A simple model for turbulence kinetic energy (TKE) and the TKE budget is presented for sheared convective atmospheric conditions based on observations from the Boundary Layer Late Afternoon and Sunset Turbulence (BLLAST) field campaign. It is based on an idealized mixed-layer approximation and a simplified near-surface TKE budget. In this model, the TKE is dependent on four budget terms (turbulent dissipation rate, buoyancy production, shear production and vertical transport of TKE) and only requires measurements of three available inputs (near-surface buoyancy flux, boundary layer depth and wind speed at one height in the surface layer) to predict vertical profiles of TKE and TKE budget terms.

This simple model is shown to reproduce some of the observed
variations between the different studied days in terms of
near-surface TKE and its decay during the afternoon transition
reasonably well. It is subsequently used to systematically study the
effects of buoyancy and shear on TKE evolution using idealized
constant and time-varying winds during the afternoon
transition. From this, we conclude that many different TKE decay
rates are possible under time-varying winds and that generalizing
the decay with simple scaling laws for near-surface TKE of the form

The model's errors result from the exclusion of processes such as elevated shear production and horizontal advection. The model also produces an overly rapid decay of shear production with height. However, the most influential budget terms governing near-surface TKE in the observed sheared convective boundary layers are included, while only second-order factors are neglected. Comparison between modeled and averaged observed estimates of dissipation rate illustrates that the overall behavior of the model is often quite reasonable. Therefore, we use the model to discuss the low-turbulence conditions that form first in the upper parts of the boundary layer during the afternoon transition and are only apparent later near the surface. This occurs as a consequence of the continuous decrease in near-surface buoyancy flux during the afternoon transition. This region of weak afternoon turbulence is hypothesized to be a “pre-residual layer”, which is important in determining the onset conditions for the weak sporadic turbulence that occur in the residual layer once near-surface stratification has become stable.

The daytime atmospheric boundary layer (ABL) is characterized
by unstable stratification, turbulent mixing of momentum, heat, scalars and
buoyancy-driven eddies. These large eddies are generated by a strong surface
heat flux but are also influenced by wind shear

During the course of any day, atmospheric boundary layer turbulence will naturally respond to different levels of shear and buoyancy production, directly influencing the level of turbulence kinetic energy (TKE). In addition, transport and dissipation of TKE can change substantially from hour to hour as well as on shorter and longer timescales, thereby influencing the level of TKE at specific heights in the ABL. Modeling the time evolution of the boundary layer for growth and decay phases of turbulence under unstable conditions can be a very challenging task, but it is important for many applications (e.g., dispersion of pollutants).

Several important earlier modeling studies of the daytime unstable ABL should
be mentioned. The early work of

For the BLLAST field campaign, several LES studies

In this paper, we present a simple one-dimensional TKE budget model based on
the analysis presented in Part 1 and assumptions about approximate height
dependencies of TKE budget terms in the mixed layer. We use this model to
carry out simulations for nine IOP days where near-surface measurements and
TKE budget estimates for both morning and afternoon periods were available.
In this way, we can compare our simulated TKE at different heights to
observations and discuss directly how the estimated budget terms act in the
model to underestimate or overestimate TKE at specific times. We want to
stress that this model has been developed with the aim of aiding in the
understanding of the most important processes that govern TKE evolution for
sheared convective situations, but it should not be regarded as a complete
description of the complex reality. As will be discussed further in the text,
the model does not include processes such as elevated shear production and
horizontal advection of TKE, which may be important at specific times. We use
observations from several different land cover types to explore the
sensitivity of the modeled boundary layer dissipation rates in relationship
to those observed over the heterogeneous BLLAST field campaign landscape.
This heterogeneity challenges some of our modeling assumptions. We insist on
carrying out the study with a simple model for near-surface TKE and TKE
budget terms because it is an important first step before more complexity and
processes may be added. Compared to the model proposed in

