Analysis of the Influence of the Length Scales in a Boundary-Layer Model

Abstract

We consider the Janjic (NCEP Office Note 437:61, 2001) boundary-layer model, which is one of the most widely used in numerical weather prediction models. This boundary-layer model is based on a number of length scales that are, in turn, obtained from a master length multiplied by constants. We analyze the simulation results obtained using different sets of constants with respect to measurements using sonic anemometers, and interpret these results in terms of the turbulence processes in the atmosphere and of the role played by the different length scales. The simulations are run on a virtual machine on the Chameleon cloud for low-wind-speed, unstable, and stable conditions.

Full text

Boundary-Layer Meteorology (2021) 179:385–401
https://doi.org/10.1007/s10546-020-00602-0
RESEARCH ARTICLE
Analysis of the Influence of the Length Scales in a
Boundary-Layer Model
Enrico Ferrero1
·Massimo Canonico1
Received: 2 February 2020 / Accepted: 29 December 2020 / Published online: 11 February 2021
© The Author(s) 2021
Abstract
We consider the Janjic (NCEP Office Note 437:61, 2001) boundary-layer model, which is
one of the most widely used in numerical weather prediction models. This boundary-layer
model is based on a number of length scales that are, in turn, obtained from a master length
multiplied by constants. We analyze the simulation results obtained using different sets of
constants with respect to measurements using sonic anemometers, and interpret these results
in terms of the turbulence processes in the atmosphere and of the role played by the different
length scales. The simulations are run on a virtual machine on the Chameleon cloud for
low-wind-speed, unstable, and stable conditions.
Keywords Boundary-layer model ·Length scales ·Mellor–Yamada model
1 Introduction
The modelling of turbulent flow is a difficult task because of the complexity of the physical
processes involved and the non-linearity of the governing equations. Turbulence is therefore
generally described by statistical methods that involve solving a set of dynamical equations
for the moments of the joint probability density functions (p.d.f.) of the velocity components
and temperature. These methods are based on the so-called Reynolds-averaged Navier–Stokes
models, which were first proposed by Reynolds. Similar models have been developed in many
different fields, ranging from stellar astrophysics (Canuto 1992; Kupka and Montgomery
2002; Kupka 2003; Kupka and Robinson 2006), to oceanography (Canuto et al. 2001,2002,
2007), and the planetary boundary layer (PBL), starting from the work of Mellor and Yamada
(1974) to more complex higher order models (Canuto and Cheng 1994; Ferrero and Racca
2004; Cheng et al. 2005; Ferrero 2005; Gryanik et al. 2005; Ferrero and Colonna 2006;
Ferrero et al. 2009,2011).
BEnrico Ferrero
enrico.ferrero@uniupo.it
Massimo Canonico
massimo.canonico@uniupo.it
1Dipartimento di Scienze e Innovazione Tecnologica, Università del Piemonte Orientale, Viale Teresa
Michel, 11, Alessandria, Italy
123
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386 E. Ferrero, M. Canonico
Different closure relations can be found in the literature, and these can be divided into two
categories: local and non-local models. The first category consists of models that include
equation for the second-order moments and that either neglect or adopt a parametrization
for the third-order moments (Mellor and Yamada 1982; Byggstoyl and Kollmann 1986;
Jones and Musange 1988; Durbin 1993; Canuto and Cheng 1994; Speziale et al. 1996;
Canuto et al. 1999; Trini Castelli et al. 2001,2005). Unfortunately, local models are not
sufficient to describe complex mixing processes such as those that occur in convection (Moeng
and Wyngaard 1989; Holstag and Boville 1993), and locality is an approximation that is
only justified when the turbulent mixing length is significantly smaller than the length scale
of heterogeneity in the mean flow. In fact, turbulence in the convective boundary layer is
associated with the non-local properties of the boundary layer as a whole (Holstag and
Moeng 1991; Moeng and Sullivan 1994; Schmidt et al. 2006), which means that mixing is
also present in regions of the boundary layer where there is no local production of turbulence
due to the non-local effects of the vertical transport. In previous studies (Ferrero and Racca
2004; Ferrero 2005; Ferrero et al. 2009), we have shown that non-locality plays an important
role in the shear-driven and stable boundary layers, despite the smaller-scale structures present
in this kind of flow.
Non-locality has been included in local models either by means of parametrizations (Dear-
dorff 1972; Holstag and Moeng 1991; Wyngaard and Weil 1991; Canuto et al. 2005), or by
considering the dynamical equations for the third-order moments (Canuto 1992; Canuto and
Cheng 1994; Zilitinkevich et al. 1999;Canutoetal.2001; Cheng et al. 2005; Gryanik et al.
