# Lagrangian and Eulerian Accelerations in Turbulent Stratified Shear Flows

Frank G. Jacobitz<EMAIL_ADDRESS>Mechanical Engineering Department,
Shiley-Marcos School of Engineering, University of San Diego, 5998 Alcalá
Park, San Diego, CA 92110, USA  Kai Schneider<EMAIL_ADDRESS>Aix-
Marseille Université, CNRS, Centrale Marseille, Institut de Mathématiques de
Marseille (I2M), 39 rue Joliot-Curie, 13453 Marseille Cedex 13, France

###### Abstract

The Lagrangian and Eulerian acceleration properties of fluid particles in
homogeneous turbulence with uniform shear and uniform stable stratification
are studied using direct numerical simulations. The Richardson number is
varied from $Ri=0$, corresponding to unstratified shear flow, to $Ri=1$,
corresponding to strongly stratified shear flow. The probability density
functions (pdfs) of both Lagrangian and Eulerian accelerations have a
stretched-exponential shape and they show a strong and similar influence on
the Richardson number. The extreme values of the Eulerian acceleration are
stronger than those observed for the Lagrangian acceleration. Geometrical
statistics explain that the magnitude of the Eulerian acceleration is larger
than its Lagrangian counterpart due to the mutual cancellation of the Eulerian
and convective acceleration, as both vectors statistically show an anti-
parallel preference. A wavelet-based scale-dependent decomposition of the
Lagrangian and Eulerian accelerations is performed. The tails of the
acceleration pdfs grow heavier for smaller scales of turbulent motion. Hence
the flatness increases with decreasing scale, indicating stronger
intermittency at smaller scales. The joint pdfs of the Lagrangian and Eulerian
accelerations indicate a trend to stronger correlations with increasing
Richardson number and at larger scales of the turbulent motion. A
consideration of the terms in the Navier–Stokes equation shows that the
Lagrangian acceleration is mainly determined by the pressure-gradient term,
while the Eulerian acceleration is dominated by the nonlinear convection term.
A similar analysis is performed for the Lagrangian and Eulerian time-rates of
change of both fluctuating density and vorticity. The Eulerian time-rates of
change are observed to have substantially larger extreme values than those of
their Lagrangian counterparts due to the the advection terms in the advection-
diffusion equation for fluctuating density and in the vorticity equation,
respectively. The Lagrangian time-rate of change of fluctuating vorticity is
mainly determined by the vortex stretching and tilting term in the vorticity
equation. Since the advection-diffusion equation for fluctuating density lacks
a quadratic term, the Lagrangian time-rate of change pdfs of fluctuating
density show a more Gaussian shape, in particular for large Richardson
numbers. Hence, the Lagrangian acceleration and time-rates of change of
fluctuating density and vorticity reflect the dominant physics of the
underlying governing equations, while the Eulerian acceleration and time-rates
of change are mainly determined by advection.

###### pacs:

47.27.Ak, 47.27.E-, 47.27.ek, 47.27.er, 47.27.Gs

††preprint: Submitted to Physical Review Fluids

##  I Introduction

An understanding of the Lagrangian acceleration properties of a fluid particle
in turbulent motion is of fundamental importance and numerous applications
exist in geophysical, environmental, and engineering flows. It aids in the
study of transport and mixing, as well as in the characterization of geometric
properties and intermittency at various scales of turbulent motion. As
proposed in Tsinober  _et al._ (2001) and Tsinober (2001), the Lagrangian
description of turbulent flows may be a more natural approach to the study of
turbulence, as it is more directly related to the dynamics of fluid particles,
which are subjected to different forces, e.g., due to the fluctuating pressure
gradient, buoyancy, viscous stresses, or other forcing terms.

Studying Lagrangian acceleration has some history. After early work by
Heisenberg (1948) and Yaglom (1949), more recent studies range from
theoretical investigations (e.g. Tsinober  _et al._ (2001); Tsinober (2001))
to applications such as the modeling of particle dispersion (e.g. Pope (1994))
highly relevant to turbulent combustion. Such studies are carried out using
both experimental (e.g. La Porta  _et al._ (2001)) as well as computational
(e.g. Yeung and Pope (1989); Yeung (2002) or Toschi and Bodenschatz (2009))
approaches.

The majority of previous investigations focused on Lagrangian properties of
isotropic turbulence. The Lagrangian acceleration was found to be strongly
intermittent and heavy tails were observed in its probability density
functions (pdf). For example, extreme values as high as 1,500 times the
acceleration of gravity were observed for the Lagrangian acceleration of fluid
particles by La Porta  _et al._ (2001) and numerical simulations by Toschi and
Bodenschatz (2009) confirmed these results.

Acceleration fluctuations and the different contributions have been studied in
Pinsky  _et al._ (2000); Tsinober  _et al._ (2001) in isotropic turbulence.
Their work is motivated by the random Taylor hypothesis or sweeping
decorrelation hypothesis stating that ‘small eddies in turbulent flow being
swept past a stationary Eulerian observer’ Tsinober  _et al._ (2001). It is
based on the prediction of Tennekes (1975) that states that the Lagrangian
acceleration must be small, justified by considering Eulerian and Lagrangian
time scales. He predicted that the rms value of the Lagrangian acceleration is
a factor $Re_{\lambda}^{-1/2}$ smaller than the Eulerian value. Lin (1953)
showed that there is no general justification to extend Taylor’s hypothesis to
turbulent shear flow. He gives some perspectives that this may still hold for
large wavenumbers (small scales), which will be the topic of future work.

In Tsinober  _et al._ (2001) direct numerical simulation data of isotropic
turbulence were analyzed for different Reynolds numbers and the Lagrangian
acceleration, called total acceleration in Tsinober’s work, was decomposed
into the Eulerian acceleration (called local acceleration in Tsinober  _et
al._ (2001)) and the convective contribution. Possible cancellation properties
between the Eulerian and convective contributions may yield reduced values of
the Lagrangian acceleration. The authors found that the variance of Lagrangian
acceleration is much smaller than the Eulerian (local) acceleration and the
advection term due to their strong negative alignment (or correlation) for
sufficiently high $R_{\lambda}$, here 140. They also observed that the
Lagrangian acceleration is strongly correlated with the pressure gradient.
Their results are thus in support with the random Taylor hypothesis.

Note that the convective contribution of the acceleration becomes large when
the flow is non-uniform, i.e., if the velocity changes along a streamline. The
convective acceleration term is nonlinear, which causes mathematical
difficulties in flow analysis; also, even in steady flow (which is perfect for
Taylor’s hypothesis), the convective acceleration can be large if spatial
gradients of velocity are large. In case it is anti-aligned with the local
acceleration, it can be balanced and the total acceleration can still be
small. This implies that the rate of Eulerian decorrelation is higher than
that of Lagrangian decorrelation, which is crucial for two-point closures, see
also the review on space-time correlations in turbulence by He  _et al._
(2017).

Many applications of Lagrangian dynamics target the transport and mixing of
natural and anthropogenic substances in the geophysical environment. Such
flows are often characterized by the presence of shear and stratification.
Homogeneous turbulent stratified shear flows with constant vertical
stratification rate $S_{\rho}=\partial\varrho/\partial y$ and constant
vertical shear rate $S=\partial U/\partial y$ represents the simplest flow
configuration in order to study the competing effects of shear and
stratification. This flow has been investigated extensively in the past.
Experimental studies include work by Komori  _et al._ (1983), Rohr  _et al._
(1988), Piccirillo and Van Atta (1997), and Keller and Van Atta (2000).
Numerical simulations were performed by Gerz  _et al._ (1989), Holt  _et al._
(1992), Jacobitz  _et al._ (1997), Jacobitz (2002), and Portwood (2019).
Hanazaki and Hunt (2004) analyzed this flow using linear theory. More
recently, the mixing properties of turbulent stratified shear flow have been
considered by, for example, Salehipour  _et al._ (2015) and Venayagamoorthy
and Koseff (2016). For a review, we refer to Gregg  _et al._ (2018).

More recently, Jacobitz  _et al._ (2016) considered Lagrangian and Eulerian
accelerations in rotating and sheared homogeneous turbulence. It was found
that the Lagrangian acceleration was mainly determined by the pressure-
gradient term in the Navier–Stokes equation, while the Eulerian acceleration
shows stronger tails due to the advection term. In the case of strong
rotation, linear effects are dominant and the Lagrangian acceleration pdf
takes an approximately Gaussian shape. A comparison of linear theory with
direct numerical simulation of rotating and sheared homogeneous turbulence was
performed by Salhi  _et al._ (2014).

The goal of this work is to investigate the acceleration statistics and to
analyze the different contributions to the acceleration in turbulent
stratified shear flows using direct numerical simulations. A key question is
the understanding of the properties of Lagrangian acceleration fluctuations
and their Eulerian counterpart and the influence of the Richardson number.

In the following, the numerical approach taken in this study is introduced
first. Then the Richardson number dependence of the Lagrangian and Eulerian
acceleration pdfs are presented and geometrical statistics of the alignment
angles of the different contributions. Using a wavelet-based scale-dependent
decomposition, the Lagrangian and Eulerian accelerations are studied at
various scales of the turbulent motion and their spatial fluctuations are
analyzed. The corresponding Lagrangian and Eulerian time-rates of change pdfs
for the fluctuating density are discussed. Finally, a summary and conclusion
of the present work is provided. Results for the Lagrangian and Eulerian time-
rates of change for the fluctuating vorticity as well as the Lagrangian and
Eulerian acceleration component are discussed in the appendix.

##  II Approach

In this section, the equations of motion and their direct numerical solution
are described, variance estimates for the Lagrangian and Eulerian
accelerations are given, the wavelet-based scale-dependent decomposition of
the accelerations is introduced, and geometrical statistics to study the
alignment of the different acceleration contributions are motivated.

###  II.1 Equations of Motion

The mean flow with velocity $(U,V,W)$ and density $\varrho$ considered in this
study has a constant vertical shear rate $S=\partial U/\partial y$ and a
constant vertical stratification rate $S_{\rho}=\partial\varrho/\partial y$,
respectively:

$U=Sy,\qquad V=W=0,\qquad\varrho=\rho_{0}+S_{\rho}y,$ (1)

where $\rho_{0}$ is the ambient density.

This study is based on the incompressible Navier–Stokes equations for the
fluctuating velocity and an advection-diffusion equation for the fluctuating
density:

$\nabla\cdot\bm{u}=0$ (2) $\displaystyle\frac{\partial\bm{u}}{\partial
t}+\bm{u}\cdot\nabla\bm{u}+Sy\frac{\partial\bm{u}}{\partial x}+Sv\bm{e}_{x}$
$\displaystyle=-\frac{1}{\rho_{0}}\nabla
p-\frac{g}{\rho_{0}}\rho\bm{e}_{y}+\nu\nabla^{2}\bm{u}$ (3)
$\frac{\partial\rho}{\partial
t}+\bm{u}\cdot\nabla\rho+S_{\rho}v=\alpha\nabla^{2}\rho$ (4)

Here, $\bm{u}=(u,v,w)$ is the fluctuating velocity, $p$ the fluctuating
pressure, $\rho$ the fluctuating density, $\nu$ the kinematic viscosity, and
$\alpha$ the scalar diffusivity. Taking the curl of the momentum equation (3)
leads to the vorticity equation:

$\displaystyle\frac{\partial\bm{\omega}}{\partial
t}+\bm{u}\cdot\nabla\bm{\omega}+\nabla\times\left(Sy\frac{\partial\bm{u}}{\partial
x}+Sv\bm{e}_{x}\right)$
$\displaystyle=\bm{\omega}\cdot\nabla\bm{u}-\nabla\times\left(\frac{g}{\rho_{0}}\rho\bm{e}_{y}\right)+\nu\nabla^{2}\bm{\omega}$
(5)

###  II.2 Numerical Approach

For their numerical solution, the equations of motion (2-4) are transformed
into a frame of reference moving with the mean velocity (see Rogallo (1981)).
This transformation enables the application of periodic boundary conditions
for the fluctuating components of velocity and density. A spectral collocation
method is used for the spatial discretization and the solution is advanced in
time with a fourth-order Runge–Kutta scheme.

