Lecture on Oceanography by Victor Zhurbas
Lecture 10. Study of surface oceanic circulation and lateral mixing using
drifters
1. Introduction
During the last three decades, satellite-tracked drifters have become a powerful tool to
study the surface oceanic circulation over a broad range of scales including large-scale
currents and mesoscale eddy fields.
satellite
(locates co-ordinates of buoys
several times a day)
transmitter
buoy
15 m
underwater sail (drag)
Fig. 0a. Drifter design
Currently the Global Drifter Program/Surface Velocity Program (GDP/SVP) data set
contains a total of more than three thousand buoy-years of data, or, more specifically, 4.6
million estimates of drifter coordinates and velocity at 15 m depth at 6-hourly intervals in the
Pacific between 60 S and 60 N obtained from 3098 drifters during the period 1979-1999. In
comparison with the Pacific Ocean, statistics of drifters in the Atlantic Ocean are considerably
poorer (1174 drifters, 1026 drifter years of data for a period 1989-1999 in the Atlantic versus
3098 drifters, 3150 drifter years of data for a period 1979-1999 in the Pacific).
Lecture on Oceanography by Victor Zhurbas
Fig. 0b. Summary plot of free-drifting buoy trajectories, 1991-1997, in the East Sea and the
NW Pacific Ocean.
2
Lecture on Oceanography by Victor Zhurbas
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Fig. 0c. Map of mean surface currents in the Pacific Ocean derived from drifters
Kuroshio Current/Kuroshio Extension, California Current, North Equatorial Current,
Equatorial Counter-current, South Equatorial Current (two branches), East Australian Current,
Indonesian Throughflow, Antarctic Circumpolar Current
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4
Gulf Stream and
North Atlantic
Current;
West Greenland
Current;
Azores Current;
Canary Current;
North Equatorial
Current;
Equatorial CounterCurrent
South Equatorial
Current;
Guiana Current;
North Brazil
Current;
South Brazil
Current;
Brazil-Malvinas
Confluence;
Agulhas
Retroflexion;
Benguela
Current;
Malvinas Current;
Antarctic
Circumpolar
Current
Fig. 0d. Map of mean surface currents in the Atlantic Ocean derived from drifters
Using drifter data one can reconstruct not only circulation patterns but mixing
parameters as well. Approaches to calculate lateral eddy diffusivity are outlined in Appendix
in the end of the lecture
2. Maps of lateral diffusivity in relation to surface circulation pattern
The GDP/SVP data set was used to calculate estimates of lateral diffusivity and the
Lagrangian length/time scales for 5 5 bins in the whole Pacific and Atlantic oceans with
2.5 offset in the x and y directions. The data consist of arrays of latitude and longitude and
drifter velocity components at 6-hourly intervals obtained by the procedure of objective
interpolation [Hansen and Poulain, 1996]. A detailed description of the approach used to
compute diffusivity is given in [Zhurbas and Oh, 2003], so we have taken the liberty not
going beyond a short description of the main idea in Appendix. All drifter data obtained for
periods 1979-1999 in the Pacific and 1989-1999 in the Atlantic have been used, and the
approach implies that not only mesoscale variability, but also seasonal and even inter-annual
variability contribute to the lateral diffusivity. Therefore, diffusivity estimates we obtain are
to be treated as climate estimates. In principle, the approach can be easily modified to resolve
the inter-annual and seasonal variability of K, but we do not have enough drifter data to do
that over the whole extent of the Pacific and/or Atlantic Oceans.
Lecture on Oceanography by Victor Zhurbas
5
2.1. Pacific Ocean
A map of the lateral diffusivity in the Pacific combined with the mean currents in the
surface layer is given in Figure 1. The mean currents were calculated in 2º 2º bins from
drifters. The diffusivity map is identical to that presented in Zhurbas and Oh [2003] except
for better resolution near to the continents achieved due to a 2.5º, instead of 5º, offset in the xdirection.
