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GRAFF ET AL.
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Relative Dispersion in the Atmosphere from Reanalysis Winds
L. S. GRAFF
Department of Geosciences, University of Oslo, and Norwegian Meteorological Institute, Oslo, Norway
S. GUTTU
Department of Geosciences, University of Oslo, and Norwegian Environment Agency, Oslo, Norway
J. H. LACASCE
Department of Geosciences, University of Oslo, Norway
(Manuscript received 25 July 2014, in final form 20 March 2015)
ABSTRACT
The dispersion of pairs of synthetic particles, advected with ECMWF winds, is examined. The particles were
deployed at three latitudes and on three potential temperature surfaces in both hemispheres. Separation
statistics are calculated and evaluated in relation to 2D turbulence theory and to Eulerian structure functions
calculated directly from the wind data.
At the smallest sampled scales (100–1000 km), the pair-separation velocities are correlated, and the dispersion is laterally isotropic, at least at the higher latitudes. At larger scales, the dispersion is zonally anisotropic, and the pair velocities are uncorrelated. In all cases, the dispersion grows exponentially in time, and
the second-order Eulerian structure functions consistently increase as separation squared. This implies
nonlocal dispersion, which obtains with energy spectra at least as steep as K23.
Regional variations are seen in the parameters however. The e-folding times and the maximum scales for
exponential growth are significantly larger on the 430-K surface than on the 315-K surface, and the dispersion
is anisotropic at low latitudes, even at the smallest scales. Therefore, 2D homogeneous turbulence theory is
applicable at best at subdeformation scales at the higher latitudes.
1. Introduction
Isentropic transport is important for many processes
in the atmosphere: the redistribution of water vapor
(Kelly et al. 1991; Yang and Pierrehumbert 1994;
Fueglistaler et al. 2005), ozone (Leovy et al. 1985;
Holton et al. 1995), airborne particulates, such as volcanic ash (Heffter and Stunder 1993; Prata et al. 2007),
and active tracers, such as heat and potential vorticity
(Hoskins et al. 1985; Haynes and McIntyre 1987; Juckes
and McIntyre 1987). Such transport can be studied using
particle (Lagrangian) dispersion. This includes the study
of single particles (absolute dispersion) and groups of
particles (relative dispersion). Relative dispersion is also
of interest, as its properties depend on the kinetic energy
Corresponding author address: Joe LaCasce, Department of
Geosciences, University of Oslo, P.O. Box 1022 Blindern, N-0315
Oslo, Norway.
E-mail: j.h.lacasce@geo.uio.no
DOI: 10.1175/JAS-D-14-0225.1
Ó 2015 American Meteorological Society
spectrum (Bennett 1984). As such, pair dispersion gives
insight into the Eulerian velocity statistics.
Relative dispersion was studied on large scales in the
1970s using constant-level balloons in the lower stratosphere. Two such experiments, conducted in the Southern Hemisphere (SH), were the EOLE (Morel and
Bandeen 1973) and Tropical Wind, Energy Conversion,
and Reference Level Experiment (TWERLE; Jullian
et al. 1977) campaigns. In both experiments, pair separations were found to grow exponentially in time, with an
e-folding scale of roughly 1 day, up to separations of
1000–2000 km (Morel and Larcheveque 1974; Er-El and
Peskin 1981). Such exponential growth implies steep
(nonlocal) energy spectra (Bennett 1984). The behavior
at larger scales was less clear; Morel and Larcheveque
(1974) found diffusive dispersion, while Er-El and
Peskin (1981) observed faster (superdiffusive) growth.
The EOLE dispersion was reexamined by Lacorata
et al. (2004), who employed a different measure [the
finite-scale Lyapunov exponent (FSLE; Aurell et al.
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1997)]. While the authors also identified exponential
growth at separations below 100 km and diffusive growth
at large scales, they suggested that the intermediate-scale
(100–1000 km) dispersion exhibited a power-law dependence. This implies a shallower energy spectrum, and,
in their case, one that was consistent with spectra derived
from aircraft measurements (Nastrom and Gage 1985).
As balloons are expensive, field experiments involve
limited numbers of instruments (of order 500 balloons in
the EOLE experiment), and the statistics are often
noisy. Large numbers of measurements can be obtained,
though, using synthetic particles advected numerically.
Although the advecting wind fields do not resolve
smaller scales, the increased statistical aspects are appealing: error bars can be reduced by increasing the
numbers of pairs.
Huber et al. (2001) studied the relative dispersion of
synthetic particles advected by European Centre for
Medium-Range Weather Forecasts (ECMWF) winds.
They found that dispersion varied qualitatively with
latitude; while it grew exponentially in time in the
tropics, it exhibited a power-law growth in the mid- and
high latitudes (and not greatly different than linear in
time). This implies a corresponding variation in the kinetic energy spectra with latitude.
LaCasce (2010) also studied the EOLE balloon data,
using probability density functions (PDFs) of pair separations. The dispersion, the second moment of the
separations, derives from the PDF, but the PDF indicates additional aspects, such as the frequency of large
separations. The PDFs in the EOLE experiment were
consistent with exponential growth at separations below
1000 km. Again, though, statistical significance was
hampered by the relatively small number of pairs
available.
Herein, we use PDFs to reexamine the relative dispersion of synthetic particles advected by ECMWF
winds. The goal is to obtain a clearer picture of the
dispersion at the smallest scales captured by the winds.
