# Hidden Lagrangian coherence and memory effects
in the statistics of Hamiltonian motions

Madalina Vlad, Dragos Iustin Palade, Florin Spineanu<EMAIL_ADDRESS>National Institute of Laser, Plasma and Radiation Physics, Magurele, Romania

##  I
1\. Introduction

Turbulence is a complex nonlinear process, which appears in many domains as
fluid mechanics, plasma physics, astrophysics, atmosphere and ocean sciences,
chemistry Falkovich -Itoh . One of the main difficulty in understanding the
dynamics of turbulence is the complicated combination of stochastic and quasi-
coherent aspects that is typical for the strongly nonlinear regimes.

Quasi-coherence or order appears at the basic level of tracer trajectories in
smooth stochastic velocity fields with finite correlation lengths $\lambda$
and times $\tau_{c}.$ Trajectory coherence (or Lagrangian coherence) is
usually a transitory process that lasts during the time of flight over
$\lambda$ with the amplitude $V$ of the stochastic velocity,
$\tau_{fl}=\lambda/V.$ It exists only for slow time variation of the velocity
with $\tau_{c}>\tau_{fl}$. In the case of two-dimensional incompressible
velocity fields, a much stronger Lagrangian coherence appears that is due to
trajectory eddying or trapping. It generates vortical structures of
trajectories, which produce non-Gaussian statistics of tracer displacements
and strongly modified transport kraichnan -VS17 . The order of the tracer
trajectories determines much more complicated effects on turbulence evolution,
which essentially amplifies the degree of quasi-coherence. The turbulence that
is dominantly two-dimensional has a self-organizing character Montgomery
-Weisse , which consists of the generation of quasi-coherent large scale
structures SV .

This paper is focused on the coherent effects that appear in tracer statistics
in two-dimensional incompressible turbulence in the presence of an average
velocity $\mathbf{V}_{d}$. We show that $\mathbf{V}_{d}$ determines strong
modifications of the transport and trajectory statistics, which are
essentially caused by hidden coherent components of the motion.

The results are based on the numerical simulation of the trajectories and
consist of a conditional statistical analysis adapted to the special
properties of the two-dimensional incompressible velocity fields.

The formulation of the problem and the simulation method are presented in
Section 2. The motion is of Hamiltonian type with the Hamiltonian function
$\phi_{t}$ composed of the stochastic and average potentials. The trajectories
evolve on the contour lines of $\phi_{t}(\mathbf{x})$ in static (frozen)
potentials ($\tau_{c}\rightarrow\infty),$ and they remain strongly correlated
to these lines for slow time variation (large $\tau_{c})$. We consider first
(Sections 3-5) frozen potentials. We discuss the configuration of the contour
lines of the potential and the main features of tracer advection in Section 3.
The space of trajectories is organized in two categories: trapped and free.
The statistics of the Lagrangian velocity and of the trajectories are examined
for each category, and their specific contributions to the global statistics
(on the whole set of trajectories) are identified (Section 4). We show that
quasi-coherent Lagrangian velocities parallel to $\mathbf{V}_{d}$ are
generated for both categories. They are hidden in the global statistics, in
the sense that their contributions compensate each other. However, they
provide explanations for the nonstandard transport produced in these
conditions. A deeper examination of the coherence induced by the average
velocity is presented in Section 5, where we determine the Lagrangian
statistics for the trajectories of each category that evolve on contour lines
of the potential with the same value of $\phi_{t}.$ These results reveal other
(hidden) coherent elements of the motion, and provide important properties
that are used for the understanding of the effects of the average velocity on
the statistics of trajectories and transport. Sections 6 and 7 deal with time
dependent potentials. The Lagrangian statistics conditioned by the value of
the potential shows that the order found in frozen potentials does not decay
due to the random time variation of the potential, as expected. On the
contrary, important quasi-coherent elements significantly increase (Section
6). Explanations are provided in Section 7. They are essentially related to
the constraint of invariance of the potential, which are approximately valid
at large $\tau_{c},$ and to the slow transition of the trajectories between
the two categories. Long memory effects are identified and their effects is
discussed. A short summary of the results and the conclusions are presented in
Section 8.

##  II 2\. The problem and the simulation method

Tracer trajectories in two-dimensional stochastic velocity fields are obtained
from

$\frac{d\mathbf{x}}{dt}=\mathbf{v}(\mathbf{x,}t)=\widetilde{\mathbf{v}}(\mathbf{x,}t)+V_{d}\mathbf{e}_{2},~{}\
\ $ (1)

where $\mathbf{e}_{1},$ $\mathbf{e}_{2}$ are the unit vectors in the plane of
the motion $\mathbf{x=(}x_{1},x_{2}),$ $\mathbf{e}_{3}$ is perpendicular on
this plane. The velocity $\mathbf{v}(\mathbf{x,}t)$ has a stochastic component
$\widetilde{\mathbf{v}}(\mathbf{x,}t)$ superposed on a constant average
velocity $\mathbf{V}_{d}=V_{d}\ \mathbf{e}_{2}.$ The incompressibility
condition $\mathbf{\nabla\cdot\widetilde{\mathbf{v}}}(\mathbf{x,}t)=0$ of the
velocity field is equivalent with the representation of
$\widetilde{\mathbf{v}}(\mathbf{x,}t)$ by a stochastic potential (or stream
function) $\phi(\mathbf{x,}t)$

$\widetilde{\mathbf{v}}(\mathbf{x,}t)=-\mathbf{\nabla}\phi(\mathbf{x,}t)\times\mathbf{e}_{3}.$
(2)

The equation of motion is of Hamiltonian type, with $x_{1}$ and $x_{2}$ the
conjugate variables and $\phi_{t}(\mathbf{x,}t)=\phi(\mathbf{x,}t)+x_{1}V_{d}$
the Hamiltonian function.

Dimensionless quantities are used in Eq. (1) with the potential normalized by
its amplitude $\Phi$, the distances by $\lambda_{0}$ that is of the order of
the correlation lengths, the velocities (including $V_{d})$ by
$V_{0}=\Phi/\lambda_{0}$ and the time by $\tau_{0}=\lambda_{0}/V_{0}.$

The potential is represented by a homogeneous and stationary Gaussian
stochastic field. Its Eulerian correlation (EC)
$E(\mathbf{x,}t)\equiv\left\langle\phi(\mathbf{x}_{0}\mathbf{,}t_{0})~{}\phi(\mathbf{x}_{0}+\mathbf{x,}t_{0}+t)\right\rangle$
in dimensionless quantities is modeled in the simulations presented here by

$E(\mathbf{x,}t)\equiv\exp\left(-\frac{x_{1}^{2}}{2\lambda_{1}^{2}}-\frac{x_{2}^{2}}{2\lambda_{2}^{2}}-\frac{t^{2}}{2\tau_{c}^{2}}\right),$
(3)

where $\lambda_{i}$ are the correlation lengths of the 2-dimensional potential
and $\tau_{c}$ is the correlation time. The EC’s of the velocity components
$E_{ii}(\mathbf{x,}t)\equiv\left\langle
v_{i}(\mathbf{x}_{0}\mathbf{,}t_{0})~{}v_{i}(\mathbf{x}_{0}+\mathbf{x,}t_{0}+t)\right\rangle$
are

$E_{11}(\mathbf{x,}t)=-\partial_{2}\partial_{2}E(\mathbf{x,}t),\
E_{22}(\mathbf{x,}t)=-\partial_{1}\partial_{1}E(\mathbf{x,}t),$ (4)

which determine the normalized amplitudes of the velocity fluctuations
$V_{1}=$ $\sqrt{E_{11}(\mathbf{0,}0)}=1/\lambda_{2},$ $V_{2}=$
$1/\lambda_{1}.$

The statistical properties of the trajectories obtained from Eq. (1) are
numerically analyzed. More precisely, we determine the statistics of the
trajectories and of the Lagrangian velocity, and a class of conditional
Lagrangian correlations that reveal the quasi-coherent components of the
motion and their properties.

We use statistical averages, which consists of generating a large number of
realizations ($r$) of the stochastic Gaussian potential and of determining the
trajectory with the initial condition $\mathbf{x}(0)=0$ in each $r$ by
effectively computing the velocity on the trajectory at each time step,
$\mathbf{v}(\mathbf{x}(t_{i}),t_{i}).$ However, the analysis of the results is
connected to the equivalent space averaging procedure. This corresponds to the
statistical ensemble of trajectories obtained in a single typical realization
of the potential by different initial conditions
$\mathbf{x}(0)=\mathbf{x}_{0}^{r},$ where the points $\mathbf{x}_{0}^{r}$ are
uniformly distributed in a very large domain.

We use the simulation code presented in PV20 , which is based on a fast
generator of Gaussian fields with prescribed spectra. In the present work, we
have implemented the so called FRD representation

$\phi(\mathbf{X})=\sum\limits_{i=1}^{N_{c}}\sqrt{S(\mathbf{K}_{i})}\sin\left(\mathbf{K}_{i}\mathbf{X}+\frac{\pi}{4}\zeta_{i}\right),$
(5)

where $\mathbf{X}\equiv(\mathbf{x,}t)$ is the three-dimensional space-time and
$\mathbf{K}_{i}\equiv(\mathbf{k}_{\perp}^{i}\mathbf{,}\omega^{i})$ are the
$N_{c}$ discrete values of the wave numbers $\mathbf{k}_{\perp}^{i}$ and
frequencies $\omega^{i}.$ $S(\mathbf{K})$ is the spectrum of the stochastic
potential, the Fourier transform of the EC (3). This representation is
different of the usual discrete Fourier decomposition by the set of the values
of $\mathbf{K}_{i}$ that are not the fixed points of a three-dimensional mesh,
but random values with uniform distribution. Also, the random phases
$\zeta_{i}$ have not continuous distributions, but discrete values $\pm 1$
(with equal probabilities).

Each set of the $N_{c}$ random values of $\mathbf{K}_{i}$ and $\zeta_{i}$
determines a realization $r$ of the potential and a trajectory (solution of
Eq. (1) with initial condition $\mathbf{x}(0)=\mathbf{0}$). The statistical
ensemble $R$ consists of a number $M$ of these sets.

