# Scaling laws of two-dimensional incompressible turbulent transport

D. I. Palade<EMAIL_ADDRESS>National Institute for Laser, Plasma and
Radiation Physics, Măgurele, Romania  Faculty of Physics, University of
Bucharest, Romania  L. M. Pomârjanschi  National Institute for Laser, Plasma
and Radiation Physics, Măgurele, Romania  Faculty of Physics, University of
Bucharest, Romania  M. Ghiță  Faculty of Physics, University of Bucharest,
Romania

###### Abstract

The diffusive transport in two-dimensional incompressible turbulent fields is
investigated with the aid of high-quality direct numerical simulations. Three
classes of turbulence spectra that are able to capture both short and long-
range time-space correlations and oscillating features are employed. We report
novel scaling laws that depart from the $\gamma=7/10$ paradigm of percolative
exponents and are dependent on the features of turbulence. A simple relation
between diffusion in the percolative and frozen regimes is found. The
importance of discerning between differential and integral characteristic
scales is emphasized.

turbulent transport, incompressible, Kubo, scaling

##  I Introduction

The problem of turbulence remains one of the most elusive topics of modern
physics. It is of immediate interest for a variety of systems such as
(geophysical) fluids [1], astrophysical systems [2], magnetically confined
plasmas [3, 4], magnetic field line dynamics [5], quantum systems [6, 7, 8],
etc. While being a compact problem, it can be formally split into two
complementary parts: the characterization of turbulence and the
characterization of transport.

The first part, which is a broad topic beyond the scope of the present work,
is related to the representation of a turbulent field
$\mathbf{v}(\mathbf{x},t)$. This is, in general, a difficult task which may
require refined solutions of non-linear fluid [9] or kinetic approaches [10,
11]. It is often the case that the resulting field
$\mathbf{v}\left(\mathbf{x},t\right)$ exhibits chaotic features that emerge,
develop and saturate as turbulent states. These features can be captured via
their Eulerian statistical properties: average, distribution and spectra.

The second part is the main concern of the present paper and can be stated as
it follows: _Given the statistical properties of a turbulent velocity flow
$\mathbf{v}\left(\mathbf{x},t\right)$, characterize the transport coefficients
(convection $\mathbf{V}(t)$ and diffusion $\hat{D}(t)$) of passive scalars
advected by this field_.

The advection is apparent at the level of a continuity-like equation
$\partial_{t}n+\nabla\left(n\mathbf{v}\right)=0$ for a scalar field
$n(\mathbf{x},t)$. In this Eulerian view, the transport coefficients
$\mathbf{V}(t),\hat{D}(t)$ appear in the Fick’s law for the scalar flux
$\mathbf{\Gamma}=\langle
n\mathbf{v}\rangle\equiv\mathbf{V}(t)n+\hat{D}(t)\nabla n$, where
$\langle\cdot\rangle$ is the statistical averaging operation over realizations
of turbulent flows $\mathbf{v}(\mathbf{x},t)$. From a Lagrangian perspective,
one can show that, under the assumption of scale-separation [4], the transport
coefficients are related to statistical averages
$\mathbf{V}(t)=d_{t}\langle\mathbf{x}(t)\rangle$,
$2\hat{D}(t)=d_{t}\left(\langle\mathbf{x}^{2}(t)\rangle-\langle\mathbf{x}(t)\rangle^{2}\right)$
over ensembles of trajectories $\\{\mathbf{x}(t)\\}$. The latter are driven by
the corresponding ensemble of stochastic velocity fields via
$\dot{\mathbf{x}}(t)=\mathbf{v}(\mathbf{x}(t),t)$, known as a V-Langevin
equation [12, 13].

The mean square displacement
$\sigma(t)=\langle\mathbf{x}^{2}(t)\rangle-\langle\mathbf{x}(t)\rangle^{2}$,
in the asymptotic limit, exhibits algebraic behaviour
$\lim\limits_{t\to\infty}\sigma(t)\sim t^{\alpha}$. Depending on the features
of turbulence, one can get $\alpha=1$ (normal, molecular-like diffusion),
$\alpha=2$ (ballistic transport), $0<\alpha<1$ (sub-diffusion) or $1<\alpha$
(superdiffusion). The last two cases are coined in literature _anomalous
transport_ [4, 14]. A sufficiently time-decorrelated turbulent flow
$\mathbf{v}(\mathbf{x},t)$ can lead to normal diffusion $\alpha\to 1$, but, if
consistent spatial correlations are present, the transport is still named
_anomalous_. This is a consequence of peculiar behaviour of the asymptotic
diffusion algebraic scaling with the strength of turbulence
$V=\sqrt{\langle\mathbf{v}^{2}(\mathbf{x},t)\rangle}$ as
$D^{\infty}=D(t\to\infty)\sim V^{\gamma}$, that may be different from
$\gamma=2$ which is specific to normal (molecular) diffusion.

In the present work we investigate the scaling laws of the turbulent
transport, for the special case of two-dimensional incompressible flows
$\nabla\cdot\mathbf{v}(\mathbf{x},t)=0$. The latter is a paradigm for the
$\mathbf{E}\times\mathbf{B}$ drift motion of charged particles in magnetically
confined plasmas and for the magnetic field lines wandering in astrophysical
turbulence. Both classes of systems are notorious for their ability to
manifest anomalous transport [15, 16].

A lot of effort has been directed towards this problem in the past decades,
from Bohm’s diffusion [17], to Corssin’s approximation [18], Kraichnan
analysis [19], Isichenko’s pioneering work [20], numerical experiments [21,
22, 23, 24, 25, 26] and other modern, theoretical, approaches [27, 5] etc.
Despite such extensive endeavors, the problem is not solved yet: we do not
have simple relations between the Eulerian (measurable) characteristics of a
turbulent flow and the transport coefficients in all physical regimes of
interest. The most promising analytical result was due to Isichenko [20] who
showed, with the aid of percolation theory, that, in the strong turbulence
(percolative) regime $\gamma=7/10$, in contrast with Bohm’s [17] $\gamma=1$
value. The universality of this value has been put to the test in the
following years by semi-analytical approximations [27, 5], which did not
confirm it. On the numerical front [21, 22, 24, 28] many investigations have
found exponents close to the analytical $\gamma=7/10$ value. Unfortunately,
the results are widely scattered, with relatively large uncertainties, poor
numerical resolution and only for limited classes of turbulence spectra.

