Asymptotic diffusion coefficients and anomalous diffusion in a meandering jet flow under environmental fluctuations

Abstract

The nontrivial dependence of the asymptotic diffusion on noise intensity has been studied for a Hamiltonian flow mimicking the Gulf Jet Stream. Three different diffusion regimes have been observed depending on the noise intensity. For intermediate noise the asymptotic diffusion decreases with noise intensity at a rate which is linearly dependent to the flow's meander amplitude. Increasing the noise the fluid transport passes through a superdiffusive regime and finally becomes diffusive again at large noise intensities. The presence of inner circulation regimes in the flow has been found to be determinant to explain the observed behavior.

Full text

PHYSICAL REVIEW E 85, 017201 (2012)
Asymptotic diffusion coefficients and anomalous diffusion in a meandering jet flow
under environmental fluctuations
A. von Kameke, F. Huhn, and V. P´
erez-Mu˜
nuzuri*
Group of Nonlinear Physics, University of Santiago de Compostela, E-15782 Santiago de Compostela, Spain
(Received 26 October 2011; revised manuscript received 16 December 2011; published 27 January 2012)
The nontrivial dependence of the asymptotic diffusion on noise intensity has been studied for a Hamiltonian
flow mimicking the Gulf Jet Stream. Three different diffusion regimes have been observed depending on the
noise intensity. For intermediate noise the asymptotic diffusion decreases with noise intensity at a rate which
is linearly dependent to the flow’s meander amplitude. Increasing the noise the fluid transport passes through
a superdiffusive regime and finally becomes diffusive again at large noise intensities. The presence of inner
circulation regimes in the flow has been found to be determinant to explain the observed behavior.
DOI: 10.1103/PhysRevE.85.017201 PACS number(s): 05.45.−a, 47.51.+a, 92.10.Lq, 47.52.+j
In the ocean detailed knowledge of the dispersion of passive
tracers is crucial in order to make predictions about the trans-
port of physical, chemical, and biological tracers. Important
processes range from heat and mass transfer, transport in
global biochemical cycles, or the spread of pollutants [1]to
the transport of plankton [2] and fish larvae [3]. Especially
in coarse climate models, transport on smaller scales has to
be parametrized and statistical transport measures such as the
eddy diffusivity are common [4].
Normal diffusion is a special case of diffusive transport and
a simple model that does not always capture the behavior of
real processes correctly. In general, diffusion can be defined
in terms of the relative dispersion of diffusing particles which
has a time dependence of the form
R2(t)=|xi(t)−xj(t)|2∼tγ,(1)
at long times. For normal diffusion γ=1, while for γ>1
the Lagrangian dispersion is considered as superdiffusion
[5]. Power-law anomalous superdiffusion (γ>1) can occur
in developed turbulence [6] and in chaotic flows [7]. The
subdiffusive case (γ<1) is found in motion through highly
heterogeneous media where particles can be trapped in certain
regions for long periods of time, such as transport in porous
media, gel electrophoresis of polymers (e.g., DNA), and
soluble proteins in the nucleus of living cells [8].
It has been shown that for long times and under the
assumption of an underlying stationary stochastic process, the
presence of a weak fluctuation causes anomalous diffusion
to asymptotically become normal at crossover times tc. These
crossover times are inversely proportional to the noise intensity
ξ, i.e., tc∝ξ−βwith some exponent βof the order of unity [9].
In terms of the waiting time distribution the transition to
normal diffusion is reflected by an exponentially decreasing
distribution ψ(t)∼exp(−ξt)/tμ[9]. Consequently, as the
noise vanishes ξ→0, the exponential term becomes one and
the crossover time goes to infinity. In the limit t→∞and for
finite noise ξthe diffusion process is normal and the asymptotic
diffusion coefficient
DA=lim
t→∞
R2(t)
t(2)
*vicente.perez@cesga.es
can be determined. This coefficient has been shown to depend
on noise intensity as
DA∼1
ξα(3)
for ξ1, while α=f(β) depends on the used model [9–11].
However, taking the limit ξ→0, DAgives an infinite result.
