11institutetext: CERN

# Studying light flavour hadrons produced in the collision of different nuclei
at the LHC

Nicolò Jacazio For the ALICE Collaboration 11<EMAIL_ADDRESS>
###### Abstract

The study of identified particle production as a function of event
multiplicity is a key tool for understanding the similarities and differences
among different colliding systems. Now for the first time, we can investigate
how particle production is affected by the collision geometry in heavy-ion
collisions at the LHC. In these proceedings, we report newly obtained ALICE
results on charged and identified particle production in Pb–Pb and Xe–Xe
collision at $\sqrt{s_{\rm NN}}=5.02$ and $\sqrt{s_{\rm NN}}=5.44$ TeV,
respectively, as a function of transverse momentum ($p_{\rm T}$) and collision
centrality. Particle spectra and ratios are compared between two different
colliding systems at similar charged-particle multiplicity densities
($\langle{\rm d}N_{\rm ch}/{\rm d}\eta\rangle$), and different initial
eccentricities. We find that in central collisions, spectral shapes of
different particles are driven by their masses. The $p_{\rm T}$-integrated
particle yield ratios follow the same trends with $\langle{\rm d}N_{\rm
ch}/{\rm d}\eta\rangle$ as previously observed in other systems, further
suggesting that at the LHC energies, event hadrochemistry is dominantly driven
by the charged-particle multiplicity density and not the collision system,
geometry or center-of-mass energy.

##  1 Introduction

The ultimate goal of heavy-ion physics is the study of the properties of the
Quark-Gluon Plasma (QGP), a deconfined and chirally restored state of matter.
The measurement of the transverse momentum ($p_{\rm T}$) spectra of identified
particles provides a way into the collective properties and of particle
production in the fireball created in heavy-ion collisions. The ALICE
experiment Aamodt:2008zz ; Abelev:2014ffa  is particularly well suited to
study the production of both identified and unidentified charged particles due
to its excellent tracking performance coupled with extensive particle
identification (PID) capabilities over a wide range of transverse momentum.
Particles are identified by combining different techniques allowing their
continuous separation over a large transverse momentum interval. In 2017 ALICE
recorded Xe–Xe collision at the highest energy per nucleon ever achieved in
the laboratory, allowing for the first time at the LHC a quantitative
comparison of different heavy-ion collision systems.

##  2 Results

We report results on the production of $\pi$, $K$, $p$ and $\phi$ measured in
Xe–Xe collisions at $\sqrt{s_{\rm NN}}\leavevmode\nobreak\
=\leavevmode\nobreak\ 5.44\leavevmode\nobreak\ {\rm TeV}$ as a function of
centrality, as discussed in ALICE:2021lsv . The data sample was recorded with
a minimum-bias trigger. The total charge collected in the V0 detectors, a set
of two scintillator detectors located in the pseudorapidity region
$2.8<\eta<5.1$ (V0A) and $-3.7<\eta<-1.7$ (V0C) and covering the full azimuth,
was used to determine the centrality of each Xe–Xe collision defined as
percentiles of the total hadronic cross section. Further details are given in
ALICE-PUBLIC-2018-003 . Contributions from weak decays of strange particles
and from particle knock-out in the material were removed with the data driven
approach described in ALICE:2021lsv . Systematic uncertainties were estimated
by varying the PID technique and the selection criteria used to define the
track sample. The amount of pile-up per event was reduced by selecting runs
with low interaction rate and by rejecting events with more than one
reconstructed vertex, resulting in a negligible effect. The evaluation of the
efficiency and acceptance corrections was performed using events simulated
with the HIJING HIJING  event generator and embedded into a detailed
description of the ALICE detector through which tracks are propagated with the
GEANT3 GEANT3  transport code.

Figure 1:  Transverse momentum distributions of $\pi^{\pm}$, $K^{\pm}$,
$p,\bar{p}$ and $\phi$ as measured in central (left) and peripheral (right)
Xe–Xe collisions at $\sqrt{s_{\rm NN}}\leavevmode\nobreak\
=\leavevmode\nobreak\ 5.44\leavevmode\nobreak\ {\rm TeV}$.

Figure 2:  Average transverse momentum of $\pi^{\pm}$, $K^{\pm}$, $p,\bar{p}$
and $\phi$ as a function of the charged-particle multiplicity density as
measured in Xe–Xe collisions at $\sqrt{s_{\rm NN}}\leavevmode\nobreak\
=\leavevmode\nobreak\ 5.44\leavevmode\nobreak\ {\rm TeV}$ ALICE:2021lsv  and
compared to Pb–Pb results at $\sqrt{s_{\rm NN}}\leavevmode\nobreak\
=\leavevmode\nobreak\ 5.02\leavevmode\nobreak\ {\rm TeV}$ Acharya:2019qge ;
PhysRevC.101.044907 .

Figure 3:  Left: $p/\phi$ and $p/\pi$ as a function of the transverse momentum
in 0–10% central Xe–Xe collisions at $\sqrt{s_{\rm NN}}\leavevmode\nobreak\
=\leavevmode\nobreak\ 5.44\leavevmode\nobreak\ {\rm TeV}$ ALICE:2021lsv
compared to 10–20% central Pb–Pb collisions at $\sqrt{s_{\rm
NN}}\leavevmode\nobreak\ =\leavevmode\nobreak\ 5.02\leavevmode\nobreak\ {\rm
TeV}$ Acharya:2019qge ; PhysRevC.101.044907  to match the charged particle
multiplicity in the two systems. The same is shown in the right panel for
peripheral collisions.