In general, the mid-latitudes of the western part of the Pacific are characterized by higher
values of K than those of the eastern part. There is a tongue of high lateral diffusivity
( 1 10 4 m 2 s -1 K 2 10 4 m 2 s -1 ) which begins off the eastern coast of Japan elongating
eastward between 31º and 40ºN. The tongue is undoubtedly related to the Kuroshio
Current/Kuroshio Extension System which is a well-known generator of warm- and cold-core
ring-eddies. A high level of K around 50ºS is likely to have been caused by baroclinic
instability in the Antarctic Circumpolar Current (e.g., Visbeck et al., [1997]).
It can be clearly seen from Figure 1 that extremely high values of K (up to 3 10 4 m 2 s -1 ) in
the eastern equatorial Pacific are attributed to the South Equatorial Current (SEC). Usually the
SEC is portrayed as two separated westward currents on both sides of the equator, in which
the northern branch does not extend west of the date line, while the southern branch centered
at 5ºS extends to the vicinity of New Guinea. The two-branch structure of the SEC in the
eastern equatorial Pacific is strongly subjected to seasonal variability, so that in April there is
an opposite, eastward flow in between the branches, while in October the branches are almost
merged into a single westward flow [Reverdin et al., 1994]. Such drastic seasonal changes in
the two-branch structure of the SEC may largely contribute to lateral mixing.
Note that the maximum values of K are observed in the northern branch of the SEC and
between the branches, while the southern branch is characterized by moderate values of K. As
a result, the maximum values of K are found just north of the equator. This displacement can
be explained by lateral mixing produced by tropical instability waves. In accordance with
observations [Qiao and Weisberg, 1998], such waves can be initiated by barotropic instability
arising primarily from the cyclonic shear region of the SEC and Equatorial Undercurrent just
north of the equator.
There is a region of high K values off the Pacific coast of Central America between 7º and
15ºN. This results from warm core anticyclonic eddies which are most likely to have been
formed from conservation of potential vorticity when the North Equatorial Countercurrent
(NECC) turns northward upon approaching the shoreline [Hansen and Maul, 1991].
Very high values of K 2 10 4 m 2 s -1 are encountered in the western equatorial Pacific
between the New Guinea and the Mindanao islands. Greatly enhanced lateral mixing in this
region is determined by an extremely complicated surface circulation pattern [Godfrey, 1996]
(see Figure 1). First, the westward flowing SEC turns to the north and east to form the
Halmahera Eddy and feeds into the NECC. Second, the westward flowing North Equatorial
Current (NEC)/Mindanao Current turns to the south and then splits in two branches. One
branch turns west to feed into the Indonesian Throughflow. The second branch turns east to
form the Mindanao Eddy and feed into the NECC.
It might seem surprising that the westward-flowing NEC manifests itself by a deep
minimum of lateral diffusivity K 4 10 3 m 2 s -1 in between 10º to 15ºN. The altimetry data
reveal that the NEC between 10º to 15ºN has a relatively low eddy kinetic energy level [Qiu,
1999]. The NEC has been shown to have less eddy energy due to both its presence in a lowlatitude band and its unidirectional flow, which make it more difficult to satisfy the necessary
condition for baroclinic instability [Qiu, 1999].
The East Australian Current (EAC) is the western boundary current within the South
Pacific subtropical gyre and is therefore the South Pacific equivalent of the Kuroshio in the
Lecture on Oceanography by Victor Zhurbas
6
North Pacific and the Gulf Stream in the North Atlantic. A prominent feature of the EAC,
which set it apart from the other western boundary currents is that baroclinic eddies formed
along its path are very vigorous compared to the current; eddy mass transports can be several
time as large as the mean EAC transport [Hamon, 1965; Godfrey and Golding, 1981]. A
further distinctive feature associated with the EAC is that its surface flow displays a strong
seasonal cycle: the seasonal amplitude of the EAC is up to 6 Sv, or about the mean of 9.5 Sv
[Ridgway and Godfrey, 1997]. Note that a high lateral diffusivity tongue in the South Pacific
subtropics originates from the vicinity of the EAC separation point (the Sugarloaf Point at
32º30´S) [Godfrey et al., 1980], then propagates northeast just in accordance with the pattern
of the surface outflow from the EAC [Ridgway and Dunn, 2003], and after crossing the dateline meridian elongates zonally within 20 to 30 S as far as 120 -110 W.
