Deep-Sea Research I 58 (2011) 468–485
Contents lists available at ScienceDirect
Deep-Sea Research I
journal homepage: www.elsevier.com/locate/dsri
Instruments and Methods
Surface circulation in the Nordic Seas from clustered drifters
I. Koszalka a,, J.H. LaCasce a, M. Andersson b, K.A. Orvik b, C. Mauritzen c
a
b
c
Institute of Geosciences, University of Oslo, 1022 Blinden, 0315 Oslo, NO, Norway
Geophysical Institute, University of Bergen, Allegt. 70, 5020 Bergen, NO, Norway
Meteorological Institute, Gaustadalleen 30D, Oslo, NO, Norway
a r t i c l e i n f o
a b s t r a c t
Article history:
Received 5 October 2010
Received in revised form
16 January 2011
Accepted 18 January 2011
Available online 31 January 2011
We compare two methods for estimating mean velocities and diffusivities from surface drifter
observations, using data from the Nordic Seas. The first is the conventional method of grouping data
into geographical bins. The second relies on a ‘‘clustering’’ algorithm, and groups velocity observations
according to nearest-neighbor distance. Capturing the spatial variability of the mean velocity requires
using bins with a length scale of 50 km. However, because many bins have few observations, the
statistical significance varies substantially between bins. Clustering yields sets with approximately the
same number of observations, so the significance is more uniform. At the densely sampled Svinøy
section, clusters can be used to construct the mean flow field with r 10 km resolution. Clustering also
excels at the estimation of eddy diffusivities, allowing resolution at the 20 km scale in the densely
sampled regions. Taking bathymetry into account in the clustering process further improves mean
estimates where the data is sparse.
Clustering the available surface drifter data, extended by recent deployments from the POLEWARD
project, reveals new features in the surface circulation. These are a large anticyclonic vortex in the
center of the Lofoten Basin and two anticyclonic recirculations at the Svinøy section. Clustering also
yields maps of the eddy diffusivities at unprecedented resolution. Diffusivities are suppressed at the
core of the Norwegian Atlantic Current, while they are elevated in the Lofoten Basin and along the
Polar Front.
& 2011 Elsevier Ltd. All rights reserved.
Keywords:
Surface drifters
Lagrangian methods
Binning
Clustering
The Nordic Seas
1. Introduction
The Nordic Seas, comprising the Norwegian, Iceland and
Greenland Seas (Fig. 1a) are a region of special significance in
the World ocean. The warm Atlantic waters entering in the south
at the surface cool as they flow northward, submerging and
eventually feeding what becomes the North Atlantic Deep Water.
The cooling is enhanced by eddies, which stir the waters into the
interior of the Nordic Seas, prolonging their contact with the
atmosphere. As such, the processes occurring here have potential
global impact, as they influence the Meridional Overturning
Circulation. The Nordic Seas have been the site of extensive
observational surveys and monitoring for at least a 100 years
(see for instance Helland-Hansen and Nansen, 1909; Dickson
et al., 2008; Mauritzen et al., accepted for publication).
Volume and heat transport through the Nordic Seas have
primarily been studied with hydrography and current meter
moorings (e.g., Mauritzen, 1996; Oliver and Heywood, 2003;
Corresponding author. Tel.: + 47 228 44582; fax: + 47 228 55269.
E-mail address: inga.koszalka@geo.uio.no (I. Koszalka).
URL: http://folk.uio.no/ingako/ (I. Koszalka).
0967-0637/$ - see front matter & 2011 Elsevier Ltd. All rights reserved.
doi:10.1016/j.dsr.2011.01.007
Orvik and Skagseth, 2005; Nilsen and Falck, 2006; Hansen and
Østerhus, 2007; Skagseth et al., 2008; Rossby et al., 2009a).
Several Lagrangian studies have been conducted here as well,
involving RAFOS and Argo floats (Gascard and Mork, 2008;
Søiland et al., 2008; Rossby et al., 2009b; Voet et al., accepted
for publication) and surface drifters (Poulain et al., 1996; Orvik
and Niiler, 2002; Jakobsen et al., 2003; Koszalka et al., 2009;
Andersson et al., in preparation). There is an ever-growing
collection of Lagrangian observations in the region. It is worthwhile therefore to try to improve the analysis methods, in order
to extract as much information as possible from the trajectories.
Lagrangian data is frequently used to estimate the Eulerian
mean velocities and the lateral diffusivities (Davis, 1991). These
quantities can then be used to describe the evolution of a tracer
field yðx,tÞ by an advection–diffusion equation
@
/ySþ U r/yS ¼ rðKr/ySÞ
@t
ð1Þ
where Uðx,yÞ and Kðx,tÞ are the time-mean Eulerian velocity and
the eddy diffusivity tensor. These represent advection by the mean
flow and diffusion by eddies, respectively. Usually, the means Uðx,yÞ
are estimated by grouping the Lagrangian data in geographical
bins of a specified size and averaging (e.g., Rossby et al., 1983;
I. Koszalka et al. / Deep-Sea Research I 58 (2011) 468–485
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Fig. 1. (a) Main bathymetric features and pathways of the Atlantic inflow in the Nordic Seas. BI, Bear Island; BS, Barents Sea; FI, Faroe Islands; GI, Gimsøy Island; GB,
Greenland Basin; IP, Iceland Plateau; LB, Lofoten Basin; LI, Lofoten Islands; NB, Norwegian Basin; SV, Svinøy section; VP, Vøring Plateau. The Svinøy section is also marked
with a black line. The two branches of Norwegian Atlantic current are the eastern branch (EB) and the western branch (WB); and NCC stands for the Norwegian Coastal
Current. (b) Drifter trajectories colored by deployment site: Svinøy (red), Lofoten Basin (cyan), Gimsøy and Barents Sea Opening (blue), Greenland Sea (pink), IcelandFaroe-Ridge and Iceland Plateau (orange). Drifters advected from the North Atlantic are plotted in green. (For interpretation of the references to color in this figure legend,
the reader is referred to the web version of this article.)
Swenson and Niiler, 1996; Brink et al., 2000; Fratantoni, 2001;
Jakobsen et al., 2003; Thompson et al., 2009). In this, one assumes
the statistics in the bins are stationary. The residual velocities (the
difference between the velocities and the local mean) are then used
to estimate eddy diffusivities. For example, the component of the
diffusivity tensor in the zonal direction (x) is written as
kxx ðtÞ ¼ /xL ðtÞuL ðtÞS ¼
Z
t
0
/uL ðtÞuL ðtÞS dt ¼
Z
t
Pxx ðtÞ dt
ð2Þ
0
where uL the Lagrangian velocity and PðtÞ the time-lagged Lagrangian velocity covariance.
The technique has been used previously to analyze drifter data
in the Nordic Seas, and our knowledge of the mean surface
circulation is based in part upon these results. Thus we know
that there are two branches of the Norwegian Atlantic Current
(NwAC), flanked by the Norwegian Coastal Current (NCC) on the
shelf (Fig. 1a). The outer branch of the NwAC rounds the Lofoten
Basin before rejoining the inner branch to the north (Poulain
et al., 1996; Saetre, 1999; Orvik and Niiler, 2002; Walczowski and
Piechura, 2007). There are also gyres in the Norwegian and
Greenland Basins. The circulation also varies seasonally, with
the slope- and gyre circulation intensifying during winter
(Isachsen et al., 2003; Andersson et al., in preparation).
A shortcoming with the method is that while the bins are
uniformly distributed in space, the data generally is not. This
complicates the choice of the bin size (e.g. Davis, 1991; Mariano
and Ryan, 2007; LaCasce, 2008; Koszalka and LaCasce, 2010). Bins
should be small enough to resolve the mean flow but larger than
the scale of the energetic eddies (otherwise the latter can be
subsumed into the mean). The bins should also be large enough to
yield statistically significant estimates. The Nordic Seas are
particularly problematic; the Norwegian Atlantic Current is
20–30 km wide in its core, a distance only somewhat larger than
the deformation-scale eddies (5–10 km) which are ubiquitous in
the Lofoten Basin and along the pathway of the main current
(Poulain et al., 1996; Skagseth and Orvik, 2002; LaCasce, 2005a;
Koszalka et al., 2009). Second, the data is irregularly distributed as
drifters were either deployed in selected areas or outside the
Nordic Seas altogether, moving into the region subsequently.