Karney et al. [12] showed that exchanging the limits, i.e.,
taking ξ→0 before t→∞ in Eq. (2), the value of DA
depends on the initial location of the tracers. This dependence
of DAon initial conditions is especially pronounced in flows
with circulation regimes where tracers may be trapped. In
particular, the relative distance in between these sticky trapping
regions and the position and size of initial tracer patches turn
out to influence the asymptotic value of DA[12].
In this Brief Report, we numerically study this limit and
show that Eq. (3) does not always hold. For that purpose, we
determine the effect of additive noise on the transport in a
Hamiltonian system leading to a nontrivial dependence of the
asymptotic diffusion coefficient on the noise intensity ξ.
The meandering jet flow was introduced as a simple
kinematic model for the Gulf Stream and is frequently used
to describe western boundary current extensions in the ocean
[13,14]. This flow is usually represented in a reference frame
ζ=x−cxt,δ=ymoving with the phase velocity cxof the
meander, where xand yare Cartesian coordinates, positive
eastward and northward, respectively. Then, the corresponding
nondimensional stream function can be written as
ψ(x,y)=−tanh y−Bcos κx
R1/2+Cy, (4)
where (x,y)=(ζ,δ)/λ are the dimensionless coordinates. The
velocity field 
V(x,y) in this moving frame is obtained from
the stream function as 
V(x,y)=ez×
∇ψ.
We consider an additional noise perturbation η(t) that mim-
ics environmental fluctuations, and we arrive at the following
coupled differential equations for the particle velocity:
˙
x=1
R1/2cosh2θ−C+η(t),
(5)
˙
y=−
Bκ sin κx(1 +B2κ2−Byκ2cos κx)
R3/2cosh2θ+η(t),
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BRIEF REPORTS PHYSICAL REVIEW E 85, 017201 (2012)
TABLE I. List of parameters used in the Gulf Stream model.
ψ04000 km2d−1
λ40 km
cx20 km d−1C0.2
A80 km B2.0
L266.67πkm κ0.3
with
R=1+(Bκ sin κx)2,θ=y−Bcos κx
R1/2.
Here η(t) is a Gaussian stochastic process with zero mean,
a temporal correlation function η(t)η(t)=2ξδ(t−t), and
noise intensity ξ.
The original parameters of the model are the total eastward
transport 2ψ0, the width λ, the amplitude A, and the wave num-
ber k=2π/L of a sinusoidal meander. These are converted
to dimensionless quantities B=λ−1A,κ=kλ,C=λψ−1
0cx,
and the dimensionless time t=ψ0λ−2t. The parameters of the
model are chosen such that they properly represent geophysical
features of the Gulf Stream. Table Isummarizes these values.
Equations are then integrated using the Heun scheme [15] and
a time step of t =10−3.
Figure 1shows the jet flow in the comoving frame for
this set of parameters. This flow can be divided into three
distinct regimes: a central meandering eastward jet (A), closed
circulation above meander troughs and below crests (B), and
exterior retrograde westward motion (C). Fluid particles are
restricted to each of these regimes and no cross-stream mixing
occurs. In the presence of an added temporal variability, the
boundaries between regimes break up, allowing transport and
exchange of fluid between regimes. For the jet amplitude
B>B
crit, the geometry of the flow changes drastically and
C
B
A
x
y
−15 −10 −5 0 5 10 15
−10
−5
0
5
10
FIG. 1. Stream function of the jet flow (4). Arrows indicate the
direction of motion and dots correspond to the different regimes in the
flow for the jet amplitude B<B
crit. The high density of contour lines
indicates the position of the central eastward jet. The cross indicates
the position of the stationary point inside the inner circulation core.
The domain shown in the figure is extended periodically in the x
direction. Set of parameters as in Table I.
the central meandering jet moves westward. For the set of
parameters indicated in Table I,Bcrit ≈3[14].
For the numerical experiments, initially a cluster of
N=500 particles separated from each other by a distance
δ0=10−3was located at the stationary point inside the closed
circulation core (B) (x0,y0)=(0,B −cosh−1C−1/2). Then,
the temporal evolution of the relative dispersion R2(t)(1)
of this cluster is calculated for times up to t=107.Itwas
found that the statistical properties calculated throughout this
paper do not depend on the number of tracers, provided that
the corresponding computation time is sufficiently long.