As we already mentioned, two approximately symmetrical (relative to the equator) tongues
of high K value in the western Pacific, centered at 21ºN and 24ºS respectively, are likely the
most remarkable feature of the diffusivity map in the Pacific Ocean. Zhurbas and Oh [2003]
related existence of the northern tongue to the apparently unstable NSTCC [Yoshida and
Kidokoro, 1967; Cushman-Roisin, 1984; Qiu, 1999]. However, we have failed to find in the
literature any information capable of explaining the existence of the southern tongue. It
motivated us to perform an investigation of the issue (see Section 3. Discussion and
Conclusions).
Fig.1. Map of lateral diffusivity in the Pacific Ocean derived from drifters
Lecture on Oceanography by Victor Zhurbas
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2.2. Atlantic Ocean
Moreover, the data are distributed quite non-uniformly, so that density of drifters is within
1000 to 8000 drifter days per 5 5 bin in the North Atlantic between 20 N and 60 N (the
same density of drifters is typical for the most of Pacific between 40 S and 50 N), from 500
to 2000 drifter days per 5 5 bin in the South Atlantic between 50 S and 20 S, and below
250 drifter days per 5 5 bin in the Tropical Atlantic. There are little or even no data off the
western coast of Africa between 5 no 25 N and throughout the south equatorial Atlantic till
20 S which results in gaps in the diffusivity map (Figure 2).
Enhanced values of K 4 10 3 m 2 s -1 are typical for the western part of the North Atlantic
Ocean in the Gulf Stream/North Atlantic Current System (GS/NAC) and south of it. North of
the GS/NAC, there is a localized increase of K up to 1 10 4 m 2 s -1 in the West Greenland
Current. The highest values of K (1.5 2.8) 10 4 m 2 s -1 are observed between the North
Atlantic Current and the Azores Current from 40º to 50ºW, which is known to be a highly
energetically active zone. In the eastern North Atlantic K is below 4 10 3 m 2s -1 everywhere
except for a narrow zonal trans-Atlantic belt of enhanced lateral diffusivity from 32º to 37ºN.
This belt is surely associated with the Azores Current frontal zone which is known to be a
strongly meandering and eddy shedding system [Käse and Siedler, 1982; Kielmann and Käse,
1987; Zhurbas et al., 1993; Tychensky et al., 1998]. Being narrow and strongly meandering,
the Azores Current has a weak signal in mean currents, averaged within 2 2 bins (less than
5 cm/s, see Figure 2).
Like the North Atlantic Ocean, the western part of equatorial Atlantic is characterized by
much higher values of K than the eastern part. High values of K 10 4 m 2 s -1 are observed by
the coast of North Brazil from 3ºS to 10ºN. This is 2º-3ºS, where the westward flowing South
Equatorial Current (SEC) turns northwestward along the coast as a western boundary current
called the North Brazil Current (NBC). After crossing the equator, its lower part turns back
eastward just north of the equator and feeds into the Equatorial Undercurrent, while the nearsurface part continues to 5ºN and then turns into the North Equatorial Countercurrent
(NECC). This regime is subjected to vigorous seasonal changes documented by both
observations [Richardson and Walsh, 1986; Richardson and Reverdin, 1987] and high
resolution numerical models [Philander and Pasanowski, 1986; Schott and Böning, 1991].
The NECC was shown to have its maximum in late summer and its minimum in late winter
with surface currents even reversing to westward. In summer the NBC/NECC regime between
3º and 7ºN shows a pronounced retroflection with strong oscillations and water mass transfer
northwestward through eddy shedding [Johns et al., 1990]. As a result, the eddy kinetic
energy and lateral diffusivity are strongly increased in a triangle with one side resting upon
the North Brazil shoreline between 3ºS and 10ºN, and a point (4ºN, 20Wº) situated within the
NECC. Note that enhanced values of lateral diffusivity in the equatorial zone, where quasigeostrophic eddies do not exist in view of vanishing the Coriolis parameter, can also result
from tropical wave instabilities [Proehl, 1996; Katz, 1997]
There are two locations of high values of K 8 10 3 m 2s -1 in the South Atlantic Ocean:
one from 35ºS to 45ºS and from 55ºW to 35ºW attributed to the Brazil-Malvinas Confluence,
and a second from 32ºS to 44ºS and from 6ºE to 22ºE attributed to the Agulhas Retroflection
(south of the Cape of Good Hope). Both locations are known as sites of the highest level of
eddy kinetic energy [Wilkin and Morrow, 1994; Florenchie and Verron, 1998].