Diffusivity estimates are similarly affected by bin size. One
assumes the diffusivities asymptote at long times, i.e. that
kð~
x ,tÞ-k1 ð~
x Þ as t-1. But the integration time is affected by
the individual drifter’s residence times in the bins. Moreover, one
needs a sufficient number of trajectory segments to obtain
convergence in the velocity autocorrelation; the smaller the bins,
the worse the convergence (Griesel et al., 2010; Koszalka and
LaCasce, 2010). For this reason, diffusivities in the Nordic Seas
have been mapped in relatively large bins (Poulain et al., 1996;
Andersson et al., in preparation). This improves the statistical
significance but yields a crude representation of the eddy field
which is so important to volume and heat exchange processes in
the region (Mauritzen, 1996; Gascard and Mork, 2008; Spall,
2010; Mauritzen et al., accepted for publication).
To improve the estimates, different bin sizes and shapes/
orientations have been used (e.g., Swenson and Niiler, 1996;
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I. Koszalka et al. / Deep-Sea Research I 58 (2011) 468–485
Falco et al., 2000; Poulain, 2001; Jakobsen et al., 2003). Methods
such as fitting the binned velocities with cubic splines, (Bauer et al.,
1998, 2002) and using different formulations of Eq. (2) (e.g., de
Verdiere, 1983; Zhurbas and Oh, 2003) have also been explored.
Recently, we introduced another method for extracting Eulerian estimates from Lagrangian data (Koszalka and LaCasce, 2010).
This involves a clustering algorithm to group a specified number
of nearest-neighbor observations. Because each group has
approximately the same number of observations, the statistical
significance is more uniform than with geographical bins. The
number and areal extent of the clusters depends on the chosen
number of members, and this can be adjusted to achieve a desired
statistical significance. The clusters are generally non-uniformly
spaced, but their distribution reflects the data coverage; there are
few clusters where there is little data. In addition, the diffusivities
are calculated from a specified number of segments of equal
length, clustered according to their mid-point position. This
greatly improves the convergence of the velocity autocorrelation.
Koszalka and LaCasce (2010) tested the clustering approach by
using trajectories generated with a stochastic routine. This
employed a stationary mean flow which was representative of
that in the Nordic Seas and with constant diffusivities (isotropic
and spatially uniform). A more challenging test is to use in situ
data in the same region, as the currents vary strongly in both
space and time. Furthermore the synthetic trajectories used
by Koszalka and LaCasce (2010) were deployed uniformly in
space. In situ data is generally deployed irregularly, and this also
affects the results (Davis, 1991).
We focus hereafter on surface drifter observations from the
Nordic Seas. A large portion of the data comes from the POLEWARD experiment, carried out under the International Polar Year
(IPY, 2007–2009) (Koszalka et al., 2009; Andersson et al., in
preparation). Historical data is also used, yielding a set which
spans roughly 20 years. We calculate means and diffusivities in
clusters and compare the results with those obtained with
geographical bins. We focus exclusively on the time mean fields
and leave the question of temporal variability for subsequent
work. We also consider the oceanographic implications of the
results, in the context of previous works (Poulain et al., 1996;
Orvik and Niiler, 2002; Andersson et al., in preparation). Clustering of the available drifter data reveals the features of the
surface circulation not seen before: a surface expression of
the large anticyclonic eddy at the center of the Lofoten Basin,
and two anticyclonic recirculations of the Norwegian Atlantic
Current at the Svinøy section. It also captures the crossover flow
over the Vøring Plateau and the surface inflow into the Barents
Sea. The clustering allows mapping of eddy diffusivities at
average 80 km resolution, and down to 20 km in densely
sampled areas.
The paper is organized as follows. The study region and drifter
data are described in Section 2. In Section 3 we consider mean
velocities and eddy kinetic energy. Eddy diffusivities are considered in Section 4. In Section 5 we include the bathymetry in the
analysis of the drifter data. In Section 6 we focus on the two key
regions, the Svinøy section and the Lofoten Basin. Concluding
remarks are offered in Section 7.
2. Data
The Lagrangian investigation of surface circulation in the
Nordic Seas commenced in the early 1990s with a release of
107 drifters by the SACLANT group (Poulain et al., 1996). Subsequently, the largest deployment was during the POLEWARD
project under the International Polar Year in 2007. During the
following two years, 148 surface drifters were deployed along the
pathway of the Norwegian Atlantic Current: at Svinøy, at Gimsøy
and Bear Island, at the Barents Sea Opening, and in the Lofoten
Basin (Koszalka et al., 2009). We combine these with data
available under the Global Drifter Program (GDP, http://www.
aoml.noaa.gov/phod/dac/gdp.html), yielding in total 360 trajectories in the area covering the Nordic Seas (201W–301E, 60–801N).
The instruments are standard Surface Velocity Program (SVP)
drifters (Lumpkin and Pazos, 2007). Each drifter consists of a
surface buoy, with a transmitter and a temperature sensor, and a
subsurface drogue at 15 m depth. The buoys are tracked by the
Argos satellite system, yielding positions with 150–1000 m accuracy up to 50 times a day.
Drifter positions (longitude, latitude) were quality-controlled
and interpolated via a kriging method to 6 h-intervals by the
AOML/NOAA Drifter Data Assembly Center (see Lumpkin and
Pazos, 2007), and updated through September 2009. Only trajectory segments with the drogue attached were used. Given that the
typical Lagrangian time scale here is TL 1 day (Poulain et al.,
1996; Andersson et al., in preparation), tracks shorter than 1 day
were discarded and segments for which the time interval
between two consecutive data points was longer than 1 day were
separated.1 Drifter positions were filtered with a Butterworth
filter with a 25 h window to suppress tidal and inertial motions,
yielding in total 58,348 drifter days. The zonal and meridional
velocities were derived by central differencing from these processed drifter positions. The trajectories are plotted in Fig. 1b. As
noted, the data is unevenly distributed over the region: the
Iceland-Faroe inflow and the NwAC route in the eastern part of
the domain are densely sampled whereas only a few drifters
visited the Greenland Basin and the East Greenland Current (due
to limited battery life-time as well as icing).2
3. Mean velocities and eddy kinetic energy
3.1. Methods
We focus first on estimating the mean velocities and eddy
kinetic energy. For the bins, we must choose bin sizes. Ideally, the
bins should be small enough to resolve the mean flow but larger
than the eddy scale. They should also be large enough to yield
significant estimates. The Nordic Seas are challenging in this
regard, because the mean and the eddy length scales are small
and comparable (Poulain et al., 1996; Skagseth and Orvik, 2002;
LaCasce, 2005a). Previous studies here used (21 11) bins (Poulain
et al., 1996; Saetre, 1999; Jakobsen et al., 2003), corresponding to a
length scale of 100 km. We will use this as a ‘‘benchmark’’ bin
size. However the present data coverage allows a finer grid
(11 0.51) as well, and we will also use that. Some bins have too
few observations and are excluded from the analysis. Typically this
is done by choosing a minimum number of independent observations that legitimate the bin estimate. The number of independent
observations is n ¼ n=ð2TL Þ where n is the number of drifter days
in each bin. We choose nT ¼ 5 for this threshold (e.g., Fratantoni,
2001; Thompson et al., 2009). This corresponds to nT ¼10 drifter
days given a Lagrangian timescale TL of 1 day. In addition, only bins
with data from at least two different drifters were retained.
For the clusters, we employed a k-means clustering algorithm
(Lloyd, 1982; MacKay, 2003). This iteratively partitions a set of n
observations ðx1 ,x2 , . . . ,xn Þ, into k subsets (clusters), S¼S1, S2, y, Sk,
1
Poulain et al. (1996) found TL ¼ 1–3 days here, while Andersson et al. (in
preparation) estimated TL ¼ 1.1 days. LaCasce (2005a) found that the Eulerian
integral time is 1–2 days, which would suggest an equal or shorter Lagrangian
time.
2
For more information about the POLEWARD project and surface drifters,
see Koszalka et al. (2009), Andersson et al. (in preparation) and http://folk.uio.no/
ingako/my_files/POLEWARD_WEBPAGE_MAIN.html.