The representation (5) provides a fast convergence of the Eulerian properties
of the stochastic potential. We have shown that reasonable errors in the EC
and in the probability of the potential are obtained at much smaller values of
$N_{c}$ and $M$ than in the usual fast Fourier representation (FFR). This
leads to the decrease of the computing times by roughly one order of magnitude
compared to the usual FFR method in two-dimensional potentials PV20 . Most of
the simulations analyzed here are performed with $N_{c}=500$ and $M=10^{5}.$

##  III 3\. Main features of tracer advection

The incompressibility of the two-dimensional velocity field
($\mathbf{\nabla\cdot\mathbf{v}}(\mathbf{x,}t)=0)$ determines two invariance
laws of the solutions of Eq. (1). It leads to equations of motion of
Hamiltonian type, with $x_{1}$ and $x_{2}$ conjugate variables and $\phi_{t}$
the Hamiltonian function. In the case of time independent (frozen) potentials
$\phi\left(\mathbf{x}\right)$, the trajectories are linked to the contour line
of the potential $\phi_{t}(\mathbf{x})$, which means that the Lagrangian
potential $\phi_{t}(\mathbf{x}(t))$ is invariant along each trajectory. The
other invariant law is statistical and applies to the motion in both frozen
and time dependent potentials $\phi_{t}(\mathbf{x}(t),t)$ for any value of the
correlation time $\tau_{c}$. It concerns the distribution of the Lagrangian
velocity $\mathbf{v}(\mathbf{x(}t),t),$ that is shown to be time independent,
and thus identical with the distribution of the Eulerian velocity
$\mathbf{v}(\mathbf{x},t)$. The Lagrangian potential is statistically
invariant too. This property is trivial in frozen potentials where
$\phi_{t}(\mathbf{x}(t))=\phi_{t}(\mathbf{x}(0))$ and it is similar to the
case of the velocity in time dependent potentials where
$\phi_{t}(\mathbf{x}(t),t)$ changes in time.

An example of configuration of the contour lines of the potential can be seen
in Fig. 1, where a typical realization of $\phi_{t}(\mathbf{x})$ is presented
for $V_{d}=0$ (left panel) and for $V_{d}=0.3$ (right panel). The contour
lines at $V_{d}=0$ are nested closed curves with multi-scale sizes that have
the dimensions $r_{\max}$ from $r_{\max}\ll\lambda_{i}$ to
$r_{\max}\rightarrow\infty.$ The average velocity $\mathbf{V}_{d}$ completely
changes the field lines by breaking the large size contour lines and
generating (winding) open paths along its direction. Islands of closed contour
lines remain between the network of open paths, but their average size
decreases as $V_{d}$ increases, and, for $V_{d}$ much larger than the
amplitude of the stochastic velocity, all the lines are open. The average
velocity also determines the limitation of the excursion of the contour lines
perpendicular to $\mathbf{V}_{d}.~{}$

Figure 1: Typical realization of the potential $\phi_{t}(\mathbf{x})$ for
$V_{d}=0$ (left panel) and for $V_{d}=0.3$ (right panel).

This configuration of the contour lines of $\phi_{t}(\mathbf{x})$ determines
solutions of Eq. (1) that are, in the presence of an average velocity, a
mixture of localized periodic (or trapped) trajectories that are closed, and
of free trajectories that have unlimited displacements along $\mathbf{V}_{d}$.
The space of trajectories $R$ is organized in two disjointed subensembles:
$tr$ for the trapped trajectories and $fr$ for the free ones ($R=tr\cup fr,$
$tr\cap fr=\varnothing$). The classification criterion is the periodicity of
the trajectories. A trajectory $r$ with period $T_{r}$ belongs to $tr$ if
$T_{r}$ is finite and to $fr$ otherwise. $T_{r}$ is defined as the time of the
first return in the initial point $\mathbf{x}(0)=\mathbf{0}$, and is
determined as the first solution of $r(t)=0,$ where
$r(t)=\sqrt{x_{1}^{2}(t)+x_{2}^{2}(t)}.$ Practically, a trajectory belongs to
the subensemble $tr$ when its period is smaller than the time of integration
$t_{\max}$. The size of each trajectory $r_{\max}=Max(r(t))$ is also
calculated.

For $V_{d}=0,$ all trajectories $\mathbf{x}(t)$ are closed, periodic functions
of time when $t\rightarrow\infty.$ At finite time $t$, open trajectories are
found, which correspond to large periods $T_{r}>t$ (and to large size contour
lines). As $t$ increases the fraction of free trajectories decreases, and, in
the limit $t\rightarrow\infty,$ all trajectories are trapped ($tr=R$ and
$fr=\varnothing)$. The probability of trajectory sizes $P(r_{\max},t)$ is
represented in Fig. 2 at two time moments, $t=60$ (dashed line) and $t=120$
(solid line). One can see that the time evolution of $P(r_{\max},t)$ affects
only the large distances, while the small $r_{\max}$ domain has invariant
probability. The contributions of the closed and open trajectories to
$P(r_{\max},t)$ are also represented in the figure. The closed trajectories
(red points) determine the invariant part of $P(r_{\max},t).$ The open
trajectories (green points) have large sizes and they determine the time
variation of $P(r_{\max},t).$ Their contribution move toward larger $r_{\max}$
as time increases and it decays, such that $P(r_{\max},t)$ is determined only
by closed trajectories in the limit $t\rightarrow\infty$. It is a decaying
function of $r_{\max}$ that scales as $P(r_{\max},t)\sim r_{\max}^{-1.3}$ at
large $t.$ The slow algebraic decay of the asymptotic probability shows that
the sizes of the trajectories cover multiple scales from
$r_{\max}\ll\lambda_{i}$ to $r_{\max}\rightarrow\infty.$

Thus, the invariance of the Lagrangian potential determines a process of
trajectory trapping manifested by eddying in the structure of
$\phi\left(\mathbf{x}\right).$

The average velocity $V_{d}$ that strongly modifies the structure of the field
lines of $\phi_{t}\left(\mathbf{x}\right)$ determines a significant change of
the probability of trajectory sizes. Two categories of trajectories coexist
for $V_{d}\lesssim V:$ periodic, closed trajectories situated on the islands
of closed contour lines of $\phi_{t}\left(\mathbf{x}\right)$ and non-periodic
trajectories along the open paths generated by the average potential $xV_{d}.$
The latter are free trajectories that make large displacements along
$\mathbf{V}_{d}.$ The probability $P(r_{\max},t)$ can be written as the sum of
the contributions of these two types of trajectories

$P(r_{\max},t)=n_{tr}(r_{\max},t)+n_{fr}(r_{\max},t),$ (6)

where $n_{tr},$ $n_{fr}$ are determined in the subensembles $tr$ and $fr$ at
time $t.$ $P(r_{\max},t),$ shown in Fig. 3 (left panel) has a second maximum.
It appears at a large value of $r_{\max}$ that increases with the increase of
$V_{d}.$ Also, the amplitude and the width of this peak increase with $V_{d}.$
It is determined by the free trajectories. The narrow peak
$n_{tr}(r_{\max},t)$ at small $r_{\max}$ is the contribution of the trapped,
periodic trajectories. It is represented in the right panel of Fig. 3, which
shows that both the maximum size and the amplitude of the trapped trajectories
decrease as $V_{d}$ increases. The average velocity hinders and eventually
eliminates the trapping process. The contribution $n_{tr}(r_{\max},t)$ in Eq.
(6) decreases with $V_{d}$ and become negligible at $V_{d}\gg 1.$ The
contribution of the free trajectories is negligible in this range of small
sizes, at any $V_{d}$, as shown in Fig. 3 (right panel) where the black points
for $P(r_{\max},t)$ are superposed on the red curves for $n_{tr}(r_{\max},t).$
The two contributions in Eq. (6) separates at large time.

The probability of the periods of the closed trajectories $P(T,t)$ calculated
from the trajectories $\mathbf{x}(t)$ at $t=60$ is shown in Fig. 4. One can
see that, at small $V_{d},$ this probability extends to large values of
$T\lesssim 100$ and it has a weak decay. As $V_{d}$ increases, the width of
$P(T,t)$ decreases and its decay is steeper. This behavior is in agreement
with the decay of the trajectory sizes at large $V_{d}.$ An average velocity
can be defined for the trapped trajectories as the maximum displacement over
the period, $v^{eff}=r_{\max}/T.$ Its probability is weakly dependent on the
average velocity.

Figure 2: The probability of trajectory sizes $P(r_{max},t)$ for $V_{d}=0$ at
$t=60$ (dashed black line) and $t=120$ (solid black line). Also shown are the
contributions of the trapped (red points) and free (green points) trajectories
at $t=120$.

Figure 3: The probability $P(r_{max},t)$ for several average velocities
$V_{d}$ that label the curves, at $t=60$ as function of $r_{max}$ (the curves
in the left panel and black points in the right panel) and the contribution of
the trapped trajectories $n_{tr}(r_{max},t)$ (right panel, red lines).

Figure 4: The probability of the periods of the trapped trajectories $P(T,t)$
as functions of $T$ at $t=60$ and at the values of $V_{d}$ that label the
curves.

Figure 5: Left panel: the fractions of free trajectories as function of time
for the values of $V_{d}$ that label the curves. Right panel: the asymptotic
values $n_{fr}$ and $n_{tr}$ and the average velocity of the free trajectories
(see next Section) as functions of $V_{d}$.

Figure 6: The average Lagrangian velocities (left panel) and the fluctuations
of the Lagrangian velocities (right panel) as functions of time. The dashed
lines are for the $v_{1}$ and the solid lines for the $v_{2}$. The green lines
are for the free trajectories and the red lines for the trapped trajectories,
while the black are averages on the whole statistical ensemble $R$.
$V_{d}=0.2$.

Figure 7: The amplitudes of fluctuations of the Lagrangian velocities as
functions of $V_{d}$. The dashed lines are for the $v_{1}$ and the solid lines
for the $v_{2}$. The green lines are for the free trajectories and the red
lines for the trapped trajectories, while the black are averages on the whole
statistical ensemble $R$.

Figure 8: The correlations of the Lagrangian velocity $v_{1}(t)$ (left panel)
and $v_{2}(t)$ (right panel). The correlations on the whole statistical
ensemble $L_{i}(t)$ (black lines) are compared to the subensemble correlations
$L_{i}^{tr}(t)$ (red lines) and $L_{i}^{fr}(t)$ (green lines). $V_{d}=0.2.$

Figure 9: Histograms of the Lagrangian velocities $v_{1}$ (left panel) and
$v_{2}$ (right panel) represented by the black curves and the contributions
determined by the free (green curves) and trapped (red curves) trajectories.
$V_{d}=0.4$.