For all these reasons, in the present work we intend to use high-quality
direct numerical simulations [13] to evaluate the turbulent transport in two-
dimensional incompressible fields. In particular, we search for scaling laws
of diffusive processes and their relation with statistical
spectra/correlations. The universality of Isichenko’s result is put to the
test. For this analysis, we employ three classes of spectra that are able to
capture both short and long-range correlations in space and time or the
existence of oscillating behaviour.

The paper is structured as it follows. In the Theory section (II) we present
the statistical description of turbulence, the associated families of
spectra/correlations to be investigated and the approach of direct numerical
simulations (DNS). In the Result section (III) we analyze the numerical
results on diffusion and scaling exponents. The final section (IV) is devoted
to conclusions and perspectives.

##  II Theory

We consider a two-dimensional incompressible turbulent flow
$\nabla\cdot\mathbf{v}(\mathbf{x},t)=0$ that can be expressed in terms of a
scalar field (stream-function) $\phi(\mathbf{x},t)$ as
$\mathbf{v}(\mathbf{x},t)=\hat{e}_{z}\times\nabla\phi(\mathbf{x},t)$ with
$\mathbf{x}\equiv(x,y)$. In fusion plasmas [27] the stream-function is related
to the electrostatic potential while in astrophysical magnetic turbulence to
components of the magnetic vector potential [22]. A statistical ensemble of
potentials $\\{\phi(\mathbf{x},t)\\}$ which are homogeneous Gaussian random
fields [13] with zero average $\langle\phi(\mathbf{x},t)\rangle=0$ and known
Eulerian correlation
$\mathcal{E}(|\mathbf{x}-\mathbf{x}^{\prime}|,|t-t^{\prime}|)=\langle\phi(\mathbf{x},t)\phi(\mathbf{x}^{\prime},t^{\prime})\rangle$
drives an associated ensemble of trajectories $\\{\mathbf{x}(t)\\}$ via the
V-Langevin eq:

$\displaystyle\frac{d\mathbf{x}(t)}{dt}=\mathbf{v}(\mathbf{x}(t),t)=\hat{e}_{z}\times\nabla\phi(\mathbf{x}(t),t),\hskip
28.45274pt\mathbf{x}(0)=0.$ (1)

The resulting trajectories $\\{\mathbf{x}(t)\\}$ are used to evaluate the
running diffusion coefficients as $D_{x}(t)=1/2d_{t}\langle x^{2}(t)\rangle$.
The homogeneous correlation
$\mathcal{E}(|\mathbf{x}-\mathbf{x}^{\prime}|,|t-t^{\prime}|)$ is related to
the turbulence spectrum
$S(\mathbf{k},\omega)=\langle|\tilde{\phi}(\mathbf{k},\omega)|^{2}\rangle$ via
a Fourier transform. We shall assume Gaussian turbulence, i.e. a normal
distribution for the values of the potential $\phi(\mathbf{x},t)$. The
implications of non-Gaussianity have been discussed elsewhere [29] and found
to be small.

###  II.1 Correlation functions and associated spectra

While distinct physical systems or dynamical regimes can lead to different
turbulence spectra, there are similar features shared among most cases of
interest. The spectrum $S(\mathbf{k},\omega)$ is usually peaked at or near the
$|\mathbf{k}|=0,\omega=0$ values and decays in the asymptotic limit
$|\mathbf{k}|,\omega\to\infty$, at least algebraically [30, 31, 32, 33]. In
order to encompass such gross properties, we consider here three classes of
parametric turbulent fields as separable, isotropic, correlation functions
$\mathcal{E}_{\nu/\mu}^{(j)}(\mathbf{x},t)$ and their corresponding spectra
$S_{\nu/\mu}^{(j)}(\mathbf{k},\omega)$.

The correlation $\mathcal{E}_{\nu}^{(1)}(\mathbf{x},t)$ (eq. (2)) consists of
a Gaussian space profile with an algebraic time decay. In the limit $\nu\to
1$, the time dependence is a long-range Lorentzian function, while for
$\nu\to\infty$ it becomes a short-ranged Gaussian. $K_{a}(z)$ is the modified
Bessel function of the second kind. In Figs. (1a)-(1b) we plot the correlation
$\mathcal{E}_{\nu}^{(1)}(\mathbf{0},t)$ and the spectrum
$S_{\nu}^{(1)}(\mathbf{0},\omega)$ for different parametric values $\nu$,
fixing $\tau_{c}=1$.

$S_{\nu}^{(1)}(\mathbf{k},\omega)\sim\left|\omega\tau_{c}\right|^{\nu-\frac{1}{2}}K_{\nu-\frac{1}{2}}\left(\sqrt{2\nu}\left|\omega\right|\tau_{c}\right)e^{-\frac{\mathbf{k}^{2}\lambda_{c}^{2}}{2}}\longrightarrow\mathcal{E}_{\nu}^{(1)}(\mathbf{x},t)=e^{-\frac{\mathbf{x}^{2}}{2\lambda_{c}^{2}}}(1+\frac{t^{2}}{2\tau_{c}^{2}\nu})^{-\nu}.$
(2)

(a)  (b)

Figure 1: Spectra $S_{\nu}^{(1)}(\mathbf{0},\omega)$ and associated
correlation functions $\mathcal{E}_{\nu}^{(1)}(\mathbf{0},t)$ for several
values of the parameter $\nu$ and $\tau_{c}=1,\lambda_{c}=1$. The black lines
correspond to the Gaussian limit.