I. Koszalka et al. / Deep-Sea Research I 58 (2011) 468–485
such that each observation is assigned to the nearest cluster in a
way that minimizes the sum, over all clusters, of the squared
distance between cluster members and the cluster center mi :
min
k X
X
Jxj mi J2 :
ð3Þ
i ¼ 1 xj A Si
The procedure involves a two-step assignment/update process. In
the assignment step, each data point is assigned to the nearest
center. In the update step, the cluster centers are adjusted to
match the sample means of their member data points. This is
repeated until the assignments are unchanged. The clustering
algorithm is embedded in an additional iterative procedure that
assures that the number of cluster members is as close as possible to
the prescribed m. See Koszalka and LaCasce (2010) for details.
Obtaining finer representation of the mean field requires
having more clusters. This is equivalent to having the lowest
possible number of observations m in each cluster, while still
a
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Fig. 2. Drifter-derived estimates of the mean velocity, color-coded by the speed magnitude. Top: obtained by binning the data in grids with varying bin size: 21 11 (a) and
11 0.51 (b). Bottom: obtained by clustering the data in clusters with varying prescribed number of members: m ¼48 drifter days (c), and m¼ 12 drifter days (d). The arrow
in the upper left corner marks a speed of 30 cm/s. Only significant estimates are plotted ( Z 10 drifter days, and at least two different drifters in the ensemble). (For
interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
472
I. Koszalka et al. / Deep-Sea Research I 58 (2011) 468–485
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Fig. 3. Drifter-derived estimate of the eddy kinetic energy eke ¼ 1=2/uu2 Sþ /vu2 S ðcm2 =s2 ). Top: obtained by binning the data in grids with varying bin size: 21 11
(a) and 11 0.51 (b). Bottom: obtained by clustering the data in clusters with varying prescribed number of members: m¼ 48 drifter days (c), and m¼12 drifter days (d).
Each color point marks position of the cluster center. Only significant estimates are plotted ( Z 10 drifter days, and at least two different drifters in the ensemble). The
contour lines of [ 2600 1800 1000 100] m depth depict main bathymetric features. (For interpretation of the references to color in this figure legend, the reader is
referred to the web version of this article.)
striving for statistical significance. We used m ¼12 drifter days, a
value slightly larger than nT used with the bins. We also employed
m¼24 and 48 drifter days, for comparison.3 Because of the nonuniform data distribution, not all clusters have the same m.
As with the bins, we eliminate clusters with fewer than 10 drifter
days or containing data from only one drifter.
3.2. Results
3
Koszalka and LaCasce (2010) picked m so that the mean standard error
among the clusters was the same as in the corresponding binning assignments.
Such a designation is less applicable here due to the inhomogeneous eddy field.
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Shown in Fig. 2 are the mean speeds jU^ ðx,yÞj ¼ U 2 þV 2 from
bin-averaging (panels a, b) and clustering (m¼48 and 12 are shown
on panels c, d). Eddy kinetic energy, EKE ¼ 12 ð/uu2 Sþ /vu2 SÞ, where
I. Koszalka et al. / Deep-Sea Research I 58 (2011) 468–485
uuðx,yÞ ¼ uðx,y,tÞUðx,yÞ and vuðx,yÞ ¼ vðx,y,tÞ Vðx,yÞ, is plotted
in Fig. 3. The various parameters for bins and clusters are shown
in Tables 1 and 2, respectively. The resolution is quantified by the
number of bins/clusters and by the ‘‘radius’’ (mean square area)
covered by them.
The significance of the mean estimate is determined by the
standard error, defined as
n
s^ ¼ pffiffiffiffiffi ,
ð4Þ
n
where n and n are the r.m.s. velocity and the number of
independent observations, in the bin or cluster. The percentage
of bins and clusters failing the requirement of n Z 10 drifter days
is also tabulated. The distributions of the number of observations
in bins and clusters, as well as standard error- and length scale
statistics are presented in Fig. 4.
Consider first the bin results. With the larger bin size
(21 11, Fig. 2a), one observes that the mean flows are intensified
near the Norwegian coast and off Spitsbergen and Greenland;
however, the structure of the flow is not well resolved. The
situation is better with the finer bins (11 0.51, Fig. 2b). Now
one observes the eastern and western branches of the Norwegian
Atlantic Current (NwAC), the Norwegian Coastal Current (NCC)
and the continuous flow feeding the West Spitsbergen Current.
Note too the large eddy ( 150 km diameter) at the center of the
Lofoten Basin (33 E, 703 N). This semi-permanent anticyclonic
feature has been observed with subsurface floats since 2001
(Gascard and Mork, 2008; Søiland et al., 2008; Voet et al.,
accepted for publication).
The improvement in the eddy kinetic energy fields is similar.
With the (21 11) bins (Fig. 3a), the eddy field is seen to be
intensified off Norway, and particularly in the Lofoten basin
region. The regional variability is much better resolved though
with the (11 0.51) bins (Fig. 3b). Eddy kinetic energy is clearly
enhanced north–west off the Lofoten and Vesterålen Islands
(reaching 500 cm2 s 2), and at the Mohns Ridge. It is also
intensified at the Iceland-Faroe Ridge and the Faroe-Shetland
Channel, as well as within the NCC. The Iceland Plateau, the
Table 1
Parameters of the binning assignments: bin size (long lat), radius (mean squared
area) in km, number of populated bins, percentage of bins with less than 10 drifter
days (disregarded from the further analysis), mean and std of r.m.s velocity in the
bins (directional average), mean and std of standard error in the bins (directional
average).
Code
(deg long lat) (km) no. bins % rejected /nS (cm/s) /s^ S (cm/s)
Coarse 2 1
Fine
1 0.5
91
46
341
1211
10.5
14.4
11.6 7 4.1
11.8 7 4.2
1.5 71.2
2.5 71.3
Table 2
Parameters of the clustering assignments: prescribed number of cluster members
m, radius (mean squared area covered by cluster) in km, number of clusters,
percentage of rejected clusters with less than 10 drifter days, mean r.m.s velocity
in the clusters (directional average), mean standard error in the clusters (directional average).
m
(km)
No. clusters
% rejected
/nS (cm/s)
/s^ S (cm/s)
48
24
12
75
45
28
1216
2432
4864
0.3
1.1
3.4
12.7 7 3.6
12.4 7 3.8
12.1 7 3.9
2.6 70.8
3.6 71.2
5.07 1.7
48-topo
24-topo
12-topo
30
18
11
1214
2425
4848
2.4
4.2
6.6
12.1 7 3.8
11.8 7 3.9
11.6 7 4.1
2.6 70.9
2.9 71.2
4.7 71.7
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west-south Norwegian Basin and the Barents Sea are more
quiescent ( o 100 cm2 s2 ).
The mean fields obtained by clustering are shown in Fig. 2c, d.
Notice that the clusters are ‘‘patchy’’, because the averages occur
only where there is sufficient data. With m¼48 drifter days, the
inner branch of the NwAC is resolved and is narrower than in the
(21 11) bins. However, here too the outer branch is barely
resolved. The resolution improves greatly with m ¼12 days
(Fig. 2d). All flow features captured by the (11 0.51) bins
(Fig. 2b) are present, but in much more detail. Note in particular
how the NCC joins the NwAC south of the Lofoten Islands, and
that there is an eddy in the Lofoten Basin.
The comparisons for the kinetic energy are similar (Fig. 3c, d).
With m¼48 drifter days, the resolution is like that obtained with
the (21 11) bins. The field with m¼12 days provides a high
resolution picture of the variability along the IFR and in the
Lofoten Basin.
However, one might question the significance of the finer scale
estimates. The number of observations in the bins is not uniform
(Fig. 4a) and is less so with the smaller bins. This is reflected in
the standard error (Eq. (4)), which varies substantially (Fig. 4b),
and in that a higher proportion of the bins are rejected for having
too little data (Table 1).
The clusters have a more uniform number of observations, by
design. While there are variations around the prescribed value of
m, the data nevertheless have a nearly delta-function shaped
distribution in all cases (Fig. 4a). The clusters are also of finer
scale. With m¼12, the typical cluster radius is 28 km, roughly half
the length scale of the finer bins (11 0.51, Tables 1 and 2). But the
size of clusters varies (Fig. 4c). In the densely sampled regions
along the Norwegian coast, the cluster radius is less than 10 km.
In the sparsely sampled areas, like the Fram Strait, the clusters are
larger and the resulting fields smoother.
The average standard error of the finest clustered fields
(m ¼12) is twice as high as with (11 0.51) bins (Tables 1 and 2).
