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Anomalous diffusion and anomalous stretching in vortical flows ∗
320
Fluid Dynamics
Kesearch
3 (1988) 320-326 North-Holland
Anomalous diffusion and anomalous stretching in vertical flows * Ruben
A. PASMANTER
The traJectortes of partlclea advected by non-random spatially periodic velocity fields may he chaotic, i.e.. extremely sensitive to their initial positions. In the most general case. the area of a cloud of advected particles grows like (time)” where the exponent 0 ranges between 0 and 2 (shear stretchmg). Thew ohsxvationa could explain man) known facts about dlsperbion procew\ in shallw tidal seas.
Abswact.
1. Introduction
Measurements of the dispersion of substances in the sea show an amazing variability [l]: patchiness, i.e.. parts of the pollutant cloud that disperse much slower than the rest of the cloud, as well as slopes ranging from 1 up to 3 in logglog plots of the growth rate of clouds. see fig. 1. In order to better understand the observed variability of dispersion processes in shallow seas, 1 have been studying the passive advection of particles by non random. tidal (i.e.. oscillatory) flows with spatial inhomogeneities due to the bottom topography of the basin. A typical velocity field of this class is: u\ = R,(t)
sin(k,js).
+A,(t)
(la)
r!,.=R,(t)+A,(t)sin(k,x),
(lb)
where the time dependent amplitudes R(t) and A(t) contain not only the fundamental (tidal component with frequency Jz but also harmonics: in particular. they have also time-independent components R,, and A,,: R(t)=R,,+R,(r)+R2(t)+ A(t)
=A,,+A,(t)
+/l:(r)
...
(2a)
+ .“,
(33)
A, and R, oscillate at the fundamental frequency 0, A1 and R2 at 2L’. etc. The first terms on the r.h.s. of (la) and (lb) describe a spatially homogeneous tide. while the second terms correspond to eddies that are attached to the bottom topography. The wave-vector k corresponds to the largest component of the bottom topography. The velocity field (1) is incompressible and is closely related to the ABC (Arnol’d~Beltrami&Childress) flows. The ABC flows are spatially periodic stationary solutions * Participation Fundamental
0169-5983/8X/$2.75
in this Conference Research of Matter
has been possible thanks to the financial support and the IUTAM Local Organizing Committee.
‘: 1988. The Japan
Society
of Fluid Mechanic\
of the Dutch
foundation
of
R.A. Pasmanier
/ Anomalous
diffusion and stretching in vorticaljlows
0
?d3 10
I2
321
C
2
Gxy cc t
2.07
m”
109
10*
10’ G:y [m*l
10’
106
t
70” lo5
10‘3
‘4 70
705 I5 70 -
)
7u6 TIME
70’ [s]
Fig. 1. Logarithm of the average squared size of pollutant clouds in the sea as a function of the logarithm of time. In shallow seas, (B) in English estuaries, (C) in open sea, (D) in American coastal waters, (E) in American estuaries, in the Baltic coast and (G) in Fjords. Taken from ref. [l]. Notice the different slopes.
(4 (F)
of Euler’s equations; moreover, they are Beltrami flows, i.e., the vorticity is everywhere parallel to the velocity field. Numerical calculations [2,3] indicate that the paths of particles advected by ABC flows can be chaotic; Yoshida [4] has given a system of six equations that is equivalent
R.A. Pasmanier
322
/ Anomalous
to the ABC flows. The results presented not only for the velocity
diffusion
and stretching
in vertical flows
below are, unless otherwise
field (1) but also for the ABC
indicated,
qualitative
valid
and similar flows.
2. Particles’ paths and their computation The paths covered by passive particles advected by the velocity the following system of ordinary nonlinear differential equations:
+,4.(t)
dx/dt=R,(t) dy/dt=R,,(t)+A,.(t) These
equations
subroutine.
of
sin(k,y),
(3a)
sin(k,x),
(3b)
using, e.g., a high-order
Adams
predictor-corrector
IMSL
However, every numerical method of integration introduces not just errors but errors. In our case, these errors may lead to the opening of paths that are in fact
systematic closed
can be integrated
field (1) are the solutions
and to the introduction
errors can be reduced
of spurious
sinks and sources
but at the cost of long computation
in the incompressible
flow. These
times.