The correlation $\mathcal{E}_{\mu}^{(2)}(\mathbf{x},t)$ eq. (3) exhibits space
oscillations and it is associated to a spectrum dominated by an energy-range,
thus, peaked at a non-zero $|\mathbf{k}|$ value:

$S_{\mu}^{(2)}(\mathbf{k},\omega)\sim\left|\mathbf{k}\lambda_{c}\right|^{\mu}e^{-\left|\mathbf{k}\lambda_{c}\right|^{2}(2+\mu)/4}e^{-\frac{\omega^{2}\tau_{c}^{2}}{2}}\longrightarrow\mathcal{E}_{\mu}^{(2)}(\mathbf{x},t)=e^{-\frac{t^{2}}{2\tau_{c}^{2}}}L_{-\frac{\mu}{2}-1}\left(-\frac{|\mathbf{x}|^{2}}{2\left(\frac{\mu}{2}+1\right)\lambda_{c}^{2}}\right)$
(3)

where $L_{a}(x)$ is the Laguerre polynomial. In Figs. (2a)-(2b) we plot the
correlation $\mathcal{E}_{\mu}^{(2)}(|\mathbf{r}|,0)$ and the spectrum
$S_{\mu}^{(2)}(|\mathbf{k}|,0)$. The parameter $\mu$ controls space
oscillations of $\mathcal{E}^{(2)}$ and, in the limit $\mu=0$, is equivalent
with a Gaussian correlation.

(a)  (b)

Figure 2: Spectra $S_{\mu}^{(2)}(|\mathbf{k}|,0)$ and the associated
correlation functions $\mathcal{E}_{\mu}^{(2)}(|\mathbf{r}|,0)$ for several
values of the parameter $\mu$ and $\tau_{c}=1,\lambda_{c}=1$. The black lines
correspond to the Gaussian limit.

(a)  (b)

Figure 3: Spectra $S_{\mu}^{(3)}(|\mathbf{k}|,0)$ and the correlation
functions $\mathcal{E}_{\mu}^{(3)}(|\mathbf{r}|,0)$ for several values of the
parameter $\mu$ and $\tau_{c}=1,\lambda_{c}=1$. The black lines correspond to
the Gaussian limit.

The correlation $\mathcal{E}_{\mu}^{(3)}(\mathbf{x},t)$ depicts algebraically
decaying space correlation and it is able to encompass both the short-ranged
($\mu\to\infty$) and the long-ranged ($\mu\to 1$) limits. In Figs. 3a-3b we
plot $\mathcal{E}_{\mu}^{(3)}(|\mathbf{r}|,0)$ and
$S_{\mu}^{(3)}(|\mathbf{k}|,0)$ at different $\mu$ values, setting
$\lambda_{c}=1$.

$S_{\mu}^{(3)}(\mathbf{k},\omega)\sim\left|\mathbf{k}\lambda_{c}\right|^{\mu-1}K_{\mu-1}\left(\sqrt{2\mu}\left|\mathbf{k}\lambda_{c}\right|\right)e^{-\frac{\omega^{2}\tau_{c}^{2}}{2}}\longrightarrow\mathcal{E}_{\mu}^{(3)}(\mathbf{x},t)=e^{-\frac{t^{2}}{2\tau_{c}^{2}}}(1+\frac{|\mathbf{x}|^{2}}{2\lambda_{c}^{2}\mu})^{-\mu}.$
(4)

At this end, we reiterate that
$\mathcal{E}^{(1)}_{\nu},\mathcal{E}^{(2)}_{\mu},\mathcal{E}^{(3)}_{\mu}$ are
designed to describe long-range time, oscillating space, respectively, long-
range space correlations. Supplementary, their functional forms were chosen
such that, in specific limits, the following equality holds
$\mathcal{E}^{(1)}_{\infty}(\mathbf{x},t)=\mathcal{E}^{(2)}_{0}(\mathbf{x},t)=\mathcal{E}^{(3)}_{\infty}(\mathbf{x},t)$.

###  II.2 Kubo numbers

It is customary in turbulence studies to define a so-called Kubo number
$K_{\star}$ [34, 28] as the ratio between a characteristic time-scale of
turbulence $T$ and the time-of-flight $\tau_{fl}$, $K_{\star}=T/\tau_{fl}$.
The latter is understood as the specific time in which the space-correlation
is lost and can be approximated as the ratio between a characteristic space-
scale of turbulence $L$ and the average amplitude of the velocity field $V$,
$\tau_{fl}=L/V$. The Kubo number $K_{\star}=TV/L$ is also a measure of the
turbulence amplitude, via $V$. From a physical point of view, $K_{\star}$
indicates the ability of individual particles to explore the spatial structure
of the velocity field before the latter changes beyond recognition, i.e.
becomes decorrelated.

With respect to $K_{\star}$, three distinct regimes with different behaviours
of the asymptotic diffusion $D^{\infty}$ can be defined: the quasilinear
regime (weak turbulence or high frequency) when $K_{\star}\ll 1$ and
$D^{\infty}\sim K_{\star}^{2}$, the strong regime (strong turbulence,
intermediate frequencies) when $K_{\star}\sim 1$ and $D^{\infty}\sim
K_{\star}^{1}$ and the percolative regime (strong turbulence, low-frequency)
when $K_{\star}\gg 1$ and $D^{\infty}\sim K_{\star}^{\gamma}$. For the latter
case Isichenko derived his famous result of $\gamma=7/10$ whose universality
is under scrutiny in this paper.