We deployed thousands of particles along latitude lines:
in the midlatitudes and the tropics, in the troposphere
and stratosphere, and in summer and winter seasons. We
also compare the results to the Eulerian statistics derived from the same winds.
2. Theoretical background
A necessary first step in relative dispersion is to establish when the pair velocities are correlated. When
separations exceed the scales of the energy-containing
eddies, pair statistics are equivalent to those of single
particles, and absolute dispersion suffices. Correlation
can be tested in various ways: for example, with the
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mean-square separation velocity (e.g., Babiano et al.
1990; LaCasce and Bower 2000; Koszalka et al. 2009).
Many of our expectations for relative dispersion on
synoptic scales derive from homogeneous 2D turbulence theory (e.g., Morel and Larcheveque 1974; Er-El
and Peskin 1981; Lacorata et al. 2004; Bennett 2006;
LaCasce 2008). In turbulent flows, relative dispersion is
determined by the Eulerian energy spectrum (Bennett
1984). One can distinguish two regimes. Assuming a
power-law kinetic energy spectrum
E(K) } K2a ,
(1)
the dispersion is local if 1 # a , 3 and nonlocal if a $ 3.
Under local dispersion, pair separations exhibit a powerlaw dependence on time, while the growth is exponential
under nonlocal dispersion.
Richardson (1926) proposed that pair separations
should be governed by a Fokker–Planck (F–P) equation
(see also Kraichnan 1966; Lundgren 1981; Bennett
2006). In 2D, this is
›
1 ›
›
p5
rk2 p ,
›t
r ›r
›r
(2)
where r is the separation, k2 (r) is the relative diffusivity,
and p(r, t) is the PDF of the separations. The Eulerian
statistics determine how the diffusivity k2 varies with
separation. Once k2 is known, the equation can be
solved for p(r, t).
Solutions for the 2D turbulent inertial ranges have
been derived previously (Lundgren 1981; Bennett 2006;
LaCasce 2010) and are given in the appendix. With uncorrelated pair velocities, the relative diffusivity is constant, and the asymptotic solution is self-similar,
proportional to the Rayleigh distribution. Under a turbulent energy cascade, the pair velocities are correlated,
and k2 5 br4/3 , where b is a constant proportional to the
third root of the energy dissipation rate. The solution
(herein called the ‘‘Richardson PDF’’) also asymptotically approaches a self-similar distribution. Under a
turbulent enstrophy cascade, k2 5 r2 /T, where T is a time
scale inversely proportional to the third root of the
enstrophy dissipation rate. The solution is proportional
to a lognormal distribution (herein called the ‘‘Lundgren
PDF’’). In contrast to the Rayleigh and Richardson
PDFs, the Lundgren PDF is never self-similar but becomes increasingly peaked with time.
With the PDF, one can derive the separation
moments:1
1
We use the raw moments, as separations are positive definite.
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FIG. 1. The 315-, 350-, and 430-K deployment isentropes (red lines) for 0000 UTC (a) 1 Jan and (b) 1 Sep averaged over the 19
deployments (years 1991–2009) and over all longitudes. Also plotted are the 6-, 12-, and 18-km height levels (dashed–dotted black lines).
The solid black vertical line indicates 608N.
hr n i 5 2p
ð‘
r n11 p dr .
(3)
0
The two of greatest interest are the dispersion (the
second moment) and the kurtosis (the fourth moment,
normalized by the second moment squared).
For uncorrelated motion, the asymptotic dispersion
grows linearly in time, as for a diffusive process (Taylor
1921), and the kurtosis, initially 1, asymptotically approaches 2. In an energy cascade, the asymptotic dispersion increases as t3 , now known as Richardson’s law.
The kurtosis asymptotically approaches a value of 5.6. In
an enstrophy cascade, both the dispersion and kurtosis
increase exponentially in time and at the same rate. Such
growth also pertains to spectra steeper than K23
(Bennett 1984).
There are other measures that have been used to
study relative dispersion. The finite-time Lyapunov
exponent (FTLE; e.g., Lapeyre 2002) is particularly
suitable under exponential growth. FSLE is another
measure, reflecting the time required for pair separations to grow by specified amounts. Both have been
used with synthetic data to map regions of intensified
mixing. The advantage with PDFs is that we have analytical solutions that dictate how they should evolve
in time.
3. Model
For particle advection, we used the Lagrangian Flexible
Particle dispersion model (FLEXPART; Stohl and Seibert
1998; Stohl and Thomson 1999; Stohl et al. 2005). This
solves the Lagrangian velocity relation
d
x 5 u[x(t)],
dt
(4)
where the velocity u comprises winds from gridded
forcing data and stochastic components representing
turbulent mixing. The latter components are primarily
important in the planetary boundary layer and in the
near-equatorial region.
The wind data come from ECMWF and span the period 1991–2009. The data have 3-h resolution, 18 3 18
spatial resolution, and 60 vertical hybrid levels. At 0000,
0600, 1200, and 1800 UTC, we used data from the
ECMWF interim reanalysis (Dee et al. 2011). For the
intermediate hours (0300, 0900, 1500, and 2100 UTC),
we used ECMWF analysis data. The subsequent results
are not sensitive to this choice; using only analysis or
reanalysis winds every 6 h yields nearly the same results.
The particles were deployed on latitude circles in
square clusters of four, separated by 18 of longitude. This
yields two pairs, oriented along the diagonals, with
equal longitudinal and latitudinal initial separations.