The fraction of trajectories that are not closed at time $t,$
$n_{fr}(t,V_{d})$ is obtained from the probability of the periods of the
closed trajectories (calculated at the time of integration, $t_{\max})$

$n_{fr}(t,V_{d})=1-n_{tr}(t,V_{d}),\ \ \
n_{tr}(t,V_{d})=\int_{0}^{t}P(T,t_{\max})~{}dT.$ (7)

This function decreases in time from $n_{fr}(0,V_{d})=1,$ as seen in Fig. 5
(left panel). In the case $V_{d}\neq 0,$ $n_{fr}(t,V_{d})$ saturates at a
value $n_{fr}(V_{d})$ in a time that becomes shorter at larger $V_{d}$. In the
case of $V_{d}=0,$ the decay is not limited, and it scales as $n_{fr}(t,0)\sim
t^{-0.6}.$

The results obtained for the asymptotic fraction of free trajectories
$n_{fr}(V_{d})\equiv\underset{t\rightarrow\infty}{\lim}n_{fr}(t,V_{d}),$
presented in Fig. 5 (right panel), are well approximated by

$n_{fr}(V_{d})=\left[1-\exp\left(-V_{d}^{2}\right)\right]^{1/4}.$ (8)

The fraction of trapped trajectories is $n_{tr}(t,V_{d})=1-n_{fr}(t,V_{d})$ at
any time, with the asymptotic value $n_{tr}(V_{d})=1-n_{fr}(V_{d})$ that is
also represented in Fig. 5 (right panel).

##  IV 4\. Lagrangian statistics in static potentials

Thus, the trajectories obtained in the stochastic potential
$\phi_{t}(\mathbf{x})$ were divided into two categories: trapped and free.
They have different topologies and different sizes, which suggests that their
contributions to the global statistical properties of the trajectories are
qualitatively different.

We analyze here the statistics of the Lagrangian velocity and of the
displacements of each category of trajectories. For any Lagrangian quantity
$A(\mathbf{x}(t)),$ we determine $\left\langle
A(\mathbf{x}(t))\right\rangle_{tr}$ and $\left\langle
A(\mathbf{x}(t))\right\rangle_{fr}$ that are conditional averages restricted
to the trapped and free trajectories, respectively. These are statistical
averages calculated over the subspaces $tr$ and $fr.$ The contribution of each
subensemble to the global average (over $R$) is the product of the probability
that a trajectory belongs to the subensemble multiplied by the statistical
average over the subensemble, $n_{c}(t,V_{d})\left\langle
A(\mathbf{x}(t))\right\rangle_{c},$ where $c=tr,~{}fr.$ It yields

$\left\langle A(\mathbf{x}(t))\right\rangle=n_{tr}(t,V_{d})\ \left\langle
A(\mathbf{x}(t))\right\rangle_{tr}+n_{fr}(t,V_{d})\ \left\langle
A(\mathbf{x}(t))\right\rangle_{fr}.$ (9)

The separation of the trajectories in these categories is performed at a large
time such that $n_{fr}(t,V_{d})$ is saturated (see Fig. 5, left panel).

###  IV.1 4.1 Statistics of the Lagrangian velocity

The statistical parameters of the Lagrangian velocity
$\mathbf{v}\left(\mathbf{x}(t)\right)\equiv\mathbf{v}(t)$ are shown in Fig. 6
for a stochastic potential with $\lambda_{1}=1,$ $\lambda_{2}=2$ and
$V_{d}=0.2.$ The average Eulerian velocity and fluctuation amplitudes are in
this case $\left\langle v_{1}\right\rangle=0,~{}\left\langle
v_{2}\right\rangle=V_{d},$ $V_{1}=0.5$ and $V_{2}=1,$ where
$V_{i}=\sqrt{\left\langle\widetilde{v}_{i}^{2}\right\rangle}$ are obtained
from Eq. (4).

The Lagrangian quantities maintain the Eulerian values at any time, as stated
by Lumley theorem. Besides this, the conditional average velocity and
fluctuation amplitudes are time invariant, as seen in Fig. 6, but their values
depend on the category.

It is interesting to note that the average velocity is determined only by the
free trajectories, while the trapped trajectories do not contribute
($\left\langle v_{2}(t)\right\rangle_{tr}=0$ at any time). The average
velocity of the free trajectories is larger than $V_{d},$ and it can be
approximated with

$\left\langle v_{2}(t)\right\rangle_{fr}=\frac{V_{d}}{n_{fr}}>V_{d}$ (10)

It is $\left\langle v_{2}(t)\right\rangle_{fr}=0.45$ for the example presented
in Fig. 6, left panel, obtained for $V_{d}=0.2$. The conditional average
velocity $\left\langle v_{2}(t)\right\rangle_{fr}$ is also shown in Fig. 5
(right panel) as function of $V_{d}.$ One can see that this average velocity
is significantly larger than $V_{d}$ only in the presence of trajectory
trapping (for $V_{d}\lesssim 1).$

This result shows that a supplementary ordered component of the Lagrangian
velocity appears for the free trajectories that exactly compensates the
missing contribution of the trapped particles, such that $\left\langle
v_{2}(t)\right\rangle=n_{fr}\left\langle v_{2}(t)\right\rangle_{fr}=V_{d}.$ It
seems to be a trivial consequence of $\left\langle
v_{2}(t)\right\rangle_{tr}=0$, but the underlying physical process is rather
complex. It essentially consists of generation of ordered motion from the
stochastic velocity $\widetilde{\mathbf{v}}(\mathbf{x,}t)$ for both types of
trajectories

$\left\langle\widetilde{v}_{2}(t)\right\rangle_{tr}=-V_{d},\ \
\left\langle\widetilde{v}_{2}(t)\right\rangle_{fr}=V_{d}\frac{n_{tr}}{n_{fr}}.$
(11)

The supplementary average velocity of the trapped trajectories is opposite to
$\mathbf{V}_{d}$ and exactly compensates it. The supplementary average
velocity of the free trajectories is along $\mathbf{V}_{d}$ and it contributes
to the increase of the Lagrangian over the Eulerian velocity.

Equations (11) are valid at any time, including $t=0.$ It can be interpreted
as the condition for the separation of the trajectories in the free and
trapped categories. The trapped trajectories start from the geometric locus
for which $\left\langle\widetilde{v}_{2}(\mathbf{x})\right\rangle=-V_{d}$ and
they remain in this domain, while the free trajectories are confined in the
complement of this domain. These ordered components of the motion are hidden,
in the sense that they are not ”seen” in the average velocity calculated on
the whole ensemble $R$ ($\left\langle v_{2}(t)\right\rangle=V_{d}).$ However,
as shown below, they have strong effects on the transport along
$\boldsymbol{V}_{d}$ through the modification of the correlation of the
Lagrangian velocity.

The amplitudes of velocity fluctuations around the average velocity are shown
in Fig. 6 (right panel). They are different for the two types of trajectories.
It is interesting to underline that the supplementary order that characterizes
trapped and free trajectories appears in the fluctuations of the velocity in
$R$. The average of the square velocity decomposed on $tr$ and $fr$
subensembles according to Eq. (9) for large time

$\left\langle v_{i}^{2}(t)\right\rangle=n_{tr}\left\langle
v_{i}^{2}(t)\right\rangle_{tr}+n_{fr}\left\langle
v_{i}^{2}(t)\right\rangle_{fr},$ (12)

leads to

$V_{2}^{2}=n_{tr}(V_{2}^{tr})^{2}+n_{fr}(V_{2}^{fr})^{2}+\frac{n_{tr}}{n_{fr}}V_{d}^{2},$
(13)

where

$(V_{i}^{c})^{2}\equiv\left\langle\left(v_{i}(t)-\left\langle
v_{i}(t)\right\rangle_{c}\right)^{2}\right\rangle_{c}$ (14)

are the amplitudes of the fluctuations of the velocity $\delta v_{i}(t)\equiv
v_{i}(t)-\left\langle v_{i}(t)\right\rangle_{c},$ $i=1,2,$ conditioned by the
category of trajectories $c=tr,\ fr$ (on the subensembles $tr$ and $fr).$
Thus, a contribution produced by the ordered motion appears (the last term of
Eq. (13)) besides the direct contributions of the conditional fluctuations. It
is determined by the ordered motion (11) generated by $V_{d}$ in the presence
of trapping (for $V_{d}\lesssim 1).$ The results presented in Fig. 6 (right
panel) show values $V_{2}^{tr}<V_{2}$ and $V_{2}^{fr}<V_{2},$ which reproduce
Eq. (13).

The conditioned amplitudes of the velocity fluctuations $V_{i}^{c}$ depend on
the average velocity $V_{d}.$ As seen in Fig. 7, the amplitudes of both
components of the trapped trajectory velocity (red curves) are continuously
decreasing functions of $V_{d}$. This is the effect of the decrease of the
size of the islands of closed contour lines of the potential, which, as
$V_{d}$ increases, shrink around the maxima and minima of $\phi(\mathbf{x})$
where the gradients are small. In the case of free trajectories (green lines),
the amplitudes of the Lagrangian velocities are different of $V_{i}$ only in
the range of $V_{d}$ that corresponds to the existence of islands of closed
contour lines of the potential. The perpendicular amplitude is increased
($V_{1}^{fr}>V_{1}),$ while the parallel amplitude is decreased
($V_{2}^{fr}<V_{2})$ such that the supplementary parallel velocity is
compensated (Eq. (13)).

One can deduce from these results that the EC defined on the geometric locus
of the free trajectories is different of the EC (3). As shown below (Section
6), the amplitude of the stochastic potential of the free trajectories
$\Delta$ is smaller than in the whole space ($\Delta<\Phi$). The correlation
lengths are evaluated using the amplitudes of velocity fluctuations of the
free trajectories

$\lambda_{1}^{fr}\sim\frac{\Delta}{V_{2}^{fr}}=\lambda_{1}\frac{\Delta}{\Phi}\frac{V_{2}}{V_{2}^{fr}},\
\lambda_{2}^{fr}\sim\frac{\Delta}{V_{1}^{fr}}=\lambda_{2}\frac{\Delta}{\Phi}\frac{V_{1}}{V_{1}^{fr}},$
(15)

where $\lambda_{1}=\Phi/V_{2}$ and $\lambda_{2}=\Phi/V_{1}.$ Thus, the
correlation lengths on the domain of free trajectories decrease with the
factor $\Delta/\Phi$ on both directions and are modified by the velocity
amplitudes (decreased along $\mathbf{V}_{d}$ and increased across
$\mathbf{V}_{d}$).