In order to avoid extremely long computation times, I replaced the trigonometric functions in (3) by piecewise linear sawtooth functions. The integration over each linear piece can be done
analytically;
the
remaining
numerical
polynomials. The piecewise linear (PL) obtained from the numerical integration Given
an initial position
[x(O),
task
reduces
to the
solution
of
second-order
model is used to check of (3).
the correctness
of the results
y(O)] and taking into account
the periodicity
in time of the
flow, we plot the successive positions of the same particle after one tidal period [x(l), y(l)], two tidal periods [x(2), y(2)], etc. Similarly, we exploit the spatial periodicity of the flow in order to plot the positions modulo a wave-length; in this way we can keep track of particles that
move
far away
from
their
initial
positions
without
having
to enlarge
the scale
of our
graphs. The set of tidal positions of a particle [x(O), one-dimensional curve. Notice that two different different
initial positions)
y(O)], [x(l), y(l)], etc, may form a smooth curves (corresponding to two particles with
can never intersect.
3. Main results In spite of the very simple form of the field (1) the paths of the advected particles astonishingly broad spectrum of possibilities. Figs. 2a and 2b show two characteristic the same
velocity
field, the only difference
between
show an plots for
the plots being the initial positions
of the
particles. In fig. 2a one sees a smooth curve formed by a number of islands: the particle moves from one island to the other after one tidal period. By contrast, in fig. 2b one sees a path that seems to cover a two-dimensional The most important
observed
Non dispersive and semi-dispersive
area. features
can be understood
in terms of four basic mechanisms:
patches
As already stated, different paths can never intersect. This means that the particles inside the islands in fig. 2a can never cross the boundary of such an island, i.e, they form a permanent
block or patch. It should be noted that in certain cases an area is surrounded by a smooth curve with very small openings; in these cases a particle that starts in the surrounded area needs a very long time until it hits one of the small openings and leaks out. In other words, the field can also generate leaky or semi-dispersive patches.
R.A. Pasmanter
/ Anomalous
diffusion and stretching in vertical flows
323
Fig. 2. (a) Eight hundred positions of one particle at times 1, 2,. _,800 (in units of the tidal period) form seven smooth islands. With each tidal period, the particle moves from one island to the next one. The positions are plotted modulo the wavelength; (b) Same velocity field as in a) but with a different initial position. Four thousand tidal positions tend to cover a two-dimensional area. Both in a) and b) the nonvanishing dimensionless parameters of the velocity field are R, = (8, S), R, = (0.025.0) and A, = A, = (0.9,0.9).
Moreover, patches may be drifting or nondrifting, nonchaotic) paths or both regular and chaotic ones.
they
may
contain
only
regular
(=
Chaotic trajectories with properties similar to those of a random walk As seen in fig. 2b, some paths tend to cover a two-dimensional area. These paths have another surprising characteristic: they are chaotic. This means that the separation between two
4 8C
‘I
324
168
012
144
300
457
613
0 00
098
195
233
390
488
585
6.83
Fig. 3. Logarithm of the average squared size of a cloud of four hundred particles as a function of the logarithm of time. The non-vanishing parameters of the velocity field are R, = (8, 8), R,(0.425,0) and A 1= A, = (0.9.0.9). For these values for the parameters, the chaotic region occupies a// space. The slope of the straight line is very close to 1, like in Browniantype diffusion.
R.A. Posmanter
324
/ Anomrrlous
diffusronmd stretchingin uortrral flows
particles that initially have been (infinitely) close to each other, grows exponentially be more precise, for these chaotic paths one has that lim tC’ lnD:in I+ra
o [ D( t)/D(O)]
in time. To
= p > 0.