The initial separation for each pair was 100 km, roughly
the same resolution as for the wind data. The particles
were placed at 108, 308, and 608 in both hemispheres.
The particles were also deployed on three isentropic
surfaces: 315, 350, and 430 K (Fig. 1). The latter two
surfaces are relatively flat and lie near 12- and 18-km
heights. The height of the 315-K surface varies, however,
being lower in the tropics and higher in the mid- and
high latitudes. In addition, we deployed particles on
1 January and 1 September to test seasonal variations.
The isentropes are qualitatively similar in September,
albeit somewhat higher in the SH and lower in the
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FIG. 2. Trajectories of 360 (of 1440) particles deployed in 1991 in (a) January at 315 K and 608N, (b) September at 315 K and 608N,
(c) January at 430 K and 608N, and (d) January at 315 K and 108N. For clarity, we only show the positions for the first 4 days after
deployment. Note that the minimum latitude is 308N in (a)–(c) and 108S in (d).
Northern Hemisphere (NH) (Fig. 1). The trajectories
were calculated over 1 month in both seasons, and the
process was repeated for the 19 Januaries and Septembers in the time period 1991–2009. The statistics were
then averaged for all years for each season.
To illustrate the behavior, we plot four sets of trajectories in Fig. 2. Each trajectory represents the first 4 days
of advection. At 608N, the January trajectories on the
315-K surface (Fig. 2a) are zonally oriented and wavy, as
might be expected for the wintertime storm track (e.g.,
Chang et al. 2002; Hoskins and Hodges 2002; Ulbrich
et al. 2009). The September trajectories (Fig. 2b) are less
zonal, consistent with a weaker subtropical jet and storm
track in summer. The trajectories in the stratosphere on
the 430-K surface (Fig. 2c) exhibit large, elliptical swirls,
suggesting advection by the stratospheric polar vortex.
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FIG. 3. The two-particle velocity correlation normalized by twice the single-particle variance. (a) The curves correspond to the full
horizontal velocity (black), the zonal component (green dotted–dashed), and the meridional component (blue dashed) for the January
deployments at 350 K and 608N. The meridional velocity correlation for the January (blue) and September (red) deployments on the 315(solid), 350- (dashed), and 430- (dotted) isentropes at (b) 608, (c) 308, and (d) 108N.
The 315-K particles at 108N, on the other hand, are more
zonal (Fig. 2d).
Figure 2 suggests predominantly zonal advection, with
variations in the spatial and temporal scales with latitude and height. The statistics discussed herein support
such differences, although the generic behavior at the
early periods will be seen to be similar.
4. Results
a. Correlation and isotropy
To establish the periods and scales over which pair
motion is correlated, we use the pair velocity correlation
normalized by twice the single-particle velocity
variance:
C5
hui uj ii6¼j
2n2
,
(e.g., Koszalka et al. 2009). The angle brackets represent
the average over available pairs.
Figure 3a shows the result for the particles launched in
January at 608N on the 350-K surface. The solid curve
corresponds to the full velocities, while the dashed
curves are the meridional and zonal correlations. All
three decrease with time, but while the meridional correlation falls to zero after 20–25 days, the zonal
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correlation is still roughly 0.4 after 30 days. As such, the
full velocity correlation is also nonzero at 30 days.
As the meridional curve provides the clearest indication of decorrelation, we use that in the other panels
(Figs. 3b–d). At all three latitudes, the correlations in the
troposphere (315 K) decay to zero soon after 10 days.
This is true in both winter and summer. However, the
decay on the 430-K surface takes about twice as long.
The decay on the 350-K surface is intermediate between
the two.
The decorrelation times, of course, depend on the
initial separation. Using smaller initial separations
lengthens the period. An alternative is to plot the correlations against rms separation (not shown). Doing so
reveals that the correlations fall to zero at scales exceeding several thousand kilometers.
The theoretical solutions also assume isotropic dispersion. We test this by plotting the ratio of the mean
square zonal and meridional pair separations (Morel and
Larcheveque 1974). The ratio for the NH releases at 608N
is shown in Fig. 4. The ratio is near 1 for the first 2 days
(Fig. 4a). That it is 1 initially follows from the choice of
initial condition, with the pairs aligned along the diagonals of a square. It is nearly 2 by day 5, and at day 10 it
is roughly 3. In terms of spatial scales, the ratio is 1 for
separations less than roughly 1000 km and reaches 2 by
2000 km (Fig. 4b). There is little variation with height or
season, except that the dispersion is more zonal on the
430-K surface in winter and more isotropic in summer.
At 308N, the flow becomes anisotropic earlier and at
smaller scales (Figs. 4c,d). The ratio is nevertheless
mostly between 1 and 2 in the 100–1000-km range
(Fig. 4d). Zonality is more pronounced on the 430-K
surface, where the ratio reaches 3 at 500-km separation.
The anisotropy is also greater at 108N (Fig. 4e), being
first evident at separations of only several hundred kilometers (Fig. 4f).
The response in the SH (not shown) is like that in the
NH, albeit more zonally anisotropic. The results,
moreover, are consistent with the EOLE balloon results
of Morel and Larcheveque (1974), who found that zonal
dispersion dominated at scales exceeding 1000 km. After
15 days, the EOLE zonal dispersion was roughly 4 times
larger than the meridional.
b. Dispersion at 608N
As the dispersion at 608N is the most persistently
isotropic, we focus on those deployments first. We begin
with the set launched on the 350-K surface in January.
