thorpe displacements observed at profiles could be qualitative classified in two groups. First, the Thorpe displacements under neutral and stable stratification conditions are always zero except in a region with isolated Z patterns which would correspond to discrete patches. These isolated overturns are very few well-defined sharp overturns which appear under stability conditions (at sunset, night and sunrise profiles). We really find other features that are smaller, some having an eddylike shape similar to the larger 5 disturbances, some a random mix of small scale fluctuations without sharp boundaries. These are the second group or non-zero Thorpe displacement regions with indistinct and distributed features which appear under convective and/or neutral conditions (at noon, afternoon and evening profiles). These Thorpe displacements are rarely zero for the whole profile. To verify this behaviour see López et al. (2008López et al. ( , 2015.  Figure 1 shows the time evolution of the maximum Thorpe displacement (d T ) max along a day for the three field campaigns. The scale (d T ) max is very small (approximately zero) under stable conditions from 23:00 to 06:00 GMT (between sunset and sunrise) for all the experiments. From 19:00 GMT, it is observed that scale (d T ) max decreases. 15 The greatest values of (d T ) max appears under convective conditions from 09:00 to 19:00 GMT being positive and negative. But the positive values of (d T ) max are greater than the negative ones. The positive (d T ) max has its greatest values about 950 m and the greatest negative (d T ) max are about 600 m. These results means as follows. Thorpe displacements were defined as the difference between the final height and the initial 20 height of the fluid particle, i.e.

Time evolution of maximum Thorpe displacements and Thorpe scale
, the fluid particle has to go up to reach its stable position, and if d T < 0((z m ) final < (z n ) initial ), it has to go down to reach its stable point. From Fig. 1  Therefore, the maximum Thorpe displacements is a parameter which could represent the dynamical behaviour of air particles and its relation with the stratification conditions. Finally, there is a gap in Fig. 1 due to non registered data between 13:00 and 14:00 GMT. Figure 2 shows the time evolution of the Thorpe scale, L T during a day for the three 5 field campaigns. The Thorpe scale L T has small values (and close to zero) under neutral and stable conditions (between sunset and sunrise) from 20:00 to 09:00 GMT. This scale reachs its greatest values under convective conditions from 09:00 to 19:00 GMT. There are two distinct behaviours with high (L T > 100 m) and low (L T < 100 m) Thorpe scales. In most of the turbulent patches, the Thorpe scale does not exceed several tens 10 of meters and they appear under stable and neutral stratification conditions when the Thorpe displacements are also small and related to instantaneous density gradients. In contrast, under convective conditions, Thorpe scales are relatively large, they exceed hundreds of meters and they may be related to convective bursts. Hence, the Thorpe scale L T is always greater under convective conditions than under stable ones and it 15 is a parameter which also could represent the dynamical behaviour of air particles. As in Fig. 1, there is a gap in Fig. 2 due to non registered data between 13:00 and 14:00 GMT. Both scales, the Thorpe scale L T and the maximum Thorpe displacement (d T ) max , have small values (and close to zero) under neutral and stable conditions, and their greatest values appear under convective conditions. Therefore, it is reasonable to 20 think which of the two scales could represent better the dynamical behaviour of turbulent overturns although both scales are alternative length scales that coul be used for characterizing turbulent overturns. Moreover, it is neccessary to choose an appropiate overturning scale to characterize instabilities leading to turbulent mixing, the turbulent overturning motions themselves 25 and to look for a relation with the Ozmidov scale at ABL data (Dillon, 1982;Lorke and Wüest, 2002;Fer et al., 2004). We could choose the Thorpe scale rather than the maximum Thorpe displacement because we only sample vertically while the turbulence is three dimensional and, therefore, the Thorpe scale is more likely to be a statisti- cally stable representation of the entire feature (Dillon, 1982). But the maximum of the Thorpe displacements is also considered as an appropiate measure of the overturning scale and it is always greater than L T (better detectable by a limited resolution instrument). Different researchers have found a linear relationship between L T and (d T ) max for profiles from the