(4)
where D(t) is the separation between the two particles at time t. The rate II. is called the Lyapunov exponent (or Lyapunov number); only when p is positive we say that the trajectory is chaotic. Notice that the value of the Lyapunov number may depend upon the direction of D(O). By changing the parameters R,(t), A,(t), R, and A,. a situation may be reached in which the chaotic paths cover all the available space and no patches are observed. Under these circumstances another striking manifestation of the chaotic character of these paths manifest itself: the average squared size of a cloud of particles grows linearly in time, i.e., just like in the case of a cloud of Brownian particles performing random walks. This behavior is shown in fig. 3. Effective shear When the time-independent homogeneous component of the velocity field vanishes, i.e., R, = 0, there is no long-term preferred direction so that no long-term drift is possible. When R, does not vanish, however, the symmetry of the system is broken and a long-term drift along the preferred direction of R, is possible. Since the velocity field is not homogeneous, different positions in space are not equivalent and one may observe diff erent long-term drifts associated with different initial positions. This means that in such cases one has an effective (or Lagrangian) periodic shear that leads to stretching of the cloud along the direction of Ro; the average squared cloud size along the preferred direction grows quadratically in time. It should be mentioned that all paths in an island have the same drift velocity; similarly all paths in a connected chaotic region have the same drift velocity. Consequently, when all space is occupied by chaotic trajectories and no regular ones are present, no shear is possible; this is the case in fig. 3.
Fig. 4. Logarithm
+ +
f
+a+
x
size of a cloud
^ _ : I
particles
x
”
276
5 53
829
of the average
of thousand
in the chaotic
as a
function
After
4000
of the
region
straight
lines
(squared
size along
and 0.75
(squared
perpendicular
are
of fig. (2b)
logarithm
tidal periods
squared
five hundred of time.
the slopes
approximately the direction
of the 1.4 fo R,,)
size long the direction
to R,).
R.A. Pasmanter
/ Anomalous
drffwion
and stretching in uor~rcalflows
325
Anomalous diffusion and anomalous stretching When both regular and chaotic paths are present, many interesting phenomena may take place. In particular, particles on a chaotic trajectory may come very close to a regular path and “stick” to it, i.e., remain close to it for very long periods of time. One could say that smooth regular paths act like traps for particles on chaotic trajectories. If the smooth path and the adjacent chaotic area have the same long-term drift, then the diffusion process is effectively blocked and the average squared size of a cloud of particles in the chaotic region grows slower than linearly with time. On the other hand, if the regular path and the paths in the adjacent chaotic have different drift velocities, then the cloud of particles in the chaotic area tends to stretch and the average squared size of a cloud of such particles grows faster than linearly with time (but slower than quadratically as they would in the case of real stretching). These characteristics can be seen in fig. 4.
4. Effects of (turbulent) diffusion
Since some of the properties discussed in the previous section are due to a certain instability in the dynamics, one may wonder what happens to them when random perturbations are present, e.,g., when (thermal or turbulent) fluctuations in the velocity field are taken into account. In order to study this question, we add a white-noise component to the velocity field (1) and compute the corresponding paths. The results of the previous section are modified as follows:
Patches. Particles inside a patch can now move out and reach the external chaotic area (or another patch), i.e., all patches become leaky. While the patches disappear in the long run, they may do so on a much longer time-scale than the rest (chaotic part) of the cloud.
Chaotic paths and diffusion. In some cases, chaotic paths and the diffusion process introduced by them are only slightly affected by the added noise. For example, if to the field of fig. 3 we add random perturbations, the only change is that the diffusion coefficient increases; the increment being essentially the diffusion coefficient associated with the external noise. On the other hand, when chaotic and regular areas coexist, particles initially diffusing in the chaotic area can diffuse into the patches and remain there for a long while until they leak back into the chaotic area. If the adjacent chaotic and regular areas have the same drift, this may reduce the value of the effective diffusion coefficient originally associated with the chaotic area.
Effective shear. Due to the random perturbations, a particle on a drifting path will move into another path with a different drift, the overall effect being that the particles perform a random walk on the length-scale of the spatial inhomogeneities of the field. Thus, just like the usual (Eulerian) shear, the effective shear becomes shear d$f usion. This occurs, however, only after a transient; the time-scale of the transient being the time required by the random noise to diffuse across the inhomogeneities of the field.
Anomalous diffusion. After a (long) transient both anomalous diffusion and anomalous shear go over to Hrownian-type diffusion and the average squared size of the cloud grows linearly in time.