Figure 4:  Left figure, top panel: $p/\pi$ as a function of charged particle
multiplicity density in two $p_{\rm T}$ intervals for Xe–Xe $\sqrt{s_{\rm
NN}}\leavevmode\nobreak\ =\leavevmode\nobreak\ 5.44\leavevmode\nobreak\ {\rm
TeV}$ ALICE:2021lsv  compared to Pb–Pb collisions at $\sqrt{s_{\rm
NN}}\leavevmode\nobreak\ =\leavevmode\nobreak\ 5.02\leavevmode\nobreak\ {\rm
TeV}$ PhysRevC.101.044907 ; Acharya:2019qge . Left figure, bottom panel: the
flow coefficient $v_{2}\\{2,|\Delta\eta|>2\\}$ reported for the same collision
systems Acharya:2018zuq ; Acharya:2018ihu  as a function of charged particle
multiplicity density. Right figure: $K/\pi$, $p/\pi$ and $p/\pi$ for
integrated yields as a function of the charged-particle multiplicity density
for Xe–Xe collisions at $\sqrt{s_{\rm NN}}\leavevmode\nobreak\
=\leavevmode\nobreak\ 5.44\leavevmode\nobreak\ {\rm TeV}$ ALICE:2021lsv
compared to Pb–Pb collisions at $\sqrt{s_{\rm NN}}\leavevmode\nobreak\
=\leavevmode\nobreak\ 2.76\leavevmode\nobreak\ {\rm TeV}$ Abelev:2013vea ;
Abelev:2014uua  and 5.02 TeV PhysRevC.101.044907 ; Acharya:2019qge .
Predictions from the canonical statistical model (CSM) are shown as bands
considering different correlation volumes Vovchenko:privatecomm  (based on
Vovchenko:2019kes ) and chemical freeze-out temperatures Cleymans:2020fsc .

The $p_{\rm T}$ spectra of identified $\pi$, $K$, $p$ and $\phi$ are shown for
different centrality classes in Fig. 1. A visual comparison of the spectral
shapes reveals that the spectra become harder with increasing centrality. This
hardening is found to be mass dependent with protons being more affected than
pions and experiencing a similar effect as the $\phi$ meson. This is
consistent with the presence of a collective evolution, known also as radial
flow. This effect can be better appreciated in Fig. 2 where the average
transverse momentum extracted from the spectra is shown as a function of the
charged particle multiplicity for each particle and compared to the results
obtained in Pb–Pb collisions. The increase in average $p_{\rm T}$ is the
largest for the heavier particles. In addition, it is striking to observe that
in the two systems the average $p_{\rm T}$ are similar provided that they are
measured at the same multiplicity and particles with similar mass i.e. $p$ and
$\phi$ have similar average $p_{\rm T}$. The effect of radial flow can also be
seen in the baryon/meson ratios shown in Fig. 3 as a depletion of $p/\pi$ at
low $p_{\rm T}$ and an enhancement at higher $p_{\rm T}$. It is worth noting
that the baryon/meson ratio exhibits an almost flat behavior only when
considering particles with similar mass (i.e. $p$ and $\phi$) indicating that
the hardening is driven by the mass and not by the quark content of the
hadron. The $p/\pi$ ratio is shown as a function of multiplicity in Fig. 4
(left) to further illustrate the effect of the radial flow, here it can be
seen that the amount of enhancement (depletion) at high $p_{\rm T}$ (low
$p_{\rm T}$) is compatible in Xe–Xe and Pb–Pb collisions when compared at
similar multiplicity. Conversely, this is not true for the $v_{2}$ that
exhibits a strong dependence on the centrality, indicating a dominant effect
of the initial shape of the collision. Also hadrochemistry is found to be
scaling well with the charged particle multiplicity as it can be seen in Fig.
4 (right). Here a general agreement of the results is found across different
systems and energies. Furthermore, the results are found to be in agreement
with predictions from thermal models.

##  3 Conclusions

In these proceedings, results on the $\pi^{\pm}$, $K^{\pm}$, $p,\bar{p}$ and
$\phi$ production measured as a function of centrality in Xe–Xe collisions at
$\sqrt{s_{\rm NN}}\leavevmode\nobreak\ =\leavevmode\nobreak\
5.44\leavevmode\nobreak\ {\rm TeV}$ were reported ALICE:2021lsv . For the
first time at the LHC, it was possible to compare collisions of different
nuclei. A comparison of data across different collision systems indicates that
the features of light hadron production scale with the event multiplicity and
are independent of the colliding system, collision geometry or collision
energy. In addition, the radial flow is correlated with the charged particle
multiplicity, and the anisotropic flow is affected by collision geometry. It
was also shown that the hadrochemistry does not depend significantly on the
collision system and scales with the charged particle multiplicity.

## References

  * (1) K. Aamodt et al. (ALICE), JINST 3, S08002 (2008)
  * (2) B. Abelev et al. (ALICE), Int. J. Mod. Phys. A29, 1430044 (2014), 1402.4476
  * (3) S. Acharya et al. (ALICE), Eur. Phys. J. C 81, 584 (2021), 2101.03100
  * (4) S. Acharya et al. (ALICE) (2018), ALICE-PUBLIC-2018-003
  * (5) X.N. Wang, M. Gyulassy, Phys. Rev. D 44, 3501 (1991)
  * (6) R. Brun et al., CERN Program Library Long Write-up p. No. W5013 (1994)
  * (7) S. Acharya et al. (ALICE), Phys. Lett. B 802, 135225 (2020), 1910.14419
  * (8) S. Acharya et al. (ALICE), Phys. Rev. C 101, 044907 (2020)
  * (9) S. Acharya et al. (ALICE), JHEP 09, 006 (2018), 1805.04390
  * (10) S. Acharya et al. (ALICE), Phys. Lett. B784, 82 (2018), 1805.01832
  * (11) B. Abelev et al. (ALICE), Phys. Rev. C88, 044910 (2013), 1303.0737
  * (12) B. Abelev et al. (ALICE), Phys. Rev. C 91, 024609 (2015), 1404.0495
  * (13) V. Vovchenko, B. DÃnigus, Private Communication (2021)
  * (14) V. Vovchenko, B. DÃnigus, H. Stoecker, Phys. Rev. C100, 054906 (2019), 1906.03145
  * (15) J. Cleymans, P.M. Lo, K. Redlich, N. Sharma, Phys. Rev. C 103, 014904 (2021), 2009.04844