equatorial undercurrent (Moum, 1996;Peters et al., 1995) 5 and a high linear correlation computed from the Banyoles99 field data where the ratio (d T ) max /L T is approximately equal to 3 (Piera Fernández, 2004). It must exist a correlation between L T and (d T ) max because when computing the rms of a set of Thorpe displacements with high kurtosis distributions, the final result depends on the largest values (Piera Fernández, 2004;Stansfield et al., 2001). A similar linear correlation be-10 tween L T and (d T ) max has been found by other researchers: a ratio (d T ) max /L T ≈ 3.3 is obtained in the oceanic thermocline (Moum, 1996), a ratio (d T ) max /L T ≈ 2.4 is obtained from laboratory experiments (Itsweire et al., 1993) and, finaly, the ratio (d T ) max /L T is nearly 3 in numerical simulations (Smyth and Moum, 2000). But for microstructure profiles from strongly stratified lakes, a power law -as (d T ) max ∼ (L T ) 0.85 -is found (Lorke 15 and Wüest, 2002). This relation also holds for profiles from other lakes under very different conditions of mixing and stratification with a strong correlation that holds over four orders of magnitude (Lorke and Wüest, 2002). Hence, we analyze the relation between L T and (d T ) max scales for our ABL data. Figure 3 shows the maximum Thorpe displacement versus the Thorpe scale from the 20 three field campaigns. The relationship |(d T ) max | = L T is indicated. We observe that the linear relationship |(d T ) max | = L T proposes by other authors (Moum, 1996;Peters et al., 1995;Piera Fernández, 2004;Itsweire et al., 1993;Smyth and Moum, 2000) does not verify for our ABL data.
Therefore, we could think that the nearly constant ratio (d T ) max /L T obtained in a wide 25 range of field and laboratory experiments, does not verify in our ABL data. And, hence, the shape of Thorpe displacements distribution could change at ABL. We also observe a strong correlation which holds over three orders of magnitude as in other researches from profiles in lakes (Lorke and Wüest, 2002). It is the first time that such a relation between this two overturning length scales is found for ABL data.
As other authors, we could state that this string correlation indicates that the Thorpe scale is the determined by the overturns near to the maximum Thorpe displacement. We find the following power law: which is similar to the one deduced by Lorke and Wüest (2002) from profiles in strongly stratified lakes. We realize a linear simple regression analysis. Of particular interest is the P value associated to the analysis of variance, which tests the statistical significance of the fitted model. For our case the P value is less than 0.05 (operating at the 10 95 % confidence level) which indicates that a significant linear relationship exists between |(d T ) max | and L T . Moreover, the R-squared coefficient is 96.95 which represents that the linear regression accounts for about 97 % of the variability in the maximum Thorpe displacement |(d T ) max | as a function of the Thorpe scale, L T . This relation between the maximum Thorpe displacement and the Thorpe scale by 15 a power law has been deduced for the overall data (not separating the data from the three field campaigns). But we have used three different experiments data set with different mixing conditions. SABLES98 and SABLES2006 experiments have been realized at night (turbulence by shear-driven) and ALMARAZ94-95 during a day cycle and, therefore, convective regions have not been excluded. Hence, we consider to analyze 20 if this power law is different from night to day. The objective is to study if it s possible to distinguish between the shear-driven overturns and the convective ones. First, we separate the data from the three experiments in two set: data obtained overnight (from Sables98, Sables2006 and Almaraz94-95 field campaigns) or night data set, and data which have been obtained during the day (only from Almaraz experiment) or day data 25 set. Then we realize a linear simple regression analysis with an adjustment by least squares for the two data sets. And, finally, we realize a comparison of regression lines procedure which is designed to compare the regression lines relating |(d T ) max | and L T at the two levels of our categorical factor (daytime and nighttime). Figure 4 represents the maximum Thorpe displacement vs. the Thorpe scale only for the daytime data set (from 07:00 to 19:00 GMT). As before, we observe that the linear relationship |(d T ) max | = L T does not verify for our ABL daytime data. We also observe 5 a strong correlation which holds over three orders of magnitude as it was deduced for the whole data set and in other researches from profiles in lakes (Lorke and Wüest, 2002).