Before searching for the relation between Kubo numbers and diffusion, a
question is to be asked: what is the correct way to define $K_{\star}$, or
equivalently, what are the _characteristic time-space scales_ of turbulence,
$T,L$? One possible answer is to consider that these scales are the Taylor
differential scales $(T_{D},L_{D})$ [35]:

$\displaystyle T\equiv T_{D}(\nu)$
$\displaystyle=\left(-\frac{\mathcal{E}(\mathbf{0},0)}{\partial_{t,t}\mathcal{E}(\mathbf{0},0)}\right)^{1/2},$
(5) $\displaystyle L\equiv L_{D}(\mu)$
$\displaystyle=\left(-\frac{\mathcal{E}(\mathbf{0},0)}{\partial_{x,x}\mathcal{E}(\mathbf{0},0)}\right)^{1/2}.$
(6)

In fact, the parametric dependencies of $\mathcal{E}^{(1-3)}$ were chosen in
such a way that all have the same differential scales
$L_{D}=\lambda_{c},T_{D}=\tau_{c}$ and the same Eulerian velocity field
amplitude
$V=\sqrt{\langle\mathbf{v}^{2}(\mathbf{x},t)\rangle}=\sqrt{-\nabla^{2}\mathcal{E}(\mathbf{x},t)}=\lambda_{c}^{-1}$,
consequently, the same _differential Kubo number
$K_{D}=VT_{D}/L_{D}=\tau_{c}/\lambda_{c}^{2}$_.

On the other hand, the differential $L_{D},T_{D}$, as the name suggests,
describe only the local tendency of correlation and may not say anything
relevant about the global features which might play a decisive role in strong
transport. Moreover, all functions $\mathcal{E}^{(1-3)}$, when departing from
the Gaussian limit, lose their monoscale character. For that, we define here
the integral time-space scales $(T_{I},L_{I})$ [35] as global measures of the
correlation:

$\displaystyle L\equiv L_{I}(\mu)$
$\displaystyle=\int_{0}^{\infty}\frac{\mathcal{E}(r,0)}{\mathcal{E}(\mathbf{0},0)}~{}dr$
(7) $\displaystyle T\equiv T_{I}(\nu)$
$\displaystyle=\int_{0}^{\infty}\frac{\mathcal{E}(\mathbf{0},t)}{\mathcal{E}(\mathbf{0},0)}~{}dt.$
(8)

The correlations $\mathcal{E}^{(1-3)}$ allow us to evaluate these quantities
$(T_{I},L_{I})$ analytically and see that they coincide with
$\sqrt{\pi/2}(T_{D},L_{D})$ in specific asymptotic limits:

$\displaystyle T_{I}(\nu)$
$\displaystyle=\tau_{c}\sqrt{\frac{\pi\nu}{2}}\frac{\Gamma\left(\nu-\frac{1}{2}\right)}{\Gamma(\nu)}\longrightarrow[\nu\to\infty]{}\sqrt{\frac{\pi}{2}}\tau_{c},\hskip
14.22636pt$ $\displaystyle\mathrm{for}\hskip
14.22636pt\mathcal{E}_{\nu}^{(1)}$ $\displaystyle L_{I}(\mu)$
$\displaystyle=\lambda_{c}\sqrt{\frac{(\mu+2)}{\pi\mu^{2}}}\frac{\Gamma\left(\frac{\mu+1}{2}\right)}{\Gamma\left(\frac{\mu}{2}\right)}\longrightarrow[\mu\to
0]{}\sqrt{\frac{\pi}{2}}\lambda_{c},\hskip 14.22636pt$
$\displaystyle\mathrm{for}\hskip 14.22636pt\mathcal{E}_{\mu}^{(2)}$
$\displaystyle L_{I}(\mu)$
$\displaystyle=\lambda_{c}\sqrt{\frac{\pi\mu}{2}}\frac{\Gamma\left(\mu-\frac{1}{2}\right)}{\Gamma(\mu)}\longrightarrow[\mu\to\infty]{}\sqrt{\frac{\pi}{2}}\lambda_{c},\hskip
14.22636pt$ $\displaystyle\mathrm{for}\hskip
14.22636pt\mathcal{E}_{\mu}^{(3)}.$

Thus, we can define, as an alternative to differential $K_{D}$, the integral
Kubo number $K_{I}$

$K_{I}=\frac{VT_{I}}{L_{I}}.$

###  II.3 Direct numerical simulations

Regarding the specific numerical approach on computing the diffusion
coefficients, we employ here the so-called direct numerical simulation method
(DNS) [13, 36, 37]. DNS attempts to mimic the real trajectories
$\\{\mathbf{x}(t)\\}$ at the numerical level. In practice, a finite ensemble
(dimension $N_{p}$) of random fields $\\{\phi(\mathbf{x},t)\\}$ is generated.
For each such realization, the associated trajectory $\mathbf{x}(t)$ that
obeys eq. (1) is computed. The technical method of generating a Gaussian
Random Field with given spectrum $S(\mathbf{k},\omega)$ is via a Fourier
decomposition:

$\displaystyle\phi(\mathbf{x},t)=\sqrt{\frac{2}{N_{c}}}\sum_{j}^{N_{c}}sin(\mathbf{k}_{j}\mathbf{x}-\omega_{j}t+\zeta_{j}),$
(9)

where $\zeta_{j}$ are random phases $\in(0,2\pi]$, while the wave-vectors
$\mathbf{k}_{j}$ and the frequencies $\omega_{j}$ are independent random
variables that have as PDF the turbulence spectrum $S(\mathbf{k},\omega)$. The
resulting field will be zero-averaged $\langle\phi(\mathbf{x},t)\rangle=0$ and
will have the correct correlation function
$\langle\phi(\mathbf{x},t)\phi(\mathbf{x}^{\prime},t^{\prime})\rangle=\mathcal{E}\left(\mathbf{x}-\mathbf{x}^{\prime},t-t^{\prime}\right)$
associated to the spectrum $S(\mathbf{k},\omega)$ by construction. Its
Gaussianity is guaranteed by the Central Limit Theorem for large numbers of
waves $N_{c}\to\infty$. For more details see [13].