Accepted by Physical Review Letters gray4.1,-120.750.02.0

# Universal Wind Profile for Conventionally Neutral Atmospheric Boundary
Layers

Luoqin Liu ID<EMAIL_ADDRESS>Srinidhi N. Gadde ID Richard J. A. M.
Stevens ID<EMAIL_ADDRESS>Physics of Fluids Group, Max Planck
Center Twente for Complex Fluid Dynamics, University of Twente, 7500 AE
Enschede, The Netherlands

###### Abstract

Conventionally neutral atmospheric boundary layers (CNBLs), which are
characterized with zero surface potential temperature flux and capped by an
inversion of potential temperature, are frequently encountered in nature.
Therefore, predicting the wind speed profiles of CNBLs is relevant for weather
forecasting, climate modeling, and wind energy applications. However, previous
attempts to predict the velocity profiles in CNBLs have had limited success
due to the complicated interplay between buoyancy, shear, and Coriolis
effects. Here, we utilize ideas from the classical Monin-Obukhov similarity
theory in combination with a local scaling hypothesis to derive an analytic
expression for the stability correction function $\psi=-c_{\psi}(z/L)^{1/2}$,
where $c_{\psi}=4.2$ is an empirical constant, $z$ is the height above ground,
and $L$ is the local Obukhov length based on potential temperature flux at
that height, for CNBLs. An analytic expression for this flux is also derived
using dimensional analysis and a perturbation method approach. We find that
the derived profile agrees excellently with the velocity profile in the entire
boundary layer obtained from high-fidelity large eddy simulations of typical
CNBLs.

Suggested keywords

††preprint: APS/123-QED

Introduction. For well over a century, wall-bounded turbulent flows have been
studied extensively [1]. A focus area is the derivation and characterization
of the mean velocity profile. In 1925, Prandtl [2] recognized that the
velocity profile in the inertial sub-layer is approximately logarithmic based
on his mixing length hypothesis. In the 1930s, von Kármán [3, 4] derived the
logarithmic law of the wall analytically using dimensional analysis. In 1956,
Coles [5] showed using measurement data that the velocity profile can be
described more accurately via the sum of the logarithmic law and a wake
function. Since then, the law of the wall has been the pillar of the
description of wall-bounded turbulence [6, 7, 8]. Recently, the universality
of the law of the wall has been supported by theoretical and experimental
studies [9, 10, 11].

The dynamics in atmospheric boundary layers, where most human activity and
biological processes occur, are much more complicated as turbulence is
generated by shear stress and buoyancy [12], while the Coriolis force creates
a wind veer [13]. In 1954, Monin and Obukhov [14] introduced a stability
correction function $\psi$ to account for deviations to the logarithmic wind
speed profile caused by thermal stratification. Based on the Buckingham $\Pi$
theorem, Monin and Obukhov concluded that $\psi$ is only a function of the
atmospheric stability parameter $z/L_{w}$, where $z$ is the vertical height
above the ground and $L_{w}$ is the Obukhov length based on the surface
potential temperature flux. The universality of this well known Monin-Obukhov
similarity theory (MOST) [15, 14, 16] has been established in the surface
layers of stable and convective atmospheric boundary layers in many field
experiments [17], as well as large eddy simulations (LES) [18]. Therefore, the
MOST is nowadays regarded as the starting point of modern micro-meteorology
[17].

Conventionally neutral atmospheric boundary layers (CNBLs) are also frequently
observed and are often considered in fundamental studies [19, 20, 21, 22, 23,
24, 25, 26]. In contrast to stable and convective atmospheric boundary layers,
CNBLs are characterized with zero surface potential temperature flux and
capped by an inversion of potential temperature. However, the classical MOST
is not applicable to CNBLs, because the surface potential temperature flux is
zero, due to which $L_{w}$ is no longer a relevant scale [18]. Many studies
[27, 28, 23, 29, 30, 31, 32, 33, 34, 35] have tried to predict the velocity
profile in CNBLs, but so far with limited success since none of them have
considered the effect of potential temperature flux near the capping inversion
layer. Therefore, the logarithmic law of the wall without stability correction
is still commonly used to predict the wind speed profile in CNBLs [36, 37, 38,
39, 40]. However, an analytical description of the velocity profiles in CNBLs
is of great fundamental interest and relevant for meteorological applications
[12, 35].

Theory. In this work, we derive the potential temperature flux profile using
dimensional analysis and a perturbation method approach. To account for the
deviation to the logarithmic wind speed profile we use ideas from the MOST in
combination with a local similarity hypothesis. Therefore, we introduce a
stability correction function $\psi$ that depends only on the local stability
parameter $z/L$, where $L$ is the local Obukhov length based on the local
potential temperature flux. The canonical shape of $\psi$ is determined by
asymptotic analysis. The derived universality profiles for the potential
temperature flux and the wind speed profiles are confirmed by the excellent
agreements with the results of high-fidelity LES.

Based on the dimensional analysis (a derivation can be found in section III of
the Supplementary Material [41]), the potential temperature flux can be
written as

$\frac{\beta
z_{0}q}{u_{*}^{3}}=-\Pi\left(Ro,Zi,\xi\right)=-{Ro}^{r}{Zi}^{s}\Pi_{1}(\xi).$
(1)

Here $q$ is the potential temperature flux, $u_{*}$ is the friction velocity,
$\beta$ is the buoyancy parameter, $\Pi$ and $\Pi_{1}$ are dimensionless
functions, $Ro=u_{*}/(|f|z_{0})$ is the Rossby number with $f$ the Coriolis
parameter and $z_{0}$ the roughness height, $Zi=N/|f|$ is the Zilitinkevich
number [28] with $N$ the free-atmosphere Brunt–Väisälä frequency, and $r$ and
$s$ are the power exponents for $Ro$ and $Zi$, respectively. The dimensionless
parameter $\xi=z/h^{\prime}$, where $h^{\prime}={h}/{(1-0.05^{2/3})}$ and $h$
is the boundary layer height at which the total momentum flux reaches $5\%$ of
the surface value. This conventional definition underestimates the actual
boundary layer height. The functional form of $h^{\prime}$ follows from the
fact that the dimensionless total momentum flux follows a power law with
exponent $3/2$ [42]. We note that the first equality in Eq. (1) is
unconditionally valid, although the corresponding functional form is hard to
determine analytically. The second equality is valid under the presumed power-
law dependence of $\Pi$ on the two independent dimensionless parameters $Ro$
and $Zi$. The values of $s$ and $r$ will be determined later from our high-
fidelity LES data.