Table 1 provides an overview of the simulations performed for this study. The
Richardson number $Ri=N^{2}/S^{2}$ is varied from $Ri=0$, corresponding to
unstratified shear flow, to $Ri=1$, corresponding to strongly stratified shear
flow. While both the mean shear rate $S=\partial U/\partial y$ and the mean
stratification rate $S_{\rho}=\partial\varrho/\partial y$ are constant for a
given simulation, the Richardson number variation is obtained by a change of
the Brunt–Väisälä frequency $N$ with $N^{2}=-g/\rho_{0}S_{\rho}$, while
keeping the mean shear rate $S$ constant.

The initial conditions are taken from a separate simulation of isotropic
turbulence without density fluctuations, which was allowed to develop for
approximately one eddy turnover time. The initial values of the Taylor-
microscale Reynolds number $Re_{\lambda}q\lambda/\nu=89$ and the shear number
$SK/\epsilon=2$ are fixed. Here $q$ is the rms of the fluctuating velocity
with $q^{2}=\overline{u_{i}u_{i}}$, $\lambda$ the Taylor-microscale with
$\lambda^{2}=5q^{2}\nu/\epsilon$, $K=q^{2}/2$ the kinetic energy, and
$\epsilon=\nu\overline{\partial u_{j}/\partial x_{k}\partial u_{j}/\partial
x_{k}}$ the dissipation rate.

Table 1 provides an overview of the eventual values of $Re_{\lambda}$, $q$,
and $\epsilon$ at time $St=10$. The table also lists values of a variety of
length scales, including the overturning scale $L_{overturn}=q^{3}/\epsilon$,
the Ellison scale $L_{Ellison}=\rho/S_{\rho}$, the Ozmidov scale $L_{Ozmidov}$
with $L_{Ozmidov}^{2}=\epsilon/N^{3}$, the Taylor-microscale $\lambda$, and
the Kolmogorov scale $\eta$ with $\eta^{4}=\nu^{3}/\epsilon$, indicating an
appropriate resolution of the simulations at $St=10$ at both the large and
small scales of the turbulent motion.

The simulations are performed on a parallel computer using $512\times
512\times 512$ grid points. To increase the resolution, instead of the
classical dealiasing with a cut-off at $2/3$ of the maximum wavenumber, a
cosine-filter dealiasing is applied. The cosine-filter is only applied to
wavenumbers larger than $2/3$ of the maximum wavenumber. Its transfer function
starts with one at $2/3$ of the maximum wavenumber, goes to zero at the
maximum wavenumber, and it follows the shape of the first quarter of the
cosine function period. The maximum wavenumber $k_{max}$ can be defined when
the cosine is equal to the value $1/2$. For the current resolution with
$N=512$ we thus have $k_{max}=227$ (instead of the value 170 obtained for
classical dealiasing). All simulations are well resolved and we have
$k_{max}\eta>1.2$ in the eventual evolution for the unstratified case. A
discussion on the influence of dealising in pseudo-spectral codes can be found
in Hou and Li (2007). The authors show that the classical $2/3$ rule does not
necessarily yield the best results and other filtering techniques, different
from the cosine-dealiasing used here, are more efficient, supporting our
choice.

###  II.3 Variance Estimates for the Lagrangian and Eulerian Accelerations in
Stratified Shear Flow

The Lagrangian and Eulerian accelerations are defined as

$\bm{a}_{L}=\frac{\partial\bm{u}}{\partial
t}+\bm{u}\cdot\nabla\bm{u}\quad\textrm{and}\quad\bm{a}_{E}=\frac{\partial\bm{u}}{\partial
t},$ (6)

respectively. Both accelerations are computed as a volume average at a fixed
time, which is an appropriate choice for homogeneous flows. The effects of
shear and buoyancy are considered as external forces.

In Jacobitz  _et al._ (2016) we provided estimates of the variances of the
Lagrangian and Eulerian accelerations writing the Navier–Stokes equations in
the form

$\frac{\partial\bm{u}}{\partial t}=-\bm{N}-\bm{\Pi}-\bm{\Lambda},$ (7)

where the terms on the right hand side are given by

$\displaystyle\bm{N}$ $\displaystyle=$
$\displaystyle\bm{a}_{C}=\bm{u}\cdot\nabla\bm{u}$ $\displaystyle\bm{\Pi}$
$\displaystyle=$ $\displaystyle\bm{a}_{P}=\nabla(p/\rho_{0})$
$\displaystyle\bm{\Lambda}$ $\displaystyle=$
$\displaystyle\bm{\Lambda}_{S}+\bm{\Lambda}_{B}+\bm{\Lambda}_{V}=Sv\bm{e}_{x}+\frac{g}{\rho_{0}}\rho\bm{e}_{y}-\nu\nabla^{2}\bm{u}.$
(8)

Here, $\bm{N}$ is the nonlinear or advection term, $\bm{\Pi}$ the pressure-
gradient term, and $\bm{\Lambda}$ the linear term with contributions from
shear, buoyancy, and viscous effects. The notation $\bm{a}_{C}$ for the
nonlinear term and $\bm{a}_{P}$ for the pressure gradient match the notation
in Tsinober  _et al._ (2001) to denote the convective and pressure
contributions, respectively, to the accelerations.

According to Jacobitz  _et al._ (2016) we have also in the case of stratified
shear flow the identity

$\left<\|\bm{N}+\bm{\Pi}+\bm{\Lambda}\|^{2}\right>=\left<\|\bm{N}+\bm{\Lambda}\|^{2}\right>-\left<\|\bm{\Pi}\|^{2}\right>.$
(9)

Here, $\|\cdot\|$ denotes the magnitude of a vector and $\left<\cdot\right>$
the volume average for a homogeneous field.

This directly implies the following exact identities for the variances of the
Eulerian acceleration ${\bm{a}}_{E}$ (called local acceleration in Tsinober
_et al._ (2001)),

$a_{E}^{2}\equiv\left<\|\frac{\partial\bm{u}}{\partial
t}\|^{2}\right>=\left<\|\bm{N}+\bm{\Pi}+\bm{\Lambda}\|^{2}\right>=\left<\|\bm{N}+\bm{\Lambda}\|^{2}\right>-\left<\|\bm{\Pi}\|^{2}\right>$
(10)

and of the Lagrangian acceleration ${\bm{a}}_{L}$ (called total acceleration
in Tsinober  _et al._ (2001)),

$a_{L}^{2}\equiv\left<\|\frac{\partial\bm{u}}{\partial
t}+\bm{u}\cdot\nabla\bm{u}\|^{2}\right>=\left<\|\bm{\Pi}+\bm{\Lambda}\|^{2}\right>.$
(11)

The variance estimates provided in Jacobitz  _et al._ (2016) for rotating
shear flows are now extended to stratified shear flows, again with the
underlying assumption and crucial simplification of isotropy of the flow. The
main difference arises in the linear term $\bm{\Lambda}$, which now includes a
buoyancy force, instead of a Coriolis force. Neglecting the friction force,
the variance of the linear term can be written as

$\Lambda^{2}=\frac{1}{3}S^{2}[1+3(\frac{g}{\rho_{0}})^{2}\frac{1}{S^{2}}\frac{\rho^{2}}{u^{2}}]u^{2}.$
(12)

Using the ratio of potential to kinetic energy

$\frac{K_{\rho}}{K}=\frac{-\frac{1}{2}\frac{g}{\rho_{0}}\frac{\rho^{2}}{S_{\rho}}}{\frac{1}{2}u^{2}}=-\frac{g}{\rho_{0}}\frac{1}{S_{\rho}}\frac{\rho^{2}}{u^{2}}$
(13)

the variance for the linear term can be written as

$\Lambda^{2}=\frac{1}{3}S^{2}[1+3Ri\frac{K_{\rho}}{K}]u^{2}.$ (14)

Hence, the variance estimate of the linear term retains the ratio of potential
to kinetic energy.

###  II.4 Scale-dependent decomposition of Lagrangian and Eulerian
accelerations

To gain insight into the scale dependence of the Lagrangian and Eulerian
accelerations, we decompose both accelerations into an orthogonal wavelet
series. Wavelets are well localized functions in space and in scale (or wave
number), see e.g. Mallat (1998), and different wavelet-based diagnostics,
including the scale-dependent energy distribution and its spatial
fluctuations, intermittency measures such as the scale dependent flatness and
anisotropy measures, have been proposed. For a review we refer the reader to
Farge and Schneider (2015).

We consider a generic vector field ${\bm{a}}=(a_{1},a_{2},a_{3})$ at a fixed
time instant and decompose each component $a_{\alpha}({\bm{x}})$ into an
orthogonal wavelet series,

$a_{\alpha}({\bm{x}})\,=\,\sum_{\lambda}\,\widetilde{a}^{\alpha}_{\lambda}\,\psi_{\lambda}({\bm{x}}),$
(15)

where the wavelet coefficients are given by the scalar product
$\widetilde{a}^{\alpha}=\langle a_{\alpha},\psi_{\lambda}\rangle$. The
wavelets $\psi_{\lambda}$ with the multi-index $\lambda=(j,{\bm{i}},d)$ are
well localized in scale $L_{0}2^{-j}$ (where $L_{0}$ corresponds to the size
of the computational domain), around position $L_{0}{\bm{i}}/2^{j}$, and
orientated in one of the seven directions $d=1,...,7$, respectively. The scale
is directly related to the wave number $k_{j}=k_{0}2^{j}$, where $k_{0}$ is
the centroid wave number of the chosen wavelet family. For Coiflets 12
wavelets used in the present work we have $k_{0}=0.77$. Large scales
correspond to small values of the scale index $j$ and to a well localized
wavelet in Fourier space around wavenumber $k_{j}$. In contrast for small
scales, which correspond to large values of $j$, the wavelet becomes less
localized in Fourier space around the mean wavenumber $k_{j}$.

Reconstructing the three components $a^{\alpha}$ at scale $2^{-j}$ by summing
only over the position ${\bm{i}}$ and direction $d$ indices in eq. 15 yields
the acceleration ${\bm{a}}^{j}$ at scale intex $j$. In terms of filtering the
acceleration at a given scale corresponds to a bandpass filtered field, with a
bandpass filter having constant relative bandwidth. This means that the
filterwidth becomes larger at larger wavenumber, corresponding to decreasing
scale. By construction we have ${\bm{a}}=\sum_{j}{\bm{a}}^{j}$, where the
${\bm{a}}^{j}$ are mutually orthogonal.

The scale-dependent moments, including scale-dependent flatness, and scale-
dependent pdfs, can thus be computed from ${\bm{a}}^{j}$ using classical
statistical estimators.