Similar to the North Atlantic, there is a belt of enhanced lateral diffusivity
( 4 10 3 m 2 s -1 K 8 10 3 m 2s -1 ) in between 28ºS to 35ºS crossing zonally the whole South
Atlantic Ocean. Figure 2 clearly shows that the belt originates from the Agulhas Retroflection
where the highest values of lateral diffusivity are observed ( K 2 10 4 m 2 s -1 ), and then
Lecture on Oceanography by Victor Zhurbas
8
gradually decreases northwestward. The Agulhas Retroflection is known to produce warm
rings by means of eddy shedding, which drift into the South Atlantic [e.g., Lutjeharms and
van Ballegooyen, 1988]. Because the surface dynamic height of a warm core eddy is higher
than surrounding waters, the rings are visible in satellite altimeter measurements and can be
tracked from maps of sea surface height anomaly. Byrne et al. [1995] have tracked more than
20 Angulas eddies and found that the trajectories intersected the 20ºW meridian in between
25º to 35ºS which fits well the latitudinal range of the enhanced diffusivity belt. So we have
no doubt that the belt results from eddy shedding by the Agulhas Retroflection and drifting
northwestward and westward across the South Atlantic. Comparing the diffusivity map and
the surface circulation pattern assembled from drifters (Figure 2) one may conclude that the
enhanced diffusivity belt is situated in between an eastward flow, most likely related to the
Subtropical Front, and a westward flow which may be regarded as an extension of the
Benguela Current.
It is interesting that the lateral diffusivity map of the South Atlantic Ocean is very similar
to the mean eddy kinetic energy map obtained by assimilation of TOPEX/POSEIDON and
ERS 1 altimeter data in a quasi-geostrophic model [Florenchie and Verron, 1998].
Figure 3 is a map of the Lagrangian velocity scale, Vrms , in the Atlantic Ocean derived
from drifters. Similar to the lateral diffusivity estimates, Vrms was estimated as the minor
principal component of the Largangian velocity covariance in 5 5 bins. In fact, Vrms is the
square root of EKE so it is not surprising that that there is general resemblance of Vrms and K
patterns (cf., Figures 2 and 3).
Fig.2. Map of lateral diffusivity in the Atlantic Ocean derived from drifters
Lecture on Oceanography by Victor Zhurbas
9
Fig. 1a. Comparison of maps of (left) the Lagrangian velocity scale (r.m.s. of eddy kinetic
energy density) and (right) lateral diffusivity.
3. Discussion and Conclusions
In the previous sections, results of recent oceanographic studies taken from the literature
sources were analyzed in order to understand the lateral diffusivity maps of the Pacific and
Atlantic Oceans derived from drifters. As a result, a more or less reasonable explanation was
found for all the features of the diffusivity maps except the high diffusivity tongue in the
south Pacific subtropics. Here we will focus on this puzzle.
Taking into account the review of surface circulation in the Pacific and Atlantic Oceans
presented in previous sections, there are two possible explanations for the South Pacific high
diffusivity tongue:
1) The tongue results from eddy formation in the EAC (off the eastern coast of Australia)
and northeastward and eastward drift across the South Pacific (an analogue of eddy
shedding by the Agulhas Retroflection and drifting northwestward and westward across
the South Atlantic);
2) The tongue results from eddy formation caused by baroclinic instability of a narrow,
seasonal zonal current (an analogue of the NSTSS in the north Pacific).
Option 1 implies that the formation mechanism of the high diffusivity tongue in the south
Pacific subtropics is similar to that of the trans-ocean belt of enhanced diffusivity in the South
Atlantic. Evidence in favor of this option is that the tongue seems to originate from the
vicinity of the AEC separation point, so the value of lateral diffusivity being the highest at
this point gradually decreases eastward along the tongue (see Figure 1). A weak point of
Option 1 hypothesis is that the eddies generally translate westward (not eastward) in view of
the Rossby wave dynamics.
