The focus of this paper is to analyse
the behaviour of the maximum Thorpe displacement

The atmospheric boundary layer (or ABL) is almost always turbulent. In the
absence of turbulence, atmospheric temperature profiles become increasingly
monotonic, due to the smoothing effect of molecular diffusion. Turbulence
gives rise to an effective eddy diffusivity and together with other causes
(such 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 local scales. Unfortunately,
measuring at small 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 such as lakes
or the ABL, can often be quantified by the Thorpe displacements

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

The results presented in this paper are based on three ABL field campaigns carried out in 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 the 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 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. The 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 northern 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. A detailed description can be consulted in Cuxart et al. (2000). The Sables98 field campaign only collects data over the night and, therefore, under neutral to stable conditions. The 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 to get vertical profiles up to 1000 m. As in Sables98, the Sables2006 field campaign also collects data over the night. Therefore, the Sables98 and Sables2006 experiments let us analyse the behaviour of overturns under stable conditions, while Almaraz94-95 is 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).

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

The maximum of the Thorpe displacement scale

The Thorpe scale

Because of the expensive nature of collecting data at microscale resolution,
there is a great interest in using parameterizations for small-scale dynamics
which are based on larger scales – as

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

Usually, the signature that might be expected for a large overturning eddy is
sharp 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, 2016). For our ABL studies, Thorpe displacements observed at
profiles could be qualitatively classified into two groups, as Fig. 1 shows.
The two graphs of Fig. 1 correspond to a campaign made on 25 September 1995.
The left graph of Fig. 1 is at 07:00 GMT (stable conditions) and the right
graph is at 17:00 GMT (convective conditions). The two kind of behaviours
are as follows. First, the Thorpe displacements under neutral and stable
stratification conditions are usually zero, except in a region with isolated

Left curve: Thorpe displacement

Time evolution of the maximum Thorpe displacements during a day
cycle. The symbols are as follows:

Figure 2 shows the time evolution of the maximum Thorpe displacement

Time evolution of the Thorpe scale during a day cycle. The symbols
are as follows:

Figure 3 shows the time evolution of the Thorpe scale,

Moreover, it is necessary to choose an appropriate overturning scale to
characterize instabilities leading to turbulent mixing and the turbulent
overturning motions themselves, and to look for a relationship 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
statistically stable representation of the entire feature (Dillon, 1982). But
the maximum of the Thorpe displacements is also considered to be an
appropriate measure of the overturning scale and it is always greater than

Hence, we analyse the relation between the

Absolute value of the maximum Thorpe displacement versus the Thorpe
scale. The symbols are as follows:

Therefore, we could think that the nearly constant ratio

Like other authors, we could state that this high correlation indicates that
the Thorpe scale is determined by the overturns near to the maximum Thorpe
displacement. We find the following power law:

The

The analysis of variance (ANOVA) is a statistical tool that separates the total variability of a data set into two components: random (which do not have any statistical influence on the given data set) and systematic factors (which have some statistical effect on the data). The ANOVA test is used to determine the impact independent variables have on the dependent variable in a regression analysis.

The

This relation between the maximum Thorpe displacement and the Thorpe scale by
a power law has been deduced for the overall data (not separating the data
from the three field campaigns). But we have used a data set with three
different experiments under different mixing conditions. The SABLES98 and
SABLES2006 experiments were conducted at night (turbulence by shear-driven)
and ALMARAZ94-95 during a day cycle and, therefore, convective regions have
not been excluded. Hence, we consider analysing whether this power law is
different from night to day. The objective is to study whether it is possible
to distinguish between the shear-driven overturns and the convective ones.
First, we separate the data from the three experiments into two sets: data
obtained overnight (from the Sables98, Sables2006, and Almaraz94-95 field
campaigns), or the night data set, and data which have been obtained during
the day (only from the Almaraz experiment), or the day data set. Then we
perform a linear simple regression analysis with an adjustment by least
squares for the two data sets. And, finally, we make a comparison of the
regression lines relating

Figure 5 represents the maximum Thorpe displacement versus the Thorpe scale only for the daytime data set (from 07:00 to 19:00 GMT). We observe a strong correlation which holds over 3 orders of magnitude as was deduced for the whole data set and other studies (Lorke and Wüest, 2002).

We perform the linear simple regression analysis. The

The confidence level is a measure of the reliability of a result. A confidence level of 95 % or 0.95 means that there is a probability of at least 95 % that the result is reliable.

), which indicates that the linear fit betweenAbsolute value of the maximum Thorpe displacement versus the Thorpe
scale for the daytime data set (

Figure 6 represents the maximum Thorpe displacement versus the Thorpe scale only for the nocturnal data set (from 20:00 to 06:00 GMT). We also observe a strong correlation which holds over 3 orders of magnitude as before (see Figs. 4 and 5).

Absolute value of the maximum Thorpe displacement versus the Thorpe
scale for the nighttime data set (

Finally, we perform the linear simple regression analysis. The

Therefore, we have deduced that the relation between the maximum Thorpe
displacement

These exponents are the slopes of the regression lines fitted to daytime and
nighttime data sets (see Figs. 5 and 6). To know whether they are
statistically different, we need to perform 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 levels of
our factor (day and night). This test fits two different regression lines to
the nighttime and daytime data sets and performs two analyses of variance
(one for each linear model and secondly for comparing the two regression
lines). For the first analysis, the

A two-sample

The

In hypothesis testing, the significance level is the criterion used for rejecting the null hypothesis (a hypothesis about a population parameter). The significance level is the probability of rejecting the null hypothesis given that it is true.

There is one more question, that is, to analyse whether the power law fits
the data better than a linear one in statistical terms. We have made a simple
regression analysis to construct three statistical models describing the
dependence of

For all three data sets, we got the same results. The analysis of variance
indicated that a linear model between

Finally, we deduce that the two power relations between the maximum Thorpe
displacement

As mentioned before, although both scales

The paper presents results related to the time evolutions of the ABL
turbulent parameters

The Thorpe scale

Equations (1) to (3) show that the relationship between the Thorpe scale

In the future, we will go on studying the power relationship between the
maximum Thorpe displacement and the Thorpe scale corresponding to ABL data to
verify the power law deduced in this paper. For this purpose, we will use a
new set of ABL data from new field campaigns. We will analyse the probability
density function of overturning length scales to clarify better the relation
between

Finally, there is another subject which is important to mention. In future
research, we need to study better the overturn identification
as in Piera et al. (2002). They propose a new method based
on wavelet denoising and the analysis of Thorpe displacement profiles for
turbulent patch identification. Although their method is for microstructure
profiles (which is not our case), it reduces most of the noise present in the
measured profiles (increasing the resolution of the overturn identification)
and is very efficient even at very low-density gradients for turbulent patch
identification. Another way to get overturn identification would be, for
example, to use a 3- or 4-D parameter space formed by (

In memory of my mother, Elena.

This work was financially supported by the Spanish Ministry of Education and Science (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 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. Edited by: P. Fraunie