2005; Colonna et al. 2009; Ferrero et al. 2009,2011), which, in Reynolds-stress formalism,
represent the turbulent transport in the second-order-moment equations, and hence account
for the non-locality of the turbulence. One shortcoming of these models is the large number
of equations that need to be solved numerically, and the high computational demand. This
can be a prohibitive task when the Reynolds-averaged Navier–Stokes models are included in
circulation models in order to provide a detailed description of the boundary-layer phenom-
ena. Although the PBL models entail, at most, one dynamical equation, that for turbulent
kinetic energy (TKE), they are applied in situations such as in complex orographic terrain
or the urban environment, i.e., in horizontally non-homogeneous areas, where, in princi-
ple, traditional boundary-layer parametrizations are no longer valid. In fact, deformations
in turbulent fields that are induced by inhomogeneities modify the spatial distribution of
turbulence, whose physical complexity demands the development of at least a second-order
model (Ying et al. 1994; Ying and Canuto 1995,1997; Cheng et al. 2020).
In a previous work (Ferrero et al. 2018), we analyzed the PBL schemes in the Weather
Research and Forecasting model (Skamarock and Klemp 2008). Herein, we consider the
Janjic (2001) model, which is one of the most commonly used boundary-layer models, and is
based on a number of length scales obtained from a master length scale multiplied by some
constants. We have evaluated the influence of these length scales on model performances. It
is worth noting that these length scales rely on the pressure (A1,A2)and dissipation terms
(B1,B2)in the original model equations (Mellor 1973), and account for the Rotta (1951a,b)
hypothesis of return-to-isotropy and the Kolmogorov (1941) hypothesis of local, small-scale
isotropy. The pressure term was called the energy redistribution term by Rotta (1951a,b)as
one of its functions is to partition energy among the three components while not contributing
to its total. The choice of these scales may therefore influence model performance in non-
homogeneous conditions in which the redistribution of energy can also be different along the
two horizontal components. Furthermore, the local isotropy hypothesis can be less realistic
in horizontal non-homogeneous conditions, such as those considered here.
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Analysis of the Influence of the Length Scales in a Boundary… 387
Below, model results obtained with different sets of constants are compared with sonic-
anemometer measurements. The simulations were performed using a virtual machine called
Chameleon Cloud. We introduce the formulation of the model in Sect. 2. The data used for
the comparison and the model set-up are presented in Sect. 3, while the results are shown in
Sect. 4. The main conclusions are drawn in Sect. 5.
2 Formulation of the Model
Janjic (2001) modified the Mellor–Yamada level-2.5 parametrization (Mellor and Yamada
1974,1982), in order to identify the minimum conditions required for satisfactory perfor-
mance in the full range of atmospheric forcing. This is achieved by imposing an upper limit
on the master length scale, which is proportional to the square root of TKE and a function
of the buoyancy and the wind shear. The TKE production/dissipation differential equation is
solved iteratively over one timestep. The differential equation that is obtained by linearizing
around the solution from of the previous iteration is solved at each individual iteration. Two
iterations appear to be sufficient for satisfactory accuracy, and the computational cost is minor
(Janjic 2001).
There is a set of empirical constants in the equations of the model, (A1,A2,B1,B2,C1)
that should be determined, with each of these constants in the model multiplied by the so-
called master length scale, providing a set of length scales that appear in the equations (Mellor
and Yamada 1982).
In Mellor and Yamada (1982), the constants B1and B2are defined as
B1=RBF2
B
Prt3/2
,B2=B1
RB
,(1)
where F2
B=u2
∗<θ
2>/< wθ >2=3.167441983, Prtis the turbulent Prandtl number,
and RB=B1/B2=16.6/10.1.
If we suppose (Mellor and Yamada 1982)
γ1=1
3−2A1
B1
=1
3−1
9,(2)
it is possible to calculate the other constants as
A1=B1
21
3−γ1,(3)
C1=γ1−1
(3A1)B−1/3
1,(4)
and
A2=A1(γ1−C1)
γ1Prt
.(5)
According to Mellor and Yamada (1982), the critical flux Richardson number is
Ri fc=B1−6A1
B1+3B2+12A1
.(6)
If Pbis buoyancy production and Psis shear production, the flux Richardson number is
Ri f=−Pb
Ps
.(7)
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388 E. Ferrero, M. Canonico
Table 1 Sets of constants used in
the simulations A1A2B1B2C1PrtRfc
MY82 0.92 0.74 16.6 10.1 0.008 0.8 0.19
BASE 0.66 0.66 11.9 7.2 0.001 1 0.19
TCF 2.14 0.64 35.9 61.0 0.167 0.48 0.09
PR074 1.04 0.76 18.7 11.4 0.101 0.74 0.19
Different sets of constants can be found in the literature. In this work, we consider the sets
from Mellor and Yamada (1982, MY82), Janjic (2001, BASE), and Trini Castelli et al. (1999,
2001, TCF). The values of the constants are listed in Table 1. The PR074 set of constants
is obtained by modifying those of Janjic (2001) with an imposed Prt=0.74 instead of
Prt=1. The value 0.74 was suggested by Kestin and Richardson (1963). Generally, it is
possible to obtain a set of constants for any value of Prt.