The paper is structured as follows. Firstly, in Sect. 2 we introduce the model main goal and description. Here we describe the different parts of the simple TKE model and illustrate the height dependence of model terms. In Sect. 3, we guide the reader further to the relevant data sets used in the paper and further documentation about how the data were selected and treated for our modeling effort. This is followed in Sect. 4 by evaluation of near-surface TKE and TKE budget terms for nine simulated IOP days including discussion of potential sources of errors in TKE prediction. In Sect. 5, we explore modeled dissipation rate for the boundary layer, using observed fluxes and winds from different surface land covers and an area-averaged flux, in comparison to observed dissipation rate and discuss the formation of a “pre-residual layer” during the afternoon transition. In Sect. 6, we use the model to simulate near-surface TKE for a variety of idealized afternoon conditions and discuss the results in relationship to previously proposed “decay laws” of turbulence. Finally, we conclude and summarize in Sect. 7.

In this section, we describe our simple model for the atmospheric boundary and surface layer turbulence kinetic energy. From inputs of time series of near-surface buoyancy flux, wind speed at one height in the surface layer and boundary layer depth estimates the model predicts vertical profiles of terms in the TKE budget equation as well as TKE. The model is initialized in the morning transition and gives an approximate description of the surface and boundary layer evolution in terms of TKE and its budget terms during unstable conditions until the end of the afternoon. Observations for one BLLAST case (20 June) are shown as the model terms are introduced, even if the observations are described in more details in Sect. 3. A more extensive evaluation of near-surface TKE budget terms is given in Sect. 4.

In this work, we consider a simplified budget for TKE of the following form,
assuming no advection and horizontal homogeneity:

Here, TKE (

The physical interpretation of the five terms in Eq. (

Given simple parametrization for the right-hand-side terms of Eq. (

An important driving force for the atmospheric boundary layer turbulence in
unstable conditions is the surface buoyancy flux, which controls near-surface
buoyancy production in this simple model. In first instance, we will
prescribe these, as determined in Part 1, from observations made at
3.23

Here,

In Part 1, shear production was shown to be an important source of
turbulence production, especially near the surface. To model shear
production, we use an idealized Monin–Obukhov similarity-based flux gradient
relationship

Here,

It should be noted here that a different functional form for the
non-dimensional wind gradient was found in Part 1, but here we chose to
keep the consensus value of von Kármán's constant. We consider it may be other
non-dimensional parameters than

To use the above wind speed relationship, we need to determine
a

A simple drag coefficient or CD-curve approach (

Using such a

Measured hourly averaged

In Fig.

The left panel shows the simple modeled height dependence for
the buoyancy production term in the TKE budget normalized by the
surface value (in red). A vertical green line is added at

Here, we describe the vertical height dependence that is assumed for
each of the right-hand-side budget terms of Eq. (

To describe the height variation in the boundary layer, we use idealized
linear profiles of buoyant production. These profiles are in general
agreement to the proposed shapes (based on measurements) from

In the middle and right panels of Fig.

The shear production considered in this simple model is given by

We multiply our stress profile by the wind gradient expression (given by
Eq. 3) to calculate the shear production term

Left panel: modeled (in lines) and observed (in dots) shear
production for the afternoon of 20 June. Red, green, black and
blue correspond to 12:30, 14:30, 16:30 and 17:30 UTC,
respectively. Horizontal black lines correspond to boundary layer
depth

In Fig.

Same as Fig.

Figure

The modeled transport term consists of transport due to both
buoyancy-produced TKE and shear-produced TKE. Such an approach may of course
be criticized as turbulence in reality cannot be separated in such a way, but
we are nevertheless not the first

The term

At the boundary layer height

This also determines the height of no turbulence

We shall soon determine

Near-surface transport fraction defined as minus the ratio of
the total transport term to the sum of buoyancy and shear
production. Red circles show hourly averaged data from the afternoon
period from 12:00 UTC to zero buoyancy flux for

Finally, the transport fraction

Same as Fig.

Equation (

The profile of total transport is shown in Fig.

Profiles of TKE during the afternoon period on 20 June. The
color scheme is the same as in earlier figures. The measurements on
both the small tower (2–8

The dissipation rate of TKE is calculated in the model using the TKE length
scale parametrization presented in Part 1:

The modeled profiles of dissipation for the afternoon of 20 June are shown in
Fig.