R.A. Pasmanter
326
/ Anomalous
diffusion and stretchmg
in vorticalflows
5. Some comments We have checked
that the results are not affected,
other incommensurate is not expected The
to introduce
occurrence
coefficient measured calculations
wavelengths in nature
drastic
to the velocity
qualitatively
speaking,
field (1). Similarly,
by the addition
a more realistic
of
3D field
changes.
of patchiness,
i.e.,
of a strong
dependence
of the dispersion
upon position in space, as well as the variability in the growth rate of clouds in the seas [l], agree very well with the predictions of our model. Numerical of particle
and in the Wadden
paths in more realistic
velocity
Sea [6] show the characteristic
fields in the western
signature
of chaotic
Scheldt
paths:
estuary
oscillations
[5] on
length-scales much smaller than the smallest scale of the velocity field. For laboratory observations of chaotic streamlines in laminar flows, refer to [7].
References [l] J.W. Talbot (1974) Interpretation of diffusion data: Proceedings of the international symposium on discharge of sewage from sea outfalls, London. [2] M. H&on (1986) Sur la topologie des lignes de courant dans un cas particulier, R.A. Acad. SC. Paris A262, 312-314. [3] T. Dombre, U. Frisch, J.M. Greene, M. H&on, A. Mehr and A.M. Soward (1986) Chaotic Streamlines in the ABC flows, J. Fluid Mech. 167, 353-391. (41 Yoshida (1986) Private communication, quoted in the paper by Dombre et al. (1986). [5] G. v. Dam. (1986) Computations of particle paths and distributions in two and three dimensional velocity fields; Fysische Afd. Colloquium day. Edited by G. v. Dam (in Dutch). RWS/WL/KNMI, The Hague 19 June 1985. [6] H. Ridderinkhof (1986) private communication. [7] W.L. Chien, H. Rising and J.M. Ottino (1986) Laminar mixing and chaotic advection in several cavity flows, J. Fluid Mech. 170, 355-378. J. Chaiken, R. Chevray, M. Tabor and Q.M. Tan (1986) Experimental study of Lagrangian turbulence in a Stokes flow, Proc. Roy. Sot. A408, 165-174.
Fluid Dynamics
Kesearch
3 (1988) 320-326 North-Holland
Anomalous diffusion and anomalous stretching in vertical flows * Ruben
A. PASMANTER
The traJectortes of partlclea advected by non-random spatially periodic velocity fields may he chaotic, i.e.. extremely sensitive to their initial positions. In the most general case. the area of a cloud of advected particles grows like (time)” where the exponent 0 ranges between 0 and 2 (shear stretchmg). Thew ohsxvationa could explain man) known facts about dlsperbion procew\ in shallw tidal seas.
Abswact.
1. Introduction
Measurements of the dispersion of substances in the sea show an amazing variability [l]: patchiness, i.e.. parts of the pollutant cloud that disperse much slower than the rest of the cloud, as well as slopes ranging from 1 up to 3 in logglog plots of the growth rate of clouds. see fig. 1. In order to better understand the observed variability of dispersion processes in shallow seas, 1 have been studying the passive advection of particles by non random. tidal (i.e.. oscillatory) flows with spatial inhomogeneities due to the bottom topography of the basin. A typical velocity field of this class is: u\ = R,(t)
sin(k,js).
+A,(t)
(la)
r!,.=R,(t)+A,(t)sin(k,x),
(lb)
where the time dependent amplitudes R(t) and A(t) contain not only the fundamental (tidal component with frequency Jz but also harmonics: in particular. they have also time-independent components R,, and A,,: R(t)=R,,+R,(r)+R2(t)+ A(t)
=A,,+A,(t)
+/l:(r)
...
(2a)
+ .“,
(33)
A, and R, oscillate at the fundamental frequency 0, A1 and R2 at 2L’. etc. The first terms on the r.h.s. of (la) and (lb) describe a spatially homogeneous tide. while the second terms correspond to eddies that are attached to the bottom topography. The wave-vector k corresponds to the largest component of the bottom topography. The velocity field (1) is incompressible and is closely related to the ABC (Arnol’d~Beltrami&Childress) flows. The ABC flows are spatially periodic stationary solutions * Participation Fundamental
0169-5983/8X/$2.75
in this Conference Research of Matter
has been possible thanks to the financial support and the IUTAM Local Organizing Committee.