3 Data for Comparison and Model Set-up
3.1 Data for Comparison
For the sake of comparison, we have used the Urban Turbulence Project (UTP) (Mortarini et al.
2013; Trini Castelli et al. 2014) dataset, based on an experiment conducted on the outskirts
of Turin, Italy, providing sonic-anemometer data at three different heights (5, 9, 25 m) above
ground. A pre-processing procedure was carried out to give hourly mean wind speed and
direction, as well as temperature and standard deviations of the velocity fluctuations, which
were collected for more than one year. The period from 0600 UTC on 26 January to 1800
UTC on 31 January 2008 was considered, and the comparison was performed with data at a
height of 25 m.
The site is located on the western edge of the Po Valley (northern Italy) at about 220
m above mean sea level (a.m.s.l.). A hill chain (maximum altitude of about 700 m a.m.s.l.)
surrounds Turin on the eastern sector, while the Alps (whose crest line is about 100 km away)
encircle it in the other three sectors. The city of Turin covers an area of about 130 km2and has
a population of about 900,000 inhabitants. The respective figures for the Turin conurbation
area are 1100 km2and 1,700,000 inhabitants. The area in which the instruments are located
is in the southern outskirts of the town on grassy, flat terrain surrounded by buildings. There
are buildings of about 30 m in height located 150 m away in the north-north-east direction,
and sparse lower buildings (heights varying from 4 to 18 m) at a distance of 70–90 m in
the other sectors. The site is characterized by horizontally non-homogeneous conditions.
Surface roughness was estimated to be in the rough and very rough categories, depending on
the incoming wind direction, and the urban fabric can be classified as being in the low-density
class. Turin is characterized by frequent occurrences of low-wind-speed conditions. More
than 80% of the data indicate a mean wind speed of <1.5 m s−1at 5 m, which is probably
due to the shielding effect of the surrounding mountain and hill chains. The few cases of
high wind speeds are related to the dry downslope slope flow from the Alps, the foehn,
which occurs in the cold season typically about 15 times per year. The highest measurements
almost correspond to the mean level of the surrounding roofs, and are thus only likely to be
influenced a small amount by the buildings.
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Analysis of the Influence of the Length Scales in a Boundary… 389
3.2 Simulations
The WRF model was run with five nested grids with horizontal spacings of 30000, 10000,
3333.33, 1111.11, 370.37 m, and 133 ×133 grid points. All the grids have 35 vertical levels,
up to an altitude of about 30000 m. The first vertical levels were at 0.2, 54, 132, 234, and 363
m, and there were eight levels in the first 1000 m in order provide enough resolution for com-
parisons between the model results and the observations, making it consistent with previous
work (see for instance Kleczek et al. 2014; Ferrero et al. 2018). For the microphysics, we
choose the WRF single-moment class 3 model (Hong et al. 2004), while the rapid radiative
transfer model (Mlawer et al. 1997) was selected for the longwave radiation, and the Dudhia
option (Dudhia 1989) was used for the shortwave radiation. The surface layer was simu-
lated using the Janjic–Monin–Obukhov Eta (Janjic 1996) parametrization, while the Unified
Noah land-surface model was used for the surface physics with the Kain–Fritsch (new eta)
parametrization, (Kain 2004) for the cumulus parametrization, only to the first and second
grids. No data were assimilated and the boundary conditions were provided by the National
Oceanic and Atmospheric Administration Global Data Assimilation System reanalysis. The
comparison with the observed data was carried out for a winter period of about 6 days. The
prevailing meteorological conditions in winter are stable and are often characterized by low
wind speed; and meteorological models generally have problems performing satisfactorily
in these conditions. While the experimental campaign provides data at three different levels,
these levels are too close to each other and to the ground to allow the vertical gradients to be
analyzed. The meteorological fields were saved hourly. The meteorological variables were
extracted from the finer grid fields and interpolated, both vertically and horizontally, to the
sonic-anemometer position. In this way, we obtained a time series that could be compared
with the observations.