In practice, we use high-fidelity simulations with unprecedented numerical
resolution. The dimension of the statistical ensemble is set to $N_{p}=10^{7}$
trajectories and $N_{c}=10^{3}$ partial waves are used. In this way we ensure
proper Gaussianity of the turbulent field, a good convergence to the
homogeneous correlation, the lack of spurious correlations or periodicity, as
well as a good convergence of the Lagrangian ensemble of trajectories over
large times. The latter are computed solving the eq. (1) with a 4th order
Runge-Kutta method and a fixed time-step of $\delta t\approx 0.05$. This is
enough to achieve the stability of trajectories and their Hamiltonian
character (when present) over long periods of time, which usually are set to
$t_{max}\sim 5-10\tau_{c}$. Note that, in order to simulate different values
of Kubo numbers (either differential $K_{D}$ or integral $K_{I}$), we keep
$\lambda_{c}=1\implies V=1$ in all simulations, but vary the $\tau_{c}$
parameter. Using dedicated programming procedures, typical simulations on
several processor servers require in terms of CPU time: $t_{CPU}\sim
1\mathrm{min}$ for $\tau_{c}\sim 1$ and up-to $t_{CPU}\sim 10^{2}\mathrm{min}$
for $\tau_{c}\sim 10^{3}$.

##  III Results

###  III.1 Generic features of diffusion processes

The diffusion coefficients $D_{x}(t)$ are intimately related to the mean
square displacement $\sigma_{x}(t)$ and the Lagrangian velocity
autocorrelation $\mathcal{L}_{x}(t)=\langle v_{x}(0)v_{x}(t)\rangle$ through
iterative time-differentiation
$\mathcal{L}_{x}(t)=d_{t}D_{x}(t)=1/2d_{t,t}\sigma_{x}(t)$. Consequently,
algebraic dependencies are expected as long-time behaviour
$\lim\limits_{t\to\infty}\sigma_{x}(t)\sim t^{\alpha},D_{x}(t)\sim
t^{\alpha-1},\mathcal{L}_{x}(t)\sim t^{\alpha-2}$. The turbulent transport in
2D incompressible fields lies between two limits: _frozen_ and _quasilinear_.

(a)  Figure 4: Typical trajectories $\mathbf{x}(t)$ in the same field
$\phi(\mathbf{x},t/\tau_{c})$ with $\tau_{c}=\\{\infty,1,0.1\\}$ (red, blue,
brown).

The _frozen turbulence_ limit corresponds to stationary fields, i.e.
$\mathbf{v}(\mathbf{x},t)\equiv\mathbf{v}(\mathbf{x})$,
$\phi(\mathbf{x},t)\equiv\phi(\mathbf{x})$, when $\tau_{c}\to\infty$. Due to
the Hamiltonian nature of the equations of motion (1), it follows that the
trajectories are closed. This can be seen in Fig. (4a), where a generic
stationary trajectory is drawn in red. At small times, the Lagrangian
velocities experienced by particles are roughly
$\mathbf{v}\left(\mathbf{x}(t),t\right)\approx\mathbf{v}\left(\mathbf{x}(0),0\right)$
which enables us to show that $\sigma_{x}(t)\approx V^{2}t^{2},D_{x}(t)\approx
V^{2}t,\mathcal{L}_{x}(t)\approx V^{2}$. At long times, since all particles
are trapped in equipotential field-lines, their asymptotic displacement
$\lim\limits_{t\to\infty}\sigma_{x}(t)$ is finite, thus, the diffusion
coefficient $\lim\limits_{t\to\infty}D_{x}(t)$ and the Lagrangian correlation
$\mathcal{L}_{x}(t)$ decay to zero. At intermediate times, at the so called
$t_{peak}\approx\tau_{fl}$, the Lagrangian correlation has a first zero which
corresponds to a maximum in diffusion $D_{peak}$. All these features can be
observed in Figs. (5a),(5b) for the red lines corresponding to frozen running
diffusion $D_{x}(t)$ (left) and Lagrangian correlation $\mathcal{L}_{x}(t)$
(right) in the case of $\mathcal{E}^{(1)}_{\nu}$. The numerical simulations
confirm the anomalous algebraic dependence $D_{x}(t)\sim t^{\alpha-1}$ with
the exponent $\alpha\approx 0.6$.

The existence of decorrelation mechanisms (through a finite $\tau_{c}$)
induces a continuous change on the landscape of the stream-function
$\phi(\mathbf{x},t)$ which is equivalent, at the level of individual
trajectories, with long jumps between equipotential lines (Fig. (4a), blue
line). This implies that the particles are no longer trapped, and their
spreading is continuously increasing. The Lagrangian correlation
$\mathcal{L}_{x}(t)$ decays strongly to $0$ which makes the process diffusive,
i.e. $\alpha\to 1$ and $D_{x}^{\infty}$ finite. These features can be seen in
(5a),(5b) for the blue lines which correspond to finite $K_{\star}$ regimes.

The _quasilinear_ limit is characterized by a small Kubo number $K_{\star}\ll
1$ (or decorrelation time $\tau_{c}\ll t_{peak}$). The particles are driven by
fields that change so rapidly in time that their motion is almost unaffected
by the space structure of the fields. The trajectories exhibit coloured-noise
driven features (Fig. (4a) brown line) and the diffusion coefficient can be
estimated as the product between field strength $V^{2}$ and the integral time
scale $T_{I}$:

$D_{x}^{\infty}=\int_{0}^{\infty}\mathcal{L}_{x}(t)~{}dt=\int_{0}^{\infty}\langle
v_{x}(\mathbf{x}(0),0)v_{x}(\mathbf{x}(t),t)\rangle\approx\int_{0}^{\infty}\langle
v_{x}(\mathbf{0},0)v_{x}(\mathbf{0},t)\rangle=V^{2}T_{I}.$

(a)  (b)

Figure 5: Running diffusion coefficient $D_{x}(t)$ (left, a)) and Lagrangian
correlation $\mathcal{L}_{x}(t)$ (right, b)) for the correlation
$\mathcal{E}_{\nu}^{(1)}(\mathbf{x},t)$ at $\nu=5,\lambda_{c}=1$ and two
different values of $\tau_{c}=3,330$ (blue line) and $\tau_{c}=\infty$ (red
line).