For instance the $q$-th order moment of ${\bm{a}}^{j}({\bm{x}})$ can be
defined by,

$M_{q}[{\bm{a}}^{j}]=\langle({\bm{a}}^{j})^{q}\rangle,$ (16)

and by construction the mean value vanishes, $\langle{\bm{a}}^{j}\rangle=0$.
The moments are thus central moments. These scale-dependent moments are
directly related to the $q$-th order structure functions Schneider  _et al._
(2004) where the increment size is $\propto 2^{-j}$.

The scale-dependent flatness, which measures the intermittency of
${\bm{a}}^{j}$ at scale $2^{-j}$, is defined by

$Fl[{\bm{a}}^{j}]=\frac{M_{4}[{\bm{a}}^{j}]}{\left(M_{2}[{\bm{a}}^{j}]\right)^{2}}.$
(17)

For a Gaussian distribution the flatness equals three at all scales.

###  II.5 Geometrical statistics

To understand the magnitude of Eulerian and Lagrangian accelerations, we
statistically assess, following Tsinober  _et al._ (2001), the alignment
properties of ${\bm{a}}_{E}$, ${\bm{a}}_{C}=\bm{N}$ and its sum corresponding
to the Lagrangian acceleration ${\bm{a}}_{L}={\bm{a}}_{E}+{\bm{a}_{C}}$. For
convenience, we partly use the notation introduced in Tsinober  _et al._
(2001) in this section. When the vectors of the Eulerian acceleration
${\bm{a}}_{E}$ and the convective terms ${\bm{a}}_{C}$ are anti-parallel, then
the magnitude of the Lagrangian acceleration ${\bm{a}}_{L}$, is small compared
to those of the Eulerian and convective contribution, since

$\langle{\bm{a}}_{L}{\bm{a}}_{L}\rangle=\langle{\bm{a}}_{E}+{\bm{a}}_{C},{\bm{a}}_{E}+{\bm{a}}_{C}\rangle=\langle{\bm{a}}_{E},{\bm{a}}_{E}\rangle+\langle{\bm{a}}_{C},{\bm{a}}_{C}\rangle+2\cos({\bm{a}}_{E},{\bm{a}}_{C})\,||{\bm{a}}_{E}||\,||{\bm{a}}_{C}||.$
(18)

If ${\bm{a}}_{E}$ and ${\bm{a}}_{C}$ are anti-aligned the cosine is negative
and the norm of ${\bm{a}}_{E}+{\bm{a}}_{C}$ is minimal. To verify the random
Taylor-hypothesis Tsinober  _et al._ (2001) computed the cosine of the angle
of the Eulerian acceleration and the convective term, motivated by the
prediction of Tennekes (1975) that the Lagrangian acceleration must be small
so that the hypothesis holds. We expect this result to hold with modification
due to shear and stratification.

For the pressure gradient term ${\bm{a}}_{P}=\bm{\Pi}$, the alignment with the
Eulerian and Lagrangian acceleration can be likewise assessed. For
sufficiently high Reynolds numbers we anticipate a strong anti-alignment of
${\bm{a}}_{P}$ with ${\bm{a}}_{L}$ showing that the flow is driven by the
pressure gradients and that linear effects are negligible. However, buoyancy
may change this result for strong stratification and its impact will be
assessed using the simulation results.

For Gaussian divergence free random fields Tsinober  _et al._ (2001) found
similar alignment properties and they concluded that the cancellation of
$\bm{a}_{E}$ and $\bm{a}_{C}$ is mostly a kinematic effect and not due to
Navier–Stokes dynamics. A justification for these findings is given by
Millionshchikow’s zero-fourth cumulant hypothesis (Monin and Yaglom, 1971)
which decomposes fourth order moments into a series of second order moments.

##  III Results

In this section, the flow evolution is briefly described first. Then, results
for the probability density functions (pdfs) of the Lagrangian and Eulerian
accelerations are provided, related to the remaining terms in the
Navier–Stokes equations, and their scale-dependent properties are presented. A
similar analysis is performed for the Lagrangian and Eulerian time-rates of
change of fluctuating density and in appendix A also for fluctuating
vorticity. In the following, the accelerations and time-rates of change are
analyzed at the instant $St=10$. Table 1 provides an overview of the series of
five simulations performed.

###  III.1 Turbulence Evolution

In order to provide a context for the present study, the energetics of the
flow is briefly discussed. More details on turbulent stratified shear flows
can be found in Jacobitz  _et al._ (1997) and Jacobitz (2002).

Figure 1 (left) shows the evolution of the turbulent kinetic energy normalized
by its initial value $K/K_{0}$. All cases result in an initial decay phase due
to the isotropic initial conditions. Then, as the Richardson number $Ri$ is
increased, the eventual evolution of the turbulent kinetic energy changes from
growth to decay with a critical value of $Ri_{cr}\approx 0.15$.

The normalized transport equation for the turbulent kinetic energy evolution
can be written as:

$\gamma=\frac{1}{SK}\frac{dK}{dt}=\frac{P}{SK}-\frac{B}{SK}-\frac{\epsilon}{SK}$
(19)

Here, $\gamma$ is the growth rate of the turbulent kinetic energy, $P/(SK)$ is
the normalized production term with $P=-S\overline{u_{1}u_{2}}$, $B/(SK)$ is
the normalized buoyancy flux with $B=g/\rho_{0}\overline{u_{2}\rho}$, and
$\epsilon/(SK)$ is the normalized dissipation rate.

Figure 1 (right) shows the dependence of $P/(SK)$, $B/(SK)$, $\epsilon/(SK)$,
and $\gamma$ on the Richardson number $Ri$ at nondimensional time $St=10$. The
normalized production rate $P/(SK)$ decreases with increasing Richardson
number $Ri$ and it assumes a slightly negative value for large $Ri$ cases,
indicating a positive Reynolds shear stress (or counter-gradient flux). The
normalized buoyancy flux $B/(SK)$ remains relatively small and it converts
kinetic to potential energy for most of the $Ri$ range. The normalized
dissipation rate $\epsilon/(SK)$ remains relatively unaffected by the $Ri$
variation. The growth rate $\gamma$ follows the trend of the normalized
production rate $P/(SK)$, offset by the contributions of $B/(SK)$ and
$\epsilon/(SK)$. Note that positive values of $\gamma$ correspond to a growth
of $K$, while a negative value of $\gamma$ indicates decay of the turbulent
kinetic energy.

The evolution of the ratio of potential to kinetic energy is given in figure 2
(left). The simulations are initialized without potential energy and a strong
initial growth is observed. The ratio of potential to kinetic energy
eventually reaches an approximately constant value, which still depends on the
Richardson number $Ri$. This dependence of the ratio $K_{\rho}/K$ on $Ri$ at
nondimensional time $St=10$ is presented in figure 2 (right). The ratio of
$K_{\rho}/K$ first increases strongly and then reaches a maximum of
$K_{\rho}/K\approx 0.3$ for $Ri=1$.

###  III.2 Lagrangian and Eulerian Accelerations

Figure 3 (top) shows the probability distribution functions (pdfs) of the
Lagrangian acceleration $\bm{a}_{L}$ (left) and of the Eulerian acceleration
$\bm{a}_{E}$ (right). The pdfs of both accelerations have stretched-
exponential shapes and they exhibit a strong and similar influence on the
Richardson number $Ri$. For small $Ri$, the extreme values of the Eulerian
acceleration are above those of the Lagrangian acceleration, which is
consistent with previous observations for sheared and rotating turbulence
(Jacobitz  _et al._ , 2016) and likewise observed for isotropic turbulence by
Tsinober  _et al._ (2001).

Figure 3 (bottom) shows the pdfs normalized with the corresponding standard
deviations of the two accelerations. For a core region of about five standard
deviations, both the Lagrangian and Eulerian accelerations show approximately
the same shape. The tails of the pdfs of both accelerations weaken with
increasing $Ri$. For small $Ri$, the tails of the Lagrangian acceleration are
heavier than the tails of the Eulerian acceleration.

Tables 2 and 3 provide statistical information of the Lagrangian and Eulerian
accelerations as a function of the Richardson number at nondimensional time
$St=10$. The magnitudes (rms values) of both accelerations decrease with
increasing $Ri$ and the magnitude of $\bm{a}_{E}$ remains always larger than
the magnitude of $\bm{a}_{L}$, also observed for isotropic turbulence Tsinober
_et al._ (2001). At small $Ri$, the heavier tails observed for the normalized
pdfs of $\bm{a}_{L}$ as compared to $\bm{a}_{E}$ result in a larger flatness
of the Lagrangian acceleration pdf as compared to its Eulerian counterpart.
The flatness values of both accelerations generally decrease with increasing
$Ri$, indicating a decreased importance of nonlinear effects which is related
to the decreasing Reynolds number. However, unlike in the presence of strong
rotation considered in Jacobitz  _et al._ (2016), the flatness values do not
reach a value close to three, characteristic for a Gaussian distribution, in
the case of strong stratification. For $Ri=0$ the flatness values of
$\bm{a}_{L}$ and $\bm{a}_{E}$ are comparable with DNS data of isotropic
turbulence Tsinober (2001) at $Re_{\lambda}=140$, where the values of $24.4$
and $12.8$ were found, while we find respectively 27.81 and 14.41 in the case
of pure shear. Note that for the variances Tsinober  _et al._ (2001) found the
values (normalized with $\epsilon^{3/2}\nu^{-1/2}=24.84$, where
$\epsilon=1.20$ and $\nu=0.0028$) of $2.75$ for $a_{L}$ and $8.19$ for $a_{E}$
(and 10.96 for $a_{C}$), while we find respectively the values 8.20 for
$a_{L}$ and 14.81 for $a_{E}$ (and 22.39 for $a_{C}$) in the case of pure
shear using the same normalization with $\epsilon^{3/2}\nu^{-1/2}$. This shows
that the order is consistent and the values are comparable.

Figure 4 shows pdfs of the shear term (top, left), the buoyancy term (top,
right), the pressure-gradient term (bottom, left), and the advection term
(bottom, right) in the Navier–Stokes equation. The shear and buoyancy terms
depend linearly on fluctuating velocity components and density and their pdfs
have hence a Gaussian shape. While the magnitude of the shear term pdf
decreases with increasing $Ri$, the magnitude of the buoyancy term pdf
increases. The pdfs of the pressure-gradient and advection terms show a
stretched-exponential shape due to the quadratic nature of the terms. The
magnitudes of both terms decrease with increasing $Ri$. For small $Ri$, the
pressure-gradient and advection terms clearly dominate the shear and buoyancy
terms, but this dominance somewhat diminishes with increasing $Ri$. Hence, the
pressure-gradient term is the generally dominant contribution to the
Lagrangian acceleration, while the advection term is important for the
Eulerian acceleration.

Table 4 provides the variances of the contributions to the linear term from
the shear term $\Lambda_{S}^{2}$, the buoyancy term $\Lambda_{B}^{2}$, and the
viscous term $\Lambda_{V}^{2}$. An estimate for the variance of the linear
term $\Lambda_{DNS}^{2}$ is computed using the triangle inequality. The table
also provides the variance of the velocity $q^{2}$, the ratio of potential to
kinetic energies $K_{\rho}/K$, and a theoretical estimate for the linear term
$\Lambda^{2}$ based on equation 14 at nondimensional time $St=10$. The
variances of the linear term computed from the simulation results
$\Lambda_{DNS}^{2}$ and the theoretical estimate $\Lambda^{2}$ agree well,
despite the assumption of isotropy used in the derivation of equation 14. The
variance of the linear term decreases with increasing Richardson number $Ri$.
The Taylor-microscale Reynolds number $Re_{\lambda}$ given in table 1 yields a
measure for the general importance of nonlinear effects in a turbulent flow.
$Re_{\lambda}$ decreases with increasing $Ri$. The results suggest that
nonlinear effects contribute the least to the turbulence evolution for the
case with a Richardson number $Ri=1$.