In order to run the WRF model we used Cloud Computing, which has been defined as “a
model that can provide convenient, on-demand network access to a shared pool of config-
urable computing resources (e.g., networks, servers, storage, applications, and services) that
can be rapidly provisioned and released with minimal management effort or service-provider
interaction” (Mell and Grance 2011). In other words, Cloud Computing can easily satisfy
the huge amount of storage and computational power required by modern applications. This
computational paradigm is now used in many fields, and users do not need to have any com-
puting skill to use the cloud infrastructure itself. Moreover, in order to exploit different cloud
infrastructures concurrently, the scientific literature has proposed various projects (Canonico
et al. 2013; Canonico and Monfrecola 2016), some of which have made their source code
available and have simple text–user interfaces, making the interactions with the major cloud
infrastructures simple (The UPO’s Distributed Computing Systems group 2020; Anglano
et al. 2020). The advantages of using this system include flexibility and the possibility of
setting up a virtual machine with specific characteristics and capacity needed for specific
case. Furthermore, it is a low-cost and low-impact system. One aim of this work is, in fact,
to investigate how well numerical weather prediction models run on this system. For our
simulation, we used a virtual machine that runs Ubuntu 16.04 as its operating system and
was equipped with 32 GB of system memory, eight cores, and 160 GB of storage.
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390 E. Ferrero, M. Canonico
4 Results
Below, the results are divided according to the weather and turbulence conditions in the cases
of low-wind-speed, unstable, and stable conditions. Low-wind-speed conditions are defined
as the cases in which the wind speed is <1.5 m s−1(Mortarini et al. 2013), while stability
conditions are determined on the basis of the Obukhov length. It can be observed that the
three data subsets are mutually exclusive when stable and unstable conditions are considered,
but not concerning a low wind speed, which can arise in various stability conditions.
4.1 Low-Wind-Speed Conditions
Quantile–Quantile (Q–Q) plots for the various parametrizations shown in Table 1for the
wind speed, temperature, and TKE, in the case of low wind speed, can be observed in Fig. 1a.
Although all of the models reproduce the distribution of wind speeds at low values, none are
able to correctly reproduce the entire data distribution. The models provide wind speeds >
1.5 m s−1for a considerable number of cases, where the corresponding measured values are
lower than this threshold. As far as temperature is concerned, a general underestimation can be
observed in all the models, without there being particular differences in one parametrization
compared with the others. This underestimation has also been found previously (see for
instance Kleczek et al. 2014; Ferrero et al. 2018), and may be attributed to limitations in the
PBL models that do not explicitly determine the heat flux. This is usually corrected through
the assimilation of measured data, which, in our case, was not carried out, as our aim is to
verify the performance of the PBL models. The results obtained for TKE are perhaps even
more interesting. In this case, we can observe how all the parametrizations, with the exception
of the TCF model, underestimate the measured value. In fact, the TCF model satisfactorily
reproduces the distribution of the observations.
The constant B1is used in the following definition of the dissipation rate,
=q2
B1l,(8)
where q2is twice the TKE and lis the master length scale. Table 1shows that the value of B1
prescribed by the TCF model is about double that found in the other sets of constants, which
means that the dissipation of TKE is lower than in the other models, and that, consequently,
more TKE is generated.
These results are summarized in Table 2which shows the values obtained for the statistical
indices. The calculated and simulated average values, the bias b, and the correlation coefficient
Rhave been taken into consideration in this analysis. The metrics have also been calculated
for the wind direction and can be seen in the table. Since it is a cyclic variable, we checked
whether the bvalues correspond to a real physical difference. All models overestimate the
wind speed (negative bias) and underestimate the temperature by 6 K. Wind direction also
shows a similar bias for all the models except for the TCF model, which yields a lower value.
As for the TKE, a positive bias affects all the models but is lower for the TCF model. The
correlation coefficient is very low for all of the variables and models, while Ris slightly better
for temperature.
