Single-particle Lagrangian statistics from direct numerical simulations of rotating-stratified turbulence
Dhawal Buaria Alain Pumir Fabio Feraco Raffaele Marino Annick Pouquet Duane Rosenberg Leonardo Primavera























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 ⟨ϵ⟩
τ_η
𝐱
𝐮
𝐘
Φ









[]dhawal.buaria@ds.mpg.de
(present address: New York University, NY 11201, USA; dhawal.buaria@nyu.edu)
Max Planck Institute for Dynamics and Self-Organization, 37077 Göttingen, Germany


Laboratoire de Physique, ENS de Lyon, CNRS and Université de Lyon, 69007 Lyon, France
Max Planck Institute for Dynamics and Self-Organization, 37077 Göttingen, Germany

Laboratoire de Mécanique des Fluides et d'Acoustique, École Centrale de Lyon, CNRS, 
 
Université Claude Bernard Lyon 1, INSA de Lyon, 69134 Écully, France
Dipartimento di Fisica, Universitá della Calabria, 87036, Arcavacata di Rende, Italy

Laboratoire de Mécanique des Fluides et d'Acoustique, École Centrale de Lyon, CNRS, 
 
Université Claude Bernard Lyon 1, INSA de Lyon, 69134 Écully, France



Laboratory for Atmospheric and Space Physics, University of Colorado 

and National Center for Atmospheric Research, Boulder, CO 80309, USA



288 Harper St., Louisville, CO 80027, USA

Dipartimento di Fisica, Universitá della Calabria, 87036, Arcavacata di Rende, Italy









Geophysical fluid flows are predominantly turbulent and often 
strongly affected by the Earth's rotation, 
as well as by stable density stratification. 

Using direct numerical simulations of forced Boussinesq equations,
we study the influence of these effects on the motion of fluid particles.
We perform a detailed study of Lagrangian statistics of acceleration, velocity
and related quantities,
focusing on cases where 
the frequencies associated with rotation and stratification (RaS),
f and N respectively,
are held at a fixed ratio N/f=5. 
The simulations are  performed in a periodic domain,
at Reynolds number Re ≈ 4000,
and Froude number Fr in the range
0.03 ≲ Fr ≲ 0.2
(with Rossby number Ro=5Fr).











As the intensity of RaS increases, 
a sharp transition is observed between a regime dominated 
by eddies to a regime dominated by waves,
which corresponds to Fr ≲ 0.07.
For the given runs, this transition to a wave-dominated regime 
can also be seemingly described by
simply comparing the time scales 1/N  and τ_η,
the latter being the Kolmogorov time scale 
based on the mean kinetic energy dissipation.
Due to the known anisotropy 
induced by RaS, we consider separately
the motion in the horizontal and vertical directions.
In the regime Nτ_η < 1,
acceleration statistics 
exhibit well known characteristics of isotropic
turbulence in both directions,
such as probability density functions (PDFs) with wide tails
and acceleration variance approximately scaling as per Kolmogorov's theory.
In contrast for Nτ_η > 1, they 
behave very differently, experiencing the direct influence
of the imposed rotation and stratification.
On the other hand, the Lagrangian velocity statistics
exhibit visible anisotropy for all runs; nevertheless
the degree of anisotropy becomes very strong in the regime
Nτ_η > 1.

We observe
that in the regime Nτ_η < 1, rotation enhances the 
mean-square displacements in horizontal
planes in the ballistic regime at short times, but
suppresses them in the diffusive regime at longer times.
This suppression of the horizontal displacements becomes 
stronger in the regime Nτ_η>1,
with no clear diffusive behavior.
In contrast, the displacements in the vertical direction are always
reduced. This inhibition is extremely strong 
in the Nτ_η > 1 regime, leading
to a scenario where particles almost appear to be trapped in 
horizontal planes. 


Single-particle Lagrangian statistics from 
direct numerical simulations 
 of rotating-stratified turbulence
    L. Primavera
    December 30, 2023
============================================================================================================







§ INTRODUCTION



Transport of material substances plays a crucial role in  
many geophysical processes <cit.>, e.g. 
dispersion of pollutants
and contaminants <cit.>,
droplet dynamics in clouds <cit.>, 
mixing of planktons and other biomatter in the oceans <cit.>.
The Lagrangian viewpoint
following the motion of fluid particles <cit.>
or analogous entities such as Brownian or inertial particles
<cit.>,
provides a natural description of such transport processes.


Not only are most geophysical flows turbulent,
they are also often strongly influenced
by anisotropies due to effects such as rotation 
(Coriolis force) and stratification (buoyancy force) 
or presence of magnetic fields <cit.>.
While particle dispersion has been studied extensively
in isotropic turbulence, e.g. see reviews 
<cit.>, 
it has only recently started receiving
attention in anisotropic flows. In particular, 
several studies have focused on flows considering either the effects
of rotation 
<cit.>
or stratification separately 
<cit.>. 
However, in many applications, adequately describing the
observed flow physics necessitates examining the combined effects of
rotation and stratification (RaS), 



e.g. in the southern abyssal 
oceans with particularly high mixing intensities
<cit.>.








Arguably the most challenging aspect of studying the
combined effects of RaS in turbulence is the 
enormous range 
of spatial and temporal scales associated with such flows.
In isotropic turbulence, typically the only 
governing parameter is the
Reynolds number (Re), which directly provides
a measure of the range of scales in the flow. 
However, at least three additional parameters
have to be considered in RaS turbulence,
namely, the Rossby number (Ro) and the Froude
number (Fr) which respectively measure the strength of 
RaS, and the
Prandtl number (Pr) which is the 
ratio of the fluid viscosity to the thermal diffusivity
(see Section <ref> for precise definitions).
This leads to consideration of additional
length and time scales 
– beyond the already wide range existing
in isotropic turbulence –
and makes the flow dynamics far more involved. In particular,
it has been recognized that the interaction between linear 
and non-linear processes in such flows leads to 
a rich variety of complex behavior, 
such as spontaneous generation of helicity, 
dual cascading of energy and many more <cit.>. 
Advances in computing power in the last decade, has allowed 
for significant progress in understanding such flows, although
mostly from the Eulerian perspective 
<cit.>.


In this work,
utilizing direct numerical simulations (DNS) of Boussinesq equations,
our objective is to investigate the dispersion of
fluid particles in RaS turbulence.
In particular, we focus on the motion of single individual particles.
In view of the very large parameter space associated with 
RaS turbulence, we limit our investigation
to the case where Ro/Fr=5, i.e., the strength of stratification is
five times that of rotation, which is largely relevant to the southern abyssal 
oceans <cit.>.  
We use the same grid sizes for all cases,
giving Re ≈ 4000-5000, and 
Froude numbers in the range 0.03 ≲ Fr ≲ 0.2.
This regime of Fr 
corresponds to the transitional regime, where both turbulent eddies
and inertia-gravity waves are 
expected to play an important role <cit.>. 
As recently demonstrated in the case of purely stratified flows, 
the transition towards a wave
dominated regime is accompanied by 
intermittent large-scale vertical drafts <cit.>. 
We show that the presence of additional rotation
does not, in general, prevent the
manifestation of this intermittency and thereby also
investigate its effect on Lagrangian statistics. 


In studying
the motion of single particles,
in the spirit of earlier studies
<cit.>,
we investigate
the properties of both their acceleration and
velocity.
The intrinsic anisotropy of the flow makes it necessary
to distinguish throughout between motion in the horizontal plane and in
the vertical direction, which are respectively perpendicular and parallel
to the direction of imposed stratification (and also the axis of rotation).
However, the anisotropy of acceleration and velocity reveal

strikingly different properties.
For acceleration statistics, which are reflective of small-scales
of turbulence, two distinct regimes are observed depending on the 
strength of imposed RaS.
At weak or moderate RaS, the properties of acceleration
are qualitatively similar to those documented in isotropic turbulence
(with minor quantitative deviations), suggesting that
the imposed anisotropy at large scales does not significantly affect the small-scales.







On the other hand, when RaS becomes very strong, 
acceleration statistics are significantly affected
and exhibit striking differences between 
the horizontal and vertical motions. 
In the range of parameters considered here, we observe a sharp transition
between the two regimes, which can be simply quantified with
the ratio
of the Kolmogorov time scale and the stratification time scale
(which is simply the inverse of the Brunt-Väisäla frequency).
The qualitative change in the statistical properties of acceleration 
occurs simultaneously with the appearance of intermittent bursts in the 
flow.


The Lagrangian velocity statistics, in contrast to acceleration,
are always affected by the imposed RaS, with the degree
of anisotropy expectedly increasing with the strength of the imposed RaS.
However, the transition observed for acceleration statistics 
is also visible in velocity statistics, resulting
in even stronger anisotropy in velocity statistics
when RaS is very strong. 
Accordingly, we characterize the integral time scales based on velocity 
autocorrelations and investigate the displacement of particles.
The particles move ballistically at short times.
At long times, the emergence of a diffusive regime
is evident for runs with moderate RaS. 
However for strong RaS, the dispersion, particularly
in vertical direction is strongly suppressed,
qualitatively consistent with earlier works on 
purely stratified flows.
 
 
This paper is organized as follows.
In Section <ref>, we discuss our numerical methods.
Essential features of the underlying Eulerian flow are discussed in
Section <ref>.
The statistics of acceleration are presented in Section <ref>.
Section <ref> presents our results on the Lagrangian
autocorrelation functions of velocity along
with results on how particles are displaced by the flow.
Finally, Section <ref> contains our concluding remarks.








§ NUMERICAL METHOD AND DATABASE




We simulate the Eulerian flow by numerically integrating the 
incompressible Boussinesq equations 
in a rotating frame,
with constant solid body rotation rate Ω (and frequency f = 2 Ω) 
and gravity g anti-aligned in the vertical (z) direction:


    ∂ /∂ t  + ·∇   =
    -∇ P - f 𝐞_z × - N θ𝐞_z + ν∇^2  + 
    ∂θ /∂ t  + ·∇θ   =
    N w + κ∇^2 
    ∇·   = 0

Here, =(u,v,w) is the velocity field, θ is the
temperature fluctuation (in units of velocity), P is the pressure normalized
by the background density,
ν is the kinematic viscosity, κ is the thermal diffusivity
and N is the  Brunt-Väisäla frequency, which characterizes the strength
of imposed stratification. 
We take ν = κ, assuming the Prandtl number to be unity.
A large-scale stochastic forcing term, , is utilized to achieve and maintain a 
statistically stationary state. 
The forcing is random in time and isotropic in Fourier space,
with the energy being injected in a spherical shell of
wavenumbers given by 2 < |𝐤| < 3 <cit.>,
with the characteristic forcing length scale 
L_f = 2 π/ 2.5.
While the use of isotropic forcing to simulate
innately anisotropic flows may appear unphysical in the present
context, earlier comparisons
with a quasi-geostrophic forcing
have demonstrated that the precise nature of
the forcing does not have any significant effect on the flow properties
in the stationary state, 
at least in the regimes studied here <cit.>.
