We emphasize that other definitions for the boundary layer height, which are
based on, for example, the vertical wind speed or potential temperature
profiles are also commonly used [31, 25, 35]. In particular, the boundary
layer height $h_{t}$ is defined as the height at which the potential
temperature flux reaches its minimum value. Previous studies [43, 44, 25, 35,
45] showed that the potential temperature flux in CNBLs decreases linearly
from zero at the surface to a minimum value at $z=h_{t}$, and then increases
to zero for $z\geq h^{\prime}$. As explained in section IV of the
Supplementary Material [41], the ratio $h_{t}/h^{\prime}\equiv 1-2\epsilon$ is
a function of $Zi$. However, as we will see later the dependence of $\epsilon$
on $Zi$ is limited over the parameter regime under consideration. Clearly,
$\epsilon\ll 1$ represents the half capping layer thickness normalized by the
height $h^{\prime}$, where the potential temperature flux recovers steeply to
zero. Therefore, we propose the following ordinary differential equation to
model the potential temperature flux,

$-\epsilon\Pi_{1}^{\prime\prime}+\Pi_{1}^{\prime}=c_{\Pi},\quad\Pi_{1}(0)=\Pi_{1}(1)=0.$
(2)

Here $c_{\Pi}$ is the slope of the dimensionless total potential temperature
flux $-(\beta z_{0}q)/(u_{*}^{3}Zi^{s}Ro^{r})$ in the surface layers, which
can be determined from simulation or measurement data. The solution of
$\Pi_{1}$ reads

$\displaystyle\Pi_{1}=\left\\{\begin{split}&c_{\Pi}\left(\xi-\frac{e^{\xi/\epsilon}-1}{e^{1/\epsilon}-1}\right),&\quad\xi\leq
1,\\\ &0,&\quad\xi>1.\end{split}\right.$ (3)

Note that Eq. (2) and its solution Eq. (3) is reminiscent of the classical
singular perturbation method [46]: The outer solution (close to the wall) is a
linear function of $\xi$ and the inner solution (close to the capping
inversion layer) is controlled by a small parameter $\epsilon$.

In contrast to the classical MOST [14] where the normalized wind speed
gradient is assumed to be a universal function, we introduce a stability
correction function $\psi$ to account for the deviation of the logarithmic
profile. Therefore we write the wind speed profile as

$\frac{\kappa U_{\rm
mag}}{u_{*}}=\ln\left(\frac{z}{z_{0}}\right)-\psi\left(\frac{z}{L}\right),$
(4)

where $\kappa=0.4$ is the von Kármán constant, $U_{\rm mag}$ is the mean wind
speed, and $\psi$ is the stability correction function that depends only on
the dimensionless stability parameter $z/L$. According to the local scaling
hypothesis [42, 47], $L$ is defined as the local Obukhov length,

$\frac{1}{L}\equiv-\frac{\kappa\beta q}{u_{*}^{3}}.$ (5)

It is worth to point out that the dimensionless slope $(\kappa
z/u_{*})\textrm{d}U_{\rm mag}/\textrm{d}z$ is usually regarded as a universal
function of the stability parameter $z/L$ in the stable and convective
atmospheric boundary layers [48]. However, under the assumption of Eq. (4),
this slope is no longer a universal function of $z/L$.

To determine the canonical shape of $\psi$, we assume

$\psi=-c_{\psi}\left(\frac{z}{L}\right)^{p},$ (6)

where $p$ is the power exponent to be determined analytically below, and
$c_{\psi}$ is an empirical constant. Recall that very close to the wall (see
Eq. (3))

$\Pi_{1}\rightarrow
c_{\Pi}\xi=c_{\Pi}\frac{z}{h^{\prime}}\quad\text{as}\quad\frac{z}{h^{\prime}}\rightarrow
0.$ (7)

Then, from asymptotic analysis [49], we find that

$\frac{z}{L}=\frac{\kappa z}{z_{0}}\Pi\rightarrow\frac{c_{\Pi}\kappa
h^{\prime}}{z_{0}}Zi^{s}Ro^{r}\left(\frac{z}{h^{\prime}}\right)^{2}\quad\text{as}\quad\frac{z}{h^{\prime}}\rightarrow
0.$ (8)

Zilitinkevich and Esau [23] showed that in the surface layers of stable, truly
neutral, and conventionally neutral atmospheric boundary layers
$\psi=-C_{u}z/L_{M}$. Here $C_{u}$ is a dimensionless constant and $L_{M}$ is
the combined turbulent length scale, which in CNBLs can be estimated as
$|f|L_{M}/u_{*}=(1+C_{m}^{2}Zi^{2})^{-1/2}$, where $C_{m}$ is an empirical
constant ([23], see also the Supplementary Material [41]). To match with the
result of Zilitinkevich and Esau [23] in the surface layer, we find that
$p=1/2$. Clearly, the determination of $p$ is independent of the values of
$c_{\Pi}$, $c_{\psi}$, $r$, $s$, and $\epsilon$. Thus, the wind speed profile
is given by

$\displaystyle\frac{\kappa U_{\rm
mag}}{u_{*}}=\left\\{\begin{split}&\ln\left(\frac{z}{z_{0}}\right)+c_{\psi}\left(\frac{z}{L}\right)^{1/2},&\quad\xi\leq\xi_{0},\\\
&\frac{\kappa G}{u_{*}},&\quad\xi>\xi_{0}.\end{split}\right.$ (9)

Here $G$ is the geostrophic wind speed, $\xi_{0}$ is the highest intersection
point of the curves described by the upper and lower expressions in Eq. (9),
$z/L$ is the dimensionless stability parameter predicted by the potential
temperature flux model (i.e. Eqs. (1) and (3)), and $c_{\psi}$ is the
empirical constant that can be determined from simulation or measurement data.