We realize the linear simple regression analysis. The P value associated to the analysis of variance is less than 0.05 (operating at the 95 % confidence level) which indicates that a significant linear relationship exists between |(d T ) max | and L T as before. The R-squared coefficient represents the percentage of the variability in |(d T ) max | which has been explained by the fitted linear regression model and is about 97 %. Figure 5 represents the maximum Thorpe displacement vs. the Thorpe scale only for the nocturnal data set (from 20:00 to 06:00 GMT). The relationship |(d T ) max | = L T is indicated. As before, we observe that the linear relationship |(d T ) max | = L T does not verify for these nocturnal ABL data set. We also observe a strong correlation which holds over three orders of magnitude as in the results mentioned before (see Figs. 3 and 4).
Finally, we realize the linear simple regression analysis. The P value associated to 20 the analysis of variance is less than 0.05 (operating at the 95 % confidence level) which indicates that a significant linear relationship exists between |(d T ) max | and L T as before. Moreover, the R-squared coefficient is 95.89 which represents that the linear regression accounts for about 96 % of the variability in the maximum Thorpe displacement |(d T ) max |. 25 Therefore, we have deduced that the relation between the maximum Thorpe desplacement |(d T ) max | and the Thorpe scale L T by a power law is different from day to night. For the nighttime data set the power law is: And for the daytime data set the relation is the following:

NPGD
We observe that the kind of relation is the same (a power law) but the exponents are different. So we question if these coefficients are statistically different and if there is or 5 not a different behaviour of the overturn length scales between day and night. These exponents are the slopes of the regression lines fitted to daytime and nighttime data sets (see Figs. 4 and 5). To know if they are statistically different we need to realize a comparison of regression lines. This procedure is a test to determine whether there are significant differences between the intercepts and the slopes at the different 10 levels of our factor (day and night). This test fits two different regression lines to the nighttime data and daytime data sets and realizes two analysis of variance (one for the linear model and secondly for). For the first analysis, the P value is less than 0.05, if we operate at the 5 % significance level, and indicates that a significant relationship of the linear form exists between |(d T ) max | and L T for daytime and nighttime data sets (t- 15 statistic tests have also been made which P values are less than 0.05, 95 % confidence level, and indicate that the model coefficients are significantly different from 0). For the second analysis of variance is performed to determine whether there are significant differences between the slopes of the daytime and nighttime lines are significantly different. We find that the F test for slopes tests the null hypotheses slopes of the lines 20 are all equal. Operating at the 1 % significance level, we find a P value (for slopes) is less than 0.01, and, therefore, there are significant differences between the slopes of the daytime and nighttime lines (we get the same result for the intercepts).
Finally, we deduce that the two power relation between the maximum Thorpe desplacement |(d T ) max | and the Thorpe scale L T for nighttime data (Eq. 2) and daytime classified overturns between day and night ones, i.e., we could distinguish between convective and shear-driven mechanism originating overturns. As mentioned before, although both scales ((d T ) max and L T ) are alternative length scales to characterize turbulent overturns, it is reasonable to choose which of the two scales could represent better overturns. If there is a high linear correlation between 5 the maximum Thorpe displacement (d T ) max and the Thorpe scale L T , this could be considered a better descriptor of the overturn properties although it depends mainly of the values of (d T ) max and the relative errors from both scales are approximately equal (Piera, 2004). But we have just deduced that the relation between the maximum Thorpe displacement (d T ) max and the Thorpe scale L T is not linear at our ABL research, un-10 less a power law as other authors (Lorke and Wüest, 2002). Consequently, there would not be a constant ratio |(d T ) max |/L T which could suggest that the shape of Thorpe displacements distribution could change. Therefore, it is necessary to study the probability density functions (pdf) of the Thorpe displacements to understand better the relation between (d T ) max and L T . Moreover, the Thorpe scale is mainly determined by larger 15 overturns which are not very frequent (Stansfield et al., 2001) and it would be very useful to determine it based on the probability density function of the Thorpe displacements. This pdf study would allow us to decide which of the two overturn length scales is a more representative measure of turbulent overturns.

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The paper presents results related to the time evolutions of the ABL turbulent parameters L T and (d T ) max during a day with different levels of stability. Secondly, the paper adds insight to the problem of the relationship between these two overturning length scales at ABL.
