A Lagrangian-trajectory study of a gradually mixed ... - Kristofer Döös

Continental Shelf Research 31 (2011) 1811–1817
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Continental Shelf Research
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Research papers
A Lagrangian-trajectory study of a gradually mixed estuary
Bror F. Jönsson a, , Kristofer Döös b , Kai Myrberg c , Peter A. Lundberg b
a Princeton University, Department of Geosciences, Guyot Hall, Princeton, NJ 08544, USA
b Department of Meteorology/Physical Oceanography, Stockholm University, SE-10691 Stockholm, Sweden
c SYKE, Mechelininkatu 34a, P.O.Box 140, FI-00251 Helsinki, Finland
article info
Article history:
Received 21 October 2009
Received in revised form
11 July 2011
Accepted 14 July 2011
Available online 11 August 2011
Keywords:
Numerical modeling
Lagrangian trajectories
Estuarine mixing
Gulf of Finland
Baltic Sea
abstract
When modelling is used for investigating estuarine systems, a choice generally has to be made between
applying simple mass-balance considerations or using a process-resolving three-dimensional (3-D)
numerical circulation model. In the present investigation of the Gulf of Finland, a gradually mixed
estuary in the Baltic Sea, it is demonstrated how Lagrangian-trajectory analysis applied to the output
from a 3-D model minimizes the disadvantages associated with both of the modelling techniques
referred to above. This formalism made it possible to demonstrate that the main part of the Gulf is
dominated by water originating from the Baltic proper, and that the most pronounced mixing with
fresh water from the river Neva takes place over a limited zone in the inner part of the Gulf. Dynamical
insights were furthermore obtained by using the Lagrangian formalism to construct overturning
stream-functions for the two source waters.
& 2011 Elsevier Ltd. All rights reserved.
1. Introduction
When undertaking oceanographic analyses, one is frequently
interested in following the paths of distinct water types taking part
in the circulation. In the context of numerical modeling, trajectory
analysis has proved to be a valuable tool for determining the origin
as well as the fate of specific water masses, see, e.g. Jönsson et al.
(2004). This holds true when examining processes ranging from
those taking place over global scales (e.g. the thermohaline
circulation) down to such local phenomena as the dispersion of
pollutants from a point source. The present study will focus on
intermediate-scale processes, exemplified by those in the northeastern
part of the Baltic (an area subject to considerable ecological
threats due to its proximity to large cities and their associated
wastewater disposal). The largest single freshwater inflow to the
Baltic Sea, viz. the river Neva, is furthermore found here.
The present study is thus focused on the Gulf of Finland,
cf. Fig. 1, which is an elongated estuary (with a mean depth of only
37 m) where physical processes ranging from small-scale vortices
to the large-scale cyclonic circulation take place (Palmén, 1930;
Andrejev et al., 2004a). The mean circulation pattern is one of an
inflow to the Gulf taking place in a deep layer along the Estonian
coast whereas the outflow, primarily in the form of fresher
surface water, takes place slightly north of the longitudinal axis
of the Gulf.
Corresponding author. Tel.: þ1 617 818 1096.
E-mail address: bjonsson@princeton.edu (B.F. Jönsson).
The western end of the Gulf is a direct and gradual continuation
of the Baltic Proper without any morphological constraints.
The eastern end of the Gulf receives the discharge from river
Neva, which with a mean runoff of 2700 m 3 s 1 constitutes
15–20% of the total Baltic freshwater inflow. This leads not only
to a salinity stratification in the vertical but also to pronounced
east–west salinity gradients in which the cyclonic mean horizontal
circulation pattern is reflected (cf. Figs. 1 and 2). The stratification
is furthermore variable in both space and time, where the
large seasonal variability of the incoming solar radiation plays a
considerable role (Alenius et al., 1998). This state of affairs has, as
will be seen, significant dynamic consequences. The watermasses
originating from the two source regions also have markedly
different characteristics, not only as regards the salinity but
also the degree of anthropogenic contamination, why their
ultimate fate in the Gulf is of considerable practical environmental
interest. This general question has been dealt with in a number
of investigations, most recently in one due to Andrejev et al.
(2004b) where it was shown that the renewal time of the water
masses of the entire Gulf is around 1–2 years and that the waterage
distribution in the Gulf is spatially non-homogenous with the
highest ages ( 2 years) found in the southeastern part of the
Gulf. The scope of this study, however, had the consequence that
the ultimate fate of the Baltic and Neva source waters was not
fully ascertained by Andrejev and co-workers.