Validation. To verify the universality of the wind speed profile for CNBLs, we
perform six high-fidelity LES. In the simulations, a CNBL over a flat surface
with periodic conditions in horizontal directions is considered. The flow is
initialized with uniform geostrophic wind speed and a linear potential
temperature profile with a constant gradient [44, 35]. The simulations are
performed with an in-house code [50, 51, 52, 53, 54, 55], which employs a
pseudo-spectral discretization in the horizontal directions and a second-order
finite difference method in the vertical direction. We employ the advanced
anisotropic minimum dissipation model to parameterize the sub-grid scale shear
stress and potential temperature flux [56]. The horizontal domain size is more
than six times larger than the boundary layer height, and the grid resolution
is $288^{3}$. We ensure that all simulations have reached the quasi-stationary
state and the statistics are averaged over one inertial period [57]. A summary
of all simulated cases is presented in Table 1. The simulated Zilitinkevich
number $Zi$ and Rossby number $Ro$ range covers the values found in typical
CNBLs at mid to high latitudes [58, 59]. More details about the numerical
method and simulation setup can be found in the Supplementary Material [41].

Table 1: Summary of all simulated cases, where the $Zi$ and $Ro$ range covers the values found in typical CNBLs at mid to high latitudes [58, 59] Case no. | A | B | C | D | E | F
---|---|---|---|---|---|---
$Zi$ | 51.2 | 88.7 | 88.7 | 88.7 | 88.7 | 153.6
$Ro$ | 2.7E7 | 4.5E4 | 3.7E5 | 3.2E6 | 2.7E7 | 2.7E7
$\epsilon$ | 0.1186 | 0.1148 | 0.1191 | 0.1213 | 0.1224 | 0.1341
$c_{\Pi}$ | 0.0335 | 0.0335 | 0.0330 | 0.0329 | 0.0330 | 0.0336
\begin{overpic}[width=173.44534pt]{most-tau-2pi} \end{overpic} Figure 1:
Vertical profile of dimensionless mean total momentum flux $\tau/\tau_{w}$.
Filled symbols: LES data; solid line: theoretical curve given by
$\tau/\tau_{w}=(1-z/h^{\prime})^{3/2}$.

Figure 1 shows the vertical profile of the dimensionless mean total momentum
flux $\tau/\tau_{w}$, where $\tau$ is the total momentum flux and $\tau_{w}$
is its surface value. All different cases in Table 1 are shown in the figure
(filled symbols). Nieuwstadt [42] analytically determined that the total
momentum flux profile in stable atmospheric boundary layers scales as
$\tau/\tau_{w}=(1-z/h)^{3/2}$. In Fig. 1 we show that this expression is still
valid for CNBLs when we consider the previously introduced boundary layer
thickness $h^{\prime}={h}/{(1-0.05^{2/3})}$. The finding that the
dimensionless momentum flux profiles obtained from all LES collapse to the
theoretical curve (see Fig. 1) confirms that $h^{\prime}$ is the appropriate
boundary layer height scale to consider.

To determine the values of the power indices $r$ and $s$, we take the vertical
derivative of Eq. (1), see details in section III of the Supplementary
Material [41]. Figure 2 shows the dimensionless mean potential temperature
gradient $\ln\,(-\beta z_{0}q^{\prime}/u_{*}^{3})$ versus (a) the Rossby
number $\ln Ro$ and (b) the Zilitinkevich number $\ln Zi$ in the surface
layer, where $q^{\prime}=\textrm{d}q/\textrm{d}\xi$. The slopes of the curve
shown in the figure determine the values of the power exponents $r$ and $s$.
In the parameter regime under consideration $r=-1.002\approx-1$ and
$s=1.004\approx 1$ describe the data very well. We note that the values of
$c_{\Pi}$ can also be determined from the figure and the results are listed in
Table 1.

\begin{overpic}[width=182.1196pt]{q-r-2pi} \put(0.0,53.0){$(a)$}
\end{overpic}\begin{overpic}[width=182.1196pt]{q-s-2pi} \put(0.0,53.0){$(b)$}
\end{overpic}

Figure 2: Dimensionless mean potential temperature gradient in the surface
layer versus (a) the Rossby number $Ro$ and (b) the Zilitinkevich number $Zi$,
where $q^{\prime}=\textrm{d}q/\textrm{d}\xi$. The values of the slope are (a)
$r=-1.002\approx-1$ and (b) $s=1.004\approx 1$, which are determined by a
least-squares fitting procedure.
\begin{overpic}[width=195.12767pt]{most-q-2pi} \end{overpic} Figure 3:
Vertical profile of dimensionless mean potential temperature flux $q/|q|_{\rm
max}$. Filled circles: LES data of the present work; filled triangles:
prediction based on a turbulence closure given by Mauritsen  _et al._ [43];
filled inverted triangles: DNS data of Jonker  _et al._ [60]; filled diamonds:
LES data of Pedersen  _et al._ [44]; filled squares: LES data of Allaerts and
Meyers [25]; filled stars: LES data of Berg  _et al._ [45]; solid line:
theoretical prediction given by Eq. (3) with $\epsilon=0.12$.