The joint pdfs of the Lagrangian and Eulerian accelerations are shown in
figure 5 for two cases with Richardson numbers $Ri=0.1$ (left) and $Ri=1$
(right) at nondimensional time $St=10$. The correlation between Lagrangian and
Eulerian accelerations is observed to increase with increasing $Ri$. The
stronger correlation of the Eulerian and Lagrangian acceleration for $Ri=1$ is
due to the reduced nonlinearity.

In order to quantify this observation, the Pearson product-moment correlation
coefficient for the Lagrangian and Eulerian accelerations in dependence of the
Richardson number $Ri$ is given in the first line of table 5 at nondimensional
time $St=10$. For unstratified shear flow with $Ri=0$, the Lagrangian and
Eulerian accelerations are almost decorrelated as indicated by $r=0.0284$.
With increasing stratification strength, the Pearson product-moment
correlation coefficient increases monotonically. A high value of $r=0.6634$ is
observed for $Ri=1$. This can be explained by the decreasing importance of the
nonlinearity, quantified by the decreasing Reynolds number.

To provide further information, geometrical statistics are performed and the
vector alignment properties of the different acceleration contributions are
studied, as introduced in section II.5. For different Richardson numbers we
consider the pdf of the cosine of the angle of two acceleration vectors, shown
in Figure 6, the mean value of the cosine as quantitative measure and the
correlation coefficient, assembled in table 5. The choice of the cosine is
motivated by the fact that for random fields in 3D the cosine of the angle is
uniformly distributed and not the angle itself. Figure 6 (bottom, left) shows
that a strong anti-alignment of the Eulerian acceleration ${\bm{a}}_{E}$ and
the convective term ${\bm{a}}_{C}={\bm{N}}$ reflected in a peak in the pdf at
$\cos=-1$, corresponding to an angle of $180$ degrees. This explains why the
Lagrangian acceleration is smaller than the Eulerian one and also smaller than
the convective term, as the anti-alignment implies that the two vectors
${\bm{a}}_{E}$ and ${\bm{a}}_{C}={\bm{N}}$ are anti-parallel. The Lagrangian
acceleration ${\bm{a}}_{L}$ and the pressure gradient $\bm{\Pi}$ in Fig. 6
(bottom, right) even show a stronger anti-alignment, which confirms that the
negative pressure gradient is the driving force of the flow dynamics. In both
cases the anti-alignment is most pronounced for $Ri=0$ and becomes weaker for
increasing Richardson numbers. This can be further quantified by mean values
of the cosine of the angle and also the correlation coefficient between the
two vectors, the results are given in table 5.

The Lagrangian acceleration is positively aligned with the Eulerian one (Fig.
6, top, left) and also with the convective acceleration (Fig. 6, top, right).
For ${\bm{a}}_{E}$ this alignment becomes stronger with increasing Richardson
number, while for ${\bm{a}}_{C}$ this becomes weaker, as the nonlinear term
diminishes. These results are consistent with those in Tsinober  _et al._
(2001) obtained for isotropic turbulence, in the case of $Re_{\lambda}=141$
(as compared to our value for unstratified shear flow $157$). For example for
the average cosine of the angle between $a_{L}$ and $a_{E}$ Tsinober  _et al._
(2001) reports a value $0.105$ (for unstratified shear flow we find $0.162$),
for $a_{L}$ and $a_{C}$ 0.353 (0.420) and $a_{E}$ and $a_{C}$ -0.762 (-0.657).

Let us also mention that Tsinober  _et al._ (2001) showed that Gaussian random
fields satisfy similar alignment properties for ${\bm{a}}_{E}$ and
${\bm{a}}_{C}={\bm{N}}$ and concluded that this is essentially a kinematic
effect.

Figure 7 presents the scale-dependent pdfs of the Lagrangian acceleration
$\bm{a}_{L}$ (left) and Eulerian acceleration $\bm{a}_{E}$ (right) for two
cases with $Ri=0.1$ (top) and $Ri=1$ (bottom) at nondimensional time $St=10$.
The pdfs have stretched-exponential shapes and the tails become heavier with
increasing scale index $j$ or decreasing scale of the turbulent motion. For
the weakly stratified case with $Ri=0.1$, the tails of the Lagrangian
acceleration are generally heavier as those of the Eulerian acceleration. For
the strongly stratified case with $Ri=1$, however, the tails of the Eulerian
acceleration are generally heavier than those of their Lagrangian counterpart.
This observation reflects the trend for the total acceleration pdfs with
increasing Richardson number discussed above.

In order to quantify the above observations, scale-dependent statistics are
provided in tables 7 and 8 for two cases with Richardson numbers $Ri=0.1$ and
$Ri=1$, respectively. While the magnitude of the total Eulerian acceleration
$a_{E}$ is larger than the magnitude of the total Lagrangian acceleration
$a_{L}$, the ordering is reversed for the accelerations at some scales of the
turbulent motion. For the case with $Ri=0.1$, the original ordering holds at
the scale with the largest magnitude, which is $j=6$ for both accelerations.
At that scale, the pdfs of the total accelerations are also most similar to
the pdfs of the accelerations at that scale. For the case with $Ri=1$, the
ordering observed for the magnitudes of the total accelerations holds more
generally at different scales of the turbulent motion. The pdfs of the total
accelerations are again most similar at the scales with the largest
magnitudes, which are $j=4$ for the Lagrangian acceleration and $j=5$ for the
Eulerian acceleration. The flatness of the accelerations generally increases
with scale index $j$, indicating more intermittency at the smallest scales of
motion. Note that for the Lagrangian acceleration, flatness values close to
three are observed for the larger scales with $j=1$, indicating that the
Lagrangian acceleration at large scale is mainly determined by linear effects.

Figure 8 shows the scale-dependent joint pdfs of the Lagrangian and Eulerian
accelerations for two cases with $Ri=0.1$ (left) and $Ri=1$ (right) as well as
at large scale with scale index $j=3$ (top) and at small scale with $j=7$
(bottom) at nondimensional time $St=10$. Consistent with the observation for
the total accelerations discussed above, the correlation increases with
stratification strength at the two scales shown. In addition, the correlation
decreases with increasing scale index $j$ or decreasing scale of the turbulent
motion considered.

This observation is shown more quantitatively using the Pearson product-moment
correlation coefficient in table 6. The correlation coefficient tends to
increase with increasing Richardson number $Ri$. Similarly, at all Richardson
numbers, the correlation coefficient decreases with decreasing scale or
increasing scale index $j$. The components at the largest scale index $j$ or
smallest scale of motion are characterized by very high flatness values. This
indicates strong intermittency present in the motion with the localized
activity impacting the correlation coefficient. Note that with increasing
$Ri$, the Taylor micro-scale Reynolds number $Re_{\lambda}$ decreases.
Starting from the same initial conditions, an increase of the Richardson
number $Ri$ necessarily results in the decrease of the Taylor micro-scale
Reynolds number $Re_{\lambda}$ due to the effect of stratification. Hence it
is difficult to determine if the origin of the increased intermittency is due
to the increased stratification or decreased $Re_{\lambda}$ as the two effects
are linked.

###  III.3 Lagrangian and Eulerian Time-Rates of Change of Fluctuating Density

The time-rates of change of fluctuating density can also be defined using
Lagrangian and Eulerian approaches as

$s_{L}=\frac{\partial\rho}{\partial
t}+\bm{u}\cdot\nabla\rho\quad\textrm{and}\quad
s_{E}=\frac{\partial\rho}{\partial t},$ (20)

respectively.

Figure 9 (top) shows the pdfs of the Lagrangian time-rate of change of
fluctuating density (left) and of the corresponding Eulerian time-rate of
change (right). The difference in the pdfs of the time-rates of change is well
more pronounced than the difference obtained for the accelerations. Figure 9
(bottom) shows the normalized pdfs of the two time-rates of change. While the
shape of the Eulerian time-rate of change pdf is again found to be stretched-
exponential, the Lagrangian time-rate of change pdf has a more Gaussian shape.
The extreme values of the Eulerian time-rate of change of fluctuating density
are substantially larger than those of the Lagrangian time-rate of change.

Tables 2 and 3, respectively, provide the dependence of the magnitudes of the
Lagrangian and Eulerian time-rates of change on the Richardson number $Ri$.
Note that for $Ri=0$, the density is a passive scalar (zero gravity) with a
mean gradient. Again, the magnitude of $s_{E}$ always remains larger than the
magnitude of $s_{L}$, consistent with the findings for the accelerations.

The flatness of the Lagrangian and Eulerian time-rates of change are also
given in tables 2 and 3, respectively. The flatness of $s_{E}$ is always
larger than that of $s_{L}$ and their values generally decrease with
increasing $Ri$. For strong stratification, the flatness of the Lagrangian
time-rate of change assumes values around three for $Ri=1$, while the Eulerian
time-rate of change yields a value of $5.475$.

Figure 10 shows pdfs of the buoyancy term (left) and advection term (right) in
the advection-diffusion equation for fluctuating density. The buoyancy term
pdf has a Gaussian shape as it is linearly related to the fluctuating density.
Its variance increases with increasing $Ri$, because the stratification rate
$S_{\rho}$ increases. The more Gaussian shape of the Lagrangian time-rate of
change of fluctuating density can be explained by the lack of a quadratic term
in the advection-diffusion equation for fluctuating density. The large
difference observed between the Lagrangian and Eulerian time-rates of change
of fluctuating density is due to the advection term.

##  IV Conclusions

A series of direct numerical simulations was performed in order to study the
Lagrangian and Eulerian acceleration properties in stably stratified turbulent
shear flows. With increasing Richardson number $Ri$, the evolution of the
turbulent kinetic energy $K$ changes from growth to decay and the variances of
the Lagrangian acceleration $\bm{a}_{L}$ and the Eulerian acceleration
$\bm{a}_{E}$ decrease. The acceleration pdfs were observed to have a
stretched-exponential symmetric shape and the flatness decreases with
increasing $Ri$.

We studied the cancellation of Eulerian and convective accelerations of fluid
particle using geometrical statistics of the vector quantities. We found a
strong preference for the anti-alignment of both vectors, which decreases with
the Richardson number. This cancellation explains why the variance of the
Lagrangian acceleration is smaller than its Eulerian counterpart and it
supports, according to Tsinober  _et al._ (2001), who performed similar
analyses for isotropic turbulence, the random Taylor hypothesis for shear flow
which becomes however weaker with increasing stratification. Nevertheless, we
do not find an order of magnitude difference in the acceleration variances, as
predicted by Tennekes (1975) for isotropic turbulence and necessary so that
the random Taylor hypothesis strictly holds. These findings are in agreement
with Lin (1953), who showed that Taylor’s hypothesis does in general not hold
for shear flow. Analyzing the alignment properties of the scale-dependent
contributions of the acceleration is an interesting perspective for future
work, already in the context of isotropic turbulence. This would allow to
check if the hypothesis holds for shear flows at least as small scales.

An estimation of the variances of the Lagrangian and Eulerian accelerations
has been derived from the Navier–Stokes equations which requires the ratio of
potential to kinetic energy. A comparison of the estimation with results from
the direct numerical simulations showed good agreement for the considered
range of Richardson numbers.