‘: 1988. The Japan
Society
of Fluid Mechanic\
of the Dutch
foundation
of
R.A. Pasmanier
/ Anomalous
diffusion and stretching in vorticaljlows
0
?d3 10
I2
321
C
2
Gxy cc t
2.07
m”
109
10*
10’ G:y [m*l
10’
106
t
70” lo5
10‘3
‘4 70
705 I5 70 -
)
7u6 TIME
70’ [s]
Fig. 1. Logarithm of the average squared size of pollutant clouds in the sea as a function of the logarithm of time. In shallow seas, (B) in English estuaries, (C) in open sea, (D) in American coastal waters, (E) in American estuaries, in the Baltic coast and (G) in Fjords. Taken from ref. [l]. Notice the different slopes.
(4 (F)
of Euler’s equations; moreover, they are Beltrami flows, i.e., the vorticity is everywhere parallel to the velocity field. Numerical calculations [2,3] indicate that the paths of particles advected by ABC flows can be chaotic; Yoshida [4] has given a system of six equations that is equivalent
R.A. Pasmanier
322
/ Anomalous
to the ABC flows. The results presented not only for the velocity
diffusion
and stretching
in vertical flows
below are, unless otherwise
field (1) but also for the ABC
indicated,
qualitative
valid
and similar flows.
2. Particles’ paths and their computation The paths covered by passive particles advected by the velocity the following system of ordinary nonlinear differential equations:
+,4.(t)
dx/dt=R,(t) dy/dt=R,,(t)+A,.(t) These
equations
subroutine.
of
sin(k,y),
(3a)
sin(k,x),
(3b)
using, e.g., a high-order
Adams
predictor-corrector
IMSL
However, every numerical method of integration introduces not just errors but errors. In our case, these errors may lead to the opening of paths that are in fact
systematic closed
can be integrated
field (1) are the solutions
and to the introduction
errors can be reduced
of spurious
sinks and sources
but at the cost of long computation
in the incompressible
flow. These
times.
In order to avoid extremely long computation times, I replaced the trigonometric functions in (3) by piecewise linear sawtooth functions. The integration over each linear piece can be done
analytically;
the
remaining
numerical
polynomials. The piecewise linear (PL) obtained from the numerical integration Given
an initial position
[x(O),
task
reduces
to the
solution
of
second-order
model is used to check of (3).
the correctness
of the results
y(O)] and taking into account
the periodicity
in time of the
flow, we plot the successive positions of the same particle after one tidal period [x(l), y(l)], two tidal periods [x(2), y(2)], etc. Similarly, we exploit the spatial periodicity of the flow in order to plot the positions modulo a wave-length; in this way we can keep track of particles that
move
far away
from
their
initial
positions
without
having
to enlarge
the scale
of our
graphs. The set of tidal positions of a particle [x(O), one-dimensional curve. Notice that two different different
initial positions)
y(O)], [x(l), y(l)], etc, may form a smooth curves (corresponding to two particles with
can never intersect.
3. Main results In spite of the very simple form of the field (1) the paths of the advected particles astonishingly broad spectrum of possibilities. Figs. 2a and 2b show two characteristic the same
velocity
field, the only difference
between
show an plots for
the plots being the initial positions
of the
particles. In fig. 2a one sees a smooth curve formed by a number of islands: the particle moves from one island to the other after one tidal period. By contrast, in fig. 2b one sees a path that seems to cover a two-dimensional The most important
observed
Non dispersive and semi-dispersive
area. features
can be understood
in terms of four basic mechanisms:
patches
As already stated, different paths can never intersect. This means that the particles inside the islands in fig. 2a can never cross the boundary of such an island, i.e, they form a permanent
block or patch. It should be noted that in certain cases an area is surrounded by a smooth curve with very small openings; in these cases a particle that starts in the surrounded area needs a very long time until it hits one of the small openings and leaks out. In other words, the field can also generate leaky or semi-dispersive patches.