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Analysis of the Influence of the Length Scales in a Boundary… 391
012345
012345
Wind speed
Data (m s 1)
Model (m s1)
BASE
TCF
PR074
MY82
270 280 290 300
270 280 290 300
Temperature
Data (K)
Model (K)
0.0 0.5 1.0 1.5
0.0 0.5 1.0 1.5
TKE
Data (m2 s 2)
Model (m2 s 2)
(a) Low wind speed
012345
012345
Wind speed
Data (m s 1)
Model (m s1)
BASE
TCF
PR074
MY82
270 280 290 300
270 280 290 300
Temperature
Data (K)
Model (K)
0.0 0.5 1.0 1.5
0.0 0.5 1.0 1.5
TKE
Data (m2 s 2)
Model (m2 s 2)
(b) Unstable
012345
012345
Wind speed
Data (m s 1)
Model (m s1)
BASE
TCF
PR074
MY82
270 280 290 300
270 280 290 300
Temperature
Data (K)
Model (K)
0.0 0.5 1.0 1.5
0.0 0.5 1.0 1.5
TKE
Data (m2 s 2)
Model (m2 s 2)
(c) Stable
Fig. 1 Quantile–Quantile plots for wind speed (m s−1), temperature (K), and TKE (m2s−2), for alow-wind-
speed, bunstable, and cstable conditions. Green: BASE, black: TCF, blue: PR074, red: MY82
4.2 Unstable Conditions
The Q–Q plots that correspond to the various simulations for the unstable cases are displayed
in Fig. 1b for the wind speed, temperature and TKE. It can be seen that all of the models
correctly reproducing the distribution of the observations for wind speed, but with a slight
overestimation of the maximum values. The temperature, as in the previous case, is under-
estimated to the same extent by all of the models. However, this is not the case for the TKE,
whose Q–Q plots differ according to the set of constants used. The TCF set of constants
provides a remarkable result, with excellent agreement for all quantiles. Moreover, the result
obtained by the PR074 model is also satisfactory, despite underestimating the maxima. The
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392 E. Ferrero, M. Canonico
Table 2 Metrics for
low-wind-speed conditions and
for wind speed (U) and direction
(Dir), for temperature (T)and
TKE (e)
bR
BASE
U(ms
−1)−0.37 −0.04
Dir (◦)8−0.02
T(K)60.38
e(m2s−2)0.08 0.06
TCF
U(ms
−1)−0.32 0.035
Dir (◦)30.012
T(K)60.35
e(m2s−2)0.01 0.06
PR074
U(ms
−1)−0.43 −0.011
Dir (◦)11 0.030
T(K)60.35
e(m2s−2)0.06 0.05
MY82
U(ms
−1)−0.38 −0.011
Dir (◦)7−0.0013
T(K)60.39
e(m2s−2)0.07 0.05
performance of the two other models is worse and shows a general underestimation. As in
the case of low wind speed, it can be noted that the TCF model provided the highest value for
the B1constant. Furthermore, the PR074 set of constants includes a B1value that is higher
than those of the MY82 and BASE sets of constants.
The results for the metrics are presented in Table 3for unstable cases, illustrating that
the best performance for wind speed and direction, in terms of the bias, is provided by the
TCF set of constants, while this model gives a bias for the temperature that is slightly higher
than that of the other models. As for the TKE, the TCF model overestimates it, while the
other models underestimate the measured value. The PR074 set of constants provides the
best result.
4.3 Stable Conditions
The Q–Q plots for the stable cases are presented in Fig. 1c. The distribution of wind speeds is
well reproduced by all of the models which, conversely, underestimate that of the temperature
equally. Even the TKE is underestimated regardless of the set of constants used. The TCF
set of constants provides slightly better performance in this case. Table 4indicates that the
PR074 model overestimates the wind speed, which is, however, underestimated in the other
cases. For the wind direction, all of the sets of constants show a large bias. It is interesting to
note that the difficulty that the models have in simulating the wind direction here is neither
observed under unstable conditions nor at low wind speeds. This fact may be related to sub-
mesoscale motion, which induces horizontal oscillations in the wind direction (Cava et al.
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Analysis of the Influence of the Length Scales in a Boundary… 393
Table 3 Metrics for unstable
conditions and for wind speed
(U) and direction (Dir), for
temperature (T)andTKE(e)
bR
BASE
U(ms
−1)−0.08 −0.032
Dir (◦)−40.018
T(K)60.46
e(m2s−2)0.08 0.24
TCF
U(ms
−1)−0.07 0.043
Dir (◦)−0.13 −0.047
T(K)70.41
e(m2s−2)−0.05 0.25
PR074
U(ms
−1)−0.16 −0.032
Dir (◦)1−0.028
T(K)60.41
e(m2s−2)0.04 0.25
MY82
U(ms
−1)−0.08 −0.054
Dir (◦)−1−0.022
T(K)60.46
e(m2s−2)0.06 0.24
2019). Except for temperature, the correlation coefficient is very low for all of the variables
and models.
4.4 Bootstrap Test
In order to demonstrate that a given model configuration is better than the alternatives, atten-
tion should be paid to demonstrating the statistical significance of the results. We performed
a bootstrap test on the bias of the wind speed and TKE. We computed the metric for two
models and noted the difference. This step was then repeated 104times, and each time the
dataset was reshuffled via resampling with replacement. This bootstrap procedure yielded
a p.d.f. of the metric differences; if the value zero is not in the tails of the distribution, the
difference in skill between the two models is not significant (Wilks 2011).
The results show that the different sets of constants do not provide statistical significance
for the bias and for the different meteorological conditions (stable, unstable, and low wind
speed) for wind speed. This result is to be expected as the PBL model is the same in all
the cases, and only the scales of the turbulence can be directly influenced by modifying the
value of the constants. As for the TKE, the results of the bootstrap test are shown in Fig. 2
for low-wind-speed, unstable, and stable conditions. The p.d.f. of the difference in the bias
of the model pairs is shown with each model corresponding to a set of constants, as listed in
Table 1.