The time evolution of the relative dispersion for three noise
intensities is shown in Fig. 2(a). We estimate the power-law
growth at large times for different noise intensities. Figure 2(b)
shows the values of the exponent γin Eq. (1) as a function
of the noise intensity, calculated for two periods of time. Our
results show that the exponent γ>1 for intermediate noise
intensities, while for small and large values of noise, γ=1.
For large noise 1/ξ →0 the exponent evolves as γ→1
because inner cores become blurred by noise and diffusion
is dominant. For intermediate noise intensities some particles
intermittently enter within the inner cores where they may
remain for some time, while others travel long distances and
this causes R2(t)to increase superdiffusively [16]. The very
high values of γup to 10 for intermediate noise intensities
in Fig. 2(b) can be understood looking at the two curves (red
squares and green triangles) in Fig. 2(a). Both curves show a
type of transitory behavior with a high slope. This transition
is expected to appear for all noise values, i.e., even for an
arbitrary long integration time of the system there is always a
noise intensity, such that the transition occurs at the end of the
simulation. Thus, the peak in Fig. 2(b) shifts to lower noise
intensities (right of the figure) as integration time increases
[cf. the two curves in Fig. 2(b)]. For small noise 1/ξ →∞
the exponent evolves as γ→1 because particles have not yet
left the inner circulation cores, i.e., the typical waiting time is
longer than the simulation time.
Theoretically, it is expected that in the limit t→∞
the diffusion should become normal and γ→1 for any
noise intensity. However, we find that for intermediate noise
intensities this limit may be well beyond an experiment’s
lifetime and was not approached in this numeric study. When
the asymptotic case of normal diffusion is not reached,
DAas defined in Eq. (2) is not a constant value and we
estimate a lower bound for DAat the maximum simulation
time t=107.
The asymptotic diffusion coefficient DAmeasured as a
function of the noise intensity is shown in Fig. 3.Forvery
large diffusion (regime HN), the trivial relation DA∝ξis
observed, as the random walk of the particles is dominated by
the added noise. For intermediate noise intensities (regime IN),
DAscales with the noise intensity as in Eq. (3)[9,10]. In
between these two regimes, DA∝ξ(HN) and DA∝ξ−α(IN),
a local minimum at ξ∼10−0.5is observed. This minimum
in diffusion value is especially striking as it neither depends
on the model parameters nor on the initial particle positions.
Finally, as noise intensity decreases, DAreaches a maximum
value Dmax
Athat depends on the jet amplitude B. In the limit
of vanishing noise 1/ξ →∞,DA∝ξfor any value of B.
For computing times greater than 107, we reproduce the
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BRIEF REPORTS PHYSICAL REVIEW E 85, 017201 (2012)
10−2 102106
10−4
104
1012
(a)
t
<r2(t)>
10−2 1021061010
0
5
10
15
(b)
1/ξ
γ
FIG. 2. (Color online) (a) Relative dispersion R2(t)for three values of noise intensity. Solid lines correspond to the best fitting to Eq. (1)
to extract γ. Noise intensities: ξ=10−9(blue circles), ξ=10−5(red squares), and ξ=10−2(green triangles). (b) Exponent γas a function
of noise intensity for two intervals of time, t∈[7 ×105,1×107] (blue circles) and t∈[2 ×105,6×105] (red squares). Lines indicate normal
(- -) and ballistic (·-) dispersion. Rest of parameters as in Table I.
same qualitative behavior although the maximum DAvalue
is displaced toward small noise intensity values (shifted to the
right in Fig. 3).
Increasing the noise, the boundaries between the jet flow
regimes described in Fig. 1become permeable to particle
crossings due to the random component η(t) and η(t)in
the equations of motion. The exchange process is similar
to the one described in Ref. [17] where lobes of fluid are
entrained and detrained from the edge of the jet. The onset
for this particle exchange occurs at a finite nonzero value
of the perturbation [14] and above it anomalous transport
develops. This onset displaces toward lower nonzero values
of noise as time increases and Dmax
A∝t. This is an important
point because it implies that Eq. (3) does not hold for small
noise intensities since we find that limt,ξ DA<D
max
A,ifξ→0
before t→∞.