The pdfs of the pressure-gradient and advection terms in the Navier–Stokes
equation, which are both quadratic terms, also have stretched-exponential
shapes. The Lagrangian and Eulerian accelerations are mainly determined by the
pressure-gradient and advection terms, respectively. While the quadratic terms
are dominant for small $Ri$, their dominance is somewhat diminished for large
$Ri$. The pdfs of the shear and buoyancy terms in the Navier–Stokes equation,
which are both linear terms, were observed to have a Gaussian shape. While the
variance of the shear term decreases with $Ri$, the variance of the buoyancy
term increases with $Ri$.

In addition, the Lagrangian and Eulerian time-rates of change of fluctuating
density and of fluctuating vorticity (see appendix A) were considered. For
both quantities, the Eulerian time-rates of change showed substantially larger
extreme values than their Lagrangian counterparts. Due to a lack of a
quadratic term on the right-hand-side of the advection-diffusion equation for
fluctuating density, the pdf of the Lagrangian time-rate of change has an
almost Gaussian shape, while the pdf of the Eulerian time-rate of change was
observed to have exponential to stretched-exponential shapes. For fluctuating
vorticity we found that the Lagrangian time-rate of change is mainly
determined by the vortex streching and tilting term.

A scale-dependent analysis using orthogonal wavelet decomposition provided
insight into the intermittency of the Lagrangian and Eulerian accelerations.
At small scales of the turbulent motion, the pdfs exhibit heavy tails,
resulting in very large flatness values and corresponding intermittency. The
correlation between the Lagrangian and Eulerian accelerations has likewise
been analyzed and we found stronger correlation at large scales of turbulent
motion as well as with increasing Richardson number. At small scales this
correlation is substantially reduced.

For rotating and sheared homogeneous turbulence, Salhi  _et al._ (2014)
observed a dominance of linear terms in the cases with strong rotation and the
flatness of the Lagrangian acceleration assumes a value of about $3$. This
observation suggests that linear theory can accurately describe properties of
such flows. In the present study, however, the flatness never reaches values
close to three, even for very large Richardson numbers. Hence, linear theory
should not yield agreement with direct numerical simulation results. Indeed,
Hanazaki and Hunt (2004) found important differences between linear theory and
the fully nonlinear evolution of homogeneous turbulence in stratified shear
flows.

Perspectives for future work include a component-wise analysis of the
Lagrangian and Eulerian acceleration, a more detailed scale-wise decomposition
of the geometric properties of the accelerations, and corresponding terms for
the vorticity evolution.

###### Acknowledgements.

FGJ acknowledges the support from a University Professor Award from the
University of San Diego and the hospitality at Aix-Marseille Université. KS
acknowledges financial support from Agence Nationale de la Recherche, project
AIFIT (ANR-15-CE40-0019) and project CM2E (ANR-20-CE46-0010-01), and the
French Research Federation for Fusion Studies within the framework of the
European Fusion Development Agreement (EFDA).

##  Appendix A Lagrangian and Eulerian Time-Rates of Change of Fluctuating
Vorticity

The Lagrangian and Eulerian time-rates of change of fluctuating vorticity
$\bm{\omega}$ are defined as

$\bm{c}_{L}=\frac{\partial\bm{\omega}}{\partial
t}+\bm{u}\cdot\nabla\bm{\omega}\quad\textrm{and}\quad\bm{c}_{E}=\frac{\partial\bm{\omega}}{\partial
t},$ (21)

respectively. This definition is analogous to the definition for the
Lagrangian and Eulerian accelerations in order to enable a comparison between
the accelerations and vorticity time-rate of change statistics. Again, the
analysis is performed at the nondimensional time $St=10$.

Figure 11 (top) shows the probability distribution functions (pdfs) of the
Lagrangian time-rate of change $\bm{c}_{L}$ (left) and of the Eulerian time-
rate of change $\bm{c}_{E}$ (right). Similar to the accelerations, pdfs with
stretched-exponential shapes are observed for both time-rates of change and a
strong and similar influence on the Richardson number $Ri$ is obtained. Again,
stronger extreme values are obtained for the Eulerian time-rate of change, but
the difference to the Lagrangian time rate of change is much more pronounced
here as compared to the accelerations. Figure 11 (bottom) shows the normalized
pdfs of the two time-rates of change. Again, for a core region of about five
standard deviations, both the Lagrangian and Eulerian time-rates of change
have an approximately similar shape. For small Richardson numbers $Ri$, the
tails of the Lagrangian time-rate of change are heavier than those of their
Eulerian counterparts. However, this ordering is reversed at larger $Ri$.

The magnitudes of the Lagrangian and Eulerian vorticity time-rates of change
are given in tables 2 and 3, respectively. Similar to the magnitudes of the
accelerations, the magnitudes of both time-rates of change decrease with
increasing $Ri$ and the variance of $\bm{c}_{E}$ remains always larger than
the variance of $\bm{c}_{L}$. This difference in the magnitudes for the
vorticity time-rate of change pdfs is much more pronounced than that of the
accelerations. The heavier tails observed for the pdf of $\bm{c}_{L}$ as
compared to $\bm{c}_{E}$ at small $Ri$ results in a larger flatness of the
Lagrangian time-rate of change pdf as compared to its Eulerian counterpart.
Again, the ordering of the flatness values is reversed at larger $Ri$. While
the flatness values decrease with increasing $Ri$, the flatness is again
observed to level off at a value of approximately 5, well above the value of 3
expected for a Gaussian pdf. Hence, some nonlinearity is still present even in
the case of strongly suppressed turbulence in strongly stratified flows.

Figure 12 shows pdfs of the shear term (top, left), the buoyancy term (top,
right), the vortex tilting and stretching term (bottom, left), and the
advection term (bottom, right) in the vorticity equation. The shear and
buoyancy terms depend linearly on the curl of fluctuating velocity components
and fluctuating density, respectively. Similarly to the respective terms in
the Navier–Stokes equation, the magnitude of the shear term decreases with
increasing $Ri$ and the magnitude of the buoyancy term increases. The pdfs of
the vortex tilting and stretching term and the advection term show stretched-
exponential shapes due to the quadratic nature of the terms. The magnitudes of
both terms decrease with increasing $Ri$. For small $Ri$, the vortex tilting
and stretching term as well as the advection term clearly dominate the shear
and buoyancy terms, but this dominance again is reduced for large $Ri$. In the
case of vorticity, the vortex tilting and stretching term is the generally
dominant contribution to the Lagrangian time-rate of change, while the
advection term is important for the Eulerian time-rate of change.

##  Appendix B Lagrangian and Eulerian component pdfs

While the main manuscript exclusively discusses the properties of vector pdfs,
this appendix presents component pdfs of the Lagrangian and Eulerian
accelerations in order to address their anisotropy in turbulent stratified
shear flow. Figure 13 compares the vector pdfs with their $x-$, $y-$, and
$z-$component pdfs for the Lagrangian acceleration (left) and Eulerian
acceleration (right) for two cases with weak stratification with $Ri=0.1$
(top) and with strong stratification with $Ri=1$ (bottom). All pdfs show
similar shapes and the flow anisotropy is reflected in the variances. A
similar observation holds for the pdfs of Lagrangian and Eulerian time-rate of
change of fluctuating vorticity (not shown here).

The ratios of the component variances to the corresponding vector variances of
the Lagrangian and Eulerian accelerations are given in tables 2 and 3,
respectively. For small Richardson numbers $Ri$, the variance ratios show an
almost equipartition between the three components for both the Lagrangian and
Eulerian accelerations. For large $Ri$, however, the ratio of the vertical
variances to the vector variances gains due to the direct impact of the
buoyancy term in the vertical component of the Navier–Stokes equation. Figure
2 shows that the ratio of potential to kinetic energy increases with
increasing $Ri$. Hence the buoyancy term impacts particularly the vertical
component of the accelerations.

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Figure 1: Evolution of the turbulent kinetic energy $K$ in nondimensional time
$St$ (left) and dependence of the normalized production rate $P/(SK)$,
buoyancy flux $B/(SK)$, and dissipation rate $\epsilon/(SK)$ on the Richardson
number at $St=10$ (right).

Figure 2: Evolution of the ratio of turbulent potential to kinetic energy
$K_{\rho}/K$ in nondimensional time $St$ (left) and dependence of this ratio
on the Richardson number at $St=10$ (right).

Figure 3: Pdfs (top) and pdfs normalized with the corresponding standard
deviations (bottom) of Lagrangian acceleration $\bm{a}_{L}$ (left) and
Eulerian acceleration $\bm{a}_{E}$ (right) at nondimensional time $St=10$.
Note that all pdfs are estimated using histograms with 100 bins, and they are
plotted in log-lin representation. Note that pdfs for the vector quantities
are shown.

Figure 4: Pdfs of the shear (top, left), buoyancy (top, right), pressure-
gradient (bottom, left), and advection (bottom, right) terms in the
Navier–Stokes equations at nondimensional time $St=10$.

Figure 5: Joint pdfs of Lagrangian acceleration $\bm{a}_{L}$ and Eulerian
acceleration $\bm{a}_{E}$ for Richardson numbers $Ri=0.1$ (left) and $Ri=1$
(right) at nondimensional time $St=10$ using a linear color scale.

Figure 6: Pdfs of the cosine of the angles between different acceleration
contributions: $\cos(a_{L},a_{E})$ (top, left), $\cos(a_{L},a_{C})$ (top,
right), $\cos(a_{E},a_{C})$ (bottom, left), and $\cos(a_{L},a_{P})$ (bottom,
right) for different $Ri$ at nondimensional time $St=10$.

Figure 7: Scale-dependent normalized pdfs of Lagrangian acceleration
$\bm{a}_{L}$ (left) and Eulerian acceleration $\bm{a}_{E}$ (right) for
Richardson numbers $Ri=0.1$ (top) and $Ri=1$ (bottom) at nondimensional time
$St=10$. Note that the flat sections in the pdfs around zero are artifacts of
an even number of bins chosen in the computation of the pdfs. Additionally,
pdfs of quantities with large variances only show fewer bins in the figure.

Figure 8: Scale-dependent joint pdfs of Lagrangian acceleration $\bm{a}_{L}$
and Eulerian acceleration $\bm{a}_{E}$ for Richardson numbers $Ri=0.1$ (left)
and $Ri=1$ (right) and at large scale with scale index $j=3$ (top) and at
small scale with $j=7$ (bottom) at nondimensional time $St=10$ using a linear
color scale.

Figure 9: Pdfs (top) and normalized pdfs (bottom) of Lagrangian time-rate of
change of fluctuating density $s_{L}$ (left) and Eulerian time-rate of change
$s_{E}$ (right) at nondimensional time $St=10$.

Figure 10: Pdfs of the buoyancy (left) and advection (right) terms in the
advection-diffusion equation for fluctuating density at nondimensional time
$St=10$.

Figure 11: Pdfs (top) and normalized pdfs (bottom) of Lagrangian time-rate of
change of vorticity $c_{L}$ (left) and Eulerian time-rate of change $c_{E}$
(right) at nondimensional time $St=10$. Note that pdfs for the vector
quantities are shown.

Figure 12: Pdfs of the shear (top, left), buoyancy (top, right), vortex
tilting and stretching (bottom, left), and advection (bottom, right) terms in
the vorticity equation at nondimensional time $St=10$.

Figure 13: Comparison of normalized vector pdfs with their corresponding
normalized component pdfs for Lagrangian acceleration (left) and Eulerian
acceleration (right) for $Ri=0.1$ (top) and $Ri=1$ (bottom) at nondimensional
time $St=10$.