On the one hand, for low-wind-speed conditions (black line in Fig. 2), it can be seen that
the difference in the bias is significant for the BASE and TCF models, and the TCF and
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394 E. Ferrero, M. Canonico
Table 4 Metrics for stable
conditions and for wind speed
(U) and direction (Dir), for
temperature (T)andTKE(e)
bR
BASE
U(ms
−1)0.02 0.086
Dir (◦)−25 −0.029
T(K)70.21
e(m2s−2)0.16 −0.14
TCF
U(ms
−1)0.07 0.097
Dir (◦)−29 0.089
T(K)70.23
e(m2s−2)0.13 −0.07
PR074
U(ms
−1)−0.02 0.037
Dir (◦)−27 0.041
T(K)70.23
e(m2s−2)0.11 −0.080
MY82
U(ms
−1)0.03 0.021
Dir (◦)−31 0.052
T(K)70.22
e(m2s−2)0.16 −0.14
MY82 models, and to a lesser extent for the TCF and PR074 models. On the other hand,
the bias differences in the other models are not significant. Thus, the TCF set of constants
provides a bias whose statistics differs significantly from the bias given by all the other sets
of constants. Similar results are obtained for unstable conditions (red line in Fig. 2)andstable
conditions (blue line in Fig. 2). The only set of constants that provides significant differences
in the bias, compared with the others, is the TCF model.
4.5 Normalized Mean Square Error Versus Fractional Bias
Figure 3shows the normalized mean square error versus the fractional bias (Chang and Hanna
2004), which are often used as a single plot to indicate overall relative model performance,
for the three meteorological conditions considered. The two indexes are defined as
FB =2(Co−Cp)
(Co+Cp),(9)
NMSE =(Co−Cp)2
CpCp
,(10)
where Coare the observations, Cpthe model predictions, and the overline indicates the
average. A perfect model would have FB =NMSE =zero. The minimum NMSE value,
(i.e. without any unsystematic errors) for a given FB (Chang and Hanna 2004) is also reported
(continuous line) in Fig. 3
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Analysis of the Influence of the Length Scales in a Boundary… 395
-0.2 -0.1 0.0 0.1 0.2
0 20406080
BIAS DIFFERENCE BASE-TCF (m2s2)
p.d.f.
Low wind speed
Stable
Unstable
(a)
-0.2 -0.1 0.0 0.1 0.2
020 40 60 80
BIAS DIFFERENCE BASE-PR074 (m2s2)
p.d.f.
(b)
-0.2 -0.1 0.0 0.1 0.2
0 20406080
BIAS DIFFERENCE BASE-MY82 (m2s2)
p.d.f.
(c)
-0.2 -0.1 0.0 0.1 0.2
0 20406080
BIAS DIFFERENCE TCF-PR074 (m2s2)
p.d.f.
(d)
-0.2 -0.1 0.0 0.1 0.2
020 40 60 80
BIAS DIFFERENCE TCF-MY82 (m2s2)
p.d.f.
(e)
-0.2 -0.1 0.0 0.1 0.2
0 20406080
BIAS DIFFERENCE PR074-MY82 (m2s2)
p.d.f.
(f)
Fig. 2 Probability density functions of the difference in bias for the TKE for the model combinations a
BASE–TCF, bBASE–PR074, cBASE–MY82, dTCF–PR074, eTCF–MY82, and fPR074–MY82
NMSE =4FB2
4−FB2.(11)
It can be seen that the best performance is obtained for temperature whose values are all
close to the origin of the graph. However, it should be noted that this result does not reflect
the systematic underestimation that was previously observed. Since the values of FB and
NMSE are normalized by the average absolute temperatures in the observation and forecast
samples, their numerical values are very low. This depends merely on the measurement unit
(K) used for temperature. As far as wind speed is concerned, the results are similar under the
three meteorological conditions, except for the underestimation shown in low-wind-speed
conditions. The wind direction has a higher NMSE value in stable conditions than under
other conditions. There are no noticeable differences between the different models for wind
speed and direction, and for temperature. By contrast, there are differences in the results
provided by the different sets of constants for the TKE. The best performance is found
for TCF constants under low-wind-speed and stable conditions, while the PR074 constants
provide the best results for unstable conditions. It can be noted that all models reproduce
observations for the TKE in stable conditions with large errors, which reflects the fact that
the values of the constants are generally not calibrated for such conditions. Similarly, the
results in low-wind-speed conditions are worse than those in unstable conditions. It is worth
noting that, while the literature on turbulence in unstable or neutral conditions is very rich,
stable and low-wind-speed conditions are still challenging fields of research.