The correlations of the Lagrangian velocity are shown in Fig. 8, where the
notations are

$L_{i}(t)=\left\langle\delta v_{i}(0)\ \delta v_{i}(t)\right\rangle,\
L_{i}^{c}(t)=\left\langle\delta v_{i}(0)\ \delta v_{i}(t)\right\rangle_{c},\ \
c=tr,fr.\ $ (16)

One can see that all the conditional correlations (for both categories and
both components of the velocity) decay to zero at large $t.$ However, the
correlation of the velocity along $\mathbf{V}_{d}$ calculated on all
trajectories, $L_{2}(t),$ has a finite asymptotic value. It is determined by
the ordered components of motion produced in subensembles $tr$ and $fr.$ An
equation similar to (13) can be obtained from (9) written for $A=v_{i}(0)\
v_{i}(t)$

$L_{2}(t)=n_{tr}L_{2}^{tr}(t)+n_{fr}L_{2}^{fr}(t)+\frac{n_{tr}}{n_{fr}}V_{d}^{2},$
(17)

which shows that $L_{2}(t)$ has a finite asymptotic tail in spite of the decay
to zero of $L_{2}^{tr}(t)$ and $L_{2}^{fr}(t).$ It is determined by the
presence of trapped trajectories at small average velocity $V_{d}.$

The histograms for the Lagrangian velocity components are time invariant for
all statistical ensembles $R,$ $tr$ and $fr.$ The histogram for all
trajectories (in $R)$ is shown in Fig. 9 together with the contributions of
the trapped and free trajectories (that include the fractions of
trajectories). One can see that the distribution is Gaussian in $R$, while
significant departures from Gaussianity appear in the subensembles $tr$ and
$fr,$ especially for the velocity component $v_{2}$ (right panel). The domain
of large positive velocities is dominated by the free trajectories, while the
trapped trajectories have the main contribution for the large negative
$v_{2}.$ The most probable value of $v_{2}$ on $tr$ (that is slightly
negative) is compensated by a longer tail at positive $v_{2}.$

The non-Gaussian distribution of the Lagrangian velocity of the trapped
trajectories provides additional information on the geometric locus of this
category of trajectories. It shows that the space average of the parallel
velocity $v_{2}(\mathbf{x})$ on this locus (that is zero for any $V_{d})$
results from the elimination of the regions with large, positive
$v_{2}(\mathbf{x}).$ In other words, the regions where the stochastic velocity
is oriented parallel to $\mathbf{V}_{d}$ belong to the geometric locus $fr.$

###  IV.2 4.2 Transport and statistics of trajectories

The statistics of the displacements is strongly non-Gaussian, in spite of the
Gaussian Lagrangian velocity. Moreover, the average and mean square
displacements (calculated for all trajectories and in the subensembles $tr$
and $fr)$ have asymptotic regimes that can be linear, quadratic or saturated
functions of time, which shows that the transport has anomalous aspects.

The average displacements are in agreement with the average Lagrangian
velocities

$\displaystyle\left\langle x_{1}(t)\right\rangle$ $\displaystyle=$
$\displaystyle\left\langle x_{1}(t)\right\rangle_{tr}=\left\langle
x_{1}(t)\right\rangle_{fr}=0,$ (18) $\displaystyle\left\langle
x_{2}(t)\right\rangle$ $\displaystyle=$ $\displaystyle V_{d}t,\ \left\langle
x_{2}(t)\right\rangle_{tr}=0,\ \left\langle
x_{2}(t)\right\rangle_{fr}=\frac{V_{d}}{n_{fr}}t.$

The dispersion $\left\langle\left(\delta x_{i}(t)\right)^{2}\right\rangle,$
where $\delta x_{i}(t)=x_{i}(t)-\left\langle x_{i}(t)\right\rangle$ are shown
in Fig. 10 for $V_{d}=0$ (left panel) and for $V_{d}=0.2$ (right panel), as
functions of time.

In the absence of the average velocity ($V_{d}=0),$ the dispersions are
similar along the two directions. The curves in the left panel of Fig. 10 are
only translated due to the different amplitudes of the stochastic velocities
$V_{1}=0.5,~{}V_{2}=1.$ The dispersions are sub-diffusive, with time increase
that is slower than linear $\left\langle\left(\delta
x_{i}(t)\right)^{2}\right\rangle\sim t^{0.68}.$ The reason is the progressive
saturation of the contributions of the trajectories with small periods. At a
time $t,$ all the trajectories with $T<t$ have saturated dispersion and only
the free trajectories (that are still not closed) determine the time variation
of $\left\langle\left(\delta x_{i}(t)\right)^{2}\right\rangle.$ The latter
results from two factors with opposite effects: the fraction of free
trajectories at time $t$ and their average size. As seen in Fig. 5, left
panel, $n_{fr}(t,0)$ is a decreasing function of time, $n_{fr}(t,0)\sim
t^{-0.6}.$ The average size of the closed trajectories is an increasing
function of $t,$ because it is an increasing function of the average period.

The average velocity $V_{d}$ makes trajectory dispersion strongly non-
isotropic, as seen in Fig. 10, right panel. The dispersion
$\left\langle\left(\delta x_{i}(t)\right)^{2}\right\rangle$ for all
trajectories (black lines) are compared to the results obtained for the
trapped $\left\langle\left(\delta x_{i}(t)\right)^{2}\right\rangle_{tr}$ (red
lines) and free $\left\langle\left(\delta
x_{i}(t)\right)^{2}\right\rangle_{fr}$ (green lines) trajectories. The
dispersions across $\mathbf{V}_{d}$ (of $x_{1}(t)$) for the whole set of
trajectories and for the subensembles $tr$ and $fr$ are all saturated (the
dashed curves in Fig. 10, right panel), which corresponds to the minimum sub-
diffusive transport. This means that the average velocity completely hinders
the perpendicular transport in the case of static stochastic potentials. The
contrary happens to the transport parallel to $\mathbf{V}_{d}$: the dispersion
of the trajectories has a very fast time-increase, $\left\langle\left(\delta
x_{2}(t)\right)^{2}\right\rangle\sim t^{2},$ which correspond to the maximum
super-diffusive transport that is of the ballistic type. It appears in spite
of the much weaker transport of the trapped and free trajectories
($\left\langle\left(\delta x_{2}(t)\right)^{2}\right\rangle_{tr}$ saturates
and $\left\langle\left(\delta x_{2}(t)\right)^{2}\right\rangle_{fr}\sim t$ is
diffusive).

This super-diffusive parallel transport is the effect of the coherent parallel
motion generated by $V_{d}$, as demonstrated using Eq. (9) for
$A=x_{i}^{2}(t).$ The relations between the dispersion of all trajectories (in
$R$) and the subensemble $tr$ and $fr$ dispersions are

$\left\langle\left(\delta
x_{1}(t)\right)^{2}\right\rangle=n_{tr}\left\langle\left(\delta
x_{1}(t)\right)^{2}\right\rangle_{tr}+n_{fr}\left\langle\left(\delta
x_{1}(t)\right)^{2}\right\rangle_{fr},$ (19) $\left\langle\left(\delta
x_{2}(t)\right)^{2}\right\rangle=n_{tr}\left\langle\left(\delta
x_{2}(t)\right)^{2}\right\rangle_{tr}+n_{fr}\left\langle\left(\delta
x_{2}(t)\right)^{2}\right\rangle_{fr}+\frac{n_{tr}}{n_{fr}}V_{d}^{2}\ t^{2}.$
(20)

The last term in Eq. (20) is dominant at large time and it makes the
asymptotic regime superdiffusive of ballistic type. This term is determined by
the supplementary average velocity generated from the stochastic components
for the free and trapped trajectories, Eq. (11). It leads to the
”concentration” of the average velocity along the free trajectories Eq. (10).
Thus, the super-diffusive parallel transport is determined by the average
velocity ($V_{d}\neq 0$) only in the presence of the islands of trapped
trajectories ($n_{tr}\neq 0$), which corresponds to $V_{d}\lesssim 1.$

Figure 10:  The dispersions of the trajectories on the whole statistical
ensemble (black line) for $V_{d}=0$ (left panel) and $V_{d}=0.2$ (right panel)
as functions of time. The dashed lines are for $x_{1}(t)$ and the solid lines
for $x_{2}(t)$. The conditional dispersions for trapped (red) and free (green)
trajectories are also shown in the right panel.

The dispersions of the trajectories (Fig. 10) are connected to the
correlations of the Lagrangian velocity (Fig. 8) and to the time dependent
diffusion coefficients, defined by $2D_{i}(t)=d\left\langle\left(\delta
x_{i}(t)\right)^{2}\right\rangle/dt,$ by Taylor formulas Taylor

$D_{i}(t)=\int_{0}^{t}L_{i}(\tau)~{}d\tau,\ \ \left\langle\left(\delta
x_{i}(t)\right)^{2}\right\rangle=2\int_{0}^{t}\left(t-\tau\right)~{}L_{i}(\tau)~{}d\tau.$
(21)

Similar equations can be written for each category of trajectories (trapped or
free). Figure 11 presents the time dependent diffusion coefficients compared
to their restrictions to the trapped and free trajectories for the $x_{1}$
(left panel) and $x_{2}$ (right panel) directions. This confirms that the
perpendicular diffusion is completely hindered (even for the free
trajectories). The time integral of $L_{1}(t)$ vanishes for all categories at
a finite time. The parallel transport is ballistic $D_{2}(t)\sim t,$ in spite
of the normal diffusion of the free trajectories and of the total confinement
of the trapped ones. It is the result of the ordered parallel motion, as seen
by performing the time derivative in Eq. (20).