Figure 3 shows the vertical profile of the dimensionless mean potential
temperature flux $q/|q|_{\rm max}$, which reduces to almost zero at
$z/h^{\prime}\geq 1$. The potential temperature flux first decreases linearly
from zero at the surface to a minimum value at
$z=h_{t}\equiv(1-2\epsilon)h^{\prime}$, and then increases rapidly to zero in
a narrow region ($1-2\epsilon\leq z/h^{\prime}\leq 1$) since $\epsilon\ll 1$.
The value of $\epsilon$ is expected to depend only on $Zi$ (see the
Supplementary Material [41]). The data in Table 1 show that for the parameter
range under consideration the variation of $\epsilon$ is limited and therefore
we take $\epsilon=0.12$ to describe the data here. Evidently, all LES data of
the present work (filled symbols) collapse very well to the introduced
theoretical model (solid line), which validates the chosen approach. For
comparison, the prediction based on a turbulence closure given by Mauritsen
_et al._ [43], the direct numerical simulations (DNS) data performed by Jonker
_et al._ [60], and the LES data taken from Pedersen  _et al._ [44], Allaerts
and Meyers [25], and Berg  _et al._ [45] are also shown in the figure. The
overall agreement between the theoretical prediction and the data from
previous studies [43, 60, 44, 25, 45] is very good, which confirms the
universality of the proposed potential temperature flux profile.

Figure 4 shows the vertical profile of the dimensionless wind speed for two
typical cases, which covers the $Zi$ and $Ro$ number range of typical CNBLs at
mid to high latitudes [58, 59]. The filled symbols are LES data, the dashed
line is the theoretical prediction given by the logarithmic law, the blue line
is the prediction of Zilitinkevich and Esau [23], the yellow line is the
prediction of Gryning  _et al._ [29], the red line is the prediction of Kelly
_et al._ [35], and the black line is the prediction given by Eq. (9) with
$c_{\psi}=4.2$ where the potential temperature flux profile is modeled by Eq.
(3) with $c_{\Pi}=0.0332$ and $\epsilon=0.12$ (see Table 1). The empirical
constant $c_{\psi}$ is determined such that it can predict the wind speed
profiles of all cases in Table 1 with minimum discrepancies. The figure shows
that the logarithmic law only accurately captures the wind speed in the lower
10% of the boundary layer, also known as the surface layer (shaded region).
The theory given by Gryning  _et al._ [29] focuses on capturing the wind speed
at the top of the CNBL, but does not capture the effect of the low-level jet.
The theory given by Kelly  _et al._ [35] is focused on the lower part of the
CNBL. The predictions by Zilitinkevich and Esau [23] agree well with the LES
data in the lower part of the CNBL, but do not capture the low-level jet,
which is represented in our approach. In contrast, the agreement between the
proposed profile (9) and the LES data is nearly perfect in the entire boundary
layer and much better than all previous approaches. This excellent agreement
confirms the universality of our proposed wind profile (9) in the considered
parameter range of CNBLs.

\begin{overpic}[width=182.1196pt]{most-u-2-2pi} \put(0.0,54.0){$(a)$}
\end{overpic}\begin{overpic}[width=182.1196pt]{most-u-6-2pi}
\put(0.0,66.0){$(b)$} \end{overpic}

Figure 4: Vertical profile of mean wind speed for (a) case B and (b) case F.
Filled symbols: LES data; dashed line: prediction given by the logarithmic
law; blue line: prediction given by Zilitinkevich and Esau [23]; yellow line:
prediction given by Gryning  _et al._ [29]; red line: prediction given by
Kelly  _et al._ [35]; black line: prediction given by Eq. (9) with
$c_{\psi}=4.2$.

Summary. We propose a universal velocity profile for CNBL derived using a
local similarity hypothesis combined with ideas from the classical Monin-
Obukhov similarity theory. We introduce a stability correction function $\psi$
to account for the deviation of the logarithmic law. The canonical shape of
$\psi$ is determined theoretically as $\psi=-c_{\psi}(z/L)^{1/2}$, where
$c_{\psi}=4.2$ is an empirical constant determined from simulation data, $z$
is the vertical height above the surface, and $L$ is the local Obukhov length.
An analytical expression for the potential temperature flux profile is also
derived from dimensional analysis and perturbation method. The universality of
the proposed profile (9) has been confirmed by its excellent agreement with
high-fidelity LES results for $Ro=[4.5\times 10^{4},2.7\times 10^{7}]$ and
$Zi\in[51,154]$, where the $Zi$ and $Ro$ number range cover the range of
values observed in typical CNBLs at mid to high latitudes. Further work is
required to assess the applicability of the approach to other parameter
regimes.

###### Acknowledgements.

We appreciate very much the valuable comments of the anonymous referees. We
acknowledge Drs. K. L. Chong and Y. X. Li for insightful discussion. This work
is part of the Shell-NWO/FOM-initiative Computational sciences for energy
research of Shell and Chemical Sciences, Earth and Live Sciences, Physical
Sciences, FOM and STW, and an STW VIDI grant (No. 14868). This work was
carried out on the national e-infrastructure of SURFsara, a subsidiary of SURF
cooperation, the collaborative ICT organization for Dutch education and
research.