Thereafter, we consider the dependence on height, latitude, and season.
The dispersion for this set is shown in Fig. 5. As the
spreading is initially isotropic, we plot the total
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dispersion, the sum of zonal and meridional components. For comparison, we also plot the Lundgren
(nonlocal) and Richardson (energy) dispersion curves
[using (A22) and (A12), respectively].
Each of the theoretical curves has two unknowns: the
initial separation r0 and the parameter that determines
the growth rate—T for the Lundgren curve and b for the
Richardson. All the pairs have the same initial separation, so we take r0 to be the same. That leaves a single
unknown for each case.
We determine these by matching the theoretical
curves to the modeled dispersion at a chosen separation.
This approach is straightforward and, in the case of the
Lundgren dispersion, allows the parameter to be determined directly. Note that matching the dispersion at a
chosen time is less desirable, because the growth rates
vary with latitude and height. So we determine the time
t* at which the dispersion is a times larger than the initial
value: hr2 (t*)i 5 ar02 . Then T is given by
T5
8t*
ln[hr2 (t*)i/r02 ]
5
8t*
.
ln(a)
(5)
We take a 5 5 herein, although the results are relatively
insensitive to the exact value.
The full Richardson dispersion [(A12)] does not lend
itself to such a simple determination of b, so we adjust
the value empirically until the dispersion matches
the observed at t*. A first guess is obtained using the
dispersion’s small time asymptotic limit [(A15)].
The results are shown in Fig. 5. The modeled dispersion (dashed black curve) increases rapidly during the
first 2 days and more slowly thereafter. The dispersion is
close to the Lundgren curve (red) during the first 2 days,
suggesting exponential growth. The Richardson curve
(blue) increases somewhat faster initially and slower
afterward. By design, the three curves intersect at 5 3
104 km2, about 1 day after launch.
The PDFs are shown in Fig. 6. At 1 day, the Lundgren
PDF (red) is similar to the modeled PDF (dashed black),
with a peak near 100 km and a long tail, while the
Richardson PDF (blue) is somewhat broader. The result
at 2 days is similar. At 4 days, though, the distinction is
less clear, but the curves differ in the tails; the Lundgren
PDF indicates higher probability than the model’s PDF,
and the Richardson PDF indicates lower probability.
The differences are amplified by day 7, but this is also
when the dispersion is significantly anisotropic.
Differences in the wings of the distribution are
reflected in the kurtosis, shown in Fig. 7. The modeled
kurtosis increases rapidly during the first 4 days and
decreases thereafter, exceeding 30 at its peak. The
Lundgren kurtosis grows comparably fast during the first
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FIG. 4. The isotropy, defined as the ratio of the zonal dispersion to the meridional dispersion, for the deployments
at (a),(b) 608; (c),(d) 308; and (e),(f) 108N. The ratio is plotted as a function of time in (a),(c),(e) and as a function of
rms separation in (b),(d),(f).
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FIG. 5. Relative dispersion for the January deployments at 608N
and 350 K (dashed black line) with the theoretical solutions for the
Lundgren (red) and Richardson (blue) regimes. The values of b
and T are given in the bottom-right corner.
1.5 days and continues to increase thereafter while the
Richardson kurtosis never exceeds 5.6. As such, the
large value of the model kurtosis cannot be explained
under Richardson dispersion.
The results, thus, favor exponential growth,
implying a steep energy spectrum. To check this, we
calculated the Eulerian structure functions from the
wind data. In homogeneous, isotropic turbulence, the
Lagrangian and Eulerian structure functions are equivalent (Bennett 1984). In the Richardson regime, the
second-order structure functions (longitudinal and
transverse) exhibit a power-law dependence on separation r:
S2 (r) 5 h[u(x 1 r) 2 u(x)]2 i } r2/3 ,
(6)
which is Kolmogorov’s 2/3 law (Kolmogorov 1941;
Frisch 1995; Lindborg 1999). The longitudinal function
is essentially the inverse Fourier transform of the energy
spectrum, and the r2/3 dependence for the structure
function corresponds to the Kolmogorov K25/3 spectrum. In the Lundgren regime, the structure functions
have a steeper slope:
S2 (r) 5 h[u(x 1 r) 2 u(x)]2 i } r2
(7)
(Lundgren 1981; Lindborg 1999; LaCasce 2002). The r2
dependence implies an energy spectrum proportional to
K 23 or steeper (e.g., Bennett 1984).
We used the ECMWF winds, interpolated to
constant-height surfaces, to compute the structure
functions. Note that the latter do not correspond exactly
to the winds experienced by the particles (which change
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height). However, the 350- and 430-K surfaces are relatively flat (Fig. 1), so the differences are likely small. As
noted, the 315-K surface varies more with latitude, but
as we are focusing on the first few days, the differences
are relatively small.
The structure functions at 608N and 12-km height are
shown in Fig. 8. Both the longitudinal and transverse
structure functions exhibit an r2 dependence for separations less than 1000 km. In addition, the structure
functions level off at separations greater than roughly
2000 km, implying the latter is the scale of the energycontaining eddies.