The diffusion profiles shown in Figs. (5a) are generic for all correlation
functions $\mathcal{E}^{(1-3)}$, both in the frozen $\tau_{c}\to\infty$ or
decorrelated $\tau_{c}=$ finite regimes.

We underline that the results from all figures shown in the present work are
scaled as it follows: $t\equiv
tV/L_{D},\mathcal{L}_{x}\equiv\mathcal{L}_{x}/V^{2},D_{x}\equiv
D_{x}/(L_{D}V),L_{I}\equiv L_{I}/L_{D},T_{I}\equiv T_{I}/T_{D}$. Nonetheless,
throughout the text, all formulae contain the correct, non-scaled
dependencies.

###  III.2 Frozen turbulence

We consider now, in detail, the diffusion process in the case of frozen
turbulence, which is an ideal limit of very slowly varying velocity fields
$\tau_{c}\to\infty$. The behaviour apparent in Fig. (5a) is qualitatively
relevant for all correlations and can be observed in Figs. (6a)-(6b) where
$D_{x}(t)$ is represented for $\mathcal{E}_{\mu}^{(2)}$, respectively
$\mathcal{E}_{\mu}^{(3)}$ at different values of the $\mu$ parameter. At small
times $t\ll t_{peak}$ all profiles follow the analytical behaviour
$D_{x}(t)\approx V^{2}t^{1}$ discussed in the previous section (III.1).
Maximal values $D_{peak}$ are present at $t_{peak}$, whereupon, diffusion
decays algebraically $D_{x}(t)\sim t^{\alpha-1}$. Thus, these three quantities
$t_{peak},D_{peak},\alpha$ and their relation with the functional structure of
turbulence are of interest for the frozen regime. Since all correlations have
the same differential scales $L_{D}=\lambda_{c}$, we discern between them with
the aid of the integral scale length $L_{I}$. We represent the dependencies
$t_{peak}(L_{I})$, $D_{peak}(L_{I})$, $\alpha(L_{I})$ in Figs. (7a)-(7c).

(a)  (b)

Figure 6: Running diffusion coefficients $D_{x}(t)$ in the case of frozen
turbulence $\lambda_{c}=1,\tau_{c}\to\infty$ for the correlations
$\mathcal{E}_{\mu}^{(2)}(\mathbf{r},0)$ (left) and
$\mathcal{E}_{\mu}^{(3)}(\mathbf{r},0)$ (right) at different $\mu$ values.
Note that the black lines ($\mu=0$ (left) or $\mu=\infty$ (right)) represents
the Gaussian limit.

The time at which the maximal diffusion is reached,
$t_{peak}(L_{I})\sim\sqrt{2}\lambda_{c}/V\approx L_{I}/V$, can be viewed as
the _real time-of-flight_ and it turns out Fig. (7a) to be consistently larger
than the gross differential estimation $\tau_{fl}=L_{D}/V=\lambda_{c}^{2}\to
1$. Moreover, it appears that the oscillating nature of $\mathcal{E}^{(2)}$
has virtually no effect (apart from numerical fluctuations) on $t_{peak}$. In
contrast, the long-range character of $\mathcal{E}^{(3)}$ leads to an
approximate linear increase:

$t_{peak}\approx\frac{L_{D}}{V}\left(0.55+0.7\left(\frac{L_{I}}{L_{D}}\right)\right).$

(a)  (b)

(c)  (d)

Figure 7: $\alpha$ anomalous exponent for the correlation classes
$\mathcal{E}^{(2)}$ (red) and $\mathcal{E}^{(3)}$ (blue) vs their integral
correlation length. Note the error bars are estimated from the amplitude of
numerical fluctuations present in the tail of the diffusion coefficients.

The maximal value of the diffusion, $D_{peak}$, is influenced both by
oscillations and long-range features of the space correlations. The
dependencies represented in Fig. (7b) resemble a relatively flat quadratic
function, thus we estimate:

$D_{peak}\approx
VL_{I}\left(1.63-1.27\left(\frac{L_{I}}{L_{D}}\right)+0.3\left(\frac{L_{I}}{L_{D}}\right)^{2}\right).$

Finally, we extract from the numerical data the percolative exponent $\alpha$
which dictates the long-time behaviour of diffusion $D_{x}(t)\approx
V^{\alpha}L_{D}^{2-\alpha}t^{\alpha-1}$ and find that it has almost linear
dependencies with the integral scale $L_{I}$ and the maximal diffusion
$D_{peak}$, but with different slopes for $\mathcal{E}^{(2)}$ and
$\mathcal{E}^{(3)}$. We approximate $\alpha\approx 0.423+0.128L_{I}/L_{D}$ for
oscillating $\mathcal{E}^{(2)}$ and $\alpha\approx 0.502+0.065L_{I}/L_{D}$ for
long-range $\mathcal{E}^{(3)}$, while $\alpha\approx
0.98-0.62D_{peak}/(VL_{D})$. Note that error bars are represented in Figs.
(7c),(7d) as an estimation of the numerical error induced by the finite
statistical ensemble of trajectories and the fitting procedure.

###  III.3 Time dependent turbulence

We consider now the more complex case of time-dependent turbulent fields, when
$\tau_{c}=$ finite. The time-decorrelation leads to pure diffusive features,
i.e. $\lim\limits_{t\to\infty}D_{x}(t)=$ finite. The time-dependent $D_{x}(t)$
can be seen in Figs. (8a)-(8c) for all three correlations
$\mathcal{E}^{(1-3)}$ at different values of the parameters $\mu,\nu$ and
setting $\lambda_{c}=1$. All profiles resemble the generic data shown in Fig.
(5a). The case $\tau_{c}=500$ was chosen as representative for the strong
turbulent (percolative) regime. In practice, a large span of differential
correlations times $T_{D}=\tau_{c}\in(0.15,10^{3})$ were simulated. We show in
Fig. (9a) the diffusion coefficients $D_{x}(t)$ for the same correlation
$\mathcal{E}_{5}^{(3)}$ but for different $\tau_{c}$ values. At small times
$t\ll t_{peak}$ the diffusion is linear $D_{x}(t)=V^{2}t^{1}$ and independent
of $\tau_{c}$, while at larger times it saturates after roughly $t\sim T_{I}$.