## References

  * Smits  _et al._ [2011] A. J. Smits, B. J. McKeon, and I. Marusic, High Reynolds number wall turbulence, Annu. Rev. Fluid Mech. 43, 353 (2011).
  * Prandtl [1925] L. Prandtl, Bericht über Untersuchungen zur ausgebildeten Turbulenz, Z. Angew. Math. Mech. 5, 136 (1925).
  * von Kármán [1930] T. von Kármán, Mechanische Ähnlichkeit und Turbulenz, Nachr. Ges. Wiss. Göttingen Math. Phys. Klasse 5, 58 (1930).
  * von Kármán [1931] T. von Kármán, Mechanische Ähnlichkeit und Turbulenz, Proc. Third Internat. Congr. Appl. Mech. Stockholm 1, 85 (1931).
  * Coles [1956] D. Coles, The law of the wake in the turbulent boundary layer, J. Fluid Mech. 1, 191 (1956).
  * Tennekes and Lumley [1972] H. Tennekes and J. L. Lumley,  _A First Course in Turbulence_ (The MIT Press, Cambridge, Massachusetts, 1972).
  * Pope [2000] S. B. Pope,  _Turbulent Flows_ (Cambridge University Press, Cambridge, 2000).
  * Davidson [2004] P. Davidson,  _Turbulence: An Introduction for Scientists and Engineers_ (Oxford University Press, Oxford, 2004).
  * Marusic  _et al._ [2013] I. Marusic, J. P. Monty, M. Hultmark, and A. J. Smits, On the logarithmic region in wall turbulence, J. Fluid Mech. 716, R3 (2013).
  * Luchini [2017] P. Luchini, Universality of the turbulent velocity profile, Phys. Rev. Lett. 118, 224501 (2017).
  * Samie  _et al._ [2018] M. Samie, I. Marusic, N. Hutchins, M. K. Fu, Y. Fan, M. Hultmark, and A. J. Smits, Fully resolved measurements of turbulent boundary layer flows up to $Re_{\tau}=20000$, J. Fluid Mech. 851, 391 (2018).
  * Katul  _et al._ [2011] G. G. Katul, A. G. Konings, and A. Porporato, Mean velocity profile in a sheared and thermally stratified atmospheric boundary layer, Phys. Rev. Lett. 107, 268502 (2011).
  * Howland  _et al._ [2020] M. F. Howland, A. S. Ghate, and S. K. Lele, Influence of the geostrophic wind direction on the atmospheric boundary layer flow, J. Fluid Mech. 883, A39 (2020).
  * Monin and Obukhov [1954] A. S. Monin and A. M. Obukhov, Basic laws of turbulent mixing in the surface layer of the atmosphere, Tr. Akad. Nauk SSSR Geophiz. Inst. 24, 163 (1954).
  * Obukhov [1946] A. M. Obukhov, Turbulence in an atmosphere with inhomogeneous temperature, Trans. Inst. Teoret. Geoz. Akad. Nauk SSSR 1, 95 (1946).
  * Monin and Yaglom [1971] A. S. Monin and A. M. Yaglom,  _Statistical Fluid Mechanics, Vol. 1. Mechanics of Turbulence_ (The MIT Press, Cambridge, Massachusetts, 1971).
  * Foken [2006] T. Foken, 50 years of the Monin–Obukhov similarity theory, Boundary-Layer Meteorol. 119, 431 (2006).
  * Khanna and Brasseur [1997] S. Khanna and J. G. Brasseur, Analysis of Monin-Obukhov similarity from large-eddy simulation, J. Fluid Mech. 345, 251 (1997).
  * Brost  _et al._ [1982] R. A. Brost, D. H. Lenschow, and J. C. Wyngaard, Marine stratocumulus layers. Part 1: Mean conditions, J. Atmos. Sci. 39, 800 (1982).
  * Grant [1986] A. L. M. Grant, Observations of boundary layer structure made during the 1981 KONTUR experiment, Q. J. R. Meteorol. Soc. 112, 825 (1986).
  * Tjernström and Smedman [1993] M. Tjernström and A.-S. Smedman, The vertical turbulence structure of the coastal marine atmospheric boundary layer, J. Geophys. Res.: Oceans 98, 4809 (1993).
  * Zilitinkevich and Esau [2002] S. S. Zilitinkevich and I. N. Esau, On integral measures of the neutral barotropic planetary boundary layer, Boundary-Layer Meteorol. 104, 371 (2002).
  * Zilitinkevich and Esau [2005] S. S. Zilitinkevich and I. N. Esau, Resistance and heat-transfer laws for stable and neutral planetary boundary layers: Old theory advanced and re-evaluated, Q. J. R. Meteorol. Soc. 131, 1863 (2005).
  * Zilitinkevich  _et al._ [2007] S. S. Zilitinkevich, I. Esau, and A. Baklanov, Further comments on the equilibrium height of neutral and stable planetary boundary layers, Q. J. R. Meteorol. Soc. 133, 265 (2007).
  * Allaerts and Meyers [2015] D. Allaerts and J. Meyers, Large eddy simulation of a large wind-turbine array in a conventionally neutral atmospheric boundary layer, Phys. Fluids 27, 065108 (2015).
  * Allaerts and Meyers [2017] D. Allaerts and J. Meyers, Boundary-layer development and gravity waves in conventionally neutral wind farms, J. Fluid Mech. 814, 95 (2017).
  * Zilitinkevich  _et al._ [2002] S. S. Zilitinkevich, V. L. Perov, and J. C. King, Near-surface turbulent fluxes in stable stratification: Calculation for use in general circulation models, Q. J. R. Meteorol. Soc. 128, 1571 (2002).
  * Esau [2004] I. N. Esau, Parameterization of a surface drag coefficient in conventionally neutral planetary boundary layer, Ann. Geophys. 22, 3353 (2004).
  * Gryning  _et al._ [2007] S.-E. Gryning, E. Batchvarova, B. Brümmer, H. Jorgensen, and S. Larsen, On the extension of the wind profile over homogeneous terrain beyond the surface boundary layer, Boundary-Layer Meteorol. 124, 251 (2007).
  * Kelly and Gryning [2010] M. Kelly and S.-E. Gryning, Long-term mean wind profiles based on similarity theory, Boundary-Layer Meteorol. 136, 377 (2010).
  * Abkar and Porté-Agel [2013] M. Abkar and F. Porté-Agel, The effect of free-atmosphere stratification on boundary-layer flow and power output from very large wind farms, Energies 6, 2338 (2013).
  * Optis  _et al._ [2014] M. Optis, A. Monahan, and F. C. Bosveld, Moving beyond Monin–Obukhov similarity theory in modelling wind-speed profiles in the lower atmospheric boundary layer under stable stratification, Boundary-Layer Meteorol. 153, 497 (2014).