To characterize RaS, we introduce 
the dimensionless Rossby (Ro) and Froude (Fr) 
numbers defined as <cit.>:

    Ro = U/L f ,     Fr = U/L N

where L and U are respectively the characteristic length 
and velocity scales of the large-eddies of the flow,
defined as L = L_f,  and 
U = ⟨ |𝐮|^2 ⟩^1/2,
the mean
amplitude of the velocity fluctuations. 
An important parameter is the ratio N/f (=Ro/Fr), 
which measures the relative strength 
of RaS.
As already mentioned, we maintain N/f=5 for 
all runs, 
as relevant in some oceanographic
situations <cit.>.


In addition, we define the following dimensionless numbers

    Re = U L/ν ,    R_IB = ⟨ϵ⟩/ν N^2 ,   R_ B = 
    Re Fr^2     ,  Ro_ω = 
    ⟨ω^2⟩^1/2/f ,

where ⟨ϵ⟩ is the 
mean dissipation rate of turbulent kinetic energy
and ⟨ω^2 ⟩ is the mean enstrophy
density (with ω being the magnitude of vorticity). 
It is worth noting that 
in homogeneous turbulence, as considered in this work,
the two are related as
⟨ω^2⟩ = ⟨ϵ⟩/ν = 1/τ_η^2,
where 

    τ_η=(ν/⟨ϵ⟩)^1/2

is the Kolmogorov time scale
characterizing the motion of turbulent eddies <cit.>.
Here, Re is the Reynolds number based on large scale.
The parameter R_IB, often referred to as the
buoyancy Reynolds number, provides a  
measure of the relative importance of the waves induced
by stratification with respect to 
turbulent eddies in the flow (or alternatively,
the relative importance of waves and non-linear processes), e.g. see
<cit.>.
This is evident by considering 
the Ozmidov length scale 
ℓ_OZ = (⟨ϵ⟩/N^3)^1/2,
which characterizes the gravity waves
and the Kolmogorov length scale, η = (ν^3/⟨ϵ⟩)^1/4,
which characterizes the dissipation scale of turbulence,
thereby giving R_IB = (ℓ_OZ/η)^4/3. 
Alternatively, R_IB can be written as the ratio
of two time scales: R_IB = (T_N/τ_η)^2, where
T_N=1/N is the stratification time scale
(and  τ_η is
the Kolmogorov time scale), once again characterizing the
relative importance of stratification to that of turbulent eddies.
An alternative perspective is also offered by the asymptotic scaling analysis 
of <cit.>, which suggests that R_IB 
can be interpreted as a replacement for the classical
Reynolds number in strongly stratified turbulence – providing
a relative measure of inertial and viscous forces.
However, utilizing this interpretation in presence of
rotation requires caution, since it invalidates the said asymptotic analysis
(especially when rotation and stratification are comparable in strength).


A separate definition of the buoyancy Reynolds 
number <cit.>
is given by the parameter R_ B. 
The two parameters are related as R_IB = β R_ B, where 
β = ⟨ϵ⟩ L/U^3. In fully developed turbulence, 
β is expected to be 
constant as a consequence of the dissipation anomaly <cit.>.
However, in the regimes considered here, it can be shown that β  
depends on the strength of stratification <cit.>.
Hence in our analysis, we utilize R_IB 
to quantify the relative strength of stratification and turbulence.
The relevance of this choice 
will become evident as our results are presented in later sections.
Finally, in addition to the Rossby number Ro based on large scales, 
it is useful to define the micro-Rossby number Ro_ω,
which in contrast to R_IB, measures the relative strength of 
small-scale turbulent motions to that of the imposed rotation.
However, using ⟨ϵ⟩ = ν⟨ω^2⟩ due to
statistical homogeneity,
Ro_ω can be simply related to 
R_IB as:
Ro^2_ω/R_IB = (N/f)^2.
Since N/f is held constant in this work,
the parameters
Ro_ω and 
R_IB essentially provide the same information.

The database utilized in the current work,
along with the main simulation parameters, is
summarized in Table <ref>.
The simulations were carried out using the
Geophysical High-Order Suite for Turbulence (GHOST) code, a versatile,
highly parallelized, pseudo-spectral code, 
utilizing hybrid MPI-OpenMP programming model, 
with second order
explicit Runge-Kutta time stepping <cit.>. 
All runs correspond to a (2π)^3 periodic domain with 512^3 grid points.
The parameters 
N and f are varied over a range of values 
keeping the ratio N/f fixed at 5. 
In addition, for a systematic comparison, 
we have also performed  additional simulations
by setting N=0 or f=0 or both. 
The simulation with N=f=0 (case 0)
simply corresponds to homogeneous isotropic turbulence (HIT).
All the runs reported were started from an initial condition, consisting of
a few random modes in the velocity field, 
whereas the temperature field, θ, was initialized to zero.
The Boussinesq equations 
were integrated until a statistically stationary state was reached.
For adequate statistical sampling, we considered a simulation period of
at least 10-20T_E in the stationary
state for each case, where 
T_E=L/U is the large eddy turnover time.
(Some additional details are available in the Supplementary 
Material <cit.>).






In Table <ref>, 
an important point to note is
that the values of R_ B and R_IB 
monotonically decrease as 
N and f
are increased. Values of N and f higher than those shown are
avoided, since the resulting regime would be completely dominated by waves
<cit.>.
The simulations with either N=0 or f=0 have their  
f or N value (respectively) corresponding to case 5, with one of
the largest N and f values.
However, as evident from Table <ref> the values of the Fr (and Ro)
numbers corresponding to runs 
5 and 7 (respectively runs 5 and 8) differ significantly, despite identical
values of the stratification (respectively rotation rate). 
These variations
are a consequence of our dynamical definition of U, which 
accounts for
the subtle interplay between turbulence and RaS.
In particular, they reflect the 
significant variation in U itself, which is 
discussed in Section <ref>. 
Importantly,
we observe that the value of R_IB is lower than 0.5 for Runs 4-6,
as well as for the purely stratified run, 7, suggesting a dominant role
of the waves in these runs. In contrast, R_IB > 2 in 
all other runs, possibly pointing to a prevalent role of the eddies. 
This leads to two qualitatively different behaviors, as demonstrated by 
our results in the following sections.



Finally, in order to obtain Lagrangian statistics, 
the fluid particles are tracked in time along with the Eulerian flow
in the stationary state,
according to the basic equation of motion

    d 𝐱^+(t)/dt  = 𝐮^+(t) = 𝐮( 𝐱^+(t), t)

where the superscript + denotes a Lagrangian quantity and the fluid particle
velocity is simply defined to be the Eulerian velocity field
evaluated at the instantaneous particle position.
For each run listed in Table <ref>, we additionally tracked
the motion of 1.5M particles (M=10^6), which are 
randomly distributed
throughout the entire domain at the time of injection. 
The number of fluid particles
are held constant for all cases, since given the same grid size
and approximately similar Reynolds numbers, their sampling requirements
are also approximately the same. 
Similar to the Eulerian grid, the particles are distributed among
parallel processors and tracked in time together with the velocity field
using a second order
Runge-Kutta scheme, and standard interpolation methods based
on cubic splines <cit.>.






§ EULERIAN VELOCITY FIELD




In this section, we provide a brief overview of the underlying
Eulerian flow, which establishes
the necessary framework for better understanding the Lagrangian statistics
reported subsequently.



  
*Energy spectra:


We begin by considering the kinetic energy spectra,
as shown in Fig. <ref>,
which is one of the most common descriptor 
of Eulerian dynamics in turbulent flows <cit.>.
The curves are normalized using the large scale variables U and L,
and split into two sub-figures depending on the strength of 
weak rotation and stratification (RaS). 







Fig. <ref>a shows the spectra for the runs with relatively 
weak RaS,
corresponding to Fr ≥ 0.086 and  R_IB>1, 
including the HIT and rotation-only runs.
The main observation is that all the spectra
demonstrate an approximate k^-5/3 Kolmogorov-like scaling
in an intermediate range of k. 
This expectation is justified for the HIT run, 
as well as the R_IB > 1 runs, for which there is a 
sufficient scale separation between ℓ_OZ and η
(note R_IB = (ℓ_OZ/η)^4/3 as discussed earlier),
to recover Kolmogorov-like
turbulence <cit.>;
although systematic deviations from the scaling 
are clearly evident as R_IB decreases. 
In fact, as shown later in this section,
the large scales for R_IB > 1 runs are still 
predominantly anisotropic, though their impact
on small-scales is weak.









On the other hand, in Fig. <ref>b,
the spectra for runs with stronger stratification,
corresponding to Fr ≤ 0.069 and  R_IB<1
are shown. The spectrum for the HIT run (dashed black line) 
is also included for comparison.
As evident, this regime significantly differs
from that at weaker stratification,
with the spectra exhibiting a spectral slope steeper
than k^-5/3
(since R_IB <1 and hence ℓ_OZ < η precludes
Kolmogorov's theory from being applicable).
It has been observed
in the closely related context of a time-dependent 
flow <cit.> that
the spectra followed the Bolgiano-Obhukov prediction <cit.>.
In this regime, energy transfer
is strongly affected by the temperature fluctuations (potential energy).
The straight line in Fig. <ref>b
indicates the corresponding k^-11/5 prediction.
Given the small range of spatial scales available in
our simulations, however, 
it is difficult to conclusively demonstrate that the spectra shown  
indeed conform to a k^-11/5 scaling
(and the fact that the precise exponent can depend on the Froude number <cit.>).
Nevertheless, the change in spectral slopes clearly evidences
that the energy transfer between
scales is strongly perturbed in runs with R_IB<1, and much less so 
in runs with R_IB>1.
In fact, this demarcation based on R_IB becomes increasingly prominent
as more results are discussed in later sections. 
The spectra of the temperature fluctuations (not shown)
follow similar trends as that of velocity.