Figure 11: The time dependent diffusion coeficients in the direction
perpendicular (left panel) and parallel (right panel) to the average velocity
for the whole statistical ensemble $R$ (black lines) and restricted to the
$tr$ (red lines) and $fr$ (green lines) subensembles. $V_{d}=0.2.$

Figure 12: The probabilities of the $x_{1}$ (left panel) and $x_{2}$ (right
panel) for the whole set of realizations (black lines) compared to the
contributions of the free (green lines) and trapped (red lines) trajectories
for $V_{d}=0.2$ and $t=97.$

The probability of the displacements $P(\mathbf{x},t)$ is strongly non-
Gaussian, as seen in in Fig. 12 (black curves). The contributions of the two
categories of trajectories are completely different: the trapped trajectories
determine the steep peak in $\mathbf{x}=\mathbf{0}$, and the free ones have a
large Gaussian distribution with the average parallel displacement
$\left\langle x_{2}(t)\right\rangle_{fr}=V_{d}t/n_{fr}.$ We note that the
transport, which is essentially produced by the free trajectories, results
from a Gaussian distribution.

##  V 5\. Coherence induced by an average velocity

The Hamiltonian structure of equation (1) is the origin of the order that
characterizes the two-dimensional incompressible turbulence. It determines the
strong connection between trajectories and the contour lines of the potential,
which are paths of the motion. The order (quasi-coherence) of the motion is
essentially represented by the existence of correlations between the potential
and the trajectories. They are represented by nonzero average displacements or
velocities conditioned by the (initial) potential.

Significant quasi-coherent characteristics of the transport process can be
found by analyzing statistical Lagrangian quantities restricted on the contour
lines with given potential $\phi^{0}.$ The trajectories that belong to this
class correspond to solutions of Eq. (1) that start (in
$\mathbf{x}(0)=\mathbf{0}$) from points with $\phi(\mathbf{0})=\phi^{0}.$ The
invariance of the total potential in this class gives

$\phi_{t}(\mathbf{x}(t))=\phi(\mathbf{x}(t))+x_{1}(t)V_{d}=\phi^{0}.$ (22)

The fractions of trajectories that evolve on the $\phi^{0}$ potential lines,
the average and the amplitude of fluctuations of their displacements and
Lagrangian velocities are determined below for each type of trajectories using
conditional averages.

The analysis starts from the representation (9) and introduces a supplementary
condition for the trajectories, namely that the initial potential is
$\phi^{0}$ $[\phi(\mathbf{x}(0))=\phi^{0}].$ Defining the fraction of these
trajectories by $n(\phi^{0})$ and the corresponding conditional average by
$\left\langle{}\right\rangle_{\phi^{0}},$ the average $\left\langle
A(\mathbf{x}(t))\right\rangle$ is the sum of the contributions from each value
$\phi^{0}$

$\left\langle
A(\mathbf{x}(t))\right\rangle=\int_{-\infty}^{\infty}\left\langle
A(\mathbf{x}(t))\right\rangle_{\phi^{0}}\ P(\phi^{0})~{}d\phi^{0},$ (23)

where $P\left(\phi^{0}\right)$ is the Gaussian distribution of the
(normalized) potential

$P\left(\phi^{0}\right)=\frac{1}{\sqrt{2\pi}}\exp\left(-\frac{\left(\phi^{0}\right)^{2}}{2}\right).$
(24)

Similar equations can be written for the contributions of the free and trapped
trajectories

$n_{c}~{}\left\langle
A(\mathbf{x}(t))\right\rangle_{c}=\int_{-\infty}^{\infty}\left\langle
A(\mathbf{x}(t))\right\rangle_{\phi^{0},c}~{}n^{c}(\phi^{0})~{}d\phi^{0},$
(25)

where $n^{c}(\phi^{0})$ is the fraction of trajectories that evolve on contour
lines $\phi^{0}$ and are in the category $c=tr,\ fr,$ and
$\left\langle{}\right\rangle_{\phi^{0},c}$ is the conditional average taken on
the subensemble of these trajectories. $n^{c}(\phi^{0})$ is related to
$n_{tr},$ $n_{fr}$ (defined in Section 3)

$n_{c}=\int_{-\infty}^{\infty}n^{c}(\phi^{0})\ d\phi^{0}.$ (26)

One obtains using Eq. (9)

$\left\langle A(\mathbf{x}(t))\right\rangle_{\phi^{0}}P(\phi^{0})=\left\langle
A(\mathbf{x}(t))\right\rangle_{\phi^{0},tr}\ n^{tr}(\phi^{0})+~{}\left\langle
A(\mathbf{x}(t))\right\rangle_{\phi^{0},fr}\ n^{fr}(\phi^{0}),$ (27)

which connects the contribution of all trajectories $\left\langle
A\right\rangle_{\phi^{0}}P(\phi^{0})~{}$ to the contributions of each category
$\left\langle A\right\rangle_{\phi^{0},c}n^{c}(\phi^{0}).$

Figure 13: The fraction of the trajectories that evolve on the contour lines
with potential $\phi^{0}$ for the free (green points), trapped (red points)
and for all trajectories (black points) obtained from the numerical
simulations compared to $P(\phi^{0})$ (solid black line), with Eq. (29) (solid
green line) and Eq. (31) (solid red line).

.

Figure 14: The width $\Delta$ of the initial potential of the free
trajectories (diamonds) and the fraction of free trajectories (circles) as
functions of the average velocity $V_{d}$. The numerical results are
interpolated by Eqs. (32) and (8)

The fractions of trajectories fulfil the equation

$P(\phi^{0})=n^{tr}(\phi^{0})+n^{fr}(\phi^{0}),$ (28)

which is obtained from Eq. (27) for $A=1.$ The numerical results obtained for
$n^{fr}(\phi^{0})$ and $n^{tr}(\phi^{0})$ (represented by points), are
compared to analytical approximations (solid lines) in Fig. 13. One can see
that the fraction of trajectories that evolve on $\phi^{0}$ contour lines
(black line and points in Fig. 13) reproduces Eq. (24). The fraction of the
free trajectories is narrower, but it is still Gaussian. We have found, as
seen in Fig. 13 (green curve), a good approximation of the data by

$n^{fr}(\phi^{0})=n_{fr}~{}G(\phi^{0};\Delta),$ (29)

where $G(\phi^{0};\Delta)$ is the Gaussian distribution

$G(\phi^{0};\Delta)=\frac{1}{\sqrt{2\pi}\Delta}\exp\left(-\frac{\left(\phi^{0}\right)^{2}}{2\Delta^{2}}\right)$
(30)

with a width $\Delta$ that depends on the average velocity $V_{d}.$ The
fraction of the trapped trajectories (red curve in Fig. 13), which according
to Eq. (28) is

$n^{tr}(\phi^{0})=P(\phi^{0})-n_{fr}~{}G(\phi^{0};\Delta),$ (31)

provides a good representation of the numerical results (red points).

The width $\Delta$ as function of the average velocity $V_{d}$ is shown in
Fig. 14 together with the fraction of free trajectories $n_{fr}.$ The
numerical results for $\Delta(V_{d})$ (diamonds) are well approximated by

$\Delta(V_{d})=\left[1-\exp\left(-V_{d}^{2}\right)\right]^{0.17}.$ (32)

Both functions saturate at large $V_{d}$ ($V_{d}>V_{1},V_{2}),$ and they have
power law dependence for small $V_{d}$

$n_{fr}\sim V_{d}^{0.5},\ \Delta\sim V_{d}^{0.34}.$ (33)

The asymptotic value $n_{fr}\rightarrow 1$ for $V_{d}\rightarrow\infty$
corresponds to the complete elimination of the islands of close contour lines
of the potential ($n_{tr}=0)$. The limit $\Delta\rightarrow 1$ confirms that
all trajectories are free, because $G(\phi^{0};1)=P(\phi^{0}),$ where
$P(\phi^{0})$ is the probability of the Eulerian potential.

Thus, the free trajectories are localized on the contour lines with small
values of $\left|\phi^{0}\right|\lesssim\Delta.$ The potential on the
geometrical locus of free trajectories is Gaussian with an amplitude that is
smaller than in the whole space.

The trapped (periodic) trajectories mainly have large $\left|\phi^{0}\right|$
: they completely occupy the range of large potential
$\left|\phi^{0}\right|\gg\Delta,$ but also have significant presence at small
potential $\left|\phi^{0}\right|\lesssim\Delta$ that correspond to free
trajectories.

The average displacements conditioned by the value of the initial potential
and by the category of the trajectories are shown in Fig. 15 for free (green),
trapped (red) and all (black) trajectories. The perpendicular (left panel) and
the parallel (right panel) displacements are shown at a large time $t=97,$
larger than the saturation time of $n_{fr}(t,V_{d})$ (seen in Fig. 5, left
panel). These represent quasi-coherent components of the motion, and appear
only in the presence of an average velocity. One can see that the average
conditional displacements are small for the trapped trajectories (red points),
and that significant values appear for the free trajectories in both
directions (green points). As shown in Fig. 15, these quantities can be
approximated by

$\left\langle x_{1}(t)\right\rangle_{\phi^{0},fr}=\frac{\phi^{0}}{V_{d}},\ \
\left\langle x_{1}(t)\right\rangle_{\phi^{0},tr}\cong 0,\ \ $ (34)
$\left\langle x_{2}(t)\right\rangle_{\phi^{0},fr}=\frac{V_{d}t}{n_{fr}},\ \
\left\langle x_{2}(t)\right\rangle_{\phi^{0},tr}=0,$ (35)

which are represented by red and green lines respectively. The black lines
have the equations

$\left\langle
x_{1}(t)\right\rangle_{\phi^{0}}=\frac{\phi^{0}}{V_{d}}F(\phi^{0}),\ \
\left\langle
x_{2}(t)\right\rangle_{\phi^{0}}=\frac{V_{d}t}{n_{fr}}F(\phi^{0}),$ (36)

where

$F(\phi^{0})=\frac{n^{fr}(\phi^{0})}{P(\phi^{0})}=\frac{n_{fr}}{\Delta}\exp\left(-\frac{\left(\phi^{0}\right)^{2}}{2}\frac{1-\Delta^{2}}{\Delta^{2}}\right).$
(37)

They result from Eq. (27) with $A=x_{i}(t)$ using (34) and (35), and provide,
as seen in Fig. 15, good approximations of the data for $\left\langle
x_{i}(t)\right\rangle_{\phi^{0}}$ (black points).