The TKE budget equation is used to solve for the evolution of TKE from
neutral morning conditions until the end of the afternoon. At the
beginning of the simulation, for simplicity, we therefore assume the
buoyant production term

Our treatment of initial conditions for dissipation

Our choices for initial conditions and modeling of morning transitions should
be recognized only as a very crude attempt to represent a much more complex
reality.

The calculation of TKE tendency is essential for the time evolution of TKE. It was shown in Part 1 that TKE tendency is typically much smaller than the other budget terms, but if it were completely zero there could not be any evolution of the TKE so there is a difference between true steady-state and quasi-steady conditions.

In our model the evolution of TKE is determined by a finite-difference (forward in
time) calculation with 1 s time step and 1

The model can be considered semi-analytical in the sense that it contains only a numerical finite-difference time-stepping scheme, but in all other ways is just a simple parametrization.

The resulting vertical profiles of TKE are shown in Fig.

It is clear that the model produces TKE of the right order of
magnitude and predicts the general reduction of TKE with height from
the smaller tower to the 60

The model only predicts an increase in TKE from the first model level to the
second, due to the prescribed reduced shear production at the first grid
point compared to the others. Otherwise, the model shows a decrease in TKE
with height, which is not necessarily true at all height ranges. The
measurements often show a small increase in TKE from 2.23 to 8.22

Figures

Concerning modeling of the buoyancy term in

For the shear production term

The profile of total transport shown in Figs.

Qualitatively the vertical profile of dissipation is similar in our model
compared to the model from

From Fig.

It is worth noting again that TKE tendency is assumed to be exactly zero in

Brief description of BLLAST measurement sites and data sets with the
required buoyancy flux, winds and temperature for the modeling of TKE. Also
listed are the roughness length (

Our proposed model is based on a simplified TKE budget including
idealized height-varying terms for shear production, buoyant
production, transport and dissipation. It is driven with surface
measurements (wind speed and fluxes) and boundary layer depth

The BLLAST field campaign took place in June and July of 2011 in southern
France at Plateau de Lannemezan, a plateau of about 200 km

Firstly, we will use wind speed and buoyancy flux from an 8 m tower
(referred to as the “divergence site”) and

Secondly, as an exploration of the sensitivity in modeling results, we also
use observed sensible and latent heat fluxes along with observed wind speed
and temperature from five other land surface covers (moor, corn, grass, wheat
and forest) to drive our TKE model. The fluxes are obtained from the
uniformly processed data set by

In Table

We will show results from 9 of the 10 IOP days previously considered in Part 1. This is because there were no measurements available from the divergence site before 10:00 UTC on 19 June and we chose to consistently do simulations constrained by observations from the time of positive sensible heat flux in the morning until the end of the afternoon, defined from zero-buoyancy flux. This choice is to allow for the turbulence to build and decay during a long time period of sheared convective atmospheric conditions for each day.

The data sets described above all consist of estimates of different
parameters, wind speed, buoyancy flux (or sensible and latent heat
flux plus potential temperature that can be used to estimate buoyancy
flux) and

The time from positive sensible heat flux until zero buoyancy flux was estimated for each day and the time series manually checked. A few suspicious values in various time series were removed in this process. Then, a linear interpolation to 1 s values was applied followed by a 1 h running mean smoothing of the data. This procedure was adopted as we, especially for wind speed at these sites, found rapid variations in time and our intention is to attempt to model the more general slow decay of turbulence kinetic energy related to persistent changes in surface flux forcing and the slower trends observed in wind speed.

For boundary layer depth estimates, a 1 h running mean time series was
formed in a similar way. Here, before linearly interpolating,
a representative boundary layer depth value for the morning at the start of
each simulation was subjectively estimated from the observed growing trend of

The upper row shows the modeled stability-corrected friction
velocity (black line) and observations at 3.23

For evaluation of our model the data set of UHF wind profiler data described
in Part 1 also includes estimates of TKE dissipation rate. It was available
at an average temporal resolution of 5 min and a spatial resolution of
75

For evaluation, we also compared model estimates with UHF estimates
and aircraft estimates of TKE dissipation rate from the Piper Aztec
research airplane

To display the slower trends and evolution of TKE dissipation rate in a height time representation, the 5 or 15 min averaged data sets was considered still quite scattered and a running mean value of 1 h was applied for comparison with the modeled dissipation rate. This is reasonable here because we use a 1 h smoothed wind and surface flux time series as input to force the model and hence do not model the more temporary rapid variations in TKE budget terms.