  * Kelly and Troen [2016] M. Kelly and I. Troen, Probabilistic stability and ‘tall’ wind profiles: theory and method for use in wind resource assessment, Wind Energy 19, 227 (2016).
  * Jiang  _et al._ [2018] Q. F. Jiang, S. P. Wang, and P. Sullivan, Large-eddy simulation study of log laws in a neutral Ekman boundary layer, J. Atmos. Sci. 75, 1873 (2018).
  * Kelly  _et al._ [2019] M. Kelly, R. A. Cersosimo, and J. Berg, A universal wind profile for the inversion‐capped neutral atmospheric boundary layer, Q. J. R. Meteorol. Soc. 145, 982 (2019).
  * Rossby and Montgomery [1935] C. G. Rossby and R. B. Montgomery, The layers of frictional influence in wind and ocean currents, Pap. Phys. Oceanogr. Meteor. 3, 1 (1935).
  * Blackadar [1962] A. K. Blackadar, The vertical distribution of wind and turbulent exchange in a neutral atmosphere, J. Geophys. Res. 67, 3095 (1962).
  * Blackadar and Tennekes [1968] A. K. Blackadar and H. Tennekes, Asymptotic similarity in neutral barotropic planetary boundary layers, J. Atmos. Sci. 25, 1015 (1968).
  * Tennekes [1973] H. Tennekes, The logarithmic wind profile, J. Atmos. Sci. 30, 234 (1973).
  * Horiguchi  _et al._ [2012] M. Horiguchi, T. Hayashi, A. Adachi, and S. Onogi, Large-scale turbulence structures and their contributions to the momentum flux and turbulence in the near-neutral atmospheric boundary layer observed from a 213-m tall meteorological tower, Boundary-Layer Meteorol. 144, 179 (2012).
  * [41] See the Supplemental Material at [url], which includes additional information on numerical simulations, derivation of the potential temperature flux, formulations of existing wind speed models, and Refs. [1,2,5,7,8], .
  * Nieuwstadt [1984] F. T. M. Nieuwstadt, The turbulent structure of the stable, nocturnal boundary layer, J. Atmos. Sci. 41, 2202 (1984).
  * Mauritsen  _et al._ [2007] T. Mauritsen, G. Svensson, S. S. Zilitinkevich, I. N. Esau, L. Enger, and B. Grisogono, A total turbulent energy closure model for neutrally and stably stratified atmospheric boundary layers, J. Atmos. Sci. 64, 4113 (2007).
  * Pedersen  _et al._ [2014] J. G. Pedersen, S.-E. Gryning, and M. Kelly, On the structure and adjustment of inversion-capped neutral atmospheric boundary-layer flows: Large-eddy simulation study, Boundary-Layer Meteorol. 153, 43 (2014).
  * Berg  _et al._ [2020] J. Berg, E. G. Patton, and P. P. Sullivan, Large-eddy simulation of conditionally neutral boundary layers: A mesh resolution sensitivity study, J. Atmos. Sci. 77, 1969 (2020).
  * Dyke [1975] M. V. Dyke,  _Perturbation Methods in Fluid Mechanics_ (Parabolic Press, Stanford, California, 1975).
  * Sorbjan [1986] Z. Sorbjan, On similarity in the atmospheric boundary layer, Boundary-Layer Meteorol. 34, 377 (1986).
  * Businger  _et al._ [1971] J. A. Businger, J. C. Wyngaard, Y. Izumi, and E. F. Bradley, Flux-profile relationships in the atmospheric surface layer, J. Atmos. Sci. 28, 181 (1971).
  * Lin and Segel [1988] C. C. Lin and L. A. Segel,  _Mathematics Applied to Deterministic Problems in the Natural Sciences_ (Society for Industrial and Applied Mathematics, Philadelphia, 1988).
  * Bou-Zeid  _et al._ [2005] E. Bou-Zeid, C. Meneveau, and M. B. Parlange, A scale-dependent Lagrangian dynamic model for large eddy simulation of complex turbulent flows, Phys. Fluids 17, 025105 (2005).
  * Stevens  _et al._ [2014] R. J. A. M. Stevens, M. Wilczek, and C. Meneveau, Large eddy simulation study of the logarithmic law for high-order moments in turbulent boundary layers, J. Fluid Mech. 757, 888 (2014).
  * Gadde and Stevens [2019] S. N. Gadde and R. J. A. M. Stevens, Effect of Coriolis force on a wind farm wake, J. Phys. Conf. Ser. 1256, 012026 (2019).
  * Gadde  _et al._ [2020] S. N. Gadde, A. Stieren, and R. J. A. M. Stevens, Large eddy simulations of stratified atmospheric boundary layers: comparison of different sub-grid models, Boundary-Layer Meteorol. (2020).
  * Liu and Stevens [2020] L. Liu and R. J. A. M. Stevens, Effects of two-dimensional steep hills on the performance of wind turbines and wind farms, Boundary-Layer Meteorol. 174, 61 (2020).
  * Liu  _et al._ [2020] L. Liu, S. N. Gadde, and R. J. A. M. Stevens, Geostrophic drag law for conventionally neutral atmospheric boundary layers revisited, Q. J. R. Meteorol. Soc. , 1 (2020).
  * Abkar and Moin [2017] M. Abkar and P. Moin, Large eddy simulation of thermally stratified atmospheric boundary layer flow using a minimum dissipation model, Boundary-Layer Meteorol. 165, 405 (2017).
  * Coleman  _et al._ [1992] G. N. Coleman, J. H. Ferziger, and P. R. Spalart, Direct simulation of the stably stratified turbulent Ekman layer, J. Fluid Mech. 244, 677 (1992).
  * Hess and Garratt [2002] G. D. Hess and J. R. Garratt, Evaluating models of the neutral, barotropic planetary boundary layer using integral measures: Part I. Overview, Boundary-Layer Meteorol. 104, 333 (2002).
  * Zilitinkevich  _et al._ [2012] S. S. Zilitinkevich, S. A. Tyuryakov, Y. I. Troitskaya, and E. A. Mareev, Theoretical models of the height of the atmospheric boundary layer and turbulent entrainment at its upper boundary, Izvestiya, Atmospheric and Oceanic Physics 48, 133 (2012).
  * Jonker  _et al._ [2013] H. J. J. Jonker, M. van Reeuwijk, P. P. Sullivan, and E. G. Patton, On the scaling of shear-driven entrainment: a DNS study, J. Fluid Mech. 732, 150 (2013).