10−2 10 210 61010
10−8
10−4
100
104
HN IN LN
1/ξ
DA
FIG. 3. (Color online) Asymptotic diffusion coefficient DA(2)at
time t=107as a function of noise intensity for different values of
the jet amplitude. From left to right, solid lines correspond to DA∼ξ
and DA∼ξ−αwith α=0.90 ±0.03. Dashed vertical lines separate
regions for high (HN), intermediate (IN), and low (LN) noise. Set
of parameters: B=1 (blue circles), B=3 (red squares), and B=5
(black stars). Rest of parameters as in Table I.
The effect of the jet amplitude Bon the asymptotic diffusion
coefficient Dmax
Ais analyzed in Fig. 4. Note that Dmax
Adecreases
as a power law with increasing jet amplitude Band the scaling
factor αin Eq. (3) linearly decreases with B,asshownin
Figs. 4(a) and 4(b), respectively. From panel (a) we also note
that Dmax
Agrows with time tby the same factor for any value
of B, and we find that Dmax
A∝t/Bχ(χ≈1).
Finally, the influence of the initial tracer positions (x0,y0)on
the diffusion was analyzed. As expected, for ξ→∞,DA∼ξ
independently of (x0,y0). However, as predicted by Karney
et al. [12], for ξ→0, DAdepends on the initial conditions
and attains a constant value for (x0,y0) values far from the
inner circulation core. In those regions, the laminar peripheral
westward currents do not disperse the tracers and R2(t)≈
const. For initial conditions inside the circulation core or at
the jet stream, the results do not differ qualitatively from those
described here.
To conclude, in this Brief Report we have analyzed the
nontrivial dependence of the asymptotic diffusion coeffi-
cient DAon noise mimicking environmental fluctuations in
a kinematic model of the Gulf Stream. Above a certain
nonzero value of noise intensity the closed flow regimes
become permeable to particle crossings which explains the
presence of a maximum asymptotic diffusion coefficient. For
intermediate noise intensities, DA∼ξ−αand the transport
becomes superdiffusive [9,10,16]. However, as an important
result we find that this dependence does not hold for ξ→0.
The scaling coefficient αwas found to decrease with the
jet amplitude. Finally, for large noise intensities DA∼ξas
expected. The transport at large times, diffusive for small
noise intensities, becomes superdiffusive at intermediate noise
intensities and then normal at large noise. The presence of
a trapping remnant regular region in the flow for low and
intermediate noise intensities turned out to be important to
explain the nontrivial dependence of the asymptotic transport
on the perturbation strength. For very small noise we did not
find superdiffusive behavior within the computing times used
in this Brief Report (∼104years). The results even remain
unchanged when B>B
crit and the direction of the jet is
reversed, which indicates that they might be applicable to other
flows consisting of jets and trapping regions.
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BRIEF REPORTS PHYSICAL REVIEW E 85, 017201 (2012)
0 5 10
102
104
106
(a)
B
DA
max
0 5 10
0.6
0.8
1
(b)
B
α
FIG. 4. (Color online) Logarithmic plot of the maximum value of DAat two instants of time [t=7×105(red squares) and t=1×107
(blue circles)] (a) and the scaling coefficient α,Eq.(3) (b) as a function of the jet amplitude B. Parameters as in Table I.
We are confident that in real experiments with comparably
short lifetimes, the three described diffusion regimes can be
observed. The jet stream model can not only be used to describe
the Gulf Stream, but also others like the Kuroshio current [18]
or the polar jet in the atmosphere. Our results may therefore
help to interpret and predict the transport of tracers in a jet
which is subject to environmental fluctuations.
This work was supported by Ministerio de Educaci´
on
y Ciencia and Xunta de Galicia under Research
Grants No. FIS2007-64698, No. FIS2010-21023, and No.
PGIDIT07PXIB-206077PR. The computational part of this
work was done using CESGA computer facilities. A.v.K. and
F.H. receive funding from FPU Grants No. AP-2009-0713 and
No. AP-2009-3550.
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