The classical technique to resolve this question is based on an
investigation of the tracer equation within the circulation-model
framework. This Eulerian formalism, however, has the drawback of
overestimating diffusive effects, and furthermore many circulation
0278-4343/$ - see front matter & 2011 Elsevier Ltd. All rights reserved.
doi:10.1016/j.csr.2011.07.007

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B.F. Jönsson et al. / Continental Shelf Research 31 (2011) 1811–1817
62°N
61°N
60°N
59°N
Stockholm
>6.5
Hanko
6.0
3.0 2.5 2.0
Helsinki
5.0 4.5 4.0 3.5
5.5
Tallin
2>
Neva
St Petersburg
Moshnyi
Island
Hiiuma
58°N
18°E 21°E
24°E
27°E 30°E 33°E
Fig. 1. Map of the Gulf of Finland where the thin black line represents the model coastline. The heavy grey line indicates where the ‘‘Baltic’’ trajectories are released to the
Gulf and the heavy black line where all trajectories are removed. Also included are observed surface isohalines as adapted from Jurva (1951).
0.0
-25.0
-50.0
-75.0
-100.0
-125.0
-150.0
22.0 23.0 24.0 25.0 27.0 28.0 29.0 30.0
Fig. 2. Salinity sections along the central axis of the Gulf of Finland. Colour-coded contours show a long-term average of the RCO-modelled salinity field. The green lines
represent observations as adapted from Jurva (1951). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this
article.)
models tend to be somewhat deficient in reproducing the salt
dynamics of the system under consideration.
The present investigation will instead approach the problem
from a Lagrangian standpoint, i.e. using trajectory analysis. In
next section the technical details of this procedure are outlined,
whereafter the subsequent section deals with the results concerning
mixing and water-mass composition obtained for the Gulf
of Finland. Hereafter, a discussion is undertaken of the large-scale
circulation in the Gulf based on a Lagrangian decomposition of
the overturning stream-function, this in order to estimate the
contributions of the waters originating from the river Neva and
the Baltic, respectively. The study is concluded by a review of the
overall outcome of the analysis.
2. Methods
Classical trajectory analysis, as primarily applied to the atmosphere,
mainly relied upon graphical techniques used in conjunction
with meteorological good sense and experience. Modern-day
trajectory methods are, however, based on numerical circulation
models. In the present study, velocity fields from the Rossby Centre
Ocean (RCO) model (Meier et al., 2003) serve as the basis for the
further investigations. This well-proven (Meier and Kauker, 2003)
finite-difference model has 41 depth levels (at intervals ranging
from 3 m in the surface region to 12 m at greater depths) and a
horizontal resolution of 2 4 nautical miles. It is based on
an Arakawa B-grid and is capable of resolving the meso-scale

B.F. Jönsson et al. / Continental Shelf Research 31 (2011) 1811–1817 1813
eddies of importance for the dynamics of the Baltic. For the present
investigation with focus on the north-eastern Baltic, the remotely
applied boundary conditions at the border to the North Sea do not
give rise to any spurious effects. Given 0.21 0.21 gridded standard
meteorological forcing (sea-level pressure, geostrophic 10 m wind
components, 2 m air temperature and relative humidity, precipitation,
and cloud cover) provided by the Swedish Meteorological and
Hydrological Institute (SMHI) at 3-hour intervals, the circulation
model yields the evolution in time of the velocity, temperature and
salinity fields. As an example of what can be achieved within the
RCO framework, Fig. 2 also includes a long-term average of the
modeled salinity along the central axis of the Gulf of Finland. The
two data sets show a high degree of resemblance to one another,
with the exception of a more pronounced Neva-river plume in the
observations. This discrepancy indicates that the RCO model may
have too large a diffusion (which in turn affects the distribution
of tracers).
In the present study, a trajectory scheme based on results due
to Döös (1995) as well as Blanke and Raynaud (1997) was used to
compute Lagrangian paths from the three-dimensional velocity
fields in the Gulf of Finland provided by the RCO model. The
trajectory algorithm is based on analytical calculations using a
prescribed velocity field which is five-fold more highly resolved in
time than the circulation-model data, from which it is generated
by linear interpolation. This permits analyses of the water-parcel
motion over smaller scales than the model grid by interpolation
of the zonal, meridional, and vertical velocities defined in the
corners of the grid cell.
It should be noted that true Lagrangian trajectories are
only affected indirectly by the sub-grid parameterisation of
viscosity and diffusion implemented within the circulation model.
(An alternative method of water-mass analysis is based on the
tracer equation, which includes diffusion in contrast to what is
the case for true Lagrangian trajectories. However, this diffusion is
not only physical, but also numerical due to, e.g. finite-difference
truncation error.) Following Döös and Engqvist (2007), sub-grid
turbulence affecting the trajectories has thus been parameterized
by adding a random turbulent velocity at each ‘‘high-resolution’’
time step, this in order to include a measure of turbulent diffusion
in the analysis of the present study.