However, nearly all the means (96%) are statistically significant
(Table 2). As expected, the standard error is smaller with larger
clusters (Fig. 4b). But since the clusters have approximately the
same number of samples, the standard errors primarily reflect the
variations in n.
4. Diffusivities
Previously, mapping of eddy diffusivities in the Nordic Seas
from surface drifters has been restricted to using large bins to
assure convergence of the velocity autocorrelations (Poulain et al.,
1996; Andersson et al., in preparation). Although these analyses
suggest general features of eddy variability (an intensification in
the Lofoten Basin and along the path of the NwAC), a more
detailed representation is desirable to study lateral mixing in
these regions. We will examine the eddy diffusivities obtained
with a range of bin- and cluster sizes.
To calculate the residual velocities, we use the mean velocities
obtained from the fine resolution cases, i.e. from the (11 0.51)
bins and from the m¼ 12 clusters, as these best capture the flow
structure (Fig. 2). The diffusivities are estimated from the mean
autocorrelation in bins and clusters (Eq. (2)) following Koszalka
and LaCasce (2010). For bins, we employ four bin-size classes:
(11 0.51), (21 11), (41 21) and (81 41). The smallest bins have
a length scale of 50 km while the largest bins are 360 km. For
comparison, the length scales of the bins in Poulain et al. (1996)
and Andersson et al. (in preparation) spanned 417–762 km. Only
particle segments that are inside the bin are used to evaluate
Eq. (2); the integration terminates when a particle leaves the bin.
As before, only bins with segments from at least two different
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I. Koszalka et al. / Deep-Sea Research I 58 (2011) 468–485
80
b−2
b−1
c−48
c−24
c−12
70
10
50
σ (cm/s)
% bins/clusters
60
b−2
b−1
c−48
c−24
c−12
40
1
30
20
10
0
0.1
5
10
20
50
100
drifter days
200
500
5
10
25
40
200
500
b−2
b−1
c−48
c−24
c−12
35
30
% bins/clusters
50
100
drifter days
25
20
15
10
5
0
0
50
100
150
radius (km)
200
250
Fig. 4. (a) Histograms of the number of observations (expressed in ‘‘drifter days’’) grouped into bins of different size and in clusters obtained with different parameter m.
(b) Standard error vs. number of drifter days for binning and clustering assignments. Bins/clusters with less than 10 drifter days or with data from only one drifter are not
considered. (c) Mean length scale (mean square area of bins/clusters) vs. drifter days for binning and clustering assignments. Bins/clusters with less than 10 drifter days or
with data from only one drifter are not considered.
drifters were retained. The parameters of the binning analyses are
listed in Table 3.
For clustering, we break the trajectories into segments of s¼ 10
and 20-days. The segments are clustered by the same algorithm
as with the mean flow, according to their mid-segment position.
The velocity autocorrelations for all segments in the cluster are
then averaged and integrated. So the cluster diffusivities derive
from approximately equal numbers of segments of uniform
length. For each prescribed segment length (s ¼10 and 20 days)
we consider two values for the number of segments in a cluster
(ensemble size), md ¼90 and 60. Parameters of the four clustering
assignments: (s¼10 days, md ¼ 60), (s ¼20 days, md ¼60), (s¼10
days, md ¼90), (s¼20 days, md ¼90), are summarized in Table 4.
With the ‘‘finest’’ assignment (s ¼10 days, md ¼ 60) we get 90
clusters, which is comparable to the number of bins for (41 21)
assignment (Ns ¼ 89); for the ‘‘coarsest’’ analysis (s¼20 days,
md ¼90) we get 29 clusters, approximately equal to the number
of bins in the (81 41) case (Tables 3 and 4). The mean
cluster radius ranges from 80 km in (s¼ 10 days, md ¼60) case
to 175 km in (s¼20 days, md ¼90) case. These radii are comparable to the mean distance between starting and final position
for a segment, which is 80 km for the 10-day segments and
130 km for the 20-day segments. However, in the densely
Table 3
Eddy diffusivities from binning: bin size (long lat), number of bins with data,
pffiffiffiffiffiffiffiffiffiffi
mean bin length scale (rb ¼ area), zonal and meridional k1 in (107 cm2/s2), mean
error (mean over 6–10 days and over all bins) in (107 cm2/s2) in zonal and
meridional directions, mean number of segments in bins (Ns), mean segment
length in bins (msl), number of eliminated bins NE (less than two different drifters,
or kðtÞ shorter than 6 days).
Binsize
Nb
84
42
21
1 0.5
26
89
316
1087
rb (km) /kz S
/km S
ERR
(kz=m )
360
181
91
46
2.4
2.4
2.2
2.0
4.4/2.8 102
5.1/4.7 46
5.9/5.5 22
7.3/7.3 11
2.0
2.3
2.3
2.0
Ns
msl
(days)
NE
17.2
10.5
5.8
3.0
1 (4 %)
6 (7%)
56 (22%)
601 (55%)
sampled regions the clusters can be as small as 20 km in diameter
(see below).
For each bin or cluster, we obtain time series of the diffusivity
as a function of time lag. Examples, for the zonal diffusivities, are
plotted in Fig. 5, for binning analyses (a–d) and clustering assignments (e–h). With the binned diffusivities, it is difficult to attain
convergence. Even with the largest (81 41) bins, there are curves
that span negative values or unrealistically large ones during the
first 10 days (Fig. 5a). With decreasing bin size, the diffusivity
I. Koszalka et al. / Deep-Sea Research I 58 (2011) 468–485
Table 4
Eddy diffusivities from clustering: prescribed segment length in days/number of
pffiffiffiffiffiffiffiffiffiffi
segments, total number of clusters, mean cluster radius (rc ¼ area), zonal and
meridional k1 in (107 cm2/s2), mean error in (107 cm2/s2) (mean over 6–10 days
and over all clusters) in zonal and meridional directions, number of eliminated
clusters NE (less than two different drifters, or kðtÞ shorter than 6 days). The last
three columns refer to k1 calculated from velocities projected across and along
the isobaths (/kac S and /kal S, respectively), and the mean error of these
estimates (ERR(kac=al )), see Section 5.
d/m
Nc
rc (km)
/kz S
/km S
ERR
(kz=m )
NE
/kac S
/kal S
ERR
(kac=al )
20/90
10/90
20/60
10/60
29
60
43
90
175
122
127
81
1.9
1.8
1.9
1.8
1.7
1.7
1.8
1.7
0.9/0.9
1.4/1.4
1.1/1.2
1.7/1.7
0
0
0
0
1.1
1.1
1.1
1.1
2.1
2.2
2.2
2.1
0.7/1.1
1.1/1.6
0.8/1.3
1.3/1.9
curves become shorter (as the time spent by drifters in bins is
shorter) and the spread in estimates increases (Fig. 5 b–d and
Table 3). The clustered diffusivities, having uniform length (10 or
20 days) and computed for an imposed ensemble size (60 or 90
segments), exhibit convergence in all four clustering assignments,
and in particular with the longer segments (20 days, Fig. 5 e–h).
We extract a single diffusivity estimate, k1 from these curves
by averaging them over the period from 6 to 10 days (Koszalka
and LaCasce, 2010). In cases where the autocorrelation becomes
negative, we used the ‘‘zero crossing method’’.4 The result is the
diffusivity maps shown in panels a–d in Fig. 6 for bins and panels
e–h for clusters. Only the zonal component is shown; the isotropy
of the diffusivity estimate is discussed in Section 5. The averages
over whole domain are reported in Table 3 and in Table 4 for bins
and clusters, respectively.
Note that in Figs. 5 and 6 we plot all the diffusivity estimates
from bins and clusters, including those estimates which do not
converge. Had we excluded the non-convergent diffusivities, most
of the binned estimates in Fig. 6c, d would be eliminated, while
only a few estimates from the higher resolution clustering would
be affected.
The diffusivities with the largest bins, (81 41), hardly reveal
any structure in the eddy field. The largest diffusivities are found
west from Svinøy and in the western extreme of the domain, off
Greenland. Decreasing the bin size (41 21 and 21 11, panels b
and c) reveals a more consistent picture of the variability, with
intensification in the Lofoten Basin, along the Norwegian coast
and near the Iceland-Faroe Ridge. But with smaller bins the
estimates are noisier and more bins are excluded for having
diffusivity curves shorter 6 days or for having too few drifters.