Table 1:  Overview of the simulation cases, including the Richardson number $Ri$, the shear rate $S$, the Brunt–Väisälä frequency $N$, the Taylor-micro scale Reynolds number $Re_{\lambda}$, The viscosity $\nu$, the turbulent velocity fluctuation $q$, the dissipation rate of kinetic energy $\epsilon$, the cut-off wavenumber $k_{max}\eta$, the overturning scale $L_{overturn}$, the Ellison scale $L_{Ellison}$, the Ozmidov scale $L_{Ozmidov}$, the Taylor microscale $\lambda$, and the Kolmogorov scale $\eta$. All values are given at nondimensional time $St=10$. All symbols are defined in the text.  $Ri$ | 0 | 0.1 | 0.2 | 0.5 | 1
---|---|---|---|---|---
$S$ | 5.3345 | 5.3345 | 5.3345 | 5.3345 | 5.3345
$N$ | 0.0000 | 1.6869 | 2.3856 | 3.7720 | 5.3345
$Re_{\lambda}$ | 156.90 | 104.08 | 76.60 | 42.84 | 32.83
$\nu$ | 0.0010 | 0.0010 | 0.0010 | 0.0010 | 0.0010
$q$ | 1.5856 | 1.0779 | 0.7969 | 0.4428 | 0.3028
$\epsilon$ | 1.2838 | 0.6230 | 0.3436 | 0.1048 | 0.0390
$k_{max}\eta$ | 1.1992 | 1.4368 | 1.6673 | 2.2437 | 2.8723
$L_{overturn}$ | 3.1052 | 2.0101 | 1.4728 | 0.8288 | 0.7119
$L_{Ellison}$ | 0.2619 | 0.1835 | 0.1324 | 0.0619 | 0.0312
$L_{Ozmidov}$ | N/A | 0.3603 | 0.1591 | 0.0442 | 0.0160
$\lambda$ | 0.09895 | 0.09656 | 0.09613 | 0.09675 | 0.10841
$\eta$ | 0.00528 | 0.00633 | 0.00735 | 0.00988 | 0.01265
Table 2: Lagrangian acceleration statistics at nondimensional time $St=10$ showing the rms of the Lagrangian acceleration $a_{L}$, the flatness $Fl_{a_{L}}$ and the ratio of the component-wise variances and the total variance. The variance and flatness values of the time-rates of change of fluctuating density $s_{L}$ and $Fl_{s_{L}}$ and fluctuating vorticity $c_{L}$ and $Fl_{c_{L}}$ are likewise given. $Ri$ | 0 | 0.1 | 0.2 | 0.5 | 1
---|---|---|---|---|---
$a_{L}$ | 19.427 | 9.833 | 5.519 | 1.808 | 0.942
$Fl_{a_{L}}$ | 27.814 | 26.041 | 13.115 | 9.364 | 4.111
$a^{2}_{Lx}/a^{2}_{L}$ | 0.335 | 0.318 | 0.306 | 0.320 | 0.327
$a^{2}_{Ly}/a^{2}_{L}$ | 0.325 | 0.327 | 0.331 | 0.356 | 0.390
$a^{2}_{Lz}/a^{2}_{L}$ | 0.340 | 0.355 | 0.363 | 0.324 | 0.242
$s_{L}$ | 0.00117 | 0.00070 | 0.00091 | 0.00101 | 0.00119
$Fl_{s_{L}}$ | 7.220 | 6.148 | 5.152 | 3.989 | 3.249
$c_{L}$ | 490.484 | 232.508 | 128.472 | 44.395 | 19.725
$Fl_{c_{L}}$ | 65.574 | 53.766 | 21.168 | 14.123 | 9.177
Table 3: Eulerian acceleration statistics at nondimensional time $St=10$ showing the rms of the Eulerian acceleration $a_{E}$, the flatness $Fl_{a_{E}}$ and the ratio of the component-wise variances and the total variance. The variance and flatness values of the time-rates of change fluctuating density $s_{E}$ and $Fl_{s_{E}}$ and of fluctuating vorticity $c_{E}$ and $Fl_{c_{E}}$ are likewise given. $Ri$ | 0 | 0.1 | 0.2. | 0.5. | 1
---|---|---|---|---|---
$a_{E}$ | 26.100 | 11.689 | 6.258 | 1.975 | 1.075
$Fl_{a_{E}}$ | 14.414 | 13.413 | 11.028 | 9.510 | 5.691
$a^{2}_{Ex}/a^{2}_{E}$ | 0.302 | 0.320 | 0.339 | 0.408 | 0.434
$a^{2}_{Ey}/a^{2}_{E}$ | 0.329 | 0.312 | 0.292 | 0.261 | 0.273
$a^{2}_{Ez}/a^{2}_{E}$ | 0.369 | 0.368 | 0.369 | 0.331 | 0.293
$s_{E}$ | 0.00577 | 0.00239 | 0.00231 | 0.00138 | 0.00129
$Fl_{s_{E}}$ | 24.992 | 24.624 | 21.960 | 16.053 | 5.475
$c_{E}$ | 2,020.203 | 781.238 | 372.030 | 91.830 | 34.461
$Fl_{c_{E}}$ | 37.813 | 35.619 | 27.855 | 24.893 | 17.724
Table 4: Variance of the contributions to the linear term from the shear term $\Lambda_{S}^{2}$, the buoyancy term $\Lambda_{B}^{2}$, and viscous term $\Lambda_{V}^{2}$, an estimate for the variance of the linear term using the triangle inequality $\Lambda_{DNS}^{2}$, the variance of the velocity $q^{2}$, the ratio of potential to kinetic energies $K_{\rho}/K$, and an estimate for the linear term $\Lambda^{2}$ given in equation 14 at nondimensional time $St=10$.  $Ri$ | 0 | 0.1 | 0.2 | 0.5 | 1
---|---|---|---|---|---
$\Lambda_{S}^{2}$ | 14.5746 | 6.3796 | 3.2517 | 0.8430 | 0.3596
$\Lambda_{B}^{2}$ | 0.0000 | 0.2728 | 0.5681 | 0.7767 | 0.7886
$\Lambda_{V}^{2}$ | 6.8850 | 2.5158 | 1.1384 | 0.2568 | 0.0723
$\Lambda_{DNS}^{2}$ | 21.4596 | 9.1682 | 4.9581 | 1.8764 | 1.2205
$q^{2}$ | 2.5142 | 1.1619 | 0.6350 | 0.1961 | 0.0917
$K_{\rho}/K$ | 0.0000 | 0.0825 | 0.1571 | 0.2783 | 0.3021
$\Lambda^{2}$ | 23.8427 | 11.2910 | 6.5897 | 2.6358 | 1.6579
Table 5: Mean value of the cosine of the angle $\overline{\cos}$ and Pearson product-moment correlation coefficient $r$ between the Lagrangian accelerations ${\bm{a}}_{L}$, the Eulerian acceleration ${\bm{a}}_{E}$, the convective contribution ${\bm{a}}_{C}={\bm{N}}$ and the pressure gradient ${\bm{a}}_{C}=\bm{\Pi}$ at nondimensional time $St=10$. The correlation coefficient is determined for all three components of the vector fields.  $Ri$ | 0 | 0.1 | 0.2 | 0.5 | 1
---|---|---|---|---|---
$r({\bm{a}}_{L},{\bm{a}}_{E})$ | 0.0284 | 0.0510 | 0.0882 | 0.2852 | 0.6634
$r({\bm{a}}_{L},{\bm{a}}_{C})$ | 0.5823 | 0.6205 | 0.6232 | 0.5493 | 0.2741
$r({\bm{a}}_{E},{\bm{a}}_{C})$ | -0.7961 | -0.7516 | -0.7241 | -0.6443 | -0.5378
$r({\bm{a}}_{L},{\bm{a}}_{P})$ | -0.9728 | -0.9545 | -0.9211 | -0.7014 | -0.2843
$\overline{\cos}({\bm{a}}_{L},{\bm{a}}_{E})$ | 0.1617 | 0.1957 | 0.2397 | 0.4347 | 0.7033
$\overline{\cos}({\bm{a}}_{L},{\bm{a}}_{C})$ | 0.4199 | 0.4579 | 0.4582 | 0.3749 | 0.1964
$\overline{\cos}({\bm{a}}_{E},{\bm{a}}_{C})$ | -0.6573 | -0.5945 | -0.5563 | -0.4561 | -0.3538
$\overline{\cos}({\bm{a}}_{L},{\bm{a}}_{P})$ | -0.9110 | -0.8761 | -0.8157 | -0.5273 | -0.2659
Table 6: Pearson product-moment correlation coefficient $r$ for the scale-dependent Lagrangian and Eulerian accelerations at nondimensional time $St=10$. The correlation coefficient is determined for all three components of the accelerations.  $Ri$ | 0 | 0.1 | 0.2 | 0.5 | 1
---|---|---|---|---|---
$r$ ($j=0$) | 0.8584 | 0.9716 | 0.9892 | 0.9969 | 0.9995
$r$ ($j=1$) | 0.8065 | 0.9347 | 0.9625 | 0.9957 | 0.9990
$r$ ($j=2$) | 0.5208 | 0.6709 | 0.8287 | 0.9728 | 0.9921
$r$ ($j=3$) | 0.1988 | 0.3427 | 0.5072 | 0.8257 | 0.9512
$r$ ($j=4$) | 0.0884 | 0.1380 | 0.2141 | 0.4959 | 0.7765
$r$ ($j=5$) | 0.0308 | 0.0478 | 0.0739 | 0.2002 | 0.4528
$r$ ($j=6$) | 0.0116 | 0.0196 | 0.0304 | 0.0993 | 0.2919
$r$ ($j=7$) | 0.0037 | 0.0097 | 0.0248 | 0.1248 | 0.3337
$r$ ($j=8$) | -0.0052 | 0.0200 | 0.0261 | 0.1707 | 0.3837
Table 7: Scale-dependent Lagrangian and Eulerian acceleration statistics for $Ri=0.1$ at nondimensional time $St=10$ showing the rms of the Lagrangian and Eulerian acceleration, $a_{E}$ and $a_{L}$, and the flatness, $Fl_{a_{E}}$ and $Fl_{a_{L}}$ for the total and the scale-dependent contributions at scale $2^{-j}$. $j$ | total | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8
---|---|---|---|---|---|---|---|---|---|---
$a_{L}$ | 9.833 | 0.156 | 0.361 | 0.769 | 1.787 | 3.672 | 5.666 | 5.773 | 3.491 | 1.280
$Fl_{a_{L}}$ | 26.041 | 4.933 | 3.468 | 4.706 | 4.968 | 6.844 | 10.979 | 39.138 | 119.480 | 252.821
$a_{E}$ | 11.689 | 0.150 | 0.360 | 0.682 | 1.589 | 3.477 | 6.026 | 7.137 | 5.344 | 2.364
$Fl_{a_{E}}$ | 13.413 | 4.296 | 3.304 | 4.103 | 5.061 | 5.981 | 8.114 | 12.607 | 27.600 | 65.829
Table 8: Scale-dependent Lagrangian and Eulerian acceleration statistics for $Ri=1$ at nondimensional time $St=10$ showing the rms of the Lagrangian and Eulerian acceleration, $a_{E}$ and $a_{L}$, and the flatness, $Fl_{a_{E}}$ and $Fl_{a_{L}}$. $j$ | total | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8
---|---|---|---|---|---|---|---|---|---|---
$a_{L}$ | 0.942 | 0.079 | 0.257 | 0.328 | 0.405 | 0.450 | 0.435 | 0.336 | 0.190 | 0.058
$Fl_{a_{L}}$ | 4.111 | 5.690 | 3.210 | 4.042 | 4.064 | 4.248 | 6.316 | 11.850 | 15.906 | 38.999
$a_{E}$ | 1.075 | 0.079 | 0.257 | 0.330 | 0.410 | 0.473 | 0.530 | 0.483 | 0.254 | 0.062
$Fl_{a_{E}}$ | 5.691 | 5.700 | 3.210 | 4.021 | 4.034 | 4.166 | 6.524 | 10.411 | 21.338 | 66.034


00footnotetext: Support of the research of the first author by the Czech
Science Foundation (GAČR), project 24-14386L, entitled “Representation of
algebraic semantics for substructural logics”, and by IGA, project PřF 2024
011, and of the second author by the Austrian Science Fund (FWF), project I
4579-N, entitled “The many facets of orthomodularity”, is gratefully
acknowledged.