Note too that the transverse functions are larger than
the longitudinal ones. This is as expected in isotropic
turbulence. For a 2D incompressible flow, the functions
obey
S2t 5
d
(rS2l )
dr
(8)
(Batchelor 1953; Lindborg 1999; LaCasce 2002). Thus, if
S2l is proportional to r2 , S2t is 3 times greater. In Fig. 8,
the line indicating r2 is plotted 3 times higher than the
January longitudinal function. As this coincides with the
January transverse function, the velocities are approximately isotropic.
c. Changes in latitude, changes in altitude
The initial dispersion is qualitatively similar for the
other deployments. The kurtosis always increases rapidly and reaches a large value. The maximum kurtoses
are mostly in the range 10–20, although values up to
nearly 60 are also seen. Likewise, the initial growth in
dispersion is approximately exponential in all cases.
What differs is the growth rate. Using the technique
described in section 4b, we estimate T for each of the 36
releases. On the 315-K surface (Fig. 9), T varies between
2 and 7 days, with smaller values at the higher latitudes
and larger ones at low latitudes. The corresponding
e-folding times (equal to T/8) range from 6 to 21 h. A
similar range is seen on the 350-K surface (middle panel)
and there is strikingly little variation with latitude.
Larger values are found, though, on the 430-K surface
(right panel), varying from 6 to 12 days in most cases (or
e-folding times from 18 to 36 h).
Using these values of T, we rescale time and compare
the dispersion curves. The results for the 12 releases on
the 350-K surface are shown in Fig. 10. In the period up
to roughly t 5 T/2, the dispersion curves overlap. The
composite curve is nearly exponential and is similar to
the Lundgren prediction, exp(8t/T) (black dashed
curve). The curves differ only at later times (when the
dispersion is zonally anisotropic).
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FIG. 6. Separation PDFs for the January deployments at 350 K and 608N (black dashed line) for (a) day 1, (b) day 2, (c) day 4, and (d) day 7.
Also shown are the Lundgren (red) and Richardson (blue) PDFs.
Many of the similarities and differences between the
releases can be seen in Fig. 11, which shows the structure
functions at 608N and 6 km (Fig. 11a), 608N and 18 km
(Fig. 11c), and 108N and 12 km (Fig. 11e). The dispersion
for the corresponding pair releases at 608N and 315 K
(Fig. 11b), 608N and 430 K (Fig. 11d), and 108N and
350 K (Fig. 11f) are shown on the right. In all cases, the
dispersion is nearly exponential during the first 1–3 days
and the structure functions increase as r2 at the smallest
scales. However, there are noticeable differences
as well.
As the longitudinal structure function S2l (r) is related
to the energy spectrum, having a larger S2l (r) implies
more energetic variability. Compared with that at 12 km
(Fig. 8), S2l (r) is greater at 6 km and lesser at 18 km
(Figs. 11a,c), so the energy levels decrease with height.
In other locations, the energy levels at 12 and 6 km are
similar, but the 18-km energy is usually less.
The Lundgren time scale (see bottom-right corners of
Figs. 11b,d,f) depends on the enstrophy dissipation rate
and so should be sensitive to the energy level. At 608N,
this is the case. The time scale is shortest in the troposphere and increases going up (T 5 2.73 days at 315 K,
T 5 5.14 days at 350 K, and T 5 7.86 days at 430 K).
However, despite the energy level at 108N being comparable to that at 608N at the same height, the 108N
particles have a longer time scale (T 5 5.59 days).
Therefore, T is affected by other factors at low latitudes.
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FIG. 7. Separation kurtosis for the January deployments at 608N
and 350 K (dashed black line) with the corresponding Lundgren
(solid red) and Richardson (solid blue) predictions. The upper
bound for the Richardson kurtosis, 5.6, is indicated by the dashed
blue line.
Differences are also evident in the degree of isotropy.
As noted, the r2 lines in the left panels of Fig. 11 are
plotted where the January transverse functions would lie
if the fluctuations were isotropic. At 608N and 6-km
height (Fig. 11a), the line overlies the transverse function (dashed blue curve), and the same is true for the
January S2t curve at 18 km (Fig. 11c). The line is higher
at 108N (Fig. 11e), indicating anisotropy. Indeed, the
longitudinal and transverse structure functions in summer overlie one another at 108N, implying anisotropy is
present at the smallest sampled scales.
To gauge the variations in isotropy among the deployments, we plot the ratio of S2t (r) to S2l (r) at the
smallest separation, r 5 100 km (Fig. 12a). There is a
clear variation with latitude, with the midlatitudes being
the most isotropic and the low latitudes being the most
anisotropic. Only the high latitudes are close to isotropy
at small separations; deviations are noticeable even at
308N and 308S. Anisotropy also increases to some extent
with height, being more pronounced at 18 km than
at 6 km.
These differences are mirrored in the relative dispersion (Fig. 4). The pair displacements are initially
isotropic, but anisotropy develops in all cases and at
different rates. In Fig. 12b, we plot the ratio of the
squared meridional separation to the squared zonal
separation at 1000 km.2 The parameter shows the same
variation as in Fig 12a, with more isotropic dispersion at
2
Note that this is the inverse of the measure shown in Fig. 4.
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FIG. 8. The longitudinal (solid) and transverse (dashed) secondorder structure functions from the deployments in January (blue)
and September (red) at 608N and 12 km. The longitudinal functions
derive from the zonal velocities and the transverse functions derive
from the meridional velocities; the separation is in longitude. The
functions are averaged in longitude and time. The black dashed–
dotted lines show the first maxima of the transverse structure
functions. The black solid line indicates an r2 slope. The y position
of the line is 3 times that of the longitudinal function in January
(blue solid line). That this nearly overlaps the transverse function
(blue dashed line) indicates the velocity fluctuations are nearly
isotropic.
midlatitudes and more zonal dispersion at low latitudes.