Thus 22% of the bins are eliminated in the (21 11) case, while
55% are in the (11 0.51) assignment (Table 3). Thus the estimates
become much sparser, as seen in panel d of Fig. 6.
The clustered diffusivities yield also a more coherent representation of the eddy field, clearly showing the intensification in
the Lofoten Basin and west off the Lofoten Islands. The lower the
value of s and of md, the more detailed the picture is, particularly
in the densely sampled Norwegian shelf and slope region. Larger
values of s and md yield fewer, larger clusters and consequently a
smoother representation. In general, the clusters in regions with
little data (Greenland Basin, East Greenland Current) encompass
segments from vast areas, resulting in a more smooth estimate.
4
The zero-crossing method often results in overestimated values (e.g., Griesel
et al., 2010). However, as it is a well-established procedure (Lumpkin et al., 2002),
we use it here to avoid negative diffusivity estimates. We chose not to use the
method of Garaffo et al. (2001), which involves assumptions about the eddy field
which are less straightforward to apply in inhomogeneous regions.
475
A comparison of the diffusivity statistics in bins and clusters is
shown in Fig. 7. Values of k1 in all bins and clusters are shown in
panel a. In bins we get a large spread of estimates, ranging from
values close to zero to rather large values of (12 107 cm2 s 1).
The spread of the clustered diffusivities is much smaller (reaching
5 107 cm2 s 1).
Fig. 7b shows a scatter-plot of the average segment length
used in calculating the autocorrelation for each cluster or bin vs.
the number of segments. The clusters have segment lengths of 10
and 20 days, however the prescribed number of segments is not
always achieved because of the irregular data distribution. Nevertheless, for 10-day segments in 60-member ensembles, 79%
clusters have more than 50 members. For the s¼20, md ¼90
assignment, 83% of clusters have more than 80 segments. The bins
on the other hand exhibit a wide range of values for both
parameters. Larger bins, (81 41) and (41 21), have typical
segment lengths of 17 and 10 days, but the (21 11) set (Fig. 6c)
has a mean segment length of 6 days. Only 10% of the bins have
segments longer than 10 days and none has 60 day segments. For
(11 0.51) bins, the mean segment length is 3 days, well below
what is required to obtain a reliable estimate.5
The mean errors (average over 6–10 days, and over all bins/
clusters) are plotted in 7c. As expected, the errors are largest with
(11 0.51) bins, reaching 7.2 107 (cm2 s 1) or three times the
mean value of k1 in these bins. The errors decrease with
increasing bin size, but even in the (81 41) case, the values are
twice that of the ‘‘noisiest’’ cluster case (s¼10, md ¼60, with a
typical error 1.7 107 cm2 s 1). For clusters, the error decreases
with increasing segment length and increasing number of segments (0.9 107 cm2 s 1 in s¼20, md ¼90 assignment). Thus, the
fixed-length, fixed-number of segments used in the clustering
greatly improves the estimate with respect to the binning.
We can also compare the results from clustering to those
obtained when using large regions, like those used by Poulain
et al. (1996) and Andersson et al. (in preparation). Following those
authors, we denote these as the Iceland-Faroe Frontal zone (IFF),
the Iceland Plateau (IP), the Norwegian Basin (NB), the Lofoten
Basin (LB) and the NwAC. The results for the regions are reported
in Table 5. The large number of segments (55 for IP and over 100
for the other four regions) and large mean segment length (25–80
days) assure good convergence properties and low errors
(0.2–0.8 107 cm2 s 1). But they do not resolve the spatial
variability of the eddy diffusivity. The average estimates k1 from
clusters falling in each of the domains are also listed; these values
are consistent within the error bounds.
5. Including topography in the clustering method
Where the data is sparse, the clusters are larger and the
estimates smoother. As such, even the finest mean fields
(m ¼12, Fig. 2d) underestimate the velocities in the West Spitsbergen Current in the Fram Strait (10–15 cm/s while the ADCP
measurements yield 30–40 cm/s, see e.g., Osinski et al., 2003).
However, in the Nordic Seas the surface circulation is strongly
steered by topography, as seen in previous studies (e.g., LaCasce,
2000; Isachsen et al., 2003; LaCasce, 2005a; Søiland et al., 2008).
We can exploit this by grouping the observations based on
position relative to the topography, prior to clustering them.
For this, we interpolated the 2-min Gridded Global Relief
Data (ETOPO2v2)6 onto the drifter positions and then partitioned
5
For more discussion on the effect of segment length on the diffusivity
estimate, see Griesel et al. (2010) and Koszalka and LaCasce (2010).
6
U.S. Department of Commerce, National Oceanic and Atmospheric Administration, National Geophysical Data Center, 2006. http://www.ngdc.noaa.gov/
mgg/fliers/06mgg01.html.
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I. Koszalka et al. / Deep-Sea Research I 58 (2011) 468–485
3
2
4
3
2
5
10
15
0
20
(20,90)
8
5
10
15
2 −1
7
4
3
2
1
5
4
3
2
1
0
5
10
days
15
20
2 −1
15
5
10
days
15
(10,90)
2
5
10
15
20
15
20
(10,60)
8
7
6
5
4
3
2
20
3
0
6
5
4
3
2
1
0
0
0
4
20
1
0
0
5
0
10
7
6
6
1
5
8
κ(τ) (107 cm2 s−1)
κ(τ) (10 cm s )
5
2
0
7
6
3
20
(20,60)
8
7
4
0
0
0
5
1
1
0
6
7
2 −1
5
7
κ(τ) (10 cm s )
7
4
6
7
2 −1
5
(1 x 0.5)
8
7
κ(τ) (10 cm s )
κ(τ) (10 cm s )
7
2 −1
κ(τ) (10 cm s )
6
(2 x 1)
8
7
1
κ(τ) (107 cm2 s−1)
(4 x 2)
8
7
κ(τ) (107 cm2 s−1)
(8 x 4)
8
0
5
10
days
15
0
20
5
10
days
Fig. 5. (a) Time series of the zonal diffusivity estimates in all bin/clusters. Top: obtained by the binning method for different bin sizes (from left to right): 81 41, 41 21,
21 11 and 11 0.51. Bottom: obtained for different clustering assignments (from left to right): (s¼ 20 d, m¼ 90), (s ¼ 20 d, m ¼60), (s ¼10 d, m¼ 90), (s¼ 10 d, m¼ 60). The
period of 6–10 days, over which k1 is estimated, is marked with vertical dashed lines.
(8 x 4)
(4 x 2)
(2 x 1)
(1 x 0.5)
80
80
80
80
78
78
78
78
76
76
76
76
74
74
74
74
72
72
72
72
70
70
70
70
68
66
64
62
60
–20
68
66
64
62
60
–20
68
66
64
62
60
–20
68
66
64
62
60
–20
–10
0
10
20
–10
(20,90)
0
10
20
–10
(20,60)
0
10
20
80
80
80
78
78
78
78
76
76
76
76
74
74
74
74
72
72
72
72
70
70
70
70
68
66
64
62
60
–20
68
66
64
62
60
–20
68
66
64
62
60
–20
68
66
64
62
60
–20
0
10
20
–10
0
10
20
–10
0
0
10
20
10
20
(10,60)
80
–10
–10
(10,90)
10
20
–10
0
Fig. 6. Top: maps of zonal diffusivity obtained by the binning method for different bin sizes (from left to right): 81 41, 41 21, 21 11 and 11 0.51. Bottom: obtained for
different clustering assignments (from left to right): (s¼ 20 d, m¼ 90), (s¼ 20 d, m¼ 60), (s¼ 10 d, m¼ 90), (s¼ 10 d, m¼ 60). Each point marks the mid-point position of a
segment assigned into the cluster colored by the mean value of k1 in the cluster. The contour lines of [ 2600 1800 1000 100] m depth depict main bathymetric
features. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
them into subsets according to depth, in classes bounded by
the ( 4000, 3200, 2500, 2000, 1100, 450, 250, 0 m)
isobaths. We clustered each subset separately as before (Section 3).
The mean velocity fields obtained with mtopo ¼48 and 12 are
compared to the estimates without the topography in Fig. 8, and
the analysis parameters are included in Table 2.