# A characterization of permutability of $2$-uniform tolerances on posets

Ivan Chajda and Helmut Länger

###### Abstract

Tolerance relations were investigated by several authors in various algebraic
structures, see e.g. the monograph [2]. Recently G. Czédli [6] studied so-
called $2$-uniform tolerances on lattices, i.e. tolerances that are compatible
with the lattice operations and whose blocks are of cardinality $2$. He showed
that two such tolerances on a lattice containing no infinite chain permute if
and only if they are amicable (a concept introduced in his paper). We extend
this study to tolerances on posets. Since in posets we have no lattice
operations, we must modify the notion of amicability. We modified it in such a
way that in case of lattices it coincides with the original definition. With
this new definition we can prove that two tolerances on a poset containing no
infinite chain permute if and only if they are amicable in the new sense.

AMS Subject Classification: 08A02, 08A05, 06A06, 06A11

Keywords: Tolerance relation, poset, permuting tolerances, $2$-uniform
tolerance, amicable $2$-uniform tolerance

##  1 Introduction

By a tolerance on an algebra $\mathbf{A}=(A,F)$ is meant a reflexive and
symmetric binary relation on $A$ having the Substitution Property with respect
to all operations of $F$. For the theory of tolerances see e.g. the monograph
[2]. Important are in particular tolerances on lattices since G. Czédli showed
in [5] that every lattice $\mathbf{L}=(L,\vee,\wedge)$ can be factorized in a
natural way by any tolerance $T$ on $\mathbf{L}$, i.e. the set $L/T$ of all
blocks of $T$ forms a lattice again. Such a situation is rather exceptional
and does not hold for other types of algebras in general, but many varieties
whose members have this property were described in [3].

Recall that a block of a tolerance $T$ on an algebra $(A,F)$ is a maximal
subset $B$ of $A$ satisfying $B^{2}\subseteq T$.

Recently G. Czédli proved in [6] that so-called $2$-uniform tolerances on a
lattice containing no infinite chain permute if and only if they are amicable.

The concept of a tolerance on a lattice was generalized to posets by the
present authors [4]. Despite the fact that we do not have explicit operations
on a poset, the concept was defined in a way that the blocks are convex again
and that posets can be factorized by these tolerances. For other interesting
properties of tolerances on posets the reader is referred to [4]. Hence the
natural question arises if also the result of G. Czédli concerning the
permutability of $2$-uniform tolerances can be extended to posets. This is the
topic of the present paper.

##  2 Basic concepts

At first we recall several concepts introduced in [4] and [6].

Let $\mathbf{P}=(P,\leq)$ be a poset. A tolerance on $\mathbf{P}$ is a
reflexive and symmetric binary relation $T$ on $P$ satisfying the following
conditions:

  1. (1)

If $(x,y),(z,u)\in T$ and $x\vee z$ and $y\vee u$ exist then $(x\vee z,y\vee
u)\in T$.

  2. (2)

If $(x,y),(z,u)\in T$ and $x\wedge z$ and $y\wedge u$ exist then $(x\wedge
z,y\wedge u)\in T$.

  3. (3)

If $x,y,z\in P$ and $(x,y),(y,z)\in T\neq P^{2}$ then there exist $u,v\in P$
with $u\leq x,y,z\leq v$ and $(u,y),(y,v)\in T$.

  4. (4)

If $(x,y)\in T\neq P^{2}$ then there exists some $(z,u)\in T$ with both $z\leq
x,y\leq u$ and $(v,z),(v,u)\in T$ for all $v\in P$ with $(v,x),(v,y)\in T$.

Conditions (3) and (4) are quite natural since they are satisfied by every
tolerance on a lattice. In condition (3) one can take $u:=x\wedge y\wedge z$
and $v:=x\vee y\vee z$, and in condition (4) one can take $z:=x\wedge y$ and
$u:=x\vee y$.

A block of a tolerance $T$ on $\mathbf{P}$ is a maximal subset $B$ of $P$
satisfying $B^{2}\subseteq T$. Let $P/T$ denote the set of all blocks of $T$.
Clearly, $T=\bigcup\limits_{B\in P/T}B^{2}$. In [4] we proved that every block
of $T$ is convex. According to Zorn’s Lemma every subset $A$ of $P$ satisfying
$A^{2}\subseteq T$ is contained in a block of $T$. Following [6] a tolerance
is called $2$-uniform if every of its blocks consists of exactly two elements.
Let $a,b\in P$. We call $a$ a lower $T$-neighbor of $b$ and $b$ an upper
$T$-neighbor of $a$ if $a\prec b$ and $(a,b)\in T$.

##  3 A characterization of permutability of $2$-uniform tolerances on posets

It is evident that $2$-uniform tolerances on a poset are very specific. Their
basic properties are as follows.

######  Lemma 3.1.

Let $\mathbf{P}=(P,\leq)$ be a poset, $a,b\in P$ and $T$ be a $2$-uniform
tolerance on $\mathbf{P}$. Then the following holds:

  1. (i)

The element $a$ has at most one lower $T$-neighbor and at most one upper
$T$-neighbor,

  2. (ii)

if $(a,b)\in T$ then $a=b$ or $a\prec b$ or $b\prec a$.

###### Proof.

Without loss of generality, assume $|P|>2$. Then $T\neq P^{2}$.

  1. (i)

Assume $a$ to have two distinct lower $T$-neighbors $c$ and $d$. Then
$(c,a),(a,d)\in T$. According to (3) there exists some $e\in P$ with $e\leq
c,a,d$ and $(e,a)\in T$. Let $B\in P/T$ with $e,a\in B$. Since $e\leq c,d\leq
a$ and $B$ is convex we conclude $c,d,a\in B$ contradicting $|B|=2$. This
shows that $a$ can have at most one lower $T$-neighbor. The second assertion
follows by duality.

  2. (ii)

Assume $(a,b)\in T$. First suppose $a\parallel b$. According to (4) there
exist $c,d\in P$ with $(c,d)\in T$ and $c\leq a,b\leq d$. Let $B\in P/T$ with
$c,d\in B$. Since $B$ is convex we conclude $c,a,b\in B$. Now we have
$a\parallel b$ and $c\leq a,b$ and hence $c\neq a,b$ contradicting $|B|=2$.
This shows $a\leq b$ or $b\leq a$. Let $C\in P/T$ with $a,b\in C$. Since $C$
is convex we obtain $[a,b]\subseteq C$ if $a\leq b$ and $[b,a]\subseteq C$ if
$b\leq a$. Now $|C|=2$ implies $a=b$ or $a\prec b$ or $b\prec a$.

∎

Let $\mathbf{P}$ be a poset, $a,b\in P$ and $T,S$ $2$-uniform tolerances on
$\mathbf{P}$.

In what follows we adopt some key concepts from [6] for $2$-uniform tolerances
on posets.

The element $a$ is called a split $(T,S)$-bottom if there exists some upper
$T$-neighbor $b$ of $a$ and some upper $S$-neighbor of $a$ being different
from $b$. Further, $a$ is called an adherent $(T,S)$-bottom if there exists
some common upper $T$-neighbor and $S$-neighbor of $a$. Finally, $a$ is called
a $(T,S)$-bottom if it is either a split $(T,S)$-bottom or an adherent
$(T,S)$-bottom. The notions split $(T,S)$-top, adherent $(T,S)$-top and
$(T,S)$-top are defined dually. A $T$-top ($T$-bottom) is an upper (a lower)
$T$-neighbor of some element of $P$.

The tolerances $T$ and $S$ are called amicable if the following four
conditions hold:

  1. (5)

If $a\neq b$ and there exists some lower $T$-neighbor of $a$ being a lower
$S$-neighbor of $b$ then there exists some upper $S$-neighbor of $a$ being an
upper $T$-neighbor of $b$.

  2. (6)

If $a\neq b$ and there exists some upper $T$-neighbor of $a$ being an upper
$S$-neighbor of $b$ then there exists some lower $S$-neighbor of $a$ being a
lower $T$-neighbor of $b$.

  3. (7)

if $a$ is a $(T,S)$-top and $b$ is either an upper $T$-neighbor or an upper
$S$-neighbor of $a$ then $b$ is a $(T,S)$-top, too,

  4. (8)

if $a$ is a $(T,S)$-bottom and $b$ is either a lower $T$-neighbor or a lower
$S$-neighbor of $a$ then $b$ is a $(T,S)$-bottom, too.

Let us note that any two $2$-uniform tolerances on a lattice $(L,\vee,\wedge)$
satisfy (5) and (6). Namely, G. Czédli [6] showed that if $a\neq b$ and there
exists some lower $T$-neighbor of $a$ being a lower $S$-neighbor of $b$ then
$a\vee b$ is an upper $S$-neighbor of $a$ being an upper $T$-neighbor of $b$,
and if $a\neq b$ and there exists some upper $T$-neighbor of $a$ being an
upper $S$-neighbor of $b$ then $a\wedge b$ is a lower $S$-neighbor of $a$
being a lower $T$-neighbor of $b$.

######  Remark 3.2.

Note that (7) and (8) coincide with conditions (A1) and (A2) from [6],
respectively. Hence, for lattices the concept of amicable $2$-uniform
tolerances as defined in [6] coincides with the one defined above.

The proof of our main theorem will broadly use the ideas from [6].

######  Theorem 3.3.

Let $T$ and $S$ be $2$-uniform tolerances on a poset $(P,\leq)$ containing no
infinite chain. Then $T$ and $S$ permute if and only if they are amicable.

###### Proof.

Let $a,b\in P$. First assume $T$ and $S$ to permute.

  1. (5)

Suppose $a\neq b$ and suppose $c$ to be some lower $T$-neighbor of $a$ being a
lower $S$-neighbor of $b$. Because of (ii) of Lemma 3.1 we have $a\parallel b$
and hence $(a,b)\notin S\cup T$ again according to (ii) of Lemma 3.1. Now
$(a,b)\in T\circ S=S\circ T$. Hence there exists some $S$-neighbor $d$ of $a$
being a $T$-neighbor of $b$. Because of $a\parallel b$ we have $d<a,b$ or
$d>a,b$. Now $d<a,b$ would imply $(a,b)=(a\vee d,c\vee b)\in T$ according to
(1) since $(a,c),(d,b)\in T$. But this contradicts (ii) of Lemma 3.1.
Therefore $d$ is an upper $S$-neighbor of $a$ being an upper $T$-neighbor of
$b$ proving (5).

  2. (6)

follows from (5) by duality.