In this section, we compare the simple model to measurements for nine IOP days studied in Part 1. The first objective is to investigate the simple model's ability to predict a reasonable near-surface TKE and TKE budget evolution for the diverse set of conditions that occurred on these 9 days despite its deficiencies. The second aim is to discuss why the model produces unreasonable results. This indicates potential focus areas for future model improvement.

The upper row shows the observed hourly averaged buoyancy
production at 2.23, 5.27 and 8.22

The upper row of Fig.

The middle row of Fig.

The observed hourly shear production is shown in the lower row of
Fig.

The observed and modeled near-surface buoyant production is shown in the
upper row of Fig.

The middle row of Fig.

The lower row shows the observed and modeled dissipation. The model captures
much of the day to day as well as hourly variability, but at times of strong
shear production, it underestimates at the 8.22

All these observed errors in the modeled TKE budget terms, which may at times be considered quite small, can lead to problems in the prediction of the TKE as any systematic errors can cause an accumulated effect for the TKE prediction.

The lower row shows the modeled TKE interpolated to
2.23

We find the modeled results of TKE at the 2.23

For the 2.23

At 61.4

Modeled boundary layer dissipation rate for 30 June for six
simulations over different surfaces with available near-surface
measurements during the BLLAST field campaign, as well as
a simulation driven by a

As an exploration into the sensitivity of model results to different observed
fluxes and winds over different surface types, in Fig.

The divergence site tower measurements show very similar low fluxes as
observed over grass and moor, and for this day also corn. This is in
contrast to the higher observed fluxes over forest and wheat. The surface
flux over the grass, moor, corn and divergence sites yielded the most similar
levels of dissipation rate compared to the observations on this day, whereas
other surface types lead to higher levels of dissipation rate. Based on
energy balance modeling, fluxes of urban and bare soil land covers were also
determined to be high, corresponding roughly to the forest level

We note that the model may overestimate boundary layer dissipation somewhat
for 30 June and turbulence may not be as capped in value in the model as
indicated from the UHF profiler. The simple model presented here lacks elevated
wind shear in the entrainment zone, which may lead to an underestimation of
dissipation rate and TKE in the upper parts of the boundary layer. Elevated
shear may, however, also affect the entrainment process and the entrainment
parameter which here has been simply taken as a constant value of

An apparently important result from this study is that the modeled
decay of dissipation rate and TKE occurs first at the upper part of
the boundary layer during the afternoon as a response to the
diminishing surface buoyancy flux forcing. This may, of course, in
reality be prevented by the presence of elevated wind shear, but on
most of the days it is also observed by the UHF
profiler.

The residual layer is defined as the statically neutral layer, characterized
by weak sporadic turbulence, that lies above the stable boundary layer and
below the capping inversion, separating the boundary layer flow from the free
atmosphere. By definition, it begins to develop only after the surface begins
to stably stratify. Therefore, it is useful to also name the region of weak
turbulence that exists during unstable conditions preceding the residual
layer as the

In this section, we first discuss the setup and results of modeled
near-surface TKE for the afternoon based upon complementary idealized
numerical simulations in Sects. 6.1–6.5. Secondly, we comment upon our
results in relationship to turbulence decay laws in Sect. 6.6. We also
compare our numerical model results to a simplified analytical expression
assuming quasi-stationary turbulence in Sect. 6.7. Here, we also illustrate
and discuss the added value of our modeling efforts taking into account
variations in wind or

Simulation settings for model experiment test runs.

Simulation settings for model experiment test runs.