I. Koszalka et al. / Deep-Sea Research I 58 (2011) 468–485
477
Fig. 7. (a) A spread of estimates of eddy diffusivity k1 in each bin and cluster, for different bin sizes and prescribed segment lengths, respectively (values on the abscissa
refer to varying longitudinal extend of bins, and to segment lengths in clusters in days). (b) A scatter plot showing the mean segment length vs. the number of segments
and obtained in bins for different bin sizes, and in clusters for different number of segment and segment lengths. The number of segments reported here refers to t ¼ 0 and
it falls off thereafter due to variable length of the tracks that occur in the bin. The mean values of these parameters over all bins/clusters are marked with boxes. (c) The
average error of the diffusivity estimate /dkS, obtained for different bin sizes and different prescribed segment lengths, respectively (averaged over all bins/clusters). The
errors dk are derived from errors on velocity autocorrelation given by the t-test at the 95% significance level.
Table 5
Eddy diffusivities in five regions: the region, area covered, mean number of
segments in region (Ns), mean segment length in region (msl), zonal and
meridional diffusivity (107 cm2/s2), mean error (average over 6–10 days and over
all bins) in (107 cm2/s2) in zonal and meridional directions, the range obtained
with four clustering analyses, average estimate for clusters inside any of the large
domains.
Region rb (km) Ns
msl
/kz S
(days)
/km S ERR
(kz=m )
/kz S
/km S
IFF
IP
NB
LB
NwAC
37.4
80.1
31.9
24.8
42.3
1.5
0.6
2.3
2.3
2.5
1.5–2.0
0.4–0.6
1.3–1.9
2.6–3.0
2.0–2.2
1.0–1.4
0.5/–/0.8
1.5–1.9
2.3–2.4
2.1–2.2
417
500
500
443
762
127
55
167
233
428
2.2
0.6
1.2
3.3
2.1
0.5/0.4
0.2/0.2
0.4/0.5
0.8/0.6
0.3/0.3
Including the bathymetry improves the estimate in the peripheral regions, but more fine structure is revealed in other areas as
well. The mtopo ¼48 clusters (panel c) have mean radius of 30 km,
a resolution comparable to the m¼12 analysis clustered without
depth (panel b). The mtopo ¼ 48 clustering captures the Lofoten eddy
and the topographically steered flow in the Norwegian Sea (compare with panel a). For mtopo ¼12, the mean cluster radius is as
small as 11 km, which produces a more accurate representation of
the West Spitsbergen Current. It also resolves two routes of the
inflow into the Barents Sea: the first at 203 E, 72:53 N related to
the bifurcating branch of the NwAC, the North Cape Current, and
the latter at 203 E, 72:53 N associated with the NCC (Skagseth, 2008).
The crossover flow from the outer to the inner branch along the
Vøring Plateau is also visible at (2283 E, 68.51N).7
Inclusion of the topography improves the diffusivity estimation too. As the mean flow is largely parallel to the topography,
this is roughly equivalent to considering the along- and acrossstream diffusivities, as in Griesel et al. (2010). Accordingly, we use
7
The EKE is not shown as differences between the estimates obtained with
and without the topography are hard to discern.
478
I. Koszalka et al. / Deep-Sea Research I 58 (2011) 468–485
a
b
78
78
76
76
74
74
72
72
70
70
68
68
66
66
64
64
62
62
60
−20 −15 −10
−5
0
5
10
15
20
25
60
−20 −15 −10
c
d
78
78
76
76
74
74
72
72
70
70
68
68
66
66
64
64
62
62
60
−20 −15 −10
−5
0
5
10
15
20
25
60
−20 −15 −10
−5
0
5
10
15
20
25
−5
0
5
10
15
20
25
Fig. 8. Clustered estimates of the mean velocity. Vectors are color-coded by the speed. Top: obtained by clustering the data in clusters with varying prescribed number of
data (in drifter days): m¼ 48 (a), and m¼ 12 (b). Bottom: corresponding fields obtained by including the topography in the clustering: m¼ 48 (c), and m ¼12 (d). The arrow
in the upper left corner marks a speed of 30 cm/s. Only significant estimates are plotted ( Z 10 drifter days, and at least two different drifters in the ensemble). (For
interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
the method of LaCasce (2000) to project the residual velocities
along and across the depth contours.8 Maps of the resulting
diffusivities in zonal, meridional, across- and along-isobath directions are shown in Fig. 9.
Note that the pattern of the eddy diffusivity in geographical
coordinates and in the along-isobath direction strongly depends
8
LaCasce (2000) notes that the decomposition requires some degree of
smoothing of the bathymetry in calculation of its gradient. We use a Gaussian
filter with a length scale of 10 km (that is, comparable to the internal radius of
deformation in the Nordic Seas) that brings about largest anisotropy between the
displacements in both directions.
on the clustering parameters, with high values where the mean
flow is strong. This is because the zonal and meridional diffusivities are affected by shear in the mean field (LaCasce, 2008;
Griesel et al., 2010). In contrast, the across-isobath diffusivity is
robust to the change in parameters, showing elevated values near
the western branch of the NwAC, at the Barents Sea Opening, and
in particular in the Lofoten Basin, where the highest diffusivities
( 3 107 cm2 s1 ) are found north–east of the Lofoten Escarpment. Note that the across-isobath diffusivities have lower magnitudes in the inner branch of the NwAC and are also reduced in
the NCC south of 683 N. This suggests that the along-isobath
diffusivities are affected by the shear in the NwAC while the
I. Koszalka et al. / Deep-Sea Research I 58 (2011) 468–485
479
κ (107 cm2 s–1)
0
0.5
1
(20, 90)
1.5
2
(20, 90)
2.5
3
(20, 90)
(20, 90)
80
80
80
80
78
78
78
78
76
76
76
76
74
74
74
74
72
72
72
72
70
70
70
70
68
68
68
68
66
64
62
60
–20
66
64
62
60
–20
66
64
62
60
–20
66
64
62
60
–20
–10
0
10
20
(10, 60)
80
–10
0
10
20
(10, 60)
80
–10
0
10
20
(10, 60)
80
78
78
78
76
76
76
76
74
74
74
74
72
72
72
72
70
70
70
70
68
68
68
68
66
64
62
60
–20
66
64
62
60
–20
66
64
62
60
–20
66
64
62
60
–20
0
10
20
–10
0
10
20
–10
0
10
20
0
10
20
(10, 60)
80
78
–10
–10
–10
0
10
20
Fig. 9. Diffusivity maps obtained from two different clustering assignments. Top: (20 d, m¼ 90). Bottom: (10 d, m¼ 60). From left to right: zonal diffusivity, meridional, projected
across- and along the topography. Each color point marks the mid-point position of a segment assigned into the cluster. The contour lines of [ 2600 1800 1000 100] m
depth depict main bathymetric features. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
across-isobath diffusivity are not. Thus the latter may provide a
most robust indication of the eddy diffusion here.
6. The Svinøy section and the Lofoten Basin
Here we will use the clustered data to take a closer look at the
circulation in two key regions: the Svinøy section and the Lofoten
Basin. Both regions were more densely sampled, so we can obtain
higher resolution of the fields.
The Svinøy section runs northwestward from the Norwegian
coast at 62:53 N and includes both branches of the Norwegian
Atlantic Current (Fig. 1a). The currents in the eastern branch have
been monitored here more or less continuously with current
meters since 1995 (Orvik and Mork, 1996; Orvik et al., 2001;
Skagseth and Orvik, 2002; Orvik and Skagseth, 2005). Monitoring
the western branch commenced recently using Seagliders and
pressure inverted echo sounders (Mauritzen et al., accepted for
publication). The Svinøy section was a primary focus of the
POLEWARD project, with the release of 91 surface drifters in
2007–2008 (Fig. 1b). These, combined with the trajectories from
the global data base, yield 9482 buoy days in the area adjacent to
the section (23 W283 E,622653 N).
We will examine the highest resolution fields, obtained
by partitioning the data into 790 clusters with m ¼12 drifter
days (Table 2). These have an average radius of 8 km, which is
approximately the deformation scale. The resulting mean flow is
shown in Fig. 10a. The eastern branch of the NwAC is seen near
the shelfbreak (at 500–1000 m depth, Orvik et al., 2001).
The drifter velocities reach 30–35 cm/s here, consistent with
averages from current meters at 20 m depth (LaCasce, 2005b).