R.A. Pasmanter
/ Anomalous
diffusion and stretching in vertical flows
323
Fig. 2. (a) Eight hundred positions of one particle at times 1, 2,. _,800 (in units of the tidal period) form seven smooth islands. With each tidal period, the particle moves from one island to the next one. The positions are plotted modulo the wavelength; (b) Same velocity field as in a) but with a different initial position. Four thousand tidal positions tend to cover a two-dimensional area. Both in a) and b) the nonvanishing dimensionless parameters of the velocity field are R, = (8, S), R, = (0.025.0) and A, = A, = (0.9,0.9).
Moreover, patches may be drifting or nondrifting, nonchaotic) paths or both regular and chaotic ones.
they
may
contain
only
regular
(=
Chaotic trajectories with properties similar to those of a random walk As seen in fig. 2b, some paths tend to cover a two-dimensional area. These paths have another surprising characteristic: they are chaotic. This means that the separation between two
4 8C
‘I
324
168
012
144
300
457
613
0 00
098
195
233
390
488
585
6.83
Fig. 3. Logarithm of the average squared size of a cloud of four hundred particles as a function of the logarithm of time. The non-vanishing parameters of the velocity field are R, = (8, 8), R,(0.425,0) and A 1= A, = (0.9.0.9). For these values for the parameters, the chaotic region occupies a// space. The slope of the straight line is very close to 1, like in Browniantype diffusion.
R.A. Posmanter
324
/ Anomrrlous
diffusronmd stretchingin uortrral flows
particles that initially have been (infinitely) close to each other, grows exponentially be more precise, for these chaotic paths one has that lim tC’ lnD:in I+ra
o [ D( t)/D(O)]
in time. To
= p > 0.
(4)
where D(t) is the separation between the two particles at time t. The rate II. is called the Lyapunov exponent (or Lyapunov number); only when p is positive we say that the trajectory is chaotic. Notice that the value of the Lyapunov number may depend upon the direction of D(O). By changing the parameters R,(t), A,(t), R, and A,. a situation may be reached in which the chaotic paths cover all the available space and no patches are observed. Under these circumstances another striking manifestation of the chaotic character of these paths manifest itself: the average squared size of a cloud of particles grows linearly in time, i.e., just like in the case of a cloud of Brownian particles performing random walks. This behavior is shown in fig. 3. Effective shear When the time-independent homogeneous component of the velocity field vanishes, i.e., R, = 0, there is no long-term preferred direction so that no long-term drift is possible. When R, does not vanish, however, the symmetry of the system is broken and a long-term drift along the preferred direction of R, is possible. Since the velocity field is not homogeneous, different positions in space are not equivalent and one may observe diff erent long-term drifts associated with different initial positions. This means that in such cases one has an effective (or Lagrangian) periodic shear that leads to stretching of the cloud along the direction of Ro; the average squared cloud size along the preferred direction grows quadratically in time. It should be mentioned that all paths in an island have the same drift velocity; similarly all paths in a connected chaotic region have the same drift velocity. Consequently, when all space is occupied by chaotic trajectories and no regular ones are present, no shear is possible; this is the case in fig. 3.
Fig. 4. Logarithm
+ +
f
+a+
x
size of a cloud
^ _ : I
particles
x
”
276
5 53
829
of the average
of thousand
in the chaotic
as a
function
After
4000
of the
region
straight
lines
(squared
size along
and 0.75
(squared
perpendicular
are
of fig. (2b)
logarithm
tidal periods
squared
five hundred of time.
the slopes
approximately the direction
of the 1.4 fo R,,)
size long the direction
to R,).
R.A. Pasmanter
/ Anomalous
drffwion
and stretching in uor~rcalflows
325
Anomalous diffusion and anomalous stretching When both regular and chaotic paths are present, many interesting phenomena may take place. In particular, particles on a chaotic trajectory may come very close to a regular path and “stick” to it, i.e., remain close to it for very long periods of time. One could say that smooth regular paths act like traps for particles on chaotic trajectories. If the smooth path and the adjacent chaotic area have the same long-term drift, then the diffusion process is effectively blocked and the average squared size of a cloud of particles in the chaotic region grows slower than linearly with time. On the other hand, if the regular path and the paths in the adjacent chaotic have different drift velocities, then the cloud of particles in the chaotic area tends to stretch and the average squared size of a cloud of such particles grows faster than linearly with time (but slower than quadratically as they would in the case of real stretching). These characteristics can be seen in fig. 4.