4.6 Taylor Diagrams
Comparing the performance of the four models using a Taylor diagram, in which the radial
distance between the centre of the model symbol and the origin indicates the normalized
standard deviation may be worthwhile. The distance from the centre of the model symbol to
123
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396 E. Ferrero, M. Canonico
-1.0 -0.5 0.0 0.5 1.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0
Low wind speed
FB
NMSE
(a)
-1.0 -0.5 0.0 0.5 1.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0
Unstable
FB
NMSE
(b)
-1.0 -0.5 0.0 0.5 1.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0
Stable
FB
NMSE
(c) Wind speed
Temperature
TKE
BASE
TCF
PR074
MY82
Fig. 3 Plots of NMSE versus the fractional bias FB based on the wind speed (squares), temperature (triangles),
and TKE (diamonds) for alow-wind-speed, bunstable, and cstable conditions. The continuous line represents
the minimum possible NMSE value for a given value of FB
the large solid circle indicates normalized root-mean-square error. The cosine of the angle
formed by the horizontal axis and segment normalized standard deviation indicates the cor-
relation coefficient. A black open circle on the horizontal axis indicates the perfect model.
Figure 4shows the Taylor diagram for the different meteorological conditions and the four
models. Generally speaking, the results show very low correlation, albeit slightly better for
temperature, as indicated by the alignment of different models along the vertical axis. The
Taylor diagrams reflect the low correlation coefficient found in the statistical analyses. It is
worth noting that we have compared the model results with data from a single measurement
station, which can be very difficult to reproduce. The results can probably be improved by
assimilating the observations, as is generally done in the operative models. However, herein,
we are interested in comparing the performance of PBL models and, for this reason, we
cannot force the model results by using assimilation. Very small differences appear for the
wind speed, while some discrepancies are found for the TKE. In addition to the observations
made above, the points representing the different models are very close to each other for
wind speed and temperature. On the one hand, as has already been shown, the differences
in the bias for wind speed in the models is not significant. On the the other hand, the Taylor
diagrams for the TKE show remarkable differences in model performance. This also means
that turbulence is determined mainly by the PBL models, which include the TKE equation,
123
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Analysis of the Influence of the Length Scales in a Boundary… 397
Wind speed
Low wind speed
Normalized Standard Deviation
Normalized Standard Deviation
0.0 0.5 1.0 1.5 2.0 2.5
0.0 0.5 1.0 1.5 2.0 2.5
1
2
0.1 0.2 0.3 0.4
0.5
0.6
0.7
0.8
0.9
0.95
0.99
Correlation
(a)
BASE
TCF
PR074
MY82
Temperature
Low wind speed
Normalized Standard Deviation
Normalized Standard Deviation
0.0 0.5 1.0 1.5
0.0 0.5 1.0 1.5
0.5
1
1.5
0.1 0.2 0.3 0.4
0.5
0.6
0.7
0.8
0.9
0.95
0.99
Correlation
(b) TKE
Low wind speed
Normalized Standard Deviation
Normalized Standard Deviation
0.0 0.5 1.0 1.5
0.0 0.5 1.0 1.5
0.5
1
1.5
0.1 0.2 0.3 0.4
0.5
0.6
0.7
0.8
0.9
0.95
0.99
Correlation
(c)
Correlation
Correlation
Correlation
Coefficient
Coefficient
Coefficient
Wind speed
Unstable
Normalized Standard Deviation
Normalized Standard Deviation
0.0 0.5 1.0 1.5
0.0 0.5 1.0 1.5
0.5
1
1.5
0.1 0.2 0.3 0.4
0.5
0.6
0.7
0.8
0.9
0.95
0.99
Correlation
(a)
BASE
TCF
PR074
MY82
Temperature
Unstable
Normalized Standard Deviation
Normalized Standard Deviation
0.0 0.5 1.0 1.5
0.0 0.5 1.0 1.5
0.5
1
1.5
0.1 0.2 0.3 0.4
0.5
0.6
0.7
0.8
0.9
0.95
0.99
Correlation
(b) TKE
Unstable
Normalized Standard Deviation
Normalized Standard Deviation
0.0 0.5 1.0 1.5
0.0 0.5 1.0 1.5
0.5
1
1.5
0.1 0.2 0.3 0.4
0.5
0.6
0.7
0.8
0.9
0.95
0.99
Correlation
(c)
Tex
t
Correlation
Coefficient
Correlation
Coefficient
Coefficient
Correlation
Wind speed
Stable
Normalized Standard Deviation
Normalized Standard Deviation
0.0 0.5 1.0 1.5
0.0 0.5 1.0 1.5
0.5
1
1.5
0.1 0.2 0.3 0.4
0.5
0.6
0.7
0.8
0.9
0.95
0.99
Correlation
(a)
BASE
TCF
PR074
MY82
Temperature
Stable
Normalized Standard Deviation
Normalized Standard Deviation
0.0 0.5 1.0 1.5
0.0 0.5 1.0 1.5
0.5
1
1.5
0.1 0.2 0.3 0.4
0.5
0.6
0.7
0.8
0.9
0.95
0.99
Correlation
(b) TKE
Stable
Normalized Standard Deviation
Normalized Standard Deviation
0.0 0.5 1.0 1.5
0.0 0.5 1.0 1.5
0.5
1
1.5
0.1 0.2 0.3 0.4
0.5
0.6
0.7
0.8
0.9
0.95
0.99
Correlation
(c)
Correlation
Correlation
Correlation
Coefficient
Coefficient
Coefficient
Fig. 4 Taylor diagrams; green: BASE, black: TCF, blue: PR074, red: MY82
and less by the mean quantities. The TCF set of constants shows the largest normalized
standard deviation and the closest to the perfect model value of unity under all atmospheric
conditions. The lowest normalized root-mean-square error is found for the BASE constants,
and found to be between 1.0 and 1.5.