The means suggest that the current accelerates in regions
where the isobaths pinch together, near 5E. The western branch
is seen as a more diffuse flow near the 2000 m isobath (Fig. 10a
and c), with velocities on the order of 20–25 cm/s. Interestingly,
this flow is flanked by two large ( 50 km) anti-cyclonic
recirculations.
The inshore (eastern) recirculation is associated with the
crossover flow from the western to the inner branch the NwAC
at 64:63 N. Examining individual drifter trajectories, we find that
this feature was present at least during the years 2007–2008,
sampled by 15 POLEWARD drifters (examples are shown in
Fig. 10b). In the earlier period, one drifter in 01/1992 and one in
09/1993 clearly seem also to be trapped in this eddy, while two
other drifters in December 1992 crossed through the site without
anticyclonic looping. There is no more drifter data available in the
1990s and early 2000s. The recirculation and related crossover
flow has been seen though with RAFOS floats in 2004–2005
(Rossby et al., 2009b, see their Fig. 7).
The offshore (western) recirculation, centered at (13 E, 653 N)
was sampled by 14 drifter tracks throughout the 20-year period
(1991–1992, 1996, 2000, 2004, 2007–2008). Thus this feature also
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65.5
65.5
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7
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0.5
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3
Fig. 10. The flow at the Svinøy section obtained from the drifter data gathered in clusters with m¼12 drifter days. (a) Mean velocity vectors. Arrows plotted in cyan
indicate speeds greater than 20 cm/s, and these plotted in red indicate speeds greater than 30 cm/s. (b) Mean velocity vectors with superimposed selected drifter tracks.
(c) Mean velocity vectors superimposed on color-coded bathymetry field (2-min ETOPO data set). (d) Mean velocity vectors and eddy kinetic energy re-sampled on a
regular grid (long lat) ¼ (13 0:53 ). (e) Across-isobath diffusivity obtained by clustering 60 segments of 10-day track length. Each point marks the mid-point position of a
segment assigned into the cluster colored by the mean value of k1 in the cluster. (For interpretation of the references to color in this figure legend, the reader is referred to
the web version of this article.)
appears to be semi-permanent. To our knowledge there are no
other observations to verify this.
The eddy kinetic energy field obtained with the clusters is
shown in Fig. 10d. The highest EKE coincide with the eastern
branch, most likely due to meandering and eddy generation
(Orvik et al., 2001). Individual drifter tracks also indicate eddying
motion here (Fig. 10b). The western branch is also a site of
elevated EKE, particularly near the bifurcation point at 33 W,
64:63 N. Higher EKE levels are also found at the centers of the
two large recirculations.
The across-isobath diffusivity (Section 4) is shown in Fig. 10e.
This has some interesting properties. For one, it is elevated near
the two anticyclonic recirculations pointing to their importance in
mixing of water properties.
In addition, the diffusivity is suppressed in the vicinity of the
eastern branch—despite that the EKE is elevated here (Fig. 10d). The
weak diffusivities could reflect suppression of eddy transport by
the steep topography, or perhaps by the mean flow (e.g., Ferrari
and Nikurashin, 2010). So eddy diffusion in the Svinøy region is
intensified off the slope.
I. Koszalka et al. / Deep-Sea Research I 58 (2011) 468–485
481
coast. The two currents are in close proximity as they pass over
the Lofoten Islands. The western branch, flowing along the
western slope of the Vøring Plateau, splits at (13 E, 683 N) into
two streams. One flows along the western side of the basin,
and the second crosses over the Vøring Plateau to join the
eastern branch. There is an indication of a weak mean current
( 10 cm=s) in the basin interior at (83 E, 68:53 N), where the
Vøring Plateau rejoins the continental slope. A large anticyclonic
eddy ( 150 km in diameter) is seen in the western part of the
basin, trapped over the 3200 m-isobath; this has typical velocities
of 25 cm/s. On its western side there is a smaller cyclonic feature
related to a complex topography there (Fig. 11a and c).
The eddy kinetic energy is shown in Fig. 11d. There is a region
of enhanced EKE off the Lofoten Islands, with maximum values of
590 cm2 s 2. The drifter tracks show strong meandering here,
The Lofoten Basin is the major heat reservoir for the Nordic
Seas, where intense cooling and densification of the Atlantic
Water occur (Orvik, 2004; Nilsen and Falck, 2006; Gascard et al.,
2004; Rossby et al., 2009a; Isachsen et al., 2007). It is also a region
of significant eddy activity. The Lofoten Basin was targeted for a
specific deployment late in POLEWARD, with 15 instruments
deployed in July 2009 (Fig. 1b, cyan color). The mean flow vectors
obtained with m¼ 24-topo clustering is shown in panel a
of Fig. 11 and on panel b with superimposed examples of drifter
trajectories. The results from the m¼12-topo clustering are
similar, but the coarser resolution field is preferable for visualization since the plots cover a larger area than the Svinøy section.
The means show a swift along-slope current off the Lofoten
Islands, with typical velocities of 30–40 cm/s and maximum
velocities of 50 cm/s. This flow is flanked by the NCC near the
71
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8 9 10 11 12 13 14 15 16 17 18
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-2000
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200
300
[cm2 s–2]
400
500
72
71
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67
−2 0
2
4
6
8 10 12 14 16 18 20
κ (107 cm2 s–1)
0
0.5
1
1.5
2
2.5
3
Fig. 11. The flow at the Lofoten Basin obtained from m¼ 24 clustering. (a) Mean velocity vectors. Arrows plotted in cyan indicate speeds greater than 20 cm/s, and these
plotted in red indicate speeds greater than 30 cm/s. (b) Mean velocity vectors at the Lofoten Escarpment with superimposed examples of the drifter tracks. (c) Mean
velocity vectors superimposed on bathymetry field (2-min ETOPO data set). (d) Eddy kinetic energy re-sampled on a regular grid (long lat)¼ (11 0.51). (e) Across-isobath
diffusivity obtained by clustering 60 segments of 10-day track length. Each point marks the mid-point position of a segment assigned into the cluster colored by the mean
value of k1 in the cluster. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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I. Koszalka et al. / Deep-Sea Research I 58 (2011) 468–485
often breaking away into the interior (Fig. 11b). Note that the
slope is particularly steep here (Fig. 11c). West off this steepest
continental slope (113 E, 693 N), an intense looping of the trajectories, and high levels of EKE, occur, seemingly related to the
topography relief (Fig. 11c). Vigorous looping of subsurface RAFOS
floats has also been seen here (Rossby et al., 2009b, their Fig. 10).
A tongue of high EKE extends from the slope toward the anticyclone at the basin center, suggesting a connection between the
slope eddies and the vortex. The EKE is also high where the
western branch bifurcates (13 E, 683 N).
As in the Svinøy section, the across-isobath diffusivity is
elevated not over the slope but offshore, in the eastern part of
the basin (Fig. 10e). The highest values ( 3 107 cm2 s1 ) occur
where the EKE is also large (113 E, 693 N). The values of k are lower
over the slope, in contrast to the EKE. This suggests that even
though the eddies are likely generated at the slope current, they
are most effective in diffusing water properties in the basin
interior.
7. Summary and discussion
We have used a clustering algorithm to calculate pseudoEulerian mean velocities and eddy diffusivities from surface
drifter data from the Nordic Seas. Clustering involves grouping a
specified number of observations into spatially localized subsets
(Koszalka and LaCasce, 2010). This is in contrast to the more
traditional binning method, in which observations are grouped
into geographical bins of a specified size, as has been done previously in the Nordic Seas (Poulain et al., 1996; Jakobsen et al., 2003;
Andersson et al., in preparation). We used data comprising 58,348
drifter days spanning a 20-year period (1991–2009). This is the first
application of clustering to the analysis of drifter data from the
ocean.
In comparing the results obtained by binning and clustering,
we find that the two approaches yield comparable results for the
mean velocities at the 50 km scale. With larger bins and larger
clusters, the means are more significant but also overly smooth,
particularly in a region such as this, where there are significant
small scale variations in the field. Clusters however provide more
uniform errors as there are approximately the same number of
observations in each; the number of samples in the bins varies
substantially. In a densely sampled region, such as the Svinøy
region here, clusters can provide a very detailed picture of the
flow ( 10 km). Clusters are also perhaps more ‘‘honest’’ in that
they provide estimates only where there is data.
