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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

Prt3/2

,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

21

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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