  
*Anisotropy of velocity fluctuations:

The above transition between strong hydrodynamic turbulence and regimes
dominated by RaS, as evidenced from
the energy spectra, clearly warrant further inspection.  
An important aspect of the flows in 
this study is the existence of a strong anisotropy of velocity 
fluctuations. Namely, the presence of RaS induces
very different properties of the velocity fluctuations in the horizontal 
plane and in the vertical direction.
This anisotropy has been investigated in many earlier studies,
in particular by analyzing the 
longitudinal and transverse energy spectra 
<cit.>. In the 
present work, we limit our discussion to  
the velocity components u_⊥ and u_, respectively in the 
horizontal and vertical directions  (perpendicular
and parallel to gravity).
The statistics involving u_⊥ are obtained by averaging the
two components in the x-y horizontal
plane (which is also the plane of rotation), whereas 
u_ simply corresponds to the vertical z-component.
An obvious manifestation of the anisotropy between the two directions
is in variances of u_⊥ and u_, as shown in Table <ref>.
It is evident that apart from the HIT run (case 0) and to a large extent
the rotation only run (case 8), the variances for
both components are very different.
To precisely quantify the anisotropy, we consider the ratio
ξ^u = [ ⟨ u_⊥^2 ⟩/⟨ u_^2 ⟩ ]^1/2,
also shown in Table <ref>.
Expectedly, this ratio is unity for case 0 and very close to unity for case 8.
However, for all runs with stratification, the ratio is always greater than 1.
For cases 1-3, the ratio increases slowly with decreasing Fr, whereas
a sharp increase occurs for cases 4-6 (and also case 7).
Clearly, this transition corresponds to the change in spectral slope
observed for the spectra in Fig. <ref>.
The runs with smaller ξ^u correspond to spectra which show
k^-5/3 scaling, whereas runs with sharp increase of ξ^u
correspond to spectra with steeper Bolgiano-like scaling.
In these flows, the 
fluctuations of u_ are essentially suppressed when the imposed
RaS are strong.
We note that the values shown here are consistent with 
the results of some earlier runs <cit.>. 







To further investigate the anisotropy,
we next consider higher order moments of the velocity components.
Since the probability density functions (PDFs)
of individual velocity
components are symmetric
(due to underlying symmetry of 
Boussinesq equations),
we note that the values of the third moments of the
velocity fluctuations are expected to be zero. 
Our results, not shown here, are compatible with this.
Thus, we consider the fourth-order moment,
through the kurtosis, defined as
K_u_i = ⟨ u_i^4 ⟩/⟨ u_i^2 ⟩^2.
It is known that the PDFs
of individual velocity
components in HIT are approximately 
Gaussian, 
which implies that the kurtosis is 
approximately 3. 
Consistent with that, we find that the value of the
kurtosis is very close to 3 in the case HIT,
(see Table <ref>).
In the case of a purely stratified
flow, however, it was found that the distribution of the velocity component 
parallel to the direction of stratification, u_, could be significantly
wider than Gaussian <cit.>. 
The strong deviations from a Gaussian distribution
have been shown to originate from intermittent vertical drafts <cit.>.
On the other hand, the kurtosis of u_⊥ is always found to be 
approximately 3.
Our values, as shown in Table <ref>, are 
consistent with these trends.
We find that 
K_u_⊥ is slightly smaller than 3 at low 
stratification,
and slightly larger than 3 for runs with strong stratification 
(Fr ≲ 0.07), with PDFs (not shown) being approximately Gaussian. 
In the vertical direction, K_u_ 
increases  very significantly beyond 3
as strength of RaS increases.


It is worthwhile to compare the dependence of K_u_
in our RaS runs with the stratification-only runs 
reported in <cit.> (see in particular 
their Fig.5).
While the sharp transition in K_u_ appears to be qualitatively 
similar, quantitatively the large values of 
K_u_ for RaS runs
occur at slightly lower Fr values than in <cit.>.
For even smaller Fr (not considered in this work), 
in a wave-dominated regime, 
one can expect that K_u_ for RaS runs
will again become 3 <cit.>, as it was the case
for the stratification-only run (case 7) discussed here.
Additional discussion of K_u_ will be presented in the following
section, in relation
with acceleration statistics.







§ ACCELERATION STATISTICS





The acceleration experienced by a fluid element, 
defined by the rate of change of velocity in the Lagrangian frame, i.e., 
𝐚^+ = d𝐮^+/dt (where 𝐮^+ is defined in
Eq. (<ref>))
and resulting from the
balance of forces acting on it, is arguably
the simplest descriptor of its motion, as also directly reflected 
in the governing fluid equations.
In addition to its
fundamental importance in turbulence theory <cit.>,
a key motivation 
to study acceleration comes 
from its central role in stochastic modeling 
of turbulent dispersion <cit.>.
In the following sub-sections, we study 
different aspects related to acceleration, namely,
acceleration variance and kurtosis, 
the probability density functions and Lagrangian
autocorrelations and frequency spectra.
As already mentioned, the anisotropy of the flow makes it necessary 
to distinguish between
the horizontal and vertical components of acceleration, denoted by
a^+_⊥ and a^+_ respectively.
In this section, we demonstrate that the anisotropy properties of 
the acceleration components strongly differ from those of the velocity
discussed in the previous section.
For convenience, we will henceforth omit the superscript '+'
from our notation,
since our subsequent results only involve Lagrangian
quantities.




 §.§ Acceleration variance







Extending the earlier analysis of the Eulerian velocity field,
we first examine the anisotropy in acceleration by considering the
second and fourth order moments, namely, the variance
and kurtosis respectively.
The moments are listed in Table <ref>. 
Due to the underlying symmetry of the Boussinesq equations, 
the third and all other odd moments of acceleration components
are zero
(similar to the velocity components). 
As can been seen from cases 1-6, the properties of a_⊥ and a_
show striking differences.
While the variances of a_ rapidly decrease 
with decreasing Ro (and Fr), the variances of 
a_⊥ do not change that much.
To better understand this behavior, we consider the
anisotropy ratio  
ξ^a = [ ⟨ a_⊥^2⟩/⟨ a_^2 ⟩ ]^1/2.
For the HIT run (case 0), the ratio, as expected, is equal to 1, 
whereas for rot-strat cases two separate behaviors are visible.



The first corresponds to 
cases 1-3 with relatively weak rotation and stratification (RaS),
where the ratio ξ^a is approximately constant, and slightly 
smaller than 1.
The ratio being close to unity suggests that the effect of anisotropy
on small scales is only minor. 
Furthermore, a simple explanation for the ratio slightly smaller
than unity can be provided based on the knowledge that
the relative strength of stratification is significantly
stronger compared to that of rotation (since N/f=5).
For that reason, the acceleration
variance in the vertical direction is likely to be 
slightly more enhanced (due to stratification) than
that in the horizontal direction (due to rotation).
On the other hand, runs 4-6 correspond to significantly 
stronger RaS, which leads to
a different behavior, with ξ^a > 1 and further increasing with 
decreasing Fr (and Ro).  
Since runs 4-6 correspond to R_IB<1, they are in a regime
where waves play a stronger role, compared to (turbulent) eddies.
With strong stratification, the motion in the
vertical direction is strongly suppressed, as also
reflected in ξ^u values (see Table <ref>).
With relative weaker effect of rotation, the acceleration
in the horizontal direction is still affected by turbulent eddies,
resulting in ξ^a to become greater than 1. 
Thereafter, the effect becomes even more pronounced with further
increase in strength of RaS.


It is instructive to compare the acceleration variances  
for case 5 (with strong RaS), to 
case 7, with exactly the same stratification, N, but no rotation 
and to case 8, with the same rotation, f, but no stratification. 
The variances of a_⊥ and a_ are stronger in the presence of 
both RaS, than with either of the two effects taken separately. 
The anisotropy ratio ξ^a also shows a very striking difference, when 
comparing runs 5, 7 and 8. For rotation-only (run 8), 
we observe ξ^a ≈ 1, implying little to no small-scale anisotropy. This can be 
viewed as consistent with
the observation that the flow spectrum for run 8 is overall comparable to 
the Kolmogorov
spectrum (see Fig. <ref>a), so the flow is dominated by eddies.
On the other hand, the anisotropy 
ratio is ξ^a < 1 (0.66) for run 7, implying stronger acceleration 
variances in the
vertical than in the horizontal direction, but an opposite situation
in the presence of both RaS: ξ^a > 1 (2.1).
This difference can be traced back to the much larger
value of ⟨ a_⊥^2 ⟩ in the presence of rotation, a
consequence of the enhancement of the horizontal component of acceleration
due to the Coriolis force.

In comparing the anisotropy of acceleration components
with that of velocity (see Table <ref>),
we find that ξ_a is always smaller 
than ξ_u, and
additionally very close to unity when R_IB>1.  
This can be explained by realizing that velocity is sensitive
to the large scales of the flow and is directly influenced
by the imposed anisotropy. On the other hand,
acceleration mostly samples the small scales,
and is not likely sensitive to the imposed anisotropy (at large scales), 
provided the scale separation between the large and small
scales is sufficiently wide – consistent
with the notion of local isotropy as dictated
by Kolmogorov's theory <cit.>.
In the current scenario, this is realized
when R_IB>1, resulting in 
ξ^a being nearly unity
(and also a k^-5/3 range
in the intermediate k range as
demonstrated earlier).
Whereas for R_IB<1, local isotropy is
clearly violated, but still the anisotropy
of acceleration is significantly weaker than that of velocity.

Additional insight can be obtained by
considering the ratio ξ^u/ξ^a, also shown in Table <ref>.
For HIT, this ratio is unity, but with increasing
strength of RaS, we observe ξ_u/ξ_a increases, peaking
at around 6 for case 4, and thereafter decreasing slightly
with further increase in strength of RaS.
A limiting value can be obtained in the case where
RaS is very strong, such that
the flow
is dominated by linear processes
(with turbulent eddies playing a very weak role).
This would lead to ξ^u/ξ^a ≈ N/f, by assuming
that a_⊥∼ u_⊥ f and a_∼ u_ N.
The value for case 6, with ξ^u/ξ^a = 18.4/3.8 ≈ 5,
appears to be consistent with this  consideration,
though it remains to be further tested for different N/f values.






Given that the statistics of acceleration 
appear to conform with local isotropy for R_IB>1,
we consider
the following relation from classical turbulence:

    ⟨ a^2 ⟩ = a_0 ⟨ϵ⟩^3/2ν^-1/2

which results from the application of 
Kolmogorov's similarity theory to acceleration variance <cit.>.
Here, a_0 is a dimensionless constant, which is plausibly universal
for HIT.
Empirical evidence based on studies of isotropic turbulence
has shown that a_0 increases slowly 
with Reynolds numbers <cit.>, which can be possibly 
viewed as a manifestation of small-scale intermittency, 
unaccounted for in
Kolmogorov's 1941 theory. 
In contrast, for RaS turbulence, one may expect
additional deviations because of the imposed anisotropy, especially 
if Re is not very large. 

