In this paper we propose two approaches to estimating the turbulent kinetic
energy (TKE) dissipation rate, based on the zero-crossing method by

Despite the fact that turbulence is one of the key physical
mechanisms responsible for many atmospheric phenomena, information on
the turbulent kinetic energy (TKE) dissipation rate

In the literature, there exist several different methods to estimate

Interestingly,

The new approaches are tested on velocity signals obtained during
the Physics of Stratocumulus Top (POST) research campaign, which was designed to investigate
the marine stratocumulus clouds and the details of vertical structure of the
stratocumulus-topped boundary layer (STBL;

The present paper is structured as follows.
In Sect.

The need to estimate the turbulent kinetic energy dissipation rate

Within the validity of Taylor's hypothesis,
Eq. (

Alternatively, one can consider the

Another method, also based on Eq. (

The signal

Using the inverse Fourier transform,
the

Based on Eqs. (

At the same time the data available from the POST measurements can only
account for a small part of the total dissipation spectrum, shown
qualitatively as a shaded region in Fig.

If one was to use the zero-crossing method (Eq.

Let us consider a signal

In the above derivation we assumed that the applied filter is rectangular in the
frequency space. The issue of frequency response characteristics of
a filter will be discussed further in Sect.

In this method we attempt to account for the
impact of the missing part of the dissipation spectrum
by introducing a correcting factor to the number of zero crossings per unit length

The corresponding one-dimensional spectrum

The value of

As a starting point for this procedure,
a first guess for the Kolmogorov length

Procedure of iterative

Use Eq. (

It should be noted that in this approach we do not have the empirical
inertial range constant

In order to present the more detailed properties of the procedure, we used the velocity signal from one of the horizontal flight
segments that took place within the turbulent atmospheric boundary layer. This segment was a part of flight

We have estimated the dissipation rate based on the number of zero crossings,
according to the methods outlined in Sect.

Before applying the threshold-crossing procedures the signal had to be filtered in order to eliminate
errors due to large-scale tendencies as
well as small-scale measurements noise. For this purpose we used
the sixth-order low-pass Butterworth filter ^{®}.
Figure

Measured velocity fluctuations:

The original formula of

It is worth noting that the spectra (

In order to use the method based on successive signal filtering we filtered
the signal with different values of

Using Eq. (

Scaling of

Even if the local isotropy assumption of

In order to quantify the errors resulting from the finite
sampling frequency and finite time window
and to test the performance of the proposed method, we performed the simulation analysis

To test the performance of the new proposals
we generated a number of artificial velocity signals
with frequency spectra and two point correlation functions
prescribed by the

We used artificial signals with

We observe an increase in

Values of the dissipation rate from simulation analysis
as a function of
higher value of the fitting range

Next, we tested the influence of the finite temporal window on
the calculated statistics. We generated ^{®} with a

We first decreased the time window, taking each time
only

Estimated values of

Finally, we would like to address the issue of larger scatter observed for

TKE dissipation rate estimates from simulation analysis for synthetic signals sampled with

The measurement signal used in Sect.

With such preparation we applied the iterative procedure, as described in Sect.

Next we considered Eq. (

It is worth noting that the proposed method is accounting for a dominant (and not directly measured) part of the spectrum based on
the theoretical knowledge about its shape. This knowledge is simply reduced to the form of the correcting factor

One-dimensional spectra:
black solid line – measured part; dashed magenta line – recovered part
with

The result of application of this method

We finally checked the estimates of the second method using
synthetic signals as described in
Sect.

Following the findings presented in the previous section both proposed methods were tested on much larger collection of data. For this purpose we used
velocity signals also obtained during the POST research campaign. We have chosen horizontal segments at various levels within the boundary layer from flights
TO10 and TO13. These flights were investigated in detail by

The dissipation rates of turbulent kinetic energy estimated from
the standard structure function method

We believe that the there is a fair consistency in those results because one should take into account
that the standard frequency spectra and structure function methods
calculate approximate values of

Dissipation rate of the kinetic energy estimated from the
structure function method

In the present work we proposed two novel modifications of the zero-crossing method, such that it can be applied to moderate-resolution measurements. The turbulent kinetic energy dissipation rates obtained using the proposed methods show fair agreement with results of the standard power spectrum and structure function approaches.

We note that the standard structure function and
power spectra methods are often used simultaneously, for better

From the perspective of practical applications we can think of several possible advantages of the zero-crossing methods.
First, the number of signal zero crossings can be calculated without
difficulty and
the proposed procedures are easy to implement.
Other advantages follow from the results of the simulation analysis
performed in Sect.

Moreover, we argue that the number-of-crossings method applied to the
fully resolved signals has become a fairly standard tool for

There are several perspectives for further work. First, the proposed methods
could be tested for a wider range of signals (e.g., from Eulerian measurements within the boundary layer adopting
the Taylor hypothesis),
characterized by different resolutions and obtained under varying
atmospheric conditions, to assess the scope of their applicability.
Second, as far as the model spectrum is concerned,
comparison with fully resolved experimental signals or direct numerical simulations data
would be valuable to test different forms of the model spectra
from

The Matlab^{®} code written for the purpose of this study is
available from the authors upon request.

POST data are available in the open database at

The authors declare that they have no conflict of interest.

We acknowledge financial support of the National Science Centre, Poland: Marta Wacławczyk was supported through the project 2014/15/B/ST8/00180, and Yong-Feng Ma and Szymon P. Malinowski were supported through the project 2013/08/A/ST10/00291. The POST field campaign was supported by US National Science Foundation through grant ATM-0735121 and by the Polish Ministry of Science and Higher Education through grant 186/W-POST/2008/0. Edited by: Ad Stoffelen Reviewed by: two anonymous referees