Option 2 implies that the formation mechanism of the high diffusivity tongue in the south
Pacific subtropics is similar to that of its counterpart in the north Pacific. Evidence in favor of
Option 2 is the striking symmetry of the northern and southern tongues relative to the equator.
The following reasoning seems useful to resolve the issues of the two options. Even if the
formation of eddies in the EAC does have a strong seasonal cycle, adopting Option 1 does not
Lecture on Oceanography by Victor Zhurbas
10
imply the considerable seasonal cycle necessary for the EKE and lateral diffusivity values
within the tongue, for it would take years for eddies to be advected throughout the huge zonal
extension of the tongue (about 9000 km). To the contrary, Option 2 implies a strong seasonal
cycle for the EKE and lateral diffusivity values within the tongue which was undoubtedly
proven in the case of its northern counterpart [Qiu, 1999]. That is why it seems worthwhile
analyzing the seasonal cycle of the EKE within the tongues of high lateral diffusivity derived
from drifters.
Figure 4 is a time series of bimonthly values of the EKE within the northern and southern
tongues of high lateral diffusivity in the subtropical Pacific derived from drifters. The EKE
estimates were obtained by combined ensemble-time-averaging the drifter velocities within
rectangles shown in Figure 1, and at successive bimonthly time intervals. Estimates are
presented only for those bimonthly periods when no less than 10 drifters spent more than 300
drifter-days within the rectangles; Figure 4 also presents a time series of monthly values of
the Southern Oscillation Index (SOI). It is clearly seen from Figure 4 that the seasonal cycle is
the main contributor to temporal variability of the EKE in both northern and southern
tongues. Inter-annual variability of the EKE is less pronounced and does not seem to have any
considerable correlation with the SOI. However, the time series of EKE is too short (near 7
years) to come to any reliable conclusion about its correlation with the SOI.
More clearly, the seasonal cycle of the EKE in the northern and southern high diffusivity
tongues of the Pacific subtropics is shown in Figure 5a. Both tongues are characterized by
well pronounced seasonal cycle of the EKE so that the difference between maximum EKE in
the spring – summer season (in April to July in the Northern Hemisphere and in September to
January in the Southern Hemisphere) and its minimum in the fall –winter season (in October
to January in the Northern Hemisphere and April to June in the Southern Hemisphere) is
about 60-70% of the annual mean. Note that in the trans-ocean belt of enhanced diffusivity in
the South Atlantic formed by the westward drift of Agulhas anticyclonic eddies, the seasonal
cycle of the EKE is not revealed at all (Figure 5b).
To learn whether there is a link between EKE changes in the high diffusivity tongues of the
Pacific subtropics and the mean currents, the monthly mean values of EKE and the zonal
component of the current, U mean , were calculated in 50 1 bins and 45 1 bins within the
rectangles situated in the northern and southern tongues, respectively (see Figure 1). The EKE
and U mean versus the month number and latitude in the high diffusivity tongues are shown in
Figure 6.
The high diffusivity tongue in the North Pacific subtropics lies at the northern periphery of
the westerly-flowing NEC, so that the year-averaged zonal current is negative everywhere
within latitude range from 18 to 30 N (i.e., this latitude range still lies to the south of the
subtropical gyre center). However, the opposite, easterly-flowing NSTCC develops in the
spring-summer season within a narrow latitudinal zone of 21 -24 N. The NEC/NSTCC
system is known to be strongly liable to baroclinic instability which explains the increase of
EKE in the spring-summer season and the existence of a high diffusivity tongue in the North
Pacific subtropics. Such a scenario is well-documented (e.g., Qiu [1999]) and corresponds
well to Figure 6.