The sensible heat flux used in these model runs are provided by
a cosine function as in

Here,

For boundary layer depth, we specify a very simple sine function
increase of

The complementary idealized numerical simulations have been performed
by systematically varying a studied parameter while keeping all the
other variables specified according to a reference simulation. We
begin by providing the details of the reference simulation. For this
simulation, we keep the wind speed constant at 2

We conduct six different types of model experiment test runs denoted
by AL (afternoon length), BLD (boundary layer depth), SH (sensible
heat flux),

The lower row shows the evolution of modeled TKE at
2

Only 2

Our BLD runs showed an increase in midday TKE from about 0.8 to
1.3

In the lower row of Fig.

Turning now to the cases of

The figure shows a comparison between the numerically
modeled TKE and the simplified analytical quasi-stationary
expression of Eq. (

Results from the slightly more complicated situation of TKE evolution
in the case of a linearly increasing wind speed during the afternoon
from zero to a specified value at the end of the afternoon
(

Finally, in the plots on the right we show the results from our

The result of an approximately linear decay of TKE in time can be very
instructive to consider in relationship to previous modeling results from

All results presented here concern the TKE decay near the surface during the
afternoon transition with still unstable conditions.

To better understand our numerical model results for TKE, near the surface we
can compare our numerical results to a simple analytical expression for TKE
which comes from assuming quasi-stationarity such that

Left: observed

Here,

For a convective boundary layer with little shear production our expression
reduces to

In Fig.

The middle panel of Fig.

The right panel of Fig.

It is not possible to conclude with the evaluation metrics used here
whether the TKE prediction was improved by including

This study presents a simple one-dimensional model to investigate
atmospheric turbulence kinetic energy in sheared convective boundary
layers. Similar to a previously proposed heuristic model for the
surface layer TKE decay

In the present work, the model was first run constrained by observations from
the BLLAST field campaign for nine IOP days with relatively successful
results for a near-surface TKE observed at 2.23

The simple TKE model with all its discussed deficiencies still often yields
quite realistic predictions of the overall evolution of boundary layer TKE
and dissipation rate throughout the ABL depth. Further evaluation and
investigation into the differences between the model and observations (as
well as differences between different instrument estimates) may be needed.
Currently, the model tends to predict quite high TKE dissipation rates when
forced by observations from a forest site and wheat field. Also, using
a

The model was used to illustrate its usefulness in understanding afternoon
transition physics. The simple model was used to identify a region of reduced
turbulence that starts in the upper parts of the boundary layer (but below
the capping inversion), which moves down with time toward the surface. This
phenomenon was conceptually described as a

The model was further used in an idealized setting to illustrate the effects
of relative amounts of shear and buoyancy for near-surface TKE at a height of
2

We simplified our numerical model results to an analytical expression for
quasi-stationary near-surface turbulence in Eq. (

In reality, atmospheric turbulence kinetic energy is governed by many
parameters, some of which have been included in the presented numerical model
for TKE and others, such as horizontal advection and elevated wind shear,
which remain to be included. In addition, TKE also has some memory of the
history of the flow that we neglect when using Eq. (

Metadata and data from the BLLAST campaign are available after registration
at:

For a qualitative comparison to our proposed height variation in TKE budget
terms we provide here the simple model from

Their expression for buoyancy production is given by

For shear production they use

For dissipation rate their expression is

In addition, for transport they use

This model from

Normalized terms in the turbulence kinetic energy equation
from

Normalized terms in the turbulence kinetic energy equation
from

The first author thanks ANR for funding this postdoctoral work and
would also like to thank Jordi Vilà-Guerau de Arellano and Arnold
Moene at Wageningen University for fruitful discussions about this
work during a research visit in December 2014. The BLLAST field
experiment was made possible thanks to the contribution of several
institutions and sources of support: INSU-CNRS (Institut National des Sciences
de l'Univers, Centre national de la Recherche Scientifique,
LEFE-IMAGO program), Météo-France, Observatoire
Midi-Pyrénées (University of Toulouse), EUFAR (EUropean
Facility for Airborne Research) BLLATE-1&2, and COST ES0802 (European
Cooperation in Science and Technology). The field
experiment would not have occurred without the contribution of all
participating European and American research groups, which all have
contributed to a significant extent. The BLLAST field experiment was
hosted by the instrumented site of Centre de Recherches
Atmosphériques, Lannemezan, France (Observatoire
Midi-Pyrénées, Laboratoire d'Aérologie). Its
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