The similarity between Figs 12a and 12b implies the
small-scale anisotropy in the structure functions manifests as growing anisotropy in the particle separations.
FIG. 9. The Lundgren time scale for all deployments, sorted by
release level: (left) 315, (middle) 350, and (right) 430 K. The symbols vary with latitude, as listed in the key. The blue markers are for
January, and the red markers are for September. In each panel, the
release cases are plotted with latitude increasing to the right.
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dispersion isotropic. At lower latitudes, the structure
functions indicate zonal anisotropy at the smallest resolved scales, and this causes the relative dispersion to
be increasingly zonal. The dispersion is also affected by
differences in the energy-containing scales, with the
largest such scales occurring in the stratosphere.
5. Comparison with previous studies
FIG. 10. Relative dispersion for all deployments at 350 K. The
Lundgren dispersion is indicated by the black dashed curve. The
dispersion has been normalized by r02 and time has been normalized
by the relevant T value for each deployment case. Line colors and
styles are as indicated in the key. Hemispheres are distinguished by
line weight: NH lines are thick and the SH lines are thin.
The energy-containing eddy scale also varies between
deployments. To gauge this, we determined the scale at
which S2t (r) flattens out [S2l (r) often continues to increase, because of persistent zonal correlations]. This
scale is about 2000 km in the troposphere at 608N
(Fig. 11a) but is greater in the wintertime stratosphere
(Fig. 11b). In most of the other cases, the scale is near
2000 km at 6 km and slightly larger at 12 km. But the
scale is considerably larger at 18 km, lying between 3000
and 4000 km. So the principal factor for the size of the
energy-containing eddies is height; latitude and season
are less important.
On the other hand, the seasonal variations are modest.
In all three cases in Fig. 11, the energy levels in winter
are greater than those in summer. The difference is most
pronounced in the stratosphere but less so at 108N and
12 km and less still at 608N and 6 km. Despite this, the
qualitative features do not change with season.
Likewise, the differences between the hemispheres
are relatively minor. In the collective plots (Figs. 9, 12),
the symbols are arranged with the SH cases on the left
and NH cases on the right. Broadly speaking, there is
little asymmetry between the hemispheres. While we
expect the dispersion to be more zonal in the SH because of the greater zonality of the mean circulation
(e.g., Hoskins and Hodges 2005; Graff and LaCasce
2014), this is not apparent in the structure functions at
the small scales.
To summarize, the initial dispersion is exponential in
all cases, and the structure functions are proportional to
r2 at small scales. But only at the high latitudes is the
The present results agree well with those of Morel and
Larcheveque (1974) and Er-El and Peskin (1981), who
observed exponential relative dispersion and isotropic
spreading at separations below 1000–2000 km in the
EOLE and TWERLE experiments. Their e-folding time
scales (1.35 and 1.2 days) are larger than that found here
for the 350-K surface (roughly 0.5 days) but are of similar magnitude. They too observed zonally anisotropic
dispersion at large scales. The papers differ on the dispersion rate at the large scales, with Morel and
Larcheveque (1974) finding diffusive growth and Er-El
and Peskin (1981) obtaining a faster rate, perhaps as fast
as t3 . But if the latter is true, it cannot reflect Richardson
dispersion, as the pair velocities are uncorrelated.
Lacorata et al. (2004) observed exponential growth in
the EOLE separations at scales below 100 km, with an
e-folding time (0.4 days) like that found here. They
suggested a transition to t3 growth occurred afterward
and persisted until 1000 km. This would be consistent
with the aircraft-derived spectra of Nastrom and Gage
(1985), which indicate a K25/3 slope at scales below a few
hundred kilometers. The dispersion in Fig. 5 arguably
exhibits a power-law increase during the period from 1.5
to 6 days, but the kurtosis (Fig. 7) greatly exceeds the
Richardson limit of 5.6 and continues growing until day
4. The ECMWF winds do not capture the K25/3 spectrum at small scales, which could well explain the difference with their results, but LaCasce (2010) suggested
the separation PDFs for the EOLE data were also
consistent with the Lundgren distribution.
Huber et al. (2001) also observed exponential dispersion, using synthetic particles and ECMWF winds,
but only at low latitudes. However, the authors chose to
fit the dispersion curves over the first 10 days. As seen
here, the pair velocities are generally decorrelated by
that time, so using the first few days is preferable. Alternately, one can identify the period over which the
dispersion increases by some factor, as we did here.
Doing so and rescaling time by the resulting Lundgren
time scale collapses the present dispersion curves onto a
single, nearly exponential curve during the initial period.
LaCasce (2010) studied the EOLE data using PDFs.
However, he obtained the Richardson parameter by
fitting the asymptotic Richardson dispersion relation
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VOLUME 72
FIG. 11. (left) The longitudinal (solid lines) and transverse (dashed) second-order structure functions for the
January (blue) and September (red) deployments at (a) 608N and 6 km, (c) 608N and 18 km, and (e) 108N and 12 km.
(right) The relative dispersion curves for the January deployments (black dashed lines) at (b) 608N and 315 K,
(d) 608N and 430 K, and (f) 108N and 350 K. The Lundgren dispersion (red) is also plotted for reference, and T is given
in the bottom-right corner.
JULY 2015
GRAFF ET AL.