Figure 6 suggests a more sophisticated scenario for the high diffusivity tongue in the South
Pacific subtropics. The SEC does not propagate from the equator as far as the NEC, and the
latitude range from 18 to 30 S is partially outside the SEC (e.g., the southern subtropical
gyre center lies within the latitude range). In the Southern Hemisphere fall-winter season
(from May to August) the westward-flowing SEC becomes weaker so that an opposite,
eastward flow is observed as close to the equator as 20 -21 S. In the Southern Hemisphere
spring-summer season the SEC becomes stronger and extends up to 26 -28 S, and this is the
Lecture on Oceanography by Victor Zhurbas
11
season when an opposite, easterly-flowing countercurrent develops in a narrow latitudinal
zone within 22 to 25 S. We failed to find any mention of this countercurrent in the literature;
though it seems quite reasonable to name it the South Subtropical Countercurrent (SSTCC).
We may suggest that, like the NEC/NSTCC, the SEC/SSTCC system is subjected to
baroclinic instability which may explain the increase of EKE in the Southern Hemisphere
spring-summer season and thereby the existence of the high diffusivity tongue in the south
Pacific subtropics.
To show the possibility of causal relationship between the SSTCC and the high diffusivity
tongue, maps of zonal component of the mean current in the surface layer have been obtained
from drifters in the south Pacific subtropics from 170 E to 100 W (Figure 7). The averaging
was done within 10 1 bins using all data (top panel), spring season data (middle panel), and
fall – early winter season data (bottom panel). We did not extend the maps to the western part
of the tongue, for the density of drifters west of 170 E was considerably poorer (below 500
drifter days per 5 5 against 1000 to 2000 drifter days per 5 5 bin east of 180 E. The
maps reveal eastward flow within 25 to 22 S that may be identified with the SSTCC. This
SSTCC is most pronounced in the spring season (the middle panel of Figure 7) when it has
the mean velocity about 10 cm/s. Note that both the SSTCC and the high diffusivity tongue
can be traced as far as 120 -110 W (cf., Figures 1 and 7), so we may suggest that the latter is
a consequence of the former.
Cornuelle and Roemmich [1990] analyzed 16 XBT sections between New Zeland and Fiji
to study the gyre-scale circulation variability in the subtropical South Pacific. They found that
the gyre center, as represented by zero velocity, is subjected to drastic migrations even within
one season which at least does not contradict the above-described interaction within the
hypothetical SEC/SSTCC system.
Ridgway and Dunn [2003] performed an extensive study to provide a much improved
picture of the mean circulation patterns in the southwestern Pacific Ocean on the base of
historical hydrography dataset. In particular, their sections of geostrophic zonal velocity
displayed topography-controlled eastward jets of the EAC outflow that can be traced as far as
180 E, and we do not exclude that the SSTCC may be considered as the extension of one of
such jets. Unfortunately, Ridgway and Dunn [2003] could not extend their analysis to the area
east of 180 E that have been poorly sampled. We hope together with Ridgway and Dunn
[2003] that future monitoring programmes such as Argo, a global array of profiling floats,
will provide the data to complete the picture of South Pacific circulation and thereby to check
the suggested mechanism for the formation of a high diffusivity tongue in the subtropical
South Pacific.
It seems worthwhile to compare our map of lateral diffusivity in the North Atlantic with
that of Lumpkin et al. [2002] and McClean et al. [2002] whose approaches to deriving
apparent diffusivity from surface drifters were different from ours. Lumpkin et al. [2002]
applied the classical approach by Taylor [1921] and computed K by integrating the
Lagrangian velocity autocorrelation for each 120-day segment of drifter trajectories up to the
first zero crossing; the mean and linear trend of each segment was removed beforehand to
discard the effect of mean velocity shear and low-frequency variability. In general, the
absolute value of K and its distribution across the North Atlantic obtained by Lumpkin et al.
[2002] is in a good accordance with our Figure 2. The only difference is that our map
provides better resolution of the enhanced diffusivity belt related to the Azores Current.
McClean et al. [2002] calculated the Davis’ diffusivity tensor [Davis, 1991] in 5 5 bins (as
we did), but did not apply the minor principal component approach to discard biases caused
by the mean velocity shear. As a result, their diffusivity maps did not resolve or only poorly
resolved such details as an enhanced diffusivity belt related to the Azores Current and their
estimates of zonal diffusivity are about twice as large as our K.