Figure 15: The conditional average displacements along $x_{1}$ axis (left
panel) and along $x_{2}$ axis (right panel) as functions of $\phi^{0}$ for the
trapped (red points), free (green points) and all (black points) trajectories,
compared to the approximations (34)-(35) (green lines) and (36) (black lines).
$V_{d}=0.3.$

Figure 16:  The contributions of the trapped (red points) and free (green
points) trajectories, to the average displacements (black points) along
$x_{1}$ (left panel) and $x_{2}$ (right panel) directions as functions of
$\phi^{0}$ for the values of $V_{d}$ label the curves. The approximations for
the free trajectories are represented by the green lines.

The contributions to the average displacements, obtained by multiplying the
conditional averages with the corresponding fractions of trajectories
($P(\phi^{0}),$ $n^{fr}(\phi^{0})$ or $n^{tr}(\phi^{0})$), are shown in Fig.
16. It appears more clearly that the coherent displacements are produced only
by the free trajectories. The black points for all trajectories are
practically superposed on the green points and they are well approximated by
the green lines that represent the contributions of the free trajectories
($\phi^{0}/V_{d}\ n^{fr}(\phi^{0})$ in the left panel and $V_{d}~{}t/n_{fr}\
n^{fr}(\phi^{0})$ in the right panel). The dependence on $\phi^{0}$ is
different across and along the average velocity $\mathbf{V}_{d}.$ In the first
case, it is an anti-symmetrical function of $\phi^{0}$ that saturates in time,
and, in the second case, it is a symmetrical Gaussian function that increases
linearly in time.

The parallel average displacement on the contour lines with initial potential
$\phi^{0},$ which increases linearly with $t$ (35), leads to an average
Lagrangian velocity

$\left\langle v_{2}(t)\right\rangle_{\phi^{0},fr}=\frac{V_{d}}{n_{fr}}.$ (38)

It is important to note that this velocity does not depend on $\phi^{0},$ and
it equals the average velocity (10). The contribution of the conditional
average velocity is determined only by the free trajectories since
$\left\langle v_{2}(t)\right\rangle_{\phi^{0},tr}=0.$ The perpendicular
average displacement of the free trajectories also determines an average
velocity, but it is transitory since $\left\langle
x_{1}(t)\right\rangle_{\phi^{0},fr}$ saturates in time.

The dispersion of the trajectories conditioned by the value of the initial
potential and by the category are shown in Fig. 17 for free (green), trapped
(red) and all (black) trajectories, in the perpendicular (left panel) and
parallel (right panel) directions. One can see that the dispersion of the free
trajectories along both directions are not dependent on the initial potential
$\phi^{0},$ and can be approximated by

$\left\langle\delta
x_{1}^{2}\right\rangle_{\phi^{0},fr}=\frac{\Delta^{2}}{V_{d}^{2}},\ \
\left\langle\delta x_{2}^{2}\right\rangle_{\phi^{0},fr}=2\ D_{2}^{fr}t$ (39)

represented by the green lines. On the contrary, the trapped trajectories have
dispersions that decay with the increase of $\phi^{0}$ (the red points). The
dispersion of the trajectories conditioned by the potential is obtained using
Eq. (27) for $A=x_{i}^{2}(t)$

$\left\langle\delta x_{i}^{2}\right\rangle_{\phi^{0}}=\left\langle\delta
x_{1}^{2}\right\rangle_{\phi^{0},fr}F+\left\langle\delta
x_{1}^{2}\right\rangle_{\phi^{0},tr}(1-F)+\left\langle
x_{i}(t)\right\rangle_{\phi^{0},fr}^{2}F(1-F),$ (40)

which depends on $\phi^{0}$ as seen in Fig. 17 (black points).

Figure 17: The conditional trajectory dispersion along $x_{1}$ axis (left
panel) and along $x_{2}$ axis (right panel) as functions of $\phi^{0}$ for the
trapped (red points), free (green points) and all (black points) trajectories,
compared to the approximations (39) (green lines). $V_{d}=0.3$ and $t=97$.

__

Thus, the analysis of the Lagrangian statistics conditioned by the initial
potential reveals a coherent component of motion perpendicular to
$\mathbf{V}_{d}$. It consists of average displacements that have the sign
correlated with the sign of $\phi^{0}.$ They appear for the free trajectories
and are hidden in the sense that the contributions of all contour lines (with
all values of $\phi^{0})$ mix to zero. The amplitude of the ordered motion is
defined by the displacements conditioned by the sign of $\phi^{0},$
$\left\langle x_{1}(t)\right\rangle_{+}$ obtained by integration over
$\phi^{0}$ on the interval $[0,\infty)$ and $\left\langle
x_{1}(t)\right\rangle_{-}$ that is the integral over $(-\infty,0].$ These are
symmetrical quantities since $\left\langle x_{1}(t)\right\rangle=\left\langle
x_{1}(t)\right\rangle_{+}+\left\langle x_{1}(t)\right\rangle_{-}=0.$ The time
derivatives of these functions determine a pair of average velocities with
opposite directions that exactly compensate each other (the hidden drifts,
HDs) that are oriented across $\mathbf{V}_{d}$. The HDs were first found in
VS18  using an approximate theoretical approach, the decorrelation trajectory
method V98 . In the presence of components of the motion that introduce a
small compressibility, an average velocity can be generated by breaking the
equilibrium of the HDs. Such effects were found in magnetically confined
plasmas VS18b -VPS21 . The HDs are transitory in frozen potentials because the
average displacements saturate $\left\langle
x_{1}(t)\right\rangle_{\phi^{0},fr}\rightarrow\phi^{0}/V_{d}.$

The analysis also shows that the parallel motion is similar on the contour
lines with different $\phi^{0},$ and that it depends only on the category of
trajectories. The asymptotic value $\left\langle
x_{1}(t)\right\rangle_{\phi^{0},fr}\rightarrow\phi^{0}/V_{d}$ represents the
centrum of the space domain that contains the trajectories, which start from
points on the line $x_{1}=0$ where the potential is $\phi^{0}.$ It is limited
by the lines $x_{1}^{-}=(\phi^{0}-\Delta)/V_{d},$
$x_{1}^{+}=(\phi^{0}+\Delta)/V_{d},$ and has infinite dimension along
$\mathbf{V}_{d}.$ Reported at this centrum, the free trajectories are
statistically identical for all values of the initial potential $\phi^{0}.$

##  VI 6\. Lagrangian statistics in time dependent potentials

The general conclusion of the analysis of trajectory statistics in frozen
potentials is the existence of a high degree of coherence, which reflects the
structure of the contour lines of $\phi_{t}(\mathbf{x})$ on which the
trajectories are bounded.

The time-dependence of the potential determines the variation of the
Lagrangian potential and the decorrelation from its initial value $\phi^{0}$.
It is expected to strengthen the random aspects of the trajectories and to
cause the elimination of the Lagrangian coherence in a time of the order of
the decorrelation time $\tau_{c}$. More precisely, the random time-variation
of the potential should vanish the averages and correlations conditioned by
$\phi^{0},$ which show the existence of hidden order. It is thus expected that
the order found in static potential is in this case only a transitory
processes with life-time $\tau_{c}.$

The trajectories are more complex than in static potentials. Closed periodic
trajectories do not exist in time dependent potentials, but trapping events
represented by almost closed eddying segments appear on all trajectories when
the decorrelation time $\tau_{c}$ is large compared to the time of flight
$\tau_{fl}=\lambda/V,$ ($\lambda=(\lambda_{1}^{2}+\lambda_{2}^{2})^{1/2}$ and
$V=(V_{1}^{2}+V_{2}^{2})^{1/2}),$ and the integration time is much longer than
$\tau_{c}.$ The trapping events are separated by long jumps, which are similar
with the free trajectories.

The separation of the trajectories in the categories $c=tr,~{}fr$ has no
meaning in time-dependent potentials. However one can define related
quantities that are not properties of the trajectories but of the contour
lines of the potential. The latter are geometric objects. The fraction of
free/trapped trajectories can be defined using the number of trajectories that
stay on open/closed contour lines of the potential at time $t.$ These
fractions do not depend on time for stationary stochastic potentials, because
the amplitude, the space correlation and the structure of the contour lines
are statistically time-invariant. They equal the asymptotic values of the time
dependent fractions $n_{c}(t,V_{d})$ obtained in static potentials from the
trajectories (Sections 3)

$n_{c}(V_{d})=\ \underset{t\rightarrow\infty}{\lim}\ n_{c}(t,V_{d}),\ $ (41)

for any $\tau_{c}$ and $c=tr,fr.$ In a similar way, the fraction of
trajectories that stay at time $t$ on contour lines of category $c$ with the
potential $\phi^{t}=\phi(\mathbf{x}(t))$ is a time-independent function of
$\phi^{t}$ and $c,$ which is the asymptotic value $n^{c}(\phi^{t})$ of the
fractions obtained in static potential ($\tau_{c}\rightarrow\infty)$ in Eqs.
(29-31). The physical meaning of these quantities will be clarified after
analyzing the significant modifications of the statistics of trajectories
produced by the time-variation of the potential.

We underline that, in time-dependent potentials, one can define the statistics
conditioned by the initial potential, but not by the category.

Our aim is to see if the special order determined by the average velocity
survives at finite $\tau_{c}.$ We analyze here the statistics on the whole set
of trajectories (in $R$), while, in the next section, the statistics
conditioned by the initial potential will be used for understanding the direct
effects of time variation on the coherent elements found here.

The time variation of $\phi$ represents a decorrelation mechanism, because it
determines the stochastic change of the Lagrangian velocity, which vanishes
the Lagrangian correlations $L_{i}(t)$ at times $t\gg\tau_{c.}$ Usually, this
produces the saturation of the time dependent diffusion coefficients
$D_{i}(t)\rightarrow D^{i}$ and diffusive transport with $\left\langle\delta
x_{i}^{2}(t)\right\rangle\rightarrow 2D^{i}t$ (as obtained from Eq. (21)). The
memory of the initial velocity is lost in a time $\tau_{c},$ which means that
the displacements at large time $t\gg\tau_{c}$ are sequences of non-correlated
random steps that yield a Gaussian distribution.

We show that this general behavior is not at all observed in the presence of
$\mathbf{V}_{d}$ at large correlation times (weak time variation). A strong
non-standard influence appears both on the transport and on the probability of
the displacements.

The Lagrangian velocity is, as expected, Gaussian at any time and for any
$\tau_{c}$, as in the static case. The time variation influences only its
correlation.