2781
FIG. 12. As in Fig. 9, but for the isotropy, as measured by (a) the ratio between the transverse and longitudinal
structure functions at r 5 100 km and (b) the ratio between the meridional and zonal relative dispersion at r 5 1000 km.
Note that the latter measure is the inverse of that shown in Fig. 4.
[(A18)]. Using the full dispersion relation [(A12)] instead, we find closer agreement between the Lundgren
and Richardson PDFs at early times. We also found that
the full Richardson dispersion does not reach the asymptotic limit before roughly 10 days (not shown), by
which time the pair velocities are decorrelated. It is
therefore preferable to use the full Richardson PDF.
6. Summary and discussion
We examined relative dispersion using pairs of synthetic particles advected with ECMWF winds in the
FLEXPART model. The particles were deployed at 108,
308, and 608 in both hemispheres on three potential
temperature surfaces (315, 350, and 430 K) and in two
seasons (January and September). The wind data had
18 3 18 resolution, so the smallest sampled scale was
roughly 100 km. We calculated various separation statistics and compared the results to Eulerian structure
functions based on expectations from homogeneous 2D
turbulence.
The pair velocities are correlated over roughly the
first decade of sampled scales (100–1000 km). The dispersion is also laterally isotropic over these scales at the
higher latitudes. At scales exceeding several thousand
kilometers, the dispersion is zonally anisotropic, and the
pair motion is uncorrelated. However, both the dispersion and the structure functions indicate zonal anisotropy down to the smallest sampled scales at low
latitudes.
For all deployments, the initial dispersion was nonlocal. The dispersion curves accordingly collapse onto
an exponential after rescaling time by the e-folding time.
The pair-separation PDFs evolve consistently with the
nonlocal solution of Lundgren (1981), and the separation kurtoses grow exponentially in time. The secondorder longitudinal structure functions exhibit an r2
dependence at the smallest scales, which is also consistent with nonlocal dispersion.
However, there are also variations. The e-folding
times increase with height, with those at 430 K being up
to 3 times larger than at the lower levels. The energycontaining scales are larger at 430 K also, implying the
exponential growth proceeds to larger scales. And the
dispersion is more anisotropic at low latitudes than at
high latitudes, as noted. Therefore, 2D turbulence theory may only be relevant at the higher latitudes.
The results do not indicate Richardson dispersion at
small scales, as would be expected from the K25/3
spectra observed with aircraft data (Nastrom and Gage
1985). This is because the ECMWF winds lack an r2/3
range at small scales. The mesoscale K25/3 range has
been observed in several high-resolution numerical
simulations (Koshyk and Hamilton 2001; Hamilton et al.
2008; Brune and Becker 2013), and it should be possible
in future to examine relative dispersion in such runs.
However, it appears that the divergent component of
the winds is important in this range (VanZandt 1982;
Cho and Lindborg 2001; Brune and Becker 2013), so a
different framework for interpreting the dispersion may
be required.
Often, as here, there are multiple dispersion regimes.
The nonlocal dispersion at small scales is succeeded by a
different type of dispersion at larger scales. Thus, the
kurtosis does not increase indefinitely, as in a pure
enstrophy cascade, but begins to fall at later times. With
additional regimes (i.e., a Richardson range at even
smaller scales), a more nuanced deployment strategy
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JOURNAL OF THE ATMOSPHERIC SCIENCES
would be desirable, comparing, for example, initial
separations of 10 and 100 km. We are currently examining this with other data.
The present study admittedly employs unrealistically
large numbers of pairs. Indeed, these calculations involve 720 pairs per deployment; repeated over 19 years,
this yields 13 680 pairs for each release—a number
greatly in excess of any in situ experiment. We found
that using groups of 200 pairs produced variations in the
dispersion curves but did not change the qualitative
behavior (not shown). The kurtosis, being a fourth-order
moment, was more sensitive to having fewer pairs. In all
cases, the kurtosis increased to large values, but the
maximum value varied.
As noted, the FLEXPART routine employs stochastic terms to represent small-scale mixing unresolved in the advecting winds. Such terms could also
be used to represent uncertainties in those winds. Removing them leads to even larger values of the maximum kurtosis, as the pairs are strained only by the
large-scale eddies. Nonlocal advection produces filaments, and, in the absence of small-scale mixing, the
filaments become progressively thinner. This is a significant issue, for example, with regard to the dispersion of volcanic ash.
Last, we did not explicitly examine the predominantly
zonal dispersion occurring at large separations because
this is an issue of absolute dispersion (as the pair velocities are uncorrelated). However, the low latitudes
exhibit zonally anisotropic, nonlocal relative dispersion;
the separations still grow exponentially in time despite
the lack of isotropy. This is interesting and perhaps
warrants further examination.
Acknowledgments. We thank Andreas Stohl and
Ignacio Pisso for sharing FLEXPART and Sabine
Eckhardt, Ignacio Pisso, and Anne Fouilloux for helping to set it up. We also thank Bjørg Rognerud for retrieving the ECMWF data and Rodrigo Caballero,
Gwendal Riviere, and three anonymous reviewers for
comments on the manuscript. JHL was supported in
part by Grant 221780 (NORSEE) from the Norwegian
Research Council.
APPENDIX
PDFs and Moments
Solutions to the Fokker–Planck equation [(2)] are
given by LaCasce (2010) and included here for convenience. The equation is solved using the Laplace transform assuming a delta-function initial condition
(implying all pairs have the same initial separation).