Where clusters really excel though is with diffusivities. There
are several reasons for this. For one, having the same number of
samples in each group greatly improves the averages of the
velocity autocorrelation. Second, using segments of uniform
length improves the convergence and also allows the user to
decide over what period the autocorrelation should be integrated.
This second point has been recognized previously (Lumpkin and
Flament, 2001; Rupolo, 2007; Griesel et al., 2010; Koszalka and
LaCasce, 2010). Third, the Eulerian means are resolved with
clusters and the estimates apply where the trajectories are. So
the residual velocities are better captured, yielding better convergence of the eddy statistics. The result is that one can produce
diffusivity maps with higher spatial resolution (typically
80–175 km, down to 20 km in the densely sampled regions), as
one has more control over the convergence of the integral. In
contrast, one requires very large (ð Z400Þ km) bins to obtain
reliable estimates of k (Andersson et al., in preparation), resulting
in an under-resolved field.
In addition, we tested how taking account of bottom topography affects the results when clustering. As the stratification is
weak in the Nordic Seas, the mean currents are strongly steered
by topography. Pre-segregating the observations by depth accordingly yields a sharper representation of the mean current, particularly in poorly sampled areas. One can also calculate the
diffusivities relative to the isobaths, which is roughly equivalent
to calculating them relative to the mean streamlines. We find that
the across-isobath diffusivity in particular is a good indicator of
mixing and is robust to changes in parameters. The along-isobath
diffusivity, like the zonal and meridional diffusivities, are evidently more affected by the mean flow. The situation is similar in
the Southern Ocean, where topographic steering is also pronounced and where a cross-isobath diffusivity was also used
by Griesel et al. (2010).
Clustering is an effective tool for analyzing Lagrangian data: it
partitions the data uniformly, improving the significance of the
mean velocity and diffusivity estimates. However, the results
must be further processed to incorporate them in a model, as
they are defined on an irregular grid. Here we employed a
combination of nearest-neighbor re-sampling and linear interpolation to obtain the residual, ‘‘eddy’’ velocities.
Our best estimates of the mean velocity, eddy kinetic energy
and eddy diffusivity in the Nordic Seas for the whole observational period (1990–2009) are presented in Fig. 12. They are
resampled on a regular grid with a cell size corresponding to the
mean cluster radius. The mean velocity (panel a) is largely
consistent with previous findings of the surface circulation
(e.g., Orvik and Niiler, 2002; Blindheim and Østerhus, 2005). The
inflow from the North Atlantic enters via the Iceland-Faroe ridge and
the Faroe-Shetland Channel, and proceeds along the western coast
of Norway in two branches, one near the shelfbreak (500–1000 m
depth) and one between the 1500 and 2000 m isobaths. The
currents merge to form a single jet off northwestern Norway. Part
of the current then enters the Barents Sea while the majority of the
flow continues along the shelfbreak west of Spitsbergen. At this
point, the drifters are presumably losing contact with the inflow
waters, as the latter are subducting under the Polar Water.
Clustering reveals several other features in the mean fields.
First, there is the large vortex located at the center of the Lofoten
Basin (2253 E, 703 N). This quasi-permanent anticyclonic eddy has
been observed previously with subsurface floats (Gascard and
Mork, 2008; Søiland et al., 2008; Voet et al., accepted for
publication), in a model (Kohl, 2007), and in sea surface height
and hydrography data (Rossby et al., 2009a). Second, there is
distinct flow from the western branch of NwAC towards the
eastern branch along the Vøring Plateau (Section 6, Fig. 11). This
flow also has a subsurface expression, as it is seen in trajectories
of 800 m-RAFOS floats (Søiland et al., 2008) and 1000/1500 m
Argo profilers (Voet et al., accepted for publication). The extensive
sampling in the Svinøy region, combined with the historical data,
allowed us to map the section with clusters as small as 8 km. We
observe two large ( 50 km), apparently semi-permanent, anticyclonic vortices, flanking the western branch of the NwAC
(Section 6, Fig. 10). The inshore vortex is associated with the
crossover flow from the outer to the inner branch, and has also
been seen with subsurface floats (Rossby et al., 2009a). These
recirculations may be formed during the interaction of the
unstable western branch with a variable topography, involving
the dynamics of the Polar Front; the surface drifter data set only
does not allow us to investigate this. Their presence likely causes
the Atlantic Water to spread offshore and thus to appear as a
wide, warm and saline wedge in the hydrographic and moored
measurements (Orvik et al., 2001). We note also that the recirculations have high eddy diffusivities suggesting that they are
regions of an intense mixing. These features may be also important for cooling of the Atlantic Water, as fluid could be trapped
here for extended periods. Finally, the trajectories indicate two
I. Koszalka et al. / Deep-Sea Research I 58 (2011) 468–485
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5
10 15 20 25
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8 10 12 14 16 18 20
0 50 100 150 200 250 300 350 400 450 500
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0
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0
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Fig. 12. Drifter-derived ‘‘best’’ estimates pertaining for the period 1990–2009 of (a) mean speed jU^ ðx,yÞj (cm/s). (b) Eddy kinetic energy eke ¼ 1=2/uu2 S þ /vu2 S ðcm2 =s2 Þ.
(c) Across-isobath diffusivity k obtained by clustering of 60 10-day long segments, re-sampled on a regular grid (long lat)¼ (11 0.51). The fields were obtained by
clustering with m¼12 and including the bathymetry in the analysis and re-sampled on a regular grid with a lengthscale corresponding to the mean cluster radius: (long,
lat) ¼(0.21 0.11) for the mean flow and eke, and (long, lat) ¼(11 0.51) for k. The contour lines of [ 2600 1800 1000 100] m depth depict main bathymetric
features.
routes into the Barents Sea, one over the slope and the second
associated with the Norwegian Coastal Current.
In terms of the eddy kinetic energy (Fig. 12b), we find elevated
values at the Iceland-Faroe Ridge (IFR), a region of strong eddy
variability (Allen et al., 1994). EKE is also high along the pathway
of NwAC and NCC. The highest values (400–500 cm2 s 2) occur in
a tongue extending from the Lofoten Islands to the center of the
Lofoten Basin. This suggests eddy shedding from the shelfbreak
and spreading into the basin. This is consistent with recent
numerical studies of Kohl (2007) and Spall (2010).
The eddy diffusivity in the Nordic Seas is best gauged using
across-isobath diffusivity, as this is less affected by the mean
shears. Elevated diffusivities occur at the IFR, in the Norwegian
Sea (where the Polar Front is located), and in particular in the
interior of the Lofoten Basin. In contrast to the EKE, the eddy
diffusivities have reduced magnitudes in the inner branch of the
NwAC and in the NCC. A similar suppression has been found
in the Antarctic Circumpolar Current (ACC) (Griesel et al.,
2010; Shuckburgh et al., 2009a, b). Thus while eddies are
generated over the slope (high EKE and strong mean flow), they
are more efficient at mixing further offshore, in the basin interior
(higher diffusivity k). Notably, we find that this area in the
Lofoten Basin (52103 E, 692723 N) features the most intense eddy
mixing in the whole Nordic Seas, as noted by Poulain et al. (1996).
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Comparing the previous figures, it is clear that there is a
complex relation between the eddy diffusivity, eddy kinetic
energy and the mean flow. A simple downgradient flux parameterization for the diffusivity would imply having large values
near the cores of the jets. We see however that the diffusivity is
often suppressed where the mean flow is strong, in line with the
recently proposed parameterization of Ferrari and Nikurashin
(2010). It is also clear that the diffusivity depends on the bottom
slope, as discussed e.g. by Isachsen (in press, and refs. therein).
Indeed, that the mixing proceeds offshore to the western branch
of the NwAC, and that the latter is steered by a submarine ridge,
implies that the topography constrains the lateral mixing. Any
parameterization of mixing here would have to take these factors
into account.
Our results support the utility of the clustering method for
analyzing drifter data. And since it is effective at resolving spatial
structures, the method should be particularly suited to the more
challenging estimation of heat fluxes.
Acknowledgments
We learned during the preparation of this article of the death
of our colleague, Volfango Rupolo. Volfango was a good friend and
a constant source of new ideas regarding Lagrangian data. We will
miss him.
The work is part of the Poleward project, funded by the Norwegian
Research Council Norklima program (grant no. 178559/S30). Details
are found on http://www.iaoos.no/ and http://folk.uio.no/ingako/
my_files/POLEWARD_WEBPAGE_MAIN.html.
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