5 Discussion and Conclusions
5.1 General Conclusion
We present the influence of length scales on the performance of the Janjic (2001) PBL models.
As these length scales are generally based on sets of constants that take different values
depending on the author, different results are obtained when varying the set of constants.
This is especially true for the TKE, which is clearly the most sensitive to the choice of PBL
123
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398 E. Ferrero, M. Canonico
model. Nonetheless, statistical analyses reveal some differences in the results obtained using
the different sets of constants for the other quantities considered (wind speed and direction,
temperature). In particular, an improved TKE can be obtained by suitably varying the length
scales. The best results for the TKE are found using the TCF set of constants, as also found
by Tomasi et al. (2019). It is worth noting that this set of constants was derived from data
measured in non-homogeneous conditions (Trini Castelli et al. 1999). The length scales rely
on the pressure (A1,A2)and dissipation terms (B1,B2)in the original model equations
(Mellor 1973), and account for the Rotta (1951a,b) hypothesis of return to isotropy and the
Kolmogorov (1941) hypothesis of local, small-scale isotropy. In the model, the larger the
A1/B1ratio, the greater the anisotropy. From Table 1, we can see that this ratio is similar for
all of the models except for the TCF set of constants, which gives a slightly higher value, and
may indicate that this set of constants is more able to take the inhomogeneities into account
than the others. The values of B1and B2, which are much higher for the TCF set of constants
than for the other sets, suggest that the dissipation, both of TKE and temperature variance,
is also higher, allowing more TKE to be produced.
5.2 Dependency on the Prandtl and Richardson Numbers
We have also found that the results depend on the value of the turbulent Prandtl number, which
in turn depends on the constants considered. The results seem to suggest that lower Prtvalues
allow better model performance to be obtained. We find that the TCF constants give the lowest
values of the critical Richardson number, Ri fc, which may indicate that turbulence is more
suppressed using this set than using the others. Nevertheless, it has been demonstrated that
turbulence also persists at higher Richardson numbers in the stable boundary layer (Canuto
et al. 2008; Ferrero et al. 2011). In Mellor and Yamada (1982), the model constant A1is
related to the momentum flux and A2to the buoyancy flux. As a consequence, if A1/A2
1, as in the TCF case, mechanical turbulence prevails over the buoyancy term, meaning
that turbulence can persist even in the presence of a negative heat flux under stable conditions.
5.3 Dependency on the Low-Wind-Speed and Stability Conditions
The differences in model skill in stable, unstable, and low-wind-speed conditions show that
unstable conditions give large standard deviations, particularly in the case of the TCF models
(Fig. 4). The largest NMSE value and bias for the TKE are found under stable conditions
(see Figs. 1,3and Tables 2,3,4). Under low-wind-speed conditions, TKE performance
is satisfactory at least when the TCF set of constants is used, although all of the models
have difficulty reproducing the wind-speed distribution (Fig. 1). It is worth noting that these
considerations may only be valid when discussing days in winter. Stable and low-wind-speed
conditions are more common in winter and these are the conditions that we have focused
on, as meteorological models generally have problems providing satisfactory performance
under these conditions.
5.4 Final Remarks
Although the experimental campaign has provided data at three different levels (5, 9, 25 m),
these are too close to the ground to allow the vertical gradients to be analyzed. Nevertheless,
the results of this work show that the choice of the set of constants can make an important
123
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Analysis of the Influence of the Length Scales in a Boundary… 399
difference. The degree of anisotropy and the dissipation rate of TKE and temperature variance
depend on the constant values whose proper selection improves the performance of the
model in conditions that are different from those assumed in the usual PBL models, such as
conditions of horizontal inhomogeneity. Finally, it is worth mentioning that these simulations
were carried out using a cloud computing system, which has many advantages, including
reduced costs and scalability.
Funding Open Access funding provided by Università degli Studi del Piemonte Orientale Amedeo Avogrado.
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which
permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give
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To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/ .
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