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12
In our previous paper [Zhurbas and Oh, 2003] a promising relationship between the
drifter-born Lagrangian length scale L K / Vrms and the first mode baroclinic radius of
deformation taken from Emery et al. [1984] of a form L Re was obtained for the midlatitudes of the North Pacific. So it seemed reasonable to bring into the analysis estimates of L
from drifters in the North Atlantic mid-latitudes. Figure 8 presents a composite relationship of
the kind consisting of data from northern mid-latitudes of both the Pacific and Atlantic. In
contrast to the North Pacific, in the North Atlantic estimates of L have proven to display poor
correlation with Re, though the values of L and Re are of the same order of magnitude. The
poorest correlation was observed east of 55 W within the NAC and the Labrador Current and
north of them (L is more then twice as large as Re), as well as in the southeast corner of the
Subtropical Gyre of the North Atlantic (Re is more then twice as large as L).
Figure 4. Time series of bimonthly mean values of the EKE within the northern (diamonds)
and southern (circles) tongues of high lateral diffusivity in the subtropical Pacific derived
from drifters and monthly mean values of the Southern Oscillation Index (asterisks).
Lecture on Oceanography by Victor Zhurbas
13
Figure 5. Seasonal cycle of EKE in the high diffusivity tongues of the subtropical Pacific (a)
and in the trans-ocean belts of enhanced diffusivity in the Atlantic (b); the diamonds and
closed circles refer to the Northern and Southern Hemispheres, respectively.
Figure 6. The mean zonal current, U mean , (left panels) and EKE (middle panels) versus the
month number and latitude in the high diffusivity tongues of the subtropical North (top
panels) and South (bottom panels) Pacific obtained from drifters. The right panels are the
year-averaged values of U mean and EKE versus latitude.
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14
Figure 7. Maps of the mean zonal current (cm/s) in the subtropical South Pacific obtained
from drifters. The averaging was done within 10 1 bins using all data (top panel), spring
season data (middle panel), and fall – early winter season data (bottom panel).
Lecture on Oceanography by Victor Zhurbas
15
Figure 8. Lagrangian length scale L K / Vrms versus the first mode baroclinic radius of
deformation Ri in the North Pacific (left) and North Atlantic (right, closed circles).
4. Appendix. Calculation of Lateral Diffusivity
There are several approaches that have been applied to calculate the diffusivity from
Lagrangian particles. Here we focus on the single particle approaches, for presently the twoparticle approach (or so-called relative diffusivity) still cannot be applied for mapping as a
result of the lack of data.
The most familiar method of extracting diffusivity from individual drifters is by integrating
the Lagrangian velocity autocorrelation [Taylor, 1921]. This approach is rather sensitive to
the asymptotic behavior of the autocorrelation at large time lags, so that very long-term
records are desirable. Even if long-term records are available, this approach does not suit the
mapping purposes very well, for it yields the diffusivity attributed to the whole drifter track
rather than the geographical region under consideration. The approach was applied to drifter
data from the northeast Pacific by Poulain and Niiler [1989] and Paduan and Niiler [1993].
To avoid difficulties with the asymptotic behavior of Lagrangian velocity autocorrelation,
Griffa et. al. [1995] proposed a parametric approach, which assumes that the Lagrangian
velocity autocorrelation has a known shape prescribed by some statistical model, so that its
value can be estimated just from several parameters of the model.
The horizontal diffusivity can be also calculated as the half growth rate of single-particle
dispersion, provided that the particles are deployed from a fixed position with sufficiently
large time intervals. To put this approach into practice, Colin de Verdiere [1983] and then
Poulain and Niiler [1989] suggested using fragments of the same drifter track as independent
tracks to achieve reliable statistics.
An advanced approach for obtaining the lateral diffusivity from drifters was developed by
Davis [1987, 1991]. He derived a generalized advection-diffusion equation for describing, in
the Eulerian frame, the evolution of the mean concentration of a passive tracer in an
inhomogeneous velocity field. In this equation, eddy transport is governed by a timedependent diffusivity tensor which is a pure single-particle statistic, since it can be calculated
from the statistics of a single Lagrangian particle. This gives us the straightforward possibility
Lecture on Oceanography by Victor Zhurbas
16
of mapping the single-particle diffusivity. Note that, in contrast to the Lagrangian velocity
autocorrelation and single-particle dispersion growth rate approaches, the Davis’ diffusivity
tensor holds true for the shear mean flow. Swenson and Niiler [1996] applied Davis’ approach
to map the lateral diffusivity in the California Current System.