Fig. <ref> shows the values of the a_0^⊥ and 
a_0^ as a function of Fr for cases 1 through 6, together with the
value obtained for case 0, in the absence of RaS.
The values of a_0^ and a_0^⊥ 
do not vary very much for 
1/Fr ≳ 12.5, 
and sharply increase, 
especially in the perpendicular direction,
at large values of 1/Fr. In particular, the value of 
a_0^⊥ at 1/Fr ≈ 16
exceeds that in the HIT case by a factor larger than  ∼ 10. 
This points to a strong difference of the small-scale properties
between runs 4-6 and the HIT run (case 0).
In comparison, and 
despite the larger value of N 
(N/f = 5 for runs 1-6), the value of a_0^ remains 
comparable to that obtained in the HIT case. 
In the purely stratified case (run 7), the value of a_0^ is 
slightly reduced compared to case 5 in the presence of both RaS
(a_0^≈ 3.1). On the other hand, the value of 
a_0^⊥≈ 1.4 for case 7 is reduced 
by over an order of magnitude compared to case 5. 
For the rotation-only run, a_0 is not very different compared to the
HIT case, suggesting that the small scales are still dominated by turbulence,
even though the large scales are highly anisotropic.
This once again demonstrates that acceleration becomes progressively dominated, as 
Ro and Fr decrease, by the frequencies associated with rotation/buoyancy.
As a consequence, the values of a_0 shown in Fig. <ref> do not
reflect only the small-scale structure of the flow, but also its global
properties.





 §.§ Acceleration PDFs






The analysis of acceleration variance can be generalized
by considering the probability density functions (PDFs) of 
acceleration components. 
In HIT,
similar to Eulerian velocity gradients, 
acceleration is also characterized by extremely
large fluctuations reflected in broad tails of the PDF,
which further broaden with increasing Reynolds number
<cit.>.
In this subsection, we study the effect of imposed RaS
on the PDFs of acceleration, especially
focusing on the long tails.  
Once again, we separate out the contributions a_ and a_⊥.


Fig. <ref>a and b respectively show the standardized PDFs of a_⊥
and a_ for all the runs listed in 
Table <ref>.
Two very distinct behaviors can be observed once again.
At moderate RaS (and for R_IB>1) 
the PDFs of acceleration exhibit very broad tails, and in this sense, they
differ only quantitatively from those obtained in the HIT case. 
This property is also reflected by the values of the kurtosis of the distributions,
shown in Table <ref>, which are all ≳ 20, 
even larger than the 
values obtained for the HIT case. 
The run corresponding to pure rotation also exhibits
very high kurtosis of the acceleration. Note that the accuracy of 
the estimates of kurtosis indicated in the table
are of the order of 10% or less, except for values marked with an asterisk,
which exhibited very large fluctuations, leading to higher error bars
for runs 3 and 8
(of about 20-30%).


In contrast for the runs with strong RaS (with R_IB<1), 
the tails of the PDFs are significantly suppressed. As shown
in Table <ref>, the corresponding kurtosis values are also
quite small, although still larger than the Gaussian value of 3.
This is particularly clear for the horizontal component a_⊥ of
acceleration, see Fig. <ref>a, and to a lesser extent, for the
vertical component a_ (Fig. <ref>b). 
The tendency of the PDFs to become narrower in runs with 
strong RaS appears to be a
consequence of the strong stratification, as the PDFs of acceleration
in the run with pure rotation are only weakly modified.
In regimes dominated by waves, weak turbulence 
theory suggests 
that the velocity and acceleration should be 
close to Gaussian <cit.>.
However, we observe that the 
kurtosis of acceleration components are significantly larger than 
the corresponding Gaussian value of 3 (with a stronger
kurtosis for the vertical, than for the horizontal component of acceleration).
This unexpected phenomenon can also be associated with the observation of bursts of
vertical velocity, manifested by the large values of K_u_ listed
in Table <ref>, particularly for runs 4-6. These bursts
are related to the phenomenon observed in <cit.> in the case
of purely stratified flows, in comparable domains of parameters. We expect
that the mechanisms are similar in the presence of a 
relatively weak rotation 
(N/f = 5), as it is the case in the present study. 







 §.§ Acceleration autocorrelation and frequency spectra




Essential information on the motion of tracers can be obtained from 
the autocorrelation function of quantities fluctuating along Lagrangian
trajectories. For example,
the autocorrelation function can be used to determine relevant time scales
and to form the frequency spectrum via Fourier transform <cit.>.
Since our analysis is concerned with statistically stationary
signals, the autocorrelation only depends on the chosen time lag.
The autocorrelation ρ^a(τ) is defined as

    C^a(τ) = ⟨ a(t+τ) a (t) ⟩    ,   ρ^a(τ) = C^a(τ)/C^a(0)

where τ is the time lag and C^a is the autocovariance. 
Statistical stationarity implies ρ^a(-τ) = ρ^a(τ)
and also imposes that the integral of the acceleration autocorrelation
must be zero, i.e., ∫_0^∞ρ^a(τ) dτ=0 <cit.>.
The frequency spectrum of acceleration E^a, which is the Fourier transform of 
C^a(τ) can be defined as 

    E^a(ω) = ∫_0^∞  C^a(τ) exp^-iωτ dτ

By definition, the integral of the spectrum gives the variance of the signal.
Yet again, we consider a_⊥ and  a_ separately.
  





  
*Perpendicular component:




Fig. <ref> shows the autororrelation function
ρ^a_⊥(τ) for all the runs with RaS.
In Fig. <ref>a, the time lag is normalized with τ_η, 
the Kolmogorov
time scale characterizing the small-scale motion.
We also include
the HIT case (dashed line) and the pure rotation case (dash-dotted line) as a reference.  

For HIT, the acceleration autocorrelation is known to decay
rapidly, becoming zero at τ≈ 2τ_η
and thereafter becoming negative and slowly approaching zero again
at very large time lags <cit.> – as readily seen in
Fig. <ref>a.
Interestingly, we observe that the runs with relatively weak RaS,
i.e., Fr ≥ 0.114, and also the run
with only rotation, behave very similarly to the HIT case
(see inset of Fig. <ref>a), albeit with minor 
variation in the zero-crossing point. 
For the rot-strat run with Fr=0.086, the autocorrelation function
shows a similar behavior up to very small time lags, but thereafter
deviates with the zero-crossing extending to τ≈ 8τ_η.
The large time behavior is markedly different with very low frequency
oscillations. 
On the other hand, for runs with strong RaS,
i.e., Fr ≤ 0.07, a very different behavior is observed
right from small time lags. The autocorrelation shows strong oscillations 
which are eventually damped out at large time lags.
Furthermore, the zero-crossing point also is strongly dependent on
Fr, moving to smaller time lags with decreasing Fr. 



Plotting ρ^a_⊥ as 
a function of τ f /2 π, see Fig. <ref>b, shows that
the period of the damped oscillations is close to 2 π/f 
(with small but significant deviation).
For comparison, we also show in Fig. <ref>b the
function ρ^a_⊥ for the case of a purely rotating flow. Somewhat 
surprisingly, the tendency to oscillate at a frequency ∼ f is not 
as strong as in the case where both RaS are present.
A similar trend was observed in <cit.>. 
We notice, however, that the stronger stratification (N=5f in runs 1-6) 
implies the presence of faster waves than in a purely rotating flow
at the same Ro number, involving in particular high values of 
k_⊥, since the dispersion law for inertia-gravity waves can be 
written as  k^2ω_IG^2=N^2k_⊥^2+f^2k_∥ <cit.>.









To better identify the frequencies 
present in the autocorrelation function, it is apt 
to take its Fourier transform to obtain the frequency spectrum.
Fig. <ref> shows the spectrum E^a_⊥.
The spectra have been made dimensionless by dividing by the mean kinetic energy dissipation 
rate ⟨ϵ⟩.
In Fig. <ref>a, the frequency is normalized by 
τ_η, whereas in Fig. <ref>b, we use the 
rotation frequency f. Note that the runs shown in 
Fig. <ref>a and b correspond to the same runs shown
respectively in Fig. <ref>a and b.
For the HIT run in Fig. <ref>a, 
Kolmogorov scaling would imply that the 
spectrum E^a is constant in the inertial range (defined as 
τ_η/T_E ≲ωτ_η≲ 1).

While the validity of Kolmogorov's similarity hypotheses to 
Lagrangian statistics is debatable, 
DNS studies suggest that the above scaling law, i.e., E^a ∼,
still may be approximately satisfied <cit.>.
Consistent with this, we find that for our HIT run, the spectrum
E^a_⊥(ω) is approximately flat over a limited range of values of 
ω. 

With addition of weak RaS, the 
changes in the structure of the spectrum 
appear to be relatively minor.
In particular, for runs with Fr ≥ 0.086, the differences
in spectra, relative to the HIT case, only become significant
for ωτ_η≲ 0.2.
For the particular case of Fr = 0.086, a small peak emerges
around ωτ_η≈ 0.1. 
With increasing strength of RaS,
the differences become more pronounced, with all the cases
for Fr ≤ 0.086 now showing a prominent peak (which is relatively broad).
This observation is consistent 
with the behavior of the autocorrelation function 
at small times, as observed in Fig. <ref>. 
In addition, 
the increase in values of E^a_⊥/⟨ϵ⟩ also 
corresponds to the strong increase in the value of a_0
for small Fr cases, as shown in Fig. <ref>
(for the perp. data points).
Interestingly, the decay of spectra for
all cases are somewhat similar at very large frequencies
(and not very different from the HIT case).  
This possibly confirms that the role of turbulent eddies
in the horizontal direction is still relevant 
as compared to the vertical direction, at least in the 
parameter range covered in this work
(note once again the fact that N/f=5, plays
an important role, since it renders the effect of rotation
relatively weaker than that of stratification).
 

Fig. <ref>b shows the spectra as a function of
ω/f. 
As expected, the peaks observed at low frequencies
in Fig. <ref>a, are all centered around ω/f ≈ 1.
For the runs with strong RaS, the frequency 
corresponding to stratification, i.e., ω/f = N/f = 5,
does not seem to play any particularly important role.



  
*Parallel component:



While the role of rotation was dominant in the 
perpendicular direction, we can in turn expect
stratification to dominate in the parallel direction.
In Fig. <ref>a and b, we show the autocorrelation
in the parallel direction ρ^a_ 
as a function of 
τ/τ_η  and of τ (N/2 π) respectively. As it was the
case for the perpendicular component of acceleration, the deviation of the
autocorrelation function from the reference 
homogeneous isotropic case, shown as a dashed line, increases gradually 
when Fr decreases. However, contrary to ρ^a_⊥, 
ρ^a_ tends to decay significantly faster than in the HIT case when 
Fr diminishes. 
In addition, for all cases, overshooting oscillations 
are clearly visible, with a  particularly strong amplitude
for cases with Fr ≤0.086.
While the zero-crossing of 
ρ^a_ for cases with weak RaS
still appears to be close to τ≈ 2τ_η 
(with minor deviations), for cases with 
strong RaS, the zero-crossing
shifts to even smaller time lags of τ≈ 0.5τ_η.
This is in contrast to ρ^a_⊥, where the zero-crossing
first shifted to higher time lags, and then moved to smaller
time lags with decreasing Fr.
Fig. <ref>b clearly
reveals that the period of the oscillations is equal to the
Brunt-Väisälä period, 2 π/N.
This strongly suggests that the motion 
in the parallel direction is dominated by stratification, and that turbulence
plays a much smaller role for the vertical motion
(in comparison to horizontal motion, where even at strongest rotation
rate, the role of turbulence still could not be ignored).
For comparison, the autocorrelation
function ρ^a_ is shown in Fig. <ref>b 
for the run with a strong stratification, 
Fr = 0.03. The tendency to oscillate is even stronger in this run, an 
effect amplified by the low value of Fr (and R_IB) in this case (see
Table <ref>).