It should furthermore be underlined that the trajectory calculations
can be run in an autonomous fashion with regard to the
circulation model, i.e. off-line. This makes it possible to carry
through the analysis without having to take recourse to excessive
computer resources.
Convection has not been taken into account in the present
study, but the effects of this process have been examined in the
course of a previous investigation (Döös, 1995) by assigning a
water parcel a random depth whenever it enters a convectively
unstable water column. (Like the velocities used to calculate the
trajectories, these convective events also originate from the
circulation model.) This study showed, however, that the effects
of convection did not affect the results to any significant degree.
To examine the behavior of the system, all water parcels
entering the Gulf of Finland from the Baltic and the river Neva
were ‘‘tagged’’ with the aim of determining the origin of the water
masses characterizing the system in ‘‘steady state’’. This refers to
when the number of water parcels released into the system equals
that exiting and when a ‘‘saturated’’ ratio between the number of
trajectories originating from the two sources is established over a
meridional transect half-way into the Gulf. These criteria were
found to be amply satisfied for numerical experiments of an
approximately 5000-day duration, well above the estimated turnover
time of the system (Andrejev et al., 2004b) being 1–2 years.
Every 6 h trajectories were seeded over the exit from the
river Neva as well as over the entire breadth and depth of the
Hanko–Hiumaa transect delimiting the Gulf of Finland from the
Baltic proper. Each trajectory was specified as representing a
volume flux of 100 m 3 s 1 , which, based on a transport of
7–8000 m 3 s 1 entering the Gulf, yielded a sufficient number of
trajectories, in this case around 80 each 6th hour. The sensitivity
was tested by stepwise increasing the volume flux associated
with each trajectory, corresponding to a decrease of the trajectory
‘‘density’’. The threshold when a further decrease of this density
had a degrading effect on the outcome was found to be around 50,
i.e. considerably below the value of 80 trajectories ‘‘seeded’’ every
6 h during our numerical simulations.
All trajectories leaving the system were removed from the
model run at a meridional transect located 5 grid-points west of
the Baltic Proper boundary where source water originally was
tagged (cf. Fig. 1). The rationale behind this procedure was to
reduce the computational costs and to, as far as possible, prevent
trajectories from recirculating, which would have entailed the
risk of multiple tagging of water parcels. When the trajectory
integrations had attained a steady state in the sense described
above, the results were analyzed using techniques to be outlined
in what follows.
3. Mixing and water mass composition
Before proceeding with a discussion of the overall results from
the numerical experiments, we examine some specific features in
order to judge whether the model can be regarded as performing
in an adequate fashion. The underlying rationale is that even if a
circulation model yields more-or-less correct overall results,
considerable local irregularities may arise, in particular adjacent
to the boundaries of the system. In the larger Baltic perspective,
the Gulf of Finland can be looked upon as a marginal area, and
thus the modelers in the course of developing the RCO formalism
did not focus specifically on this region, even though the discharge
from the river Neva potentially could create anomalies
here. The model results were thus verified by collating the timeaveraged
RCO-generated velocity fields with the results from
earlier studies. One interesting area for comparisons of this type
is located close to Moshnyi Island in the inner part of the Gulf,
where Andrejev et al. (2004b) reported exceptionally long residence
times of the water. This state of affairs also manifests itself
in the present circulation-model results, assuming the form of a
prevalence of stable, highly persistent eddies in the area. In
general, the time-averaged RCO velocity fields employed for
calculating the trajectories proved to be in agreement with
well-known transport patterns (Palmén, 1930) characterizing
the Gulf of Finland. Thus high-saline water from the Baltic Proper
enters the Gulf as a deep boundary current adjacent to the
Estonian coast, whereas transports associated with the river Neva
discharge tends to take place on the Finnish side of the Gulf.
Once each experiment had been concluded, analysis of the
trajectory behavior was undertaken by calculating a mixing ratio
R defined as the number of trajectories from Neva divided by the
total number of trajectories within each grid-box. This was done
for each cell in the Gulf of Finland modeling domain, the value
R¼1 representing a situation where all water originated from the
river Neva, the value R¼0 indicating the presence of only Baltic
water. To investigate the relative water-mass composition along
the main axis of the Gulf of Finland, this distribution was
averaged across the Gulf. By also carrying through a vertical
integration, it proved feasible to construct a Hovmøller diagram
representing the evolution of this water-mass distribution along a
section in the Gulf of Finland as a function of time, cf. Fig. 3. From
this diagram it can be concluded that a saturated ratio between
the number of trajectories originating from the two sources was

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B.F. Jönsson et al. / Continental Shelf Research 31 (2011) 1811–1817
1990
50
Year
1985
Depth (m)
100
1981
150
0% 50%
Particles orginating from Neva
100%
24°E 26°E
28°E
24°E 26°E
28°E
0% 50%
Particles orginating from Neva
100%
Fig. 3. Hovmøller diagram showing the time-evolution of the mixing between the
water masses from the river Neva (red) and those originating from the Baltic
Proper (blue) following a longitudinal transect through the Gulf of Finland. (For
interpretation of the references to color in this figure legend, the reader is referred
to the web version of this article.)