To make the classical Taylor’s approach applicable to inhomogeneous mean flow, Bauer et
al. [1998] proposed separating the large scale mean from the observed velocity with a least
squares bicubic smoothing spline interpolation scheme to minimize the velocity fluctuation
energy at low frequencies [Inoue, 1986]. Using such a procedure, they obtained reasonable
estimates of the lateral diffusivity in the tropical Pacific Ocean, which is a region with strong
currents and strong shear.
Oh et al. [2000] argued that the Davis’ diffusivity tensor calculated for a point release
experiment in a shear flow does yield the true value of diffusivity, but when it is applied to a
finite bin in a shear flow it can result in biased diffusivity estimates. To fix the problem, they
suggested using the minor principal component of both the Davis’ diffusivity tensor and the
half growth rate of the single-particle dispersion tensor for the “true” value of diffusivity.
Simulation experiments showed that the minor principal component estimates of diffusivity
computed for finite bins in a shear flow do remain practically unbiased. Oh et al. [2000]
applied their minor principal component approach to calculate lateral diffusivity in the East
Sea (the Sea of Japan) and the northwest Pacific.
Keeping in mind the task of mapping the lateral diffusivity across the whole Pacific, one
has to choose or develop a robust approach to satisfy the following requirements:
- it should contain a rigid, unambiguous algorithm for obtaining a reliable estimate of
diffusivity for a wide variety of environmental conditions without any localized fitting;
- the diffusivity estimates it yields should be in reasonably good accordance with other
authors’ estimates related to the whole area under consideration.
Such a robust approach is described below.
Following Davis [1991], the single-particle diffusivity tensor is defined as
k jk ( x, t )
v j (t 0 | x, t 0 )d k (t 0 t | x, t 0 ) ,
(1)
where v and d are the departures from the Lagrangian mean velocity and displacement,
respectively, t 0 is the label for the time origin, and angle brackets indicate averaging over the
ensemble of particles. The notation a(t | x, t 0 ) represents the value of a at time t of a particle
passing through x at time t 0 .
Lateral diffusivity can be also estimated as the half growth rate of the single-particle
dispersion tensor (or the displacement variance tensor), s jk :
1 s jk
; s jk (x, t ) d j (t 0 t | x, t 0 )d k (t 0 t | x, t 0 ) .
(2)
2 t
Tensors k jk and k *jk are functions of the Eulerian position x and time t, and the physical
k *jk (x, t )
sense of diffusivity is attributed to the asymptotic values k jk (x, ) and k *jk (x, ) .
In the case of shear mean flow, tensor k jk in its “pure” definition (i.e., when the averaging
ensemble consists of particles passing through a fixed position x at different moments) still
yields the true value of diffusivity, while components of k *jk can be biased by the shear effect.
That is why Davis [1991] advises that it is better to calculate the diffusivity directly from (1)
rather than (2). However, dealing with ocean drifters we are forced considering particles that
pass through a finite vicinity of the position x (e.g., 5 5 bins) rather than through the fixed
position x (so-called pseudotrack approach suggested by Colin de Verdiere [1983], and
Lecture on Oceanography by Victor Zhurbas
17
Swenson and Niiler, [1996]). Once the pseudotrack approach is applied, both k jk and k *jk can
be amplified (positively biased) by shear.
Simulation experiments by Oh et al. [2000] showed that in the case of an isotropic eddy
field and mean flow shear, the minor principal component of both k jk and k *jk tensors
computed with the pseudotrack approach remains practically unbiased by the shear (in
contrast to the major principal component which is greatly amplified by the shear effect). For
this reason, Zhurbas and Oh (2003) suggested using the minor principal component estimates
for a “robust” approach to get scalar lateral diffusivity for mapping purposes.
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