The corresponding frequency spectra
E^a_(ω) are plotted as a function of
ωτ_η and of ω /N 
in Fig. <ref>a and b respectively.
The spectra are again non-dimensionalized by ⟨ϵ⟩.
Unlike in the case of E^a_⊥, we observe that deviations
from the HIT run are already prominent even for runs with weak
RaS.
All spectra are characterized by the presence of a peak, with
the peak becoming more sharp and prominent as Fr decreases.
An inspection of spectra in Fig. <ref>b clearly shows
that all these peaks correspond to ω/N ≈ 1, i.e., the respective 
Brunt-Väisälä frequencies, conforming with the dominant
role of stratification in the vertical direction. 
Note the peak for the stratification-only run is even more
pronounced. 
For runs with strong RaS, i.e., Fr≤0.069,
a very minor enhancement of the spectra is visible in the band of frequencies
ω/N ≈ 0.2 (or ω/f ≈ 1). 


Another point to note is that even though
the peaks in E^a_ become sharper with decreasing Fr,
their amplitudes do not vary much, in
contrast to E^a_⊥, where the peaks vary
by more than an order of magnitude.
This can be explained by the variation of
acceleration variance as seen in Fig. <ref>.
While the a_0^⊥ sharply shoots up for
runs with Fr≤0.069, the a_0^ 
only shows a minor variation in comparison. 



The results of this Subsection complement the conclusions of
Subsections <ref> and <ref>, as well
as those of Section <ref>, which were
pointing to two qualitatively different regimes, dominated by waves for 
R_IB < 1, and by eddies R_IB > 1. Whereas in the latter regime, 
the variances and the PDFs of accelerations were showing only moderate
deviations compared to HIT flow, Fig. <ref>b 
and <ref>b reveals the role of N and f 
in the horizontal and vertical motion closer to the transition, 
when R_IB≳ 1
(R_IB = 2.69 for run 3). The transition
to a wave-dominated regime for R_IB < 1 corresponds to 
the formation of peaks, which are much more intense 
(see Fig. <ref>b) or much sharper 
(see Fig. <ref>b), leading to a qualitatively 
very different dynamics.







§ LAGRANGIAN VELOCITY AND DISPERSION STATISTICS





The spread or dispersion of a material under the action of turbulence
is of obvious importance and can be directly studied from
the investigation of the Lagrangian velocity along trajectories.
In this regard, we briefly summarize the classical theory below <cit.>.

Single-particle dispersion is best understood by considering
the mean-square displacement of a particle from its initial 
position. From Eq. (<ref>), we can write for
each direction:

    ⟨ Y_i^2 (t) ⟩ = ⟨ u_i(0)^2⟩∫_0^t ∫_0^t ρ^u_i (  t' ,  t” ) dt' dt”

where Y_i(t) = x_i(t) - x_i(0) is the displacement 
of the particle from its position x_i(0) at t=0 to 
its position x_i(t) at time t,
and ρ^u_i(t', t”) is the velocity autocorrelation.
For statistical stationarity, the autocorrelation function 
at t' and t” only depends on τ = | t' - t”|, allowing Eq. (<ref>) 
to be rewritten as

    ⟨Y_i^2 (t) ⟩ = 2 ⟨u_i^2⟩∫_0^t (t-τ) ρ^u_i (τ) dτ

Once again, we treat separately the 
horizontal and vertical components, as explained before. 
The expression for the dispersion, 
Eq. (<ref>), simplifies in the 
limit of short and long times, respectively to

    ⟨ Y_⊥, ^2 (t) ⟩   = ⟨ u_⊥, ^2⟩ t^2     for  
    t ≪τ_K   , 
    ⟨ Y_⊥, ^2 (t) ⟩   = D_⊥,  t       for   t≫ T_L^⊥, ,

where D_⊥, =2⟨ u_⊥, ^2⟩ T_L^⊥, are the 
diffusion coefficients
and 

    T_L^⊥,  = ∫_0^∞ρ^u_⊥,  (τ) dτ
 
are
the Lagrangian integral times.
In the following subsections, we discuss separately 
the behavior of the autocorrelation
functions, integral time scales  and then the mean-square displacements.






 §.§ Velocity autocorrelation and integral time scales



In this subsection, we investigate the behavior of the velocity 
autocorrelations 
and the corresponding integral time scales derived from them.


Similar to the acceleration spectra, the velocity  frequency spectra
can be obtained through a Fourier transform of the velocity autocorrelations.
However, the velocity spectra can also be simply
derived from the acceleration spectra using the simple relation
E^u(ω) = E^a(ω) /ω^2 <cit.>.



Consequently, we do not discuss the results for E^u  in this section
(since they provide similar information as E^a). Instead, we have briefly
summarized them in the Supplementary Material <cit.>. 





  
*Perpendicular component:



Fig. <ref>a-b shows the autocorrelation functions 
ρ^u_⊥(τ);


similar to results for the perpendicular component of acceleration, 
the time lag τ is normalized
first by τ_η and then by f.
The correlation function ρ^u_⊥, plotted
as a function of τ/τ_η, see Fig. <ref>a,
shows only weak deviations from
the HIT case (shown in black dashed line) 
when RaS are moderate (runs 1-3). 
We recall that in the HIT case,
the long time behavior of the velocity autocorrelation can be well
represented by an exponential functional form, 
ρ^u ∼exp^-τ/T_L <cit.>. 
We notice that the runs with RaS are fundamentally
different from the HIT run with appearance of oscillations resulting 
in negative values of the correlation function. 
These deviations become stronger with imposed RaS.




To better understand these oscillations, Fig. <ref>b  
shows the autocorrelation as a function of τ f/2π.
Although an oscillatory behavior can be seen in the runs 
with the highest values of RaS rates, the frequency
is close to, but differs from f.
On the other hand, 
ρ^u_⊥ in the flow
with rotation only (shown in black dash-dotted line 
in Fig. <ref>b), appears to be monotonically decreasing
(no oscillations).












 




  
*Parallel component:






The autocorrelations and spectra corresponding to the parallel
component of velocity are shown in Fig. <ref>.
We first consider the autocorrelation ρ^u_  as a function of
τ/τ_η in Fig. <ref>a. 
In all the cases with RaS, 
ρ^u_ strongly differs from the HIT case 
(indicated by a black dashed line).
Two patterns are distinctly visible. First, all the autocorrelation
functions
decay rapidly. In terms of the dimensionless time t/τ_η, the
decay rate becomes increasingly larger when Fr decreases
(especially in comparison to perpendicular component
shown in Fig. <ref>a).
Second, the autocorrelations all overshoot to become strongly 
negative and show distinct oscillations.
The primary period of these oscillations corresponds to the stratification 
frequency, as demonstrated by
Fig. <ref>b, which shows the autocorrelations
as a function of τ N/2π. In addition, Fig. <ref>b
also shows that all cases superpose reasonably well for the initial
decay of the autocorrelation,
These results clearly demonstrate that stratification plays a dominant
role in the vertical motion, and in comparison
turbulence plays a weaker role. 
The fast dynamics in the vertical direction 
is dominated by the strong vertical gradients 
arising from the constraint that the vertical Froude number be of 
order unity <cit.>, 
and leading to strong spatial and temporal intermittency, 
dissipation and mixing <cit.>.

This is true, even for the runs at weak RaS
(Fr ≳ 0.08). A similar behavior was obtained for
the vertical acceleration correlation function, see
Section <ref>. 
We notice that in the case of a purely
stratified run, without any rotation, the autocorrelation function 
ρ^u_ clearly oscillates, with a modulation at twice
the Brunt-Väisälä period. 







  
*Integral time scales:





As highlighted by Eqs. (<ref>) and (<ref>),
the Lagrangian integral time scale  T_L,
characterizes the particle dispersion at large times and
is simply obtained by integrating the autocorrelation function:
T_L = ∫_0^∞ρ^u dτ.
We begin by noticing that many of the correlation functions shown in 
Figs. <ref> and <ref> exhibit an oscillatory behavior.
This makes the calculation of their integrals generally prone to statistical
errors. As such, the values of T_L^⊥,, listed in 
Table <ref>, are subject to 
large relative errors, the more so as the values of T_L^⊥, 
are small. 


Over the range of Re considered here,
the variation of T_L^⊥ 
is found to be rather weak at moderate RaS (Fr ≳ 0.08).
In particular,
the ratio T_L^⊥/τ_η, is of the order of 10
for the HIT run, and remains approximately constant
for cases 1-4 and also for the run 8, which corresponds to a flow with
rotation only.
For cases 5-6 with strong RaS, the ratio sharply drops by a factor ∼ 2,
which can be attributed to a sharp decrease in the
mean dissipation ⟨ϵ⟩ (hence causing a
sharp increase in τ_η).
The above observations are consistent with earlier 
results which also highlight a sharp transition
to wave-dominated regime as R_IB significantly decreases.
We note that for the run with pure stratification (case 7), 
the velocity autocorrelation in the horizontal direction 
converges extremely slowly
to zero at large time lags, pointing to an extremely large value
of T_L^⊥. A similar behavior has been 
noticed in <cit.>.
In this regard, the presence of rotation plays a crucial 
role. In particular, the 
dispersion relation and the structure of the eigenmodes, underlying the
wave motion
is strongly modified by rotation. 

 
On the other hand, 
the integral time T_L^ is  
substantially smaller than T_L^⊥
for all runs with stratification (cases 1-7).
The ratio T_L^/τ_η is still around 10
for the HIT and purely rotating flows (cases 0 and 8 respectively),
but is greatly reduced for all runs with stratification.
This can be evidently attributed to the strongly oscillating nature
of the velocity autocorrelation functions for cases 1-7
(see Fig. <ref>b), which results in significant 
cancellation when calculating T_L^.
The ratio T_L^/τ_η drops by more than 
an order of magnitude in going from HIT to case 1, which corresponds
to the weakest RaS. Thereafter, with 
increasing strength of RaS, the 
ratio further decreases.  
Interestingly, the ratio T_L^N/2π
shows a much reduced variation, and remains in the range 0.01-0.02
for all cases with stratification. This suggests
that despite the strong oscillations in the autocorrelation
the time scale, T_L^, is not strictly zero, but 
instead scales inversely with N, consistent with earlier
predictions for purely stratified flows <cit.>.