(a)  (b)  (c)

Figure 8: Running diffusion profiles for all correlations
$\mathcal{E}^{(1-3)}$ at different $\mu,\nu$ values at $\tau_{c}=500$. (a)
Figure 9: Running diffusion profiles for different $\tau_{c}$ values in the
case of $\mathcal{E}^{(3)}_{5}$.

In many systems of interest (fusion plasmas, astrophysical magnetic fields,
etc.) there is a clear separation of scales between: the time scale $\sim
T_{I}$ of turbulent transport is much smaller than macroscopic, experimentally
relevant, time-scales of dynamics. Because of that, only the asymptotic
diffusion $D_{x}^{\infty}$ is of practical relevance. Consequently, we shall
focus on $D_{x}^{\infty}$ from now on.

(a)  (b)  (c)

Figure 10: Asymptotic diffusion coefficients $D^{\infty}_{x}$ at various
differential Kubo numbers $K_{D}$ and integral correlation lengths/times
$L_{I},T_{I}$

In Figs. (10a)-(10c) we show how $D_{x}^{\infty}/K_{D}$ is dependent on the
differential Kubo number $K_{D}$, for all three classes of correlations. The
quasilinear behaviour $D_{x}^{\infty}=V^{2}T_{I}$ can be observed in the
region $K_{D}\ll 1$ for all cases. A maximum of $D_{x}^{\infty}/K_{D}$ is
present at $K_{D}\approx T_{I}/T_{D}$, then the profile decays algebraically
in the strong turbulent regime $K_{D}\gg 1$.

Regarding the correlation $\mathcal{E}^{(1)}$ it is interesting to note that
the $D_{x}^{\infty}$ profiles are different only in the quasilinear regime,
while beyond $K_{D}>1$ become identical. This suggests that in strong
turbulence the specific details of time-decorrelation are not important, only
those of space correlations.

Regarding $\mathcal{E}^{(2)}$, it appears that the space oscillations of the
correlation function are reflected in the $D_{x}^{\infty}$ profile in a
qualitatively similar manner with the frozen $D_{x}(t)$ (Fig. (6a). The
position of the maxima is roughly stable to correlation changes, but its value
it is not. Moreover, in the strong regime, $K_{D}\gg 1$, the slope is
dependent on $L_{I}$ (or, equivalently, $\mu$). Similar observations can be
drawn for the case of $\mathcal{E}^{(3)}$ which mimics the frozen time
profiles.

(a)  (b)

Figure 11: The dependence of the anomalous exponent $\gamma$ with the times-
space integral scales. First figure is for $\mathcal{E}^{(1)}$ and the second
for $\mathcal{E}^{(2-3)}$. Figure 12: The dependence between the anomalous
exponents $\gamma$ from the time-decorrelated case and the exponent $\alpha$
from the frozen case for all correlation classes $\mathcal{E}^{(1-3)}$.

Our results confirm the algebraic behaviour of the asymptotic diffusion
(percolative) with the differential Kubo number as
$D_{x}^{\infty}=AK_{D}^{\gamma}$. Moreover, analyzing the numerical data, we
have found that the proportionality constant is exactly
$A=\lambda_{c}^{2}/\tau_{c}=L_{D}^{2}/T_{D}$. Using the exact quasilinear
result $D_{x}^{\infty}=V^{2}T_{I}$ we can provide a gross estimation for
diffusion profiles in all regimes:

$D_{x}^{\infty}(K_{D})\approx\frac{L_{D}^{2}}{T_{D}}\frac{T_{I}K_{D}^{2}}{T_{D}+T_{I}K_{D}^{-\gamma+2}}$

We explore further the dependence (universality) of the $\gamma$ exponent on
turbulence features, in particular, the integral scale length $L_{I}$ (eq.
(7)). We plot the results in Figs. (11a)-(11b), assisted by error bars that
indicate the numerical error due to statistical fluctuations and interpolation
procedures. The qualitative observation from Fig. (10a) that
$\mathcal{E}^{(1)}_{\nu}$ has no impact on the slope of
$D_{x}^{\infty}(K_{D})$ at large $K_{D}$ is confirmed in Fig. (11a) where the
value $\gamma\approx 0.718$, very close to Isichenko’s value $\gamma=7/10$,
has been found for all $T_{I}$.

The dependence of $\gamma$ with the integral scale $L_{I}$ shown in Fig. (11b)
reveals a coherent monotonically dependence $\gamma(L_{I})$. For both classes
of space correlations $\mathcal{E}^{(2-3)}$, the dependence is linear, but
with different slopes, similar with the dependence of the $\alpha(L_{I})$
exponent.

It is worth mentioning that the dependence of various quantities
$\alpha,t_{peak},\gamma$ (Figs. (7a),(LABEL:4c) and (11b)) with the integral
scale length $L_{I}$ is not coherent between $\mathcal{E}^{(2)}$ and
$\mathcal{E}^{(3)}$ (different slopes). This suggests that $L_{I}$ is not the
best universal measure of turbulent features and something more intricate
should be searched for. This could be the purpose of a future work.