The dispersions of the trajectories $\left\langle\delta
x_{i}^{2}(t)\right\rangle$ and the probabilities $P(x_{i},t)$ for a typical
case that illustrates the effects of weka time variation of the potential are
shown in Figs. 18 and 19, by the black lines for $V_{d}=0.3$ and
$\tau_{c}=33$. We also present, for comparison, two examples with $V_{d}=0.3$
(for the static case $\tau_{c}=\infty$ (red lines) and for fast time variation
$\tau_{c}=3.3$ (blue)), and two examples with $V_{d}=0$ (for $\tau_{c}=\infty$
(green) and $\tau_{c}=33$ (cyan)).

When $V_{d}=0,$ the subdiffusive transport at $\tau_{c}=\infty$ with
$\left\langle\delta x_{i}^{2}(t)\right\rangle\sim t^{0.68}$ is transformed
into normal transport ($\left\langle\delta
x_{i}^{2}(t)\right\rangle\rightarrow 2D^{i}t$) at large $\tau_{c}$ (Fig. 18,
green and cyan curves). The process appears for all finite values of
$\tau_{c},$ which only influence the diffusion coefficients $D^{i}.$ However,
the probabilities of displacements are Gaussian only for fast time variation.
In the static case, $P(x_{i},t)$ has a steep peak in $x_{i}=0,$ which
corresponds to trapped trajectories, superposed on a large Gaussian component,
which yields from the trajectories that are not closed at time $t.$ The steep
peak is flattened by time variation and the probabilities have extended
exponential shapes at large $\tau_{c}.$ When $\tau_{c}$ decreases,
$P(x_{i},t)$ evolves to Gaussian distribution, which is attained when
$\tau_{c}<\tau_{fl}$.

The average velocity makes the transport strongly anisotropic. In frozen
potential $\tau_{c}=\infty,$ the transport is ballistic in the parallel
direction and saturated perpendicular to $\mathbf{V}_{d},$ as discusses in
Section 4.2 and also shown in Figs. 18 and 19 (red curves). The normal
transport and the Gaussian probability are reached only for fast time
variation of the potential ($\tau_{c}<\tau_{fl}$), as seen in the example for
$\tau_{c}=3.3$ (blue curves). In these conditions, the motion along the
contour lines of $\phi(\mathbf{x},t)$ is completely hindered, which means that
the quasi-coherent components are eliminated.

Compared to these cases, the trajectory statistics at slow time variation in
the presence of $V_{d}$ (the black curves) is strongly anomalous with complex
behavior. Trajectory dispersion at large time has nonstandard time-dependence
in both directions

$\left\langle\delta x_{1}^{2}(t)\right\rangle\sim t^{\alpha_{1}},\ \
\left\langle\delta x_{2}^{2}(t)\right\rangle\sim t^{\alpha_{2}},$

where $\alpha_{1}<1$ and $\alpha_{2}>1,$ which corresponds to subdiffusive
perpendicular transport (but not saturated as in the static case) and
superdiffusive parallel transport (but not of ballistic type). These powers
are functions of $\tau_{c}$ and $V_{d},$ $\alpha_{i}(\tau_{c},V_{d}).$ When
$\tau_{c}$ decreases, $\alpha_{1}$ increases and saturates
$\alpha_{1}\rightarrow 1,$ while $\alpha_{2}$ decreases and saturates
$\alpha_{2}\rightarrow 1.$ A similar effect is determined by the increase of
$V_{d},$ which leads to normal transport at $V_{d}\gtrsim 1.$ In the example
presented in Fig. 18 (black curves), $\alpha_{1}(33,0.3)=0.57,$
$\alpha_{2}(33,0.3)=1.35.$ The probabilities are very large, especially in the
parallel direction, and the peak in $\mathbf{x}=0$ persists for very long time
($t=300=6\tau_{c}$ in Fig. 19).

.

Figure 18: Comparision of the dispersions of the trajectories in time-
dependent potential with the static cases along $x_{1}$ (dashed lines) and
$x_{2}$ (solid lines) directions. The values of $V_{d}$ and $\tau_{c}$ label
the curves.

Figure 19: The probabilities of the displacement along $x_{1}$ (left panel)
and $x_{2}$ (right panel) directions, show the effect of the time variation of
the potential for typical cases with the values of $V_{d}$ and $\tau_{c}$ that
label the curves.

Thus, the transport and the statistics of displacements are non-standard when
$V_{d}\lesssim 1$ and $\tau_{c}>\tau_{fl}.$ In these conditions the structure
of the contour lines of the potential shows island of closed lines in a
network of open lines. Also, the trajectories approximately follow the contour
lines for distances of the order of the correlation length before they are
removed by the time variation of the potential.

##  VII 7\. Enhanced coherence and long memory effects

###  VII.1 7.1 Hidden ordered motion

Ordered motion conditioned by the initial potential $\phi^{0}$ was found in
the presence of an average velocity $V_{d}$ for the free trajectories. It is
represented by the average displacements $\left\langle
x_{i}(t)\right\rangle_{\phi^{0},fr}$ that are conditioned by the initial
potential and by the category $c=fr$ [Eq. (36)]. These quantities obtained in
a time-dependent potential (with $\tau_{c}=33)$ are shown in Fig. (20) for
$x_{1}$ and in Fig. (21) for $x_{2}$, compared to the static case. Significant
differences appear for both directions.

One can see in Fig. (20, left panel) that the perpendicular displacements
$\left\langle x_{1}(t)\right\rangle_{\phi^{0}}$ are larger in time-depending
potential, although the calculations are at a very large time, $t=6\tau_{c}$
(where the EC of the potential (3) is negligible, $E(t)=10^{-8}$). The main
contribution comes from large values of the potential $\left|\phi^{0}\right|,$
which is negligible in the static potential. The amplitude of the ordered
motion is represented by the average displacements conditioned by the sign of
$\phi^{0},$ $\left\langle x_{1}(t)\right\rangle_{+}$ and $\left\langle
x_{1}(t)\right\rangle_{-},$ which determine by time derivative the HDs.
Surprisingly, the time evolution of these quantities shows a continuous
increase in time-dependent potential, while it saturates in the static case,
as seen in Fig. (20, right panel) for $\left\langle
x_{1}(t)\right\rangle_{+}$. This means that the hidden drifts are transitory
in static potentials, but they are long-life statistical quantities in time
dependent potential. Their amplitude decays on a long time scale, much longer
than the decorrelation time of the potential.

Figure 20: Ordered perpendicular displacements $\left\langle
x_{1}(t)\right\rangle_{\phi^{0}}$ as functions of $\phi^{0}$ at $t=200$ (left
panel) and $\left\langle x_{1}(t)\right\rangle_{+}$ as function of time (right
panel) for a time-dependent potential with $\tau_{c}=33$ (black points)
compared to the static case (dashed blue lines)

Figure 21: Ordered parallel displacements $\left\langle
x_{2}(t)\right\rangle_{\phi^{0}}$ (left panel) and average velocity
$\left\langle v_{2}(t)\right\rangle_{\phi^{0}}$ (right panel) as functions of
$\phi^{0}$ at $t=200$ for a time-dependent potential with $\tau_{c}=33$ (black
points) compared to the static case (dashed blue lines)

The time variation of the potential modifies the parallel displacements
$\left\langle x_{2}(t)\right\rangle_{\phi^{0}}$ by determining the extension
of the contribution on the whole range of $\phi^{0}$ and the dependence on
these quantities of the initial potential (Fig. (21, left panel). It is peaked
on $\phi^{0}=0$ and has a weak (algebraic) decay at large
$\left|\phi^{0}\right|,$ instead of the concentration on the domain of free
trajectories with uniform average displacement Eq. (35). However, the average
Lagrangian velocity is uniform on the whole range of $\phi^{0}$ and it has the
Eulerian value $V_{d},$ as seen in Fig. (21, right panel). The process of
concentration of the Lagrangian average velocity on the domain of free
trajectories found in static potentials is eliminated by the time-variation.

The fluctuations of the trajectories $\left\langle\delta
x_{i}^{2}(t)\right\rangle_{\phi^{0}}$ and the transport $\left\langle\delta
x_{i}(t)\delta v_{i}(t)\right\rangle_{\phi^{0}}$ conditioned by the initial
potential are all asymptotically uniform on the whole domain of $\phi^{0}$.
They reach this stage in a long time compared to the correlation time of the
potential ($t\gg\tau_{c}$) staring from the values corresponding to static
potential that are maintained at small time $t<\tau_{c}$.

These results show that the trajectories are statistically identical with the
exception of the perpendicular average displacement $\left\langle
x_{i}(t)\right\rangle_{\phi^{0}},$ which depend on $\phi^{0}.$

###  VII.2 7.2 Long memory

The persistent Lagrangian order and the non-standard characteristics of the
trajectories in the time dependent case can be understood by analyzing the
statistics of the Lagrangian potential $\phi(t)\equiv\phi(\mathbf{x}(t),t).$

The distribution of the Lagrangian potential has the same invariance property
as the Lagrangian velocity. It has the Gaussian probability of the Eulerian
potential at any time $t,$ for both the static and the time-dependent cases,
at any value of the average velocity $V_{d}.$ However, significant differences
appear between these cases concerning the correlation and the average
conditioned by the initial potential, as seen in Figs. (22) and 23.

The correlation of the Lagrangian potential
$L_{\phi}(t)=\left\langle\phi(0)~{}\phi(t)\right\rangle$ is far from the
Eulerian time-correlation, $E(\mathbf{0},t).$ Starting from the trivial case
of static potential with $V_{d}=0,$ where the invariance of the potential
implies $L_{\phi}(t)=$ $E(\mathbf{0})=1,$ in all the other cases shown in Fig.
(22), the Lagrangian correlation is stronger than the Eulerian one, as it has
a long tail with much slower decay. It demonstrates that the Lagrangian
potential has a long-time memory.

Figure 22: Correlation of the Lagrangian potential for $V_{d}=0$ (dashed
lines) and $V_{d}=0.3$ (continuous lines) for static ($\tau_{c}=\infty$), slow
($\tau_{c}=33$) and fast time-variation ($\tau_{c}=3.3$), compared to the
Eulerian correlations (dotted blue lines). Long tails with exponential decay
appear in time-dependent potentials with large $\tau_{c}$.