4. Effects of (turbulent) diffusion
Since some of the properties discussed in the previous section are due to a certain instability in the dynamics, one may wonder what happens to them when random perturbations are present, e.,g., when (thermal or turbulent) fluctuations in the velocity field are taken into account. In order to study this question, we add a white-noise component to the velocity field (1) and compute the corresponding paths. The results of the previous section are modified as follows:
Patches. Particles inside a patch can now move out and reach the external chaotic area (or another patch), i.e., all patches become leaky. While the patches disappear in the long run, they may do so on a much longer time-scale than the rest (chaotic part) of the cloud.
Chaotic paths and diffusion. In some cases, chaotic paths and the diffusion process introduced by them are only slightly affected by the added noise. For example, if to the field of fig. 3 we add random perturbations, the only change is that the diffusion coefficient increases; the increment being essentially the diffusion coefficient associated with the external noise. On the other hand, when chaotic and regular areas coexist, particles initially diffusing in the chaotic area can diffuse into the patches and remain there for a long while until they leak back into the chaotic area. If the adjacent chaotic and regular areas have the same drift, this may reduce the value of the effective diffusion coefficient originally associated with the chaotic area.
Effective shear. Due to the random perturbations, a particle on a drifting path will move into another path with a different drift, the overall effect being that the particles perform a random walk on the length-scale of the spatial inhomogeneities of the field. Thus, just like the usual (Eulerian) shear, the effective shear becomes shear d$f usion. This occurs, however, only after a transient; the time-scale of the transient being the time required by the random noise to diffuse across the inhomogeneities of the field.
Anomalous diffusion. After a (long) transient both anomalous diffusion and anomalous shear go over to Hrownian-type diffusion and the average squared size of the cloud grows linearly in time.
R.A. Pasmanter
326
/ Anomalous
diffusion and stretchmg
in vorticalflows
5. Some comments We have checked
that the results are not affected,
other incommensurate is not expected The
to introduce
occurrence
coefficient measured calculations
wavelengths in nature
drastic
to the velocity
qualitatively
speaking,
field (1). Similarly,
by the addition
a more realistic
of
3D field
changes.
of patchiness,
i.e.,
of a strong
dependence
of the dispersion
upon position in space, as well as the variability in the growth rate of clouds in the seas [l], agree very well with the predictions of our model. Numerical of particle
and in the Wadden
paths in more realistic
velocity
Sea [6] show the characteristic
fields in the western
signature
of chaotic
Scheldt
paths:
estuary
oscillations
[5] on
length-scales much smaller than the smallest scale of the velocity field. For laboratory observations of chaotic streamlines in laminar flows, refer to [7].
References [l] J.W. Talbot (1974) Interpretation of diffusion data: Proceedings of the international symposium on discharge of sewage from sea outfalls, London. [2] M. H&on (1986) Sur la topologie des lignes de courant dans un cas particulier, R.A. Acad. SC. Paris A262, 312-314. [3] T. Dombre, U. Frisch, J.M. Greene, M. H&on, A. Mehr and A.M. Soward (1986) Chaotic Streamlines in the ABC flows, J. Fluid Mech. 167, 353-391. (41 Yoshida (1986) Private communication, quoted in the paper by Dombre et al. (1986). [5] G. v. Dam. (1986) Computations of particle paths and distributions in two and three dimensional velocity fields; Fysische Afd. Colloquium day. Edited by G. v. Dam (in Dutch). RWS/WL/KNMI, The Hague 19 June 1985. [6] H. Ridderinkhof (1986) private communication. [7] W.L. Chien, H. Rising and J.M. Ottino (1986) Laminar mixing and chaotic advection in several cavity flows, J. Fluid Mech. 170, 355-378. J. Chaiken, R. Chevray, M. Tabor and Q.M. Tan (1986) Experimental study of Lagrangian turbulence in a Stokes flow, Proc. Roy. Sot. A408, 165-174.