 §.§ Mean-square displacement



Fig. <ref> shows the mean-square displacement
for horizontal and vertical directions as a function of time
(both axes normalized appropriately by
Kolmogorov scales).
At small times, a clear t^2 scaling is visible 
for all runs in both components, as expected from 
Eq.(<ref>). 
The displacement in the horizontal 
direction is enhanced by RaS,
whereas inhibited in the vertical direction.
This can be readily explained by considering the
variance of the components of the
velocity ⟨ v_⊥^2 ⟩ and ⟨ v_^2 ⟩, as shown 
in Table <ref> (see also Eq. (<ref>)). 






At long times, 
transition to a diffusive regime, given by 
Eq. (<ref>), is expectedly seen in the HIT run  <cit.>,
for both horizontal and vertical components.
A similar behavior is also visible for runs with
weak RaS (Fr ≳ 0.08), when considering the 
horizontal component. 
In comparison, the growth 
of the vertical component at long times for these runs is significantly inhibited,
nevertheless still showing a very slow approach to a diffusive regime.
Our results are generally consistent with those obtained earlier for a
comparable parameter range (N/f = 10, Fr ≈ 0.16) <cit.>.
Note the rotation-only run, behaves like the HIT run,
consistent with the small degree of anisotropy in
both velocity and acceleration statistics as noted earlier.


On the other hand, for runs with strong RaS (Fr ≲ 0.07), 
the growth of both horizontal and vertical components appears to be
slower than a linear behavior.  
This is particularly clear for the vertical displacements
⟨ Y_^2 ⟩, which appear to increase extremely slowly
with time, if at all.  
In fact, the values of ⟨ Y_^2 ⟩/η^2 remain 
small, of the order of ∼ 10, over a very long time.
The absence of a clear diffusive regime for ⟨ Y_^2 ⟩ is
arguably not very surprising, given the small values of T^L_ and 
⟨ v_^2 ⟩, hence of the diffusion coefficient 
D = ⟨ v_^2 ⟩ T_L^. 
Assuming that  T^L_
is inversely proportional to N (multiplied with a small prefactor), 
the expected
diffusive behavior will likely be visible only after an extremely 
long time. Consistent with our own results, 
the near constancy of ⟨ Y_^2 ⟩ was
also reported in purely stratified flows 
<cit.>.

Contrary to the very low values of T_L^ for runs with strong RaS
(see Table <ref>), the values of T_L^⊥ do not 
appear to be particularly small. This makes the lack of a clear diffusive regime 
for runs 4-6 for the horizontal displacement much more surprising.
It has been noticed in the context of stratified flows,
that the horizontal displacement could grow as t^2 <cit.>. 
Here, our results show that, 
in a regime dominated by waves (runs 4-6), the growth of 
⟨ Y_⊥ ^2 ⟩ is slower than t^1 over a long time.
This suggests a strong interaction between the horizontal and vertical 
directions owing to the presence of rotation 
(and inertia-gravity waves), which was also visible
in autocorrelations of acceleration and velocity. 
However, given the large values of T_L^⊥,
it can be expected that the diffusive regime is ultimately reached
at sufficiently longer times.


Finally, we note that the strong vertical drafts
responsible for high kurtosis of u_ (see Section <ref>), 
and increased mixing efficiency <cit.>, 
do not appear
to cause an appreciable growth in ⟨ Y_ ^2 ⟩
for the runs considered here.
This is not inconsistent given the 
kurtosis is a normalized fourth-order moment,
whereas ⟨ Y_ ^2 ⟩  
goes as the variance of vertical velocity
and the integral scale T_L^,
both of which are strongly suppressed.
It is likely that at higher Reynolds numbers than considered
here, the effect of the vertical drafts is felt more directly
on the vertical displacement of fluid particles.






§ DISCUSSION AND CONCLUSIONS




Motivated by the well established effect of both 
rotation and stratification (RaS) 
in numerous geophysical flows,
and by the observation
of Lagrangian tracers, such as buoys 
in the ocean <cit.>,

we have investigated the properties of particle trajectories in such flows.
We have utilized direct numerical simulations of the 
Boussinesq equations in a periodic domain,
with both stratification and the axis of rotation aligned in the vertical direction, 
in a statistically stationary regime maintained by an external forcing.
We have focused on cases where the corresponding rotation and stratification
frequencies,  f and N respectively, are held in a fixed ratio N/f=5,
as relevant in oceanographic situations <cit.>.
The usual computer limitations have restricted the Reynolds numbers
to approximately around 4000 in our runs,
which are smaller than those observed in nature,
but still large enough to allow turbulence to adequately develop.
The Froude numbers investigated are in the range 
0.03 ≲ Fr ≲ 0.2,
with corresponding Rossby number Ro=5Fr, compatible with
geophysical fluid flows.
Our results illustrate the complex physical effects involved 
in a turbulent flow under the combined action of RaS. Whereas the
corresponding physical effects have been recently investigated in an Eulerian 
context in many studies, we have documented 
the elementary properties of particle trajectories,
in particular concerning acceleration, velocity and displacement
statistics.
Due to the imposed large-scale anisotropy, 
we differentiate between motion in the horizontal plane and in the vertical direction.
We also appropriately compare with rotation-only and stratification-only runs and 
also with the well studied case of isotropic turbulence.

While the parameters Fr and Ro  
vary smoothly with 
the strength of imposed RaS, we observe a sharp change in the nature of
the flow when Fr ≲ 0.07 for the runs considered,
corresponding to a regime where waves appear to dominate over the
non-linearities, as also observed in <cit.> for stratification-only runs.
An inspection of the Eulerian energy spectra (and the temperature spectra),
shows a clear transition from a Kolmogorov k^-5/3 like behavior
to a Bolgiano k^-11/5 like behavior at intermediate wavenumbers.
The transition also corresponds to the peak of intermittent disruption 
of shear layers, 
with a maximum of the kurtosis of the vertical velocity. In this 
context, our work extends the stratification-only results of <cit.>. 
Interestingly, we find that this transition to wave-dominated regime
can be given by a simple condition of 
R_IB<1 based on the buoyancy Reynolds number R_IB.


The effect of this transition is also evident in Lagrangian statistics.
For acceleration statistics, the effect of imposed RaS appears very weak before the
transition, suggesting the imposed anisotropy does not affect the small-scale isotropy
significantly. The probability density functions (PDFs) of acceleration
in both horizontal and vertical direction display wide tails very similar 
to isotropic turbulence, 
and the variance of acceleration can be approximately scaled with 
Kolmogorov variables. 
The Lagrangian autocorrelations and frequency spectra also appear 
to behave in a similar fashion, although 
a weak effect of RaS is still visible at low frequencies
in the spectra (when R_IB≳ 1).

However, the sharp transition at R_IB≲ 1 leads to 
a very strong anisotropy between the horizontal and vertical directions,
with the particle motion strongly dominated by RaS instead of turbulent eddies.
The PDFs of acceleration show suppressed tails,
and acceleration variance shows no clear scaling.
Moreover, the autocorrelations and frequency spectra clearly reflect
the dominant roles of frequencies N and f.


On the other hand, Lagrangian velocity statistics are always
affected by the imposed RaS. Anisotropy is already evident
for weak RaS; however the degree of anisotropy becomes very strong
with the said transition at R_IB≲ 1. 
This is most readily seen in the integral time scale based on the 
autocorrelations from vertical velocity
and hence also reflected in long time behavior of 
mean-square displacement of particles in the vertical direction.
Similar effects had been observed in the 
case of purely stratified flows <cit.>. In such flows, it 
is known that stratification leads to the formation of horizontal layers, that
inhibits the 
transport in the vertical direction. 
In the presence of both RaS, slanted layers appear <cit.> 
which somewhat inhibit transport in both the vertical and the horizontal directions. 

It is interesting to consider that the transitional behavior observed 
in this work is possibly also
linked to the enhanced vertical velocities that develop in 
strongly stably stratified turbulent flows, such as in the nocturnal 
planetary boundary layer <cit.>, and warrant further examination.
The resulting dispersion of Lagrangian particles must feel these hot spots of 
strong vertical velocity and strong local dissipation linked to localized shear instabilities, 
as also observed in the ocean.
Several studies point to a local Richardson number, measuring 
stratification with respect to the internal shear associated with 
vertically sheared horizontal winds, to be mostly in the vicinity of the
value for linear instability <cit.>.
This leads to flows which are altogether close to criticality, and strongly
anisotropic, intermittent and dissipative <cit.>. 
We expect that the observed similarity between the statistical
properties of acceleration when the flow is dominated by eddies
(for R_IB > 1), and those in HIT, may be very useful in modeling 
the dispersion of particles in turbulent flows in the presence of RaS.

Whereas 
R_IB seems to
provide a clear criterion to distinguish between the different regimes 
observed here, alternative ways <cit.> have been proposed to 
describe the transition 
between wave-dominated flows, and flows where waves and eddies interact. 
Namely, it was found that the transition
occurs around R_ B∼ 10, where R_ B=Re Fr^2. In this sense, the flows studied here
are all in the regime of interacting waves and eddies, although run 6 is very
close to the transition towards a wave-dominated flow.













We conclude by mentioning that many interesting issues remain to be 
addressed for particle motion in such flows.
For example, the problem of separation of pairs, or of clusters of 
particles, both forward and backward in time  <cit.>
or considering the effects of particle inertia or molecular diffusion.
In addition, 
we stress that the picture we have drawn is based on runs at a
relative moderate Reynolds numbers.
Exploring higher Reynolds numbers, and also higher Prandtl/Schmidt numbers <cit.>, 
along with different values of
N/f will help in better understanding the parameter space and 
how the results from current work translate
to practical geophysical applications.








§ ACKNOWLEDGMENTS


The computing resources utilized in this work
were provided by
PSMN at the Ecole Normale Superieure de Lyon.
D. Buaria and A. Pumir acknowledge support from 
the European High-performance Infrastructure in Turbulence
(EuHIT) program and 
the Max Planck Society. 
A. Pumir acknowledges additional support
from the IDEXLYON project (Contract ANR-16-IDEX-0005)
under University of Lyon auspices. 
R. Marino acknowledges support from the program PALSE
(Programme Avenir Lyon Saint-Etienne) and IMPULSION of the 
University of Lyon, in the framework of the program
Investissements d'Avenir (ANR-11-IDEX-0007),
from the Inge'LySE (Lyon-Saint-Etienne) and from ERCOFTAC.
A. Pouquet gratefully acknowledges support from 
Laboratory for Atmospheric and Space Physics (LASP), 
in particular Bob Ergun, at the
University of Colorado, Boulder.