VOLUME 72
When pair motion is uncorrelated, the relative diffusivity is constant, and the solution to (2) is as follows:
p(r, t) 5
2
r r
r 1 r2
1
exp 2 0
I0 0
,
2k2 t
4pk2 t
4k2 t
(A1)
where r0 is the initial pair separation and I0 is a modified
Bessel function. At long times and large separations
(t r0 r/k2 , r r0 ), this asymptotically approaches
p(r, t) 5
1
r2
,
exp 2
4pk2 t
4k2 t
(A2)
which is proportional to the Rayleigh distribution.
The separation moments are defined in (3):
hrn i 5 2p
ð‘
r n11 p dr .
(A3)
0
With (A2), this is
n
hrn i 5 (4k2 t)n/2 G 1 1 ,
2
(A4)
where G is the gamma function. The relative dispersion
is the second moment:
hr2 i 5 2p
ð‘
0
r3 p dr 5 4k2 t .
(A5)
The linear dependence on time is as expected for a diffusive process (Taylor 1921). Furthermore, the kurtosis
(the fourth moment normalized by the square of the
second moment) is 2 in the asymptotic limit.
In a turbulent energy cascade, the pair velocities are
correlated and the diffusivity has the form
k2 5 br4/3 ,
(A6)
where b is a constant (proportional to the third root of
the energy dissipation rate). This applies both to the
forward energy cascade in 3D turbulence and the inverse cascade in 2D turbulence. The corresponding solution to (2) is
2
3
!
2/3 1 r2/3 )
9(r
9r01/3 r1/3
3
0
4
5
.
exp 2
I2
p(r, t) 5
2bt
4bt
4pbtr02/3 r2/3
(A7)
The moments of the Richardson distribution given
by LaCasce (2010) apply in the long time limit. However, one can calculate the full dispersion (i.e., from the
delta-function initial condition) as follows. Inserting
(A7) into (3),
JULY 2015
2783
GRAFF ET AL.
!ð
pffiffiffi!
2
‘
9r02/3
9r01/3 u
3
1
9u
(3n12)/2
hr i 5
exp 2
u
exp 2
du,
I
2bt
2 btr02/3
4bt 2
4bt
0
n
where u 5 r2/3 . We evaluate the integral in (A8), exploiting the following identity:
ð‘
0
pffiffiffi
xm21/2 exp(2ax)I2n (2g x) dx
2
2
G(m 1 n 1 1/2) 21
g
g
2m
g exp
5
a M2m,n
(A9)
G(2n 1 1)
2a
a
(A8)
(Gradshteyn and Ryzhik 2007). Here, M2m,n is the
Whittaker M function. An alternate form is
1
1
M2m,n (z) 5 exp 2 z zn11/2 M 1 n 1 m, 1 1 2n, z ,
2
2
(A10)
where M here is Kummer’s function (Abramowitz and
Stegun 1964). Thus,
!
!
(3n13)/2
2
9r02/3
9r02/3
3
1
4bt
G(3n/2 1 3)
M2(3n13)/2,1
hr i 5
exp 2
2 btr0 9
G(3)
8bt
4bt
!
!
3n/2
9r2/3
1
3n 1 6 4bt
6 1 3n 9r02/3
, 3,
.
exp 2 0 M
5 G
2
2
9
2
4bt
4bt
n
So the relative dispersion is
The corresponding raw moments are
!
!
9r02/3
9r02/3
5! 4bt 3
M 6, 3,
.
exp 2
hr i 5
2 9
4bt
4bt
2
(A12)
The initial dispersion, the small time limit of
(A12), can be estimated using the large argument
limit of the Kummer function (Abramowitz and
Stegun 1964):
G(b) x a2b
G(2 2 a)G(b 2 a 1 1)
lim M(a, b, x) 5
e x
.
11
x/‘
G(a)
G(1 2 a)G(b 2 a)x
(A13)
One can show that
lim hr
t/0
2
i 5 r02
!
20bt
1 1 2 /3 .
3r0
27r02/3
.
20t*
(A15)
At large separations and long times, the Richardson
dispersion in (A7) asymptotically approaches a selfsimilar form:
!
5
3
1
9r2/3
p(r, t) 5
.
exp 2
2 4p(bt)3
4bt
1 4bt 3n/2 3n 1 6
hr i 5
,
G
2 9
2
n
hr2 i 5 5:2675b3 t3 .
(A18)
The kurtosis is constant and equal to 5.6 in the asymptotic limit, and it is less than 5.6 at all times for the full
dispersion.
The other inertial range in 2D turbulence is the
enstrophy cascade range with an energy spectrum proportional to K23 . Here, the relative diffusivity is given by
k2 5
r2
,
T
(A19)
where T is a time scale inversely proportional to the
third root of the enstrophy dissipation rate. The corresponding solution to the F–P equation is
(
)
[ln(r/r0 ) 1 2t/T]2
p(r, t) 5
exp 2
(A20)
4t/T
4p3/2 (t/T)1/2 r02
1
(Lundgren 1981; Bennett 2006; LaCasce 2010). This has
separation moments of
(A16)
(A17)
with a relative dispersion
(A14)
As such, the first guess for b, referred to in section 4b, is
b5
(A11)
hr
n
i 5 r0n exp
n(n 1 2)t
.
T
(A21)
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JOURNAL OF THE ATMOSPHERIC SCIENCES
So the dispersion
8t
hr2 i 5 r02 exp
T
(A22)
grows exponentially in time. The kurtosis grows at the
same rate.
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