Finally, we can ask for the relation between asymptotic diffusion in the
decorrelated case, $\tau_{c}=$ finite, and the percolative limit,
$\tau_{c}=\infty$. If an analytical relation would be available, that could
enable the evaluation of diffusion coefficients in a large range of cases
solely from the frozen profile. Unfortunately, we have not been able to find
any simple relation between $D_{x}(t\to\infty;\tau_{c})$ and
$D_{x}(t;\tau_{c}\to\infty)$. The only observation, important as it is, can be
seen in Fig. (12) where we plot the $\gamma$ exponent for the strong turbulent
regime in the time-decorrelated case vs. the $\alpha$ exponent for the long-
time behaviour of diffusion in the frozen turbulent limit, i.e.
$D_{x}^{\infty}\sim K_{D}^{\gamma},D_{x}(t)\sim t^{\alpha-1}$. A linear
dependence is found: $\gamma\approx 0.367+0.6\alpha$. This implies that we can
estimate $D_{x}^{\infty}$ as:

$\displaystyle D_{x}^{\infty}(K_{D})$
$\displaystyle=D_{x}^{frozen}\left(t=T_{D}K_{D}^{-\alpha/\gamma}\right)$ (10)
$\displaystyle\gamma$ $\displaystyle=0.367+0.6\alpha.$ (11)

##  IV Conclusions

In the present work we have investigated the features of diffusive transport
in two-dimensional incompressible turbulent fields. The turbulence is
described by its spectrum, or, equivalently, its correlation function. In
order to encompass as general features as possible, we have considered three
distinct classes of correlations with: long-range time dependencies
$\mathcal{E}^{(1)}$, space oscillations $\mathcal{E}^{(2))}$, or long-range
space dependencies $\mathcal{E}^{(3)}$. Employing the method of direct
numerical simulations, the transport coefficients have been computed on a
large span of physical regimes. We have used unprecedented numerical
resolution in order to capture in great detail any small effects induced by
the geometrical attributes of turbulence.

For the frozen regime of time-independent turbulence we have numerically
confirmed that the diffusion has a linear behaviour at small times, while
being anomalous ($\sim t^{\alpha-1}$) in the long-time limit. We have found
that both the maximal diffusion $D_{peak}$ and the slope of the algebraic
decay $\alpha$ are dependent on the correlation function. The analytical and
approximate results for the frozen regime are summarized in Table (1).

percolative | $D_{x}(t)$ | $D_{peak}/(L_{I}V)$ | $\alpha$
---|---|---|---
$\tau_{c}=\infty$ | $V^{2}t^{1},t\ll t_{peak}$ | $1.63-1.27L_{I}+0.3L_{I}^{2}$ |  $0.423+0.128L_{I}$, for $\mathcal{E}_{\mu}^{(2)}$
| $\lambda_{c}^{2-\alpha}V^{\alpha}t^{\alpha-1},t\gg t_{peak}$ | $1.63-1.27L_{I}+0.3L_{I}^{2}$ |  $0.502+0.065L_{I}$, for $\mathcal{E}_{\mu}^{(3)}$
Table 1: Summary of analytical and approximate results for the case of frozen
turbulence.

For the time decorrelated case we have confirmed that diffusion is roughly
$V^{2}t^{1}$ at small times, then saturates to finite values at longer times.
The dependence of asymptotic diffusion $D_{x}^{\infty}$ on Kubo numbers
indicates that in the quasilinear case $D_{x}^{\infty}=V^{2}T_{I}$. In the
percolative regime, the dependence of diffusion on $K_{D}$ is algebraic
$K_{D}^{\gamma}$. The critical exponent $\gamma$ is insensitive to the
features of time decorrelation but is linearly dependent on the integral space
scale. Furthermore, $\gamma$ increases linearly with the $\alpha$ exponent.
The analytical and approximate results for this case are summarized in Table
(2).

decorrelated | $t\ll t_{peak}$ | $D_{x}^{\infty},K_{\star}\ll 1$ | $D_{x}^{\infty},K_{\star}\gg 1$ | $\gamma$
---|---|---|---|---
$\tau_{c}=$ finite | $D_{x}(t)=V^{2}t^{1}$ | $V^{2}T_{I}=L_{I}^{2}/T_{I}K_{I}^{2}$ | $L_{D}^{2}/T_{D}K_{D}^{\gamma}$ |  $0.665+0.04L_{I}$, for $\mathcal{E}_{\mu}^{(3)}$
|  | $V^{2}T_{I}=L_{D}^{2}/T_{D}^{2}T_{I}K_{D}^{2}$ |  |  $0.605+0.09L_{I}$, for $\mathcal{E}_{\mu}^{(2)}$
Table 2: Summary of analytical and approximate results for the decorrelated
regime.

Finally, simple approximate, analytical, formulas for the relation between
diffusion and Kubo numbers are shown here for practical purposes.

$\displaystyle D_{x}^{\infty}(K_{D})$
$\displaystyle=\frac{L_{D}^{2}}{T_{D}}\frac{\left(\frac{T_{I}}{T_{D}}\right)K_{D}^{2}}{1+\left(\frac{T_{I}}{T_{D}}\right)K_{D}^{2+\gamma}}$
(12) $\displaystyle D_{x}^{\infty}(K_{I})$
$\displaystyle=\frac{L_{I}^{2}}{T_{I}}\frac{K_{I}^{2}}{1+\left(\frac{T_{D}}{T_{I}}\right)^{\gamma+1}\left(\frac{L_{I}}{L_{D}}\right)^{\gamma+2}K_{I}^{2+\gamma}}.$
(13)

We acknowledge that the linear interpolation of critical exponents
$\alpha,\gamma$ as functions of $L_{I}$ (Figs. (7c), (11b)) is not fully
coherent between long-range and oscillating correlation functions
$\mathcal{E}^{(3)},\mathcal{E}^{(2)}$. This suggests that a more appropriate
global measure of Eulerian turbulent spectra is possible, beyond integral
scales $L_{I}$, and are to be sought in future works.

## Acknowledgements

This research was partially supported by Romanian Ministry of Research,
Innovation and Digitalization under Romanian National Core Program LAPLAS VII
– contract no. 30N/2023.

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