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@bib@innerbibempty

[Pedlosky(1979)]pedlosky
  author author J. Pedlosky, @noop  title Geophysical Fluid
  Dynamics (publisher Springer, year
  1979)NoStop
[Weiss and Provenzale(2007)]weiss07
  author author J. B. Weiss and author A. Provenzale, @noop  title Transport and mixing
  in geophysical flows, Vol. volume 744 (publisher Springer, year 2007)NoStop
[Wyngaard(1992)]wyngaard
  author author J. C. Wyngaard, title title Atmospheric
  turbulence, @noop  journal journal
  Annu. Rev. Fluid Mech. volume 24, pages 205–234 (year 1992)NoStop
[Stohl et al.(2002)Stohl,
  Eckhardt, Forster, James,  and Spichtinger]stohl02
  author author A. Stohl, author S. Eckhardt,
  author C. Forster, author P. James,  and author
  N. Spichtinger, title
  title On the pathways and timescales of
  intercontinental air pollution transport, @noop  journal journal J. Geophysical Research: Atmospheres  volume 107, pages ACH–6 (year 2002)NoStop
[Shaw(2003)]shaw_03
  author author R. A. Shaw, title title
  Particle-turbulence interactions in atmospheric clouds, @noop 
  journal journal Ann. Rev. Fluid Mech. volume 35, pages 183–227 (year
  2003)NoStop
[Guasto et al.(2012)Guasto,
  Rusconi, and Stocker]guasto
  author author J. S. Guasto, author R. Rusconi,   and author R. Stocker,  title title Fluid mechanics of
  planktonic microorganisms, @noop  journal journal Annu. Rev. Fluid Mech. volume 44,  pages 373–400 (year 2012)NoStop
[Monin and Yaglom(1971)]MY.I
  author author A. S. Monin and author A. M. Yaglom, @noop  title Statistical Fluid
  Mechanics. Vol. I, Vol. volume 1 (publisher
  MIT Press, year 1971)NoStop
[Buaria et al.(2016)Buaria,
  Yeung, and Sawford]BYS.2016
  author author D. Buaria, author P. K. Yeung,   and author B. L. Sawford,  title title A Lagrangian study of
  turbulent mixing: forward and backward dispersion of molecular trajectories
  in isotropic turbulence, @noop  journal journal J. Fluid Mech. volume 799, pages 352–382 (year 2016)NoStop
[Pumir and Wilkinson(2016)]pumir16
  author author A. Pumir and author M. Wilkinson, title title Collisional
  aggregation due to turbulence, @noop  journal
  journal Annu. Rev. Condens. Matter Phys. volume 7, pages 141–170 (year
  2016)NoStop
[Davidson(2013)]Davidson_13
  author author P. A. Davidson, @noop  title Turbulence in
  Rotating, Stratified and Electrically Conducting Fluids (publisher Cambridge University Press, year
  2013)NoStop
[Sawford(2001)]Sawford:01
  author author B. Sawford, title title Turbulent
  relative dispersion, @noop  journal journal Annu. Rev. Fluid Mech. volume 33,  pages 289–317 (year 2001)NoStop
[Yeung(2002)]Yeung02
  author author P. K. Yeung, title title Lagrangian
  investigations of turbulence, @noop  journal
  journal Ann. Rev. Fluid Mech. volume
  34, pages 115–142 (year 2002)NoStop
[Toschi and Bodenschatz(2009)]TB_ARFM
  author author F. Toschi and author E. Bodenschatz, title title
  Lagrangian properties of particles in turbulence, @noop 
  journal journal Annu. Rev. Fluid Mech. volume 41, pages 375–404 (year
  2009)NoStop
[DelCastello and Clercx(2011a)]DelCastello+:11a
  author author L. DelCastello and author H. J. H.  Clercx, title title
  Lagrangian acceleration of passive tracers in statistically steady
  rotating turbulence, @noop  journal journal Phys. Rev. Lett. volume 107, pages 214502 (year 2011a)NoStop
[DelCastello and Clercx(2011b)]DelCastello+:11b
  author author L. DelCastello and author H. J. H.  Clercx, title title
  Lagrangian velocity autocorrelations in statistically steady rotating
  turbulence, @noop  journal journal
  Phys. Rev. E volume 83, pages 056316
  (year 2011b)NoStop
[Naso and Godeferd(2012)]Naso:2012
  author author A. Naso and author F. S. Godeferd, title title Statistics of
  the perceived velocity gradient tensor in a rotating turbulent flow,  @noop  journal journal New J Phys  volume 125002, pages 7558–7563
  (year 2012)NoStop
[van Artrijk et al.(2008)van Artrijk, Clercx, and Winters]Artrijk:08
  author author M. van Artrijk, author H. J. H. Clercx,  and author K. B. Winters, title title
  Single-particle, particle-pair, and multiparticle dispersion of fluid
  particles in forced stably stratified turbulence, @noop 
  journal journal Phys. Fluids volume 20, pages 025104 (year
  2008)NoStop
[Brethouwer and Lindborg(2009)]Brethouwer:09
  author author G. Brethouwer and author E. Lindborg, title title Numerical
  study of vertical dispersion by stratified turbulence, @noop 
  journal journal J. Fluid Mech. volume 631, pages 149–163 (year
  2009)NoStop
[Sujovolsky et al.(2018)Sujovolsky, Mininni, and Rast]Sujovolsky:18
  author author N. E. Sujovolsky, author P. D. Mininni,  and author M. P. Rast, title title Single-particle
  dispersion in stably stratified turbulence, @noop  journal journal Phys. Rev. Fluids volume 3, pages 034603 (year
  2018)NoStop
[Garabato et al.(2004)Garabato, Polzin, King, Heywood, and Visbeck]Naveira:04
  author author A. C. N.  Garabato, author K. L.  Polzin, author B. A.  King, author K. J. Heywood,  and author M. Visbeck, title title Widespread
  intense turbulent mixing in the southern ocean, @noop  journal journal Science volume
  303, pages 210–213 (year 2004)NoStop
[Nikurashin et al.(2012)Nikurashin, Vallis, and Adcroft]Nikurashin:12
  author author M. Nikurashin, author G. K. Vallis,  and author A. Adcroft, title title Routes to
  energy dissipation for geostrophic flows in the southern ocean,  @noop  journal journal Nat. Geosci.  volume 6, pages 48–51 (year
  2012)NoStop
[van Haren and Gostiaux(2016)]vanharen_16j
  author author H. van
  Haren and author L. Gostiaux, title title Convective
  mixing by internal waves in the Puerto Rico Trench, @noop 
  journal journal J. Mar. Res. volume 74, pages 161–173 (year
  2016)NoStop
[Marino et al.(2013)Marino,
  Mininni, Rosenberg, and Pouquet]marino13b
  author author R. Marino, author P. D. Mininni,
  author D. Rosenberg,  and  author A. Pouquet, title title Emergence of helicity in rotating
  stratified turbulence, @noop  journal journal Phys. Rev. E volume 87, pages 033016 (year 2013)NoStop
[Pouquet and Marino(2013)]Marino:13
  author author A. Pouquet and author R. Marino, title title Geophysical
  turbulence and the duality of the energy flow across scales, @noop
   journal journal Phys. Rev. Lett. volume 111, pages 234501 (year
  2013)NoStop
[Haller(2015)]haller_arfm
  author author G. Haller, title title Lagrangian
  coherent structures, @noop  journal journal Annu. Rev. Fluid Mech. volume 47,  pages 137–162 (year 2015)NoStop
[Ivey et al.(2008)Ivey,
  Winters, and Koseff]ivey_08
  author author G. Ivey, author K. Winters,   and author J. Koseff,  title title Density stratification,
  turbulence but how much mixing? @noop  journal
  journal Ann. Rev. Fluid Mech. volume
  40, pages 169 (year 2008)NoStop
[Deusebio et al.(2014)Deusebio, Augier, and Lindborg]deusebio_14b
  author author E. Deusebio, author P. Augier,   and author E. Lindborg,  title title Third-order structure
  functions in rotating and stratified turbulence: a comparison between
  numerical, analytical and observational results, @noop  journal journal J . Fluid Mech. volume 755, pages 294–313 (year
  2014)NoStop
[Rosenberg et al.(2015)Rosenberg, Pouquet, Marino, and  Mininni]Rosenberg:15
  author author D. Rosenberg, author A. Pouquet,
  author R. Marino,  and author P. D. Mininni, title title Evidence for Bolgiano-Obukhov
  scaling in rotating-stratified turbulence using high-resolution direct
  numerical simulations, @noop  journal journal Phys. Fluids volume 27, pages 055105 (year 2015)NoStop
[Lindborg and Mohanan(2017)]lindborg_17
  author author E. Lindborg and author A. V. Mohanan, title title A
  two-dimensional toy model for geophysical turbulence, @noop 
  journal journal Phys. Fluids volume 29 (year 2017)NoStop
[Waite(2017)]waite_17
  author author M. L. Waite, title title Random forcing of
  geostrophic motion in rotating stratified turbulence, @noop 
  journal journal Phys. Fluids volume 29 (year 2017)NoStop
[Gregg et al.(2018)Gregg,
  D'Asaro, Riley, and Kunze]gregg_18
  author author M.C. Gregg, author E.A. D'Asaro,
  author J.J. Riley,  and  author E. Kunze, title title Mixing efficiency in the ocean,  @noop  journal journal Ann. Rev. Marine
  Sci. volume 10, pages 9.1–9.31
  (year 2018)NoStop
[Pouquet et al.(2018)Pouquet, Rosenberg, Marino, and  Herbert]pouquet_18
  author author A. Pouquet, author D. Rosenberg,
  author R. Marino,  and author C. Herbert, title
  title Scaling laws for mixing and dissipation in
  unforced rotating stratified turbulence, @noop  journal journal J. Fluid Mech. volume
  844, pages 519–545 (year 2018)NoStop
[Rorai et al.(2014)Rorai,
  Mininni, and Pouquet]Rorai:14
  author author C. Rorai, author P. D. Mininni,
   and author A. Pouquet,  title title Turbulence comes in bursts
  in stably stratified flows, @noop  journal journal Physical Review E volume 89, pages 043002 (year 2014)NoStop
[Feraco et al.(2018)Feraco, Marino, Pumir, Primavera, Mininni, Pouquet, and  Rosenberg]Feraco:18
  author author F. Feraco, author R. Marino, author A. Pumir,
  author L. Primavera, author P. D. Mininni, author A. Pouquet,  and author D. Rosenberg, title title Vertical drafts and mixing in
  stratified turbulence: sharp transition with the Froude number,  @noop  journal journal Europhys. Lett.  volume 123, pages 44002 (year 2018)NoStop
[Marino et al.(2013)Marino, Mininni, Rosenberg, and  Pouquet]Marino:13a