Figure 23: Caracterization of the memory of the Lagrangian potential. Left:
The correlations conditioned by the category $<\phi(0)\phi(t)>_{c}$ for
$V_{d}=0.1$ (dotted), $V_{d}=0.2$ (dashed), $V_{d}=0.3$ (continuous). Right:
The normalized average potential conditioned by the initial potential for
$V_{d}=0.3$, for $\tau_{c}=\infty$ (continuous), $\tau_{c}=33$ (dotted), and
the values of $\phi_{0}$ that label the curves.

The memory effect is strongest in almost static potentials (very large
$\tau_{c})$ with average velocity $V_{d}=0.$ The Lagrangian correlation
decreases much slower than the Eulerian one, and it is larger than than
$E(\mathbf{0},t)$ at any time (Fig. 22, the curve for
$V_{d}=0,~{}\tau_{c}=33)$. In this example, at $t=200\cong 6\tau_{c},$
$L_{\phi}$ decreases only at $0.4,$ while $E(\mathbf{0},t)=1.5~{}10^{-8}.$

The average velocity ($V_{d}\neq 0)$ determines a faster decrease of
$L_{\phi}(t)$ at small time that leads to smaller values compared to the case
$V_{d}=0$ (Fig. 22, the curve for $V_{d}=0.3,~{}\tau_{c}=33)$. The
decorrelation takes place on two time-scales. There is a fast decay at small
time that is followed by a slaw decrease of $L_{\phi}(t).$ The fast decay is
the same for $\tau_{c}=\infty$ and $\tau_{c}=33$ at $V_{d}=0.3$, which shows
that this process is not a consequence of the potential time variation, but
rather of the presence of $V_{d}.$

In the static case, the memory of the Lagrangian potential is infinite
($L_{\phi}(t)$ saturates). The asymptotic value is positive and it is a
decreasing function of $V_{d}$. The time-dependence of $L_{\phi}(t)$ is the
result of a selective decorrelation mechanism determined by the average
velocity. This process can be understood by analyzing the correlation of
$\phi(t)$ conditioned by the category $c=tr,~{}fr$ in the static potential
($\tau_{c}=\infty).$ As seen in Fig. 23, left panel,
$\left\langle\phi(0)~{}\phi(t)\right\rangle_{c}$ decays to zero for the free
trajectories, while it saturates for the trapped trajectories at a value that
is comparable to the conditioned amplitude
$\left\langle\phi^{2}(0)\right\rangle_{tr}$. This demonstrates that in static
potential the decorrelation affects only the free trajectories and that the
memory effect is determined by the trapped trajectories, which approximately
maintain the initial potential $\phi(0)$. The asymptotic value of the average
Lagrangian potential is thus

$\left\langle\phi(0)~{}\phi(t)\right\rangle=\left\langle\phi(0)~{}\phi(t)\right\rangle_{tr}n_{tr}=\phi^{0}n_{tr}.$
(42)

It is interesting to note that at finite $\tau_{c},$ the significant decrease
of the correction appears at any time, although it seems that the process
determined by $V_{d}$ is transitory in static potentials. This is caused by
the interaction of the effects of $V_{d}$ with the influence produced by the
time variation, which determine the two time-scale evolution of $L_{\phi}(t).$
It is clearly evidenced by the average Lagrangian potential conditioned by the
initial value $\phi^{0}$ normalized by this value,
$\left\langle\phi(t)\right\rangle_{\phi^{0}}/\phi^{0}$ represented in Fig. 23,
right panel for the static case (lines) and for $\tau_{c}=33$ (points) for
several values of $\phi_{0}.$

An important property of the Lagrangian potential in slow time-dependent
potential is that its correlation and the average conditioned by $\phi^{0}$
have the same time-decay (as seen in Fig. 22 for the case $V_{d}=0.3,$
$\tau_{c}=33$ and in the right panel of Fig. 23, all the curves have the same
behavior $\exp\left(-t/88\right)$).

The long memory of the potential and the increase of the average displacements
$\left\langle x_{1}(t)\right\rangle_{\phi^{0}}$ (Fig. 20) are the result of
the same process. It consists of the liberation of the trapped trajectories
with large $\left|\phi^{0}\right|$ followed by repeated stochastic events of
capture and release combined with the constraint of the total potential
invariance (22) that approximately holds for small time intervals. Considering
the case of the peaks of the potential, the liberation of the trapped
trajectories with large $\phi^{0}$ is produced when the time variation
determines the decrease of the potential to $\Delta$. The contour lines of the
potential that are open have the average perpendicular displacement
$\Delta/V_{d}$ and the average potential along them equal to zero (as imposed
by Eq. (22)). The stochastic recapture is uniformly distributed over the
potential and has the average perpendicular location $\Delta/V_{d}$. This
cancels asymptotically the average of the potentials on the trapping events
and leads to the average $\Delta/V_{d}$ of the positions of the trapping
events. This happens on a time scale that is much larger than $\tau_{c}$.
These released trajectories with large $\phi^{0}$ determine the slow decay of
their initial average potential and the increase of their average displacement
from zero to the largest possible value $\Delta/V_{d}$. Thus, the memory of
the Lagrangian potential and the strengthening of the coherence of the
trajectories are both determined by the slow evolution toward uniform
distribution of the trapping events on the trajectories caused by the time-
variation of the potential.

##  VIII 8\. Summary and conclusions

A detailed study of the Lagrangian coherence of the trajectories in
2-dimensional incompressible velocity fields is presented. The strong order
that appear in this kind of velocity fields is determined by the Hamiltonian
structure of the advection equations (1), (2) and by the space correlation of
the stochastic potential. The trajectories follow the contour lines of the
potential in the static case, and, for slowly varying potentials, they remain
close to them for long time intervals. This study is focused on the
identification and understanding of the order generated by an average velocity
$V_{d}$ superposed on the stochastic field. It determines an average
potential, which strongly modifies the structure of contour lines of the total
potential $\phi_{t}(\mathbf{x},t)=\phi(\mathbf{x},t)+x_{1}Vd$ by generation of
a network of open lines between islands of closed lines. As a result, two
categories of trajectories are found in static (frozen) potential: trapped
(closed, periodic) and free (with unlimited displacement in the parallel
direction to $\mathbf{V}_{d}$).

The results presented here are based on the numerical simulation of the
trajectories and on a complex statistical analysis that includes conditional
averages and correlations connected to the topology of the contour lines of
the potential $\phi_{t}.$ The statistics of displacements and of the
Lagrangian velocity are determined for the whole ensemble $R$, for the
categories trapped ($tr$) and free ($fr$), and also on sets of contour lines
of the potential conditioned by the value $\phi^{0}$ at the starting point of
the trajectories. This analysis reveals the origin of coherence and provides
explanations for the nonstandard statistics, transport and memory of this
motion.

In the case of frozen potentials, we have found that the statistical
properties determined for the two categories $tr,$ $fr$ are completely
different compared to those obtained in the whole space $R$. The average
velocity $V_{d}$ generates coherence in the Lagrangian velocity, which
acquires average components from he stochastic ones for both categories. The
supplementary coherent velocity cancels the average velocity $V_{d}$ of the
trapped trajectories, and it determines larger velocity for the free
trajectories that compensate the missing contribution of the trapped ones
($\left\langle v_{2}(t)\right\rangle_{fr}=V_{d}/n_{fr}$). Thus, the
statistical invariance of the Lagrangian velocity (Lumley theorem) is ensured
in a rather non-trivial manner that involves hidden coherent parallel
velocity.

The statistical analysis conditioned by the initial potential $\phi^{0}$
reveals additional important aspects. The free trajectories have Gaussian
distribution of $\phi^{0}$ with a width $\Delta$ that is smaller than the
dispersion of the Eulerian potential. It shows the existence of ordered
perpendicular motion (average displacements across $\mathbf{V}_{d}$ (34)) that
appear for the free trajectories and are proportional with $\phi^{0}.$ These
averages $\left\langle x_{1}(t)\right\rangle_{\phi^{0},fr}$ increase in time
from zero and saturate at $\phi^{0}/V_{d}.$ They generate the hidden drifts, a
pair of transitory average velocities perpendicular on $\mathbf{V}_{d}$
conditioned by the sign of $\phi^{0},$ which have opposite directions and
exactly compensate each other. We have also found that the Lagrangian
statistics of the free trajectories conditioned by $\phi^{0}$ depends on the
value $\phi^{0}$ only through the average perpendicular displacement. This
means that the trajectories in the category $fr$ for different values of
$\phi^{0}$ are statistically identical, and are organized in strips with
limited perpendicular extensions.

The probability of the displacements is non-Gaussian with separated
contributions of the categories: a steep peak for $tr$ and Gaussian
distribution that move along $\mathbf{V}_{d},$ but with larger velocity
$V_{d}/n_{fr}$ for the $fr$ subensemble. The time-invariant Gaussian
distribution of the Lagrangian velocity is the sum of the non-Gaussian
contributions of the two categories, which are both non-Gaussian, but
invariant.

The transport is produced only by the free trajectories. A paradoxical
behavior was found: the statistics of the trajectories is strongly non-
Gaussian, but the transport is produced by Gaussian trajectories, which in
fact yield from the non-Gaussian velocity distribution of the free
trajectories. The transport is anomalous, subdiffusive across $\mathbf{V}_{d}$
and superdiffusive of ballistic type along $\mathbf{V}_{d}.$ The latter
results from the ordered parallel motion. category.

The free trajectories can be associated to a geometrical locus on the two-
dimensional space $\left(x_{1},x_{2}\right)$, in the sense that each point is
the initial condition for such trajectory and all trajectories are confined in
this domain $fr$. The complement of $fr$ is the geometric locus of the trapped
trajectories $tr$, which is composed of the islands of closed contour lines of
the potentials. The (Eulerian) statistical characteristics of each geometrical
locus were identified.

The time-dependence of the stochastic potential produces anomalous increase of
the Lagrangian coherence, instead of the expected decay after the
decorrelation time. In particular, the perpendicular average displacements
conditioned by $\phi^{0}$ significantly increase and the transitory hidden
drifts become long-life structure that survive at $t\gg\tau_{c}.$ The enhanced
coherence is found to be associated to a long memory of the Lagrangian
potential. Also, the trajectories conditioned by the initial potential become
statistically identical for all values of $\phi^{0},$ not only on the domain
of small potential with width $\Delta,$ as in frozen potentials. These effects
are caused by the stochastic liberation by the time-variation of the potential
of the trajectories that initially are trapped, followed by repeated
stochastic captures that are constraint by of the approximate invariance of
the total potential.

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