The Thorpe scale L T and the maximum Thorpe displacement (d T ) max , have small than effective values in the stratosphere which are L T ∼ 1-1.1 m (Gavrilov, 2013), values in mixing surface layers and seasonal thermoclines which are L T ∼ 0.03-1.90 m (Dillon, 1982), values in vertical mixing process induced by internal tides which are L T ∼ 0.2-4.2 m (Kitade et al., 2003) or values in dense overflow which are L T ∼ 1-17 m (Fer et al., 2004). The greater values appear under convective conditions which could 5 generate overturns of larger scale. Under shear-driven conditions, our Thorpe scales are smaller than convective ones, ranging in (1100) m, but they are also greater if we compare them with the scales of other authors. Therefore, we deduce that there would be a relation between the ABL processes which generate mixing and the overturn size and behaviour (for example, the terrain shape interacts with the ability of the ABL to 10 produce local mixing very near the ground and this could be affect to overturns). This theme will need further field work where different conditions are met (combination of the boundary condition effects and of stability combining the 3-D and 2-D characteristics of scale to scale direct and inverse cascades, intermittency of the forcing and scale to scale stratified turbulence cascade, Vindel et al., 2008;Yagüe et al., 2006).

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Equations (1) to (3) show that the relationship between the Thorpe scale L T and the maximum Thorpe displacement (d T ) max is a power law which has been statiscally demostrated. We must therefore conclude that the linear relation proposes by other authors (Moum, 1996;Peters et al., 1995;Piera Fernández, 2004;Itsweire et al., 1993;Smyth and Moum, 2000) is not adequate for our spanish ABL data. Research will con-20 tinue on this interesting question which is related to the selection a length scale for characterizing turbulent overturns. This last problem would be better analyzed if we study the probability density function (pdf) of overturning length scales. The objective is to decide if L T is or not statistically a more appropiate length scale than (d T ) max . Moreover, it is interesting to verify the assumption that the Thorpe scales have a uni-25 versal probability density function which could be used to verify how accurately the Thorpe scales were computed and also to determine if (d T ) max is statistically better than L T as overturning length scale. It is very likely that the pdf parameters depend on the governing background conditions generating Thorpe displacements, which are Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | different in the boundary layers from those in the interior layers with intermittent mixing, or in convective conditions from shear-driven conditions. We also would like to verify if the density probability function is decaying exponentially for increasing displacement length with a separate cut-off before (d T ) max .
In the future, we will go on studying the power relationship between the maximum 5 Thorpe displacement and the Thorpe scale corresponding to ABL data to verify the power law deduced at this paper. For this purpose, we will use new set of spanish ABL data from new field campaigns. We will analyze the probability density function of overturning length scales to clarify better the relation between (d T ) max and L T and as a tool to choose the more appropiate turbulent patch length scale. Moreover, we 10 would like to study the following hypothesis if the Thorpe scale is greater than the integral scale there would be a local convective process and if it is not, there would be stratification.
Finally, there is another subject which is important to mention. At future researches, we need to study better the overturn identification as Piera et al. (2002). They propose 15 a new method based on wavelet denoising and the analysis of Thorpe displacements profiles for turbulent patch identification. Although their method is for microstructure profiles (that is not our case), it reduces most of the noise present in the measured profiles (increasing the resolution of the overturn identification) and it is very efficient even at very low-density gradients for turbulent patch identification. Another way to get 20 overturn identification would be, for example, to use a 3 or 4 dimensional parameter space formed by (L O , L T , L MO ) to locate mixing events and also to study the evolution of the processess.
Acknowledgements. In memory of my mother, Elena.
This work was financially supported by the Spanish Ministry of Education and Science

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(projects CGL2009-12797-C03-03 and CGL2012-37416-C04-02) which is gratefully acknowledged. We are also in debt to all the team participating in ALMARAZ94-95, SABLES98 and SABLES2006. The authors would also acknowledge the help of ERCOFTAC (SIG 14). We acknowledge the referees for assisting in evaluating this paper; their constructive criticism helped in improving the manuscript.