  author author R. Marino, author P. D. Mininni, author D. Rosenberg,  and author A. Pouquet, title title Inverse
  cascade in rotating stratified turbulence: fast growth of large scales,  @noop  journal journal Europhys. Lett.  volume 102, pages 44006 (year 2013)NoStop
[Pouquet et al.(2019a)Pouquet, Rosenberg, Stawarz, and Marino]pouquet_19e
  author author A. Pouquet, author D. Rosenberg,
  author J. Stawarz,  and  author R. Marino, title title Helicity dynamics, inverse, and
  bidirectional cascades in fluid and magnetohydrodynamic turbulence: A brief
  review, @noop  journal journal Earth
  Space Sci. volume 6, pages 1–19
  (year 2019a)NoStop
[Buaria et al.(2019)Buaria,
  Pumir, Bodenschatz, and Yeung]BPBY2019
  author author D. Buaria, author A. Pumir,
  author E. Bodenschatz,  and  author P. K. Yeung, title title Extreme velocity gradients in turbulent
  flows, @noop  journal journal New
  J. Phys. volume 21, pages 043004
  (year 2019)NoStop
[Marino et al.(2015)Marino,
  Rosenberg, Herbert, and Pouquet]marino_15w
  author author R. Marino, author D. Rosenberg,
  author C. Herbert,  and  author A. Pouquet, title title Interplay of waves and eddies in
  rotating stratified turbulence and the link with kinetic-potential energy
  partition, @noop  journal journal
  EuroPhys. Lett. volume 112, pages
  49001 (year 2015)NoStop
[Brethouwer et al.(2007)Brethouwer, Billant, Lindborg, and  Chomaz]Brethouwer:07
  author author G. Brethouwer, author P. Billant,
  author E. Lindborg,  and  author J. M. Chomaz, title title Scaling analysis and simulations of
  strongly stratified flows, @noop  journal journal J. Fluid Mech. volume 585, pages 343–368 (year 2007)NoStop
[Ishihara et al.(2009)Ishihara, Gotoh, and Kaneda]Ishihara09
  author author T. Ishihara, author T. Gotoh,   and author Y. Kaneda,  title title Study of high-Reynolds
  number isotropic turbulence by direct numerical simulations, @noop
   journal journal Ann. Rev. Fluid Mech.  volume 41, pages 165–80 (year 2009)NoStop
[Mininni et al.(2011)Mininni, Rosenberg, Reddy, and  Pouquet]Mininni:11
  author author P. D. Mininni, author D. Rosenberg,
  author R. Reddy,  and author A. Pouquet, title
  title A hybrid mpi-openmp scheme for scalable parallel
  pseudospectral computations for fluid turbulence, @noop 
  journal journal Parallel Comput. volume 37, pages 316–326 (year
  2011)NoStop
[sup()]supp
  @noop  journal See Supplemental Material at [URL will be
  inserted by publisher] for additional details. NoStop
[van Hinsberg et al.(2013)van
  Hinsberg, Boonkkamp, Toschi, and  Clercx]hinsberg
  journal   author author M. A. T. van Hinsberg, author J. H. M. ten Thije Boonkkamp, author
  F. Toschi,  and author
  H. J. H. Clercx, title
  title Optimal interpolation schemes for particle
  tracking in turbulence, @noop  journal journal Phys. Rev. E volume 87, pages 043307 (year 2013)NoStop
[Buaria and Yeung(2017)]buaria.cpc
  author author D. Buaria and author P. K. Yeung, title title A highly scalable
  particle tracking algorithm using partitioned global address space (PGAS)
  programming for extreme-scale turbulence simulations, @noop 
  journal journal Comput. Phys. Commun. volume 221, pages 246 – 258 (year
  2017)NoStop
[Lindborg(2006)]lindborg_06
  author author E. Lindborg, title title The energy
  cascade in a strongly stratified fluid, @noop  journal journal J. Fluid Mech. volume
  550, pages 207–242 (year 2006)NoStop
[Verma et al.(2017)Verma,
  Kumar, and Pandey]verma_17
  author author M. Verma, author A. Kumar,  and  author A. Pandey, title title Phenomenology of buoyancy-driven
  turbulence: recent results, @noop  journal journal New J. Phys. volume 19, pages 025012 (year 2017)NoStop
[Bolgiano(1959)]Bolgiano:59
  author author R. Bolgiano, title title Turbulent
  spectra in a stably stratified atmosphere, @noop  journal journal J. Geophys. Res. volume 64, pages 2226–2229 (year
  1959)NoStop
[Obukhov(1959)]obukhov_59
  author author A.M. Obukhov, title title Effect of the
  Archimedean forces on the structure of the temperature field in a turbulent
  flow, @noop  journal journal Dokl.
  Akad. Nauk SSSR volume 125, pages
  1246–1248 (year 1959)NoStop
[Kimura and Herring(2012)]kimura12
  author author Y. Kimura and author J. R. Herring, title title Energy spectra
  of stably stratified turbulence, @noop  journal
  journal J. Fluid Mech. volume 698,  pages 19–50 (year 2012)NoStop
[Wilson and Sawford(1996)]wilson96
  author author J. D. Wilson and author B. L. Sawford, title title Review of
  Lagrangian stochastic models for trajectories in the turbulent
  atmosphere, @noop  journal journal
  Bound.-Layer Met. volume 78, pages
  191–210 (year 1996)NoStop
[Sawford et al.(2003)Sawford, Yeung, Borgas, Vedula, Porta, Crawford, and  Bodenschatz]Sawford03
  author author B. L. Sawford, author P. K. Yeung,
  author M. S. Borgas, author P. Vedula, author
  A. La Porta, author
  A. M. Crawford,  and author
  E. Bodenschatz, title
  title Conditional and unconditional acceleration
  statistics in turbulence, @noop  journal journal Phys. Fluids volume 15, pages 3478–3489 (year 2003)NoStop
[Nazarenko(2011)]Nazarenko:2011
  author author S. Nazarenko, @noop  title Wave Turbulence  (publisher Springer-Verlag, year
  2011)NoStop
[Yeung and Pope(1989)]Yeung89
  author author P. K. Yeung and author S. B. Pope, title title Lagrangian
  statistics from direct numerical simulations of isotropic turbulence,  @noop  journal journal J. Fluid Mech.  volume 207, pages 531–586 (year 1989)NoStop
[Yeung et al.(2007)Yeung,
  Pope, Kurth, and Lamorgese]yeung:07
  author author P. K. Yeung, author S. B. Pope,
  author E. A. Kurth,  and  author A. G. Lamorgese,  title title Lagrangian conditional
  statistics, acceleration and local relative motion in numerically simulated
  isotropic turbulence, @noop  journal journal J Fluid Mech volume 582, pages 399–422 (year 2007)NoStop
[Sawford and Yeung(2011)]Sawford:11
  author author B. L. Sawford and author P. K. Yeung, title title Kolmogorov
  similarity scaling for one-particle Lagrangian statistics, @noop
   journal journal Phys. Fluids volume 23, pages 091704 (year
  2011)NoStop
[Billant and Chomaz(2001)]billant_01
  author author P. Billant and author J.-M. Chomaz, title title Self-similarity
  of strongly stratified inviscid flows, @noop  journal journal Phys. Fluids volume
  13, pages 1645–1651 (year 2001)NoStop
[Pouquet et al.(2019b)Pouquet, Rosenberg, and Marino]pouquet_19p
  author author A. Pouquet, author D. Rosenberg,
   and author R. Marino,  title title Linking dissipation,
  anisotropy and intermittency in rotating stratified turbulence at the
  threshold of linear shear instabilities, @noop  journal journal Phys. Fluids volume
  31, pages 105116 (year
  2019b)NoStop
[Lindborg and Brethouwer(2008)]lb08
  author author E. Lindborg and author G. Brethouwer, title title Vertical
  dispersion by stratified turbulence, @noop  journal
  journal J. Fluid Mech. volume 614,  pages 303–314 (year 2008)NoStop
[Cambon et al.(2004)Cambon, Godeferd, Nicolleau, and  Vassilicos]cambon:04
  author author C. Cambon, author F. S. Godeferd, author F. C. G. A.  Nicolleau,  and author J. C. Vassilicos, title title Turbulent diffusion in rapidly rotating flows with and
  without stable stratification, @noop  journal
  journal J. Fluid Mech. volume 499,  pages 231–255 (year 2004)NoStop
[Poje et al.(2017)Poje,
  Özgökmen, Bogucki, and Kirwan, Jr.]poje_17
  author author A. C. Poje, author T. M. Özgökmen, author D. J. Bogucki,  and author A. D. Kirwan, Jr., title title Evidence
  of a forward energy cascade and Kolmogorov self-similarity in submesoscale
  ocean surface drifter observations, @noop  journal
  journal Phys. Fluids volume 29,  pages 020701 (year 2017)NoStop
[Lenschow et al.(2012)Lenschow, Lothon, Mayor, Sullivan, and Canut]lenschow_12
  author author D. H. Lenschow, author M. Lothon,
  author S. D. Mayor, author P. P. Sullivan,  and author G. Canut, title
  title A comparison of higher-order vertical velocity
  moments in the convective boundary layer from Lidar with in situ
  measurements and Large-Eddy Simulation, @noop  journal journal Bound. Lay. Met. volume 143, pages 107–123 (year
  2012)NoStop
[Smyth et al.(2019)Smyth,
  Nash, and Moum]smyth_19
  author author W. D. Smyth, author J. D. Nash,   and author J. N. Moum,  title title Self-organized criticality
  in geophysical turbulence, @noop  journal journal Sci. Rep. volume 9, pages
  3747 (year 2019)NoStop
[Buaria et al.(2015)Buaria,
  Sawford, and Yeung]BSY.2015
  author author D. Buaria, author B. L. Sawford,
   and author P. K. Yeung,  title title Characteristics of backward
  and forward two-particle relative dispersion in turbulence at different
  Reynolds numbers, @noop  journal journal Phys. Fluids volume 27, pages 105101 (year 2015)NoStop
[Xu et al.(2015)Xu,
  Pumir, and Bodenschatz]Xu2015
  author author H. Xu, author A. Pumir,  and  author E. Bodenschatz,  title title Lagrangian view of time
  irreversibility of fluid turbulence, @noop  journal
  journal Science China Physics, Mechanics & Astronomy  volume 59, pages 614702 (year 2015)NoStop
[Buaria et al.(2020)Buaria,
  Clay, Sreenivasan, and Yeung]buaria20
  author author D. Buaria, author M. P. Clay,
  author K. R. Sreenivasan,  and  author P. K. Yeung, title title Turbulence is an ineffective mixer when
  Schmidt numbers are large, @noop  journal
  journal preprint, arXiv 2004.06202  (year
  2020)NoStop



             