  3. (7)

Suppose $b$ to be either an upper $T$-neighbor or $S$-neighbor of $a$. Without
loss of generality assume $(a,b)\in T$. First suppose $a$ to be a split
$(T,S)$-top. Let $c$ and $d$ denote the lower $T$-neighbor and $S$-neighbor of
$a$, respectively. Then $c\parallel d$. Since $(d,b)\in S\circ T=T\circ S$
there exists some $e\in P$ with $(d,e)\in T$ and $(e,b)\in S$. Now $b\leq e$
would imply $d<a<b\leq e$ which together with $(d,e)\in T$ would contradict
(ii) of Lemma 3.1. Again according to Lemma 3.1 we have $e\prec b$. Now $a$
and $e$ are a lower $T$-neighbor and $S$-neighbor of $b$, respectively,
showing $b$ to be a $(T,S)$-top. Next assume $a$ to be an adherent
$(T,S)$-top. Let $c$ denote the common lower $T$-neighbor and $S$-neighbor of
$a$. Since $(c,b)\in S\circ T=T\circ S$ there exists some $d\in P$ with
$(c,d)\in T$ and $(d,b)\in S$. Now $d\leq c$ would imply $d\leq c<a<b$ which
together with $(d,b)\in S$ would contradict (ii) of Lemma 3.1, and $b\leq d$
would imply $c<a<b\leq d$ which together with $(c,d)\in T$ would again
contradict (ii) of Lemma 3.1. Thus a further application of (ii) of Lemma 3.1
yields $c\prec d\prec b$. Now $a$ and $d$ are a lower $T$-neighbor and
$S$-neighbor of $b$, respectively, showing $b$ to be a $(T,S)$-top.

  4. (8)

follows from (7) by duality.

Hence $T$ and $S$ are amicable.
Conversely, assume $T$ and $S$ to be amicable. Suppose $(a,b)\in T\circ S$.
Then there exists some $c\in P$ with $(a,c)\in T$ and $(c,b)\in S$. We want to
show $(a,b)\in S\circ T$. If $a=b$ then $(a,b)=(a,a)\in S\circ T$. If $c=a$
then $(a,b)=(c,b)\in S$ and $(b,b)\in T$ and hence $(a,b)\in S\circ T$. If
$c=b$ then $(a,a)\in S$ and $(a,b)=(a,c)\in T$ and hence $(a,b)\in S\circ T$.
So we can assume that $a,b,c$ are mutually distinct. According to (ii) of
Lemma 3.1 we consider the following four cases.
$\underline{c\prec a\text{ and }c\prec b.}$
Because of (5) there exists a common upper $S$-neighbor $d$ of $a$ and
$T$-neighbor of $b$ and hence $(a,d)\in S$ and $(d,b)\in T$ showing $(a,b)\in
S\circ T$.
$\underline{a\prec c\text{ and }b\prec c.}$
This case is dual to the previous one.
$\underline{a\prec c\prec b.}$
Put $a_{0}:=a$, $a_{1}:=c$ and $a_{2}:=b$. For $i\geq 3$ define $a_{i}$ as
follows: If $i$ is odd and $a_{i-1}$ is a $T$-bottom then define $a_{i}$ to be
the unique upper $T$-neighbor of $a_{i-1}$. If $i$ is even and $a_{i-1}$ is an
$S$-bottom then define $a_{i}$ to be the unique upper $S$-neighbor of
$a_{i-1}$. Note that $a_{0}$ is a $T$-bottom, $a_{1}$ is the unique upper
$T$-neighbor of $a_{0}$, $a_{1}$ is an $S$-bottom and $a_{2}$ is the unique
upper $S$-neighbor of of $a_{1}$. Since $a_{2}\prec a_{3}\prec
a_{4}\prec\cdots$, but $\mathbf{P}$ has no infinite chain, there exists some
$n\geq 2$ such that $a_{2},\ldots,a_{n}$ are defined, but $a_{n+1}$ is not.
First assume $n$ to be even. Then $a_{n}$ is not a $T$-bottom. Since every
element of $P$ belongs to at least one $2$-element block of $T$, $a_{n}$ is a
$T$-top. But it is also an $S$-top and therefore a $(T,S)$-top. If $n$ is odd
then a similar reasoning shows that $a_{n}$ is a $(T,S)$-top. So $a_{n}$ is a
$(T,S)$-top in any case. Now assume $a_{n}$ to be an adherent $(T,S)$-top.
Then, according to (i) of Lemma 3.1, $a_{n-1}$ is an adherent $(T,S)$-bottom.
Now (8) shows that $a_{n-2}$ is a $(T,S)$-bottom, too. Next assume $a_{n}$ to
be a split $(T,S)$-top. Suppose $n$ is even. Then $a_{n-1}$ is the unique
lower $S$-neighbor of $a_{n}$, $a_{n}$ has a unique lower $T$-neighbor $d$ and
$a_{n-1}\neq d$. According to (6) there exists some lower $S$-neighbor $e$ of
$d$ being a lower $T$-neighbor of $a_{n-1}$. Since $a_{n-2}$ is a lower
$T$-neighbor of $a_{n-1}$, too, (i) of Lemma 3.1 yields $e=a_{n-2}$. Therefore
$a_{n-2}$ is a $(T,S)$-bottom. With roles of $T$ and $S$ interchanged, one can
prove that $a_{n-2}$ is a $(T,S)$-bottom also in the case when $n$ is odd.
Hence in any case $a_{n-2}$ is a $(T,S)$-bottom. Applying (8) finitely many
times yields that $a_{0}=a$ is a $(T,S)$-bottom, too. First assume $a$ to be a
split $(T,S)$-bottom. Then there exists some upper $S$-neighbor $f$ of $a$
being distinct from $c$. According to (5) there exists some upper $S$-neighbor
$g$ of $c$ being an upper $T$-neighbor of $f$. Since $b$ is an upper
$S$-neighbor of $c$, too, (i) of Lemma 3.1 yields $g=b$. Hence $(a,f)\in S$
and $(f,b)=(f,g)\in T$ showing $(a,b)\in S\circ T$. Now suppose $a$ to be an
adherent $(T,S)$-bottom. Then $c$ is a $(T,S)$-top. According to (7), $b$ is a
$(T,S)$-top, too. Hence $b$ has a unique lower $T$-neighbor $h$. Assume $c\neq
h$. According to (6) there exists some lower $S$-neighbor $i$ of $h$ being a
lower $T$-neighbor of $c$. Since $a$ is a lower $T$-neighbor of $c$, too, (i)
of Lemma 3.1 yields $i=a$. On the other hand, $i=a$ is a lower $S$-neighbor of
$h$. Hence $c$ and $h$ are two distinct upper $S$-neighbors of $a$
contradicting (i) of Lemma 3.1. This shows $c=h$. Now $(a,c)\in S$ and
$(c,b)=(h,b)\in T$ proving $(a,b)\in S\circ T$.
$\underline{b\prec c\prec a.}$
This case is dual to the previous one.
Hence in any case we have $(a,b)\in S\circ T$. This shows $T\circ S\subseteq
S\circ T$. By duality, we obtain $S\circ T\subseteq T\circ S$ and hence
$T\circ S=S\circ T$, i.e. $T$ and $S$ permute ∎

##  4 Examples

######  Example 4.1.

Consider the poset $\mathbf{P}$ depicted in Figure 1

$0$$a$$b$$c$$d$$e$$f$$g$$1$Fig. 1Poset

and put

$\displaystyle T$
$\displaystyle:=\\{0,a\\}^{2}\cup\\{a,b\\}^{2}\cup\\{c,e\\}^{2}\cup\\{d,g\\}^{2}\cup\\{f,1\\}^{2},$
$\displaystyle S$
$\displaystyle:=\\{0,a\\}^{2}\cup\\{a,b\\}^{2}\cup\\{c,e\\}^{2}\cup\\{d,f\\}^{2}\cup\\{g,1\\}^{2}.$

Then $T$ and $S$ are permuting $2$-uniform tolerances on $\mathbf{P}$ since

$T\circ S=T\cup S\cup\\{(0,b),(b,0),(d,1),(f,g),(g,f),(1,d)\\}=S\circ T$

and hence they are amicable according to Theorem 3.3.

######  Example 4.2.

Consider the poset $\mathbf{P}$ visualized in Figure 2

$0$$a$$b$$c$$d$$e$$f$$g$$h$$i$$j$$1$Fig. 2Poset

and put

$\displaystyle T$
$\displaystyle:=\\{0,a\\}^{2}\cup\\{a,c\\}^{2}\cup\\{b,d\\}^{2}\cup\\{d,e\\}^{2}\cup\\{f,h\\}^{2}\cup\\{g,j\\}^{2}\cup\\{i,1\\},$
$\displaystyle S$
$\displaystyle:=\\{0,b\\}^{2}\cup\\{a,d\\}^{2}\cup\\{c,e\\}^{2}\cup\\{f,g\\}^{2}\cup\\{h,j\\}^{2}\cup\\{i,1\\}^{2}.$

Then $T$ and $S$ are permuting $2$-uniform tolerances on $\mathbf{P}$ since

$\displaystyle T\circ S$ $\displaystyle=T\cup
S\cup\\{(0,d),(a,b),(a,e),(b,a),(c,d),(d,0),(d,c),(e,a),(f,j),(g,h),(h,g),$
$\displaystyle\hskip 17.07164pt(j,f)\\}=S\circ T$

and hence they are amicable according to Theorem 3.3.

######  Example 4.3.

Consider the poset $\mathbf{P}$ depicted in Figure 3

$0$$a$$b$$c$$d$$1$Fig. 3Poset

and put

$\displaystyle T$
$\displaystyle:=\\{0,a\\}^{2}\cup\\{b,d\\}^{2}\cup\\{c,1\\}^{2},$
$\displaystyle S$
$\displaystyle:=\\{0,b\\}^{2}\cup\\{a,c\\}^{2}\cup\\{d,1\\}^{2}.$

Then $T$ and $S$ do not permute since $(a,b)\in(T\circ S)\setminus(S\circ T)$,
and $T$ and $S$ are not amicable since $0$ is a lower $T$-neighbor of $a$
being a lower $S$-neighbor of $b$, but there exists no upper $S$-neighbor of
$a$ being an upper $T$-neighbor of $b$.

## References

  * [1] 9
  * [2] I. Chajda, Algebraic Theory of Tolerance Relations. Palacký University Olomouc, Olomouc 1991. ISBN 80-7067-042-8.
  * [3] I. Chajda, G. Czédli and R. Halaš, Independent joins of tolerance factorable varieties. Algebra Universalis 69 (2013), 83–92.
  * [4] I. Chajda and H. Länger, Tolerances on posets. Miskolc Math. Notes 24 (2023), 725–736.
  * [5] G. Czédli, Factor lattices by tolerances. Acta Sci. Math. (Szeged) 44 (1982), 35–42.
  * [6] G. Czédli, Permuting $2$-uniform tolerances on lattices. J. Multiple-Valued Log. Soft Computing 39 (2022), 97–104.

Authors’ addresses:

Ivan Chajda
Palacký University Olomouc
Faculty of Science
Department of Algebra and Geometry
17\. listopadu 12
771 46 Olomouc
Czech Republic
<EMAIL_ADDRESS>
Helmut Länger
TU Wien
Faculty of Mathematics and Geoinformation
Institute of Discrete Mathematics and Geometry
Wiedner Hauptstraße 8–10
1040 Vienna
Austria, and
Palacký University Olomouc
Faculty of Science
Department of Algebra and Geometry
17\. listopadu 12
771 46 Olomouc
Czech Republic
<EMAIL_ADDRESS>