Mixingfront Location
River Discharge m 3 /s
28˚E
27˚E
3000
2000
0
1 2 3 4 5 6 7 8 9 10 11 12 13
Time (Years)
Fig. 4. Diagram showing the evolution in time of the location of the zone of
maximum mixing in the Gulf of Finland (heavy black line), the Neva freshwater
discharge (dotted line), and the observed wind at the representative Landsort
meteorological station south of Stockholm in Sweden (thin black line).
established after an initial adjustment period of approximately
one year.
The most pronounced mixing between the water-masses from
the Baltic Proper and the river Neva can be expected to take place
in a zone encompassing the maximal gradient between the two
water-masses. From Fig. 3, this zone is seen to vary somewhat in
position as well as extent, although no systematic tendencies are
visible. The two most likely ‘‘suspects’’ when attempting to assign
responsibility for these irregularities are the variations of the
freshwater input from the Neva and the longer-term properties of
the overall wind conditions characterizing the Baltic region. Fig. 4
thus shows the evolution in time of the position of the mixing
zone (taken to coincide with the maximum gradient in Fig. 3), the
Neva freshwater discharge, and the observed wind at the Landsort
meteorological station south of Stockholm in Sweden (known to
be representative of the larger-scale conditions affecting the
10
8
6
Wind m/s
Fig. 5. Diagram showing the time-average (1980–1994) of the mixing between
waters from the river Neva (red) and those from the Baltic Proper (blue) following
a transect along the Gulf of Finland. (For interpretation of the references to color in
this figure legend, the reader is referred to the web version of this article.)
Baltic). From this diagram it is appears as though, in contrast to
commonly held prejudices, the river discharge does not play a
major role. Consistent with results due to Elken et al. (2003), the
wind regime, however, tends to demonstrate the same temporal
scales of variability as does the location of the mixing zone.
Although the present results do not show any pronounced
correlations, it is relevant to underline that a previous study has
suggested that the Baltic system may be quite sensitive to longterm
changes of the westerly winds (Meier and Kauker, 2003).
These questions, nevertheless, remain an unresolved issue and
further studies are needed.
Even if the mixing zone tends to wander somewhat, this
feature appears to be comparatively well-localized, with the
water masses originating from the Neva and the Baltic Proper
mainly mixing in a rather narrow zone, most frequently located
somewhere between 271E and 291E. The fact that this mixing
takes place comparatively close to the exit of the Neva also
indicates the predominance of Baltic water in the largest part of
the Gulf, a state of affairs that is easily verified on the basis of
salinity records from the area (Jurva, 1951).
In Fig. 5, the laterally averaged distribution ranging from 0 to
1 dealt with above is instead represented as a time-mean over the
entire period from 1980 to 1994. This diagram confirms the
picture of a stable sloping mixing zone located in the inner part
of the Gulf. To gain an appreciation of one of the advantages of the
Lagrangian formalism, it is of considerable interest to compare
the results in Fig. 5 with the salinity section in Fig. 2, where the
isohaline configuration only gives weak indications of where the
mixing actually takes place. Analogously Fig. 5 does not do justice
to the salinity distribution shown in Fig. 2. The Baltic surface
waters entering the Gulf over the Hanko–Hiumaa transect have
comparatively low salinities and in fact give rise to a haline
stratification very similar to that visible in the RCO-modeled
salinity section shown in Fig. 2.
4. Large-scale circulation
In a broader oceanographical context, the overall circulation in
the Gulf of Finland is also of interest. A convenient way (Blanke
et al., 1999) of representing the long-term circulation of an
estuary is to use a Lagrangian stream-function, which is calculated
by summing over selected trajectories describing water
pathways of particular interest. Hereby one can isolate the

B.F. Jönsson et al. / Continental Shelf Research 31 (2011) 1811–1817 1815
behaviour of specific water masses by following sets of trajectories
from one or more initial sections (in the present case the
mouth of the river Neva and the Hanko–Hiiumaa transect). A brief
general outline of the procedure is given in the appendix.
Applying this formalism to the previously described velocity
fields from the RCO-model, the river Neva inflow to the Gulf of
Finland is first considered. In Fig. 6(A), the resulting, laterally
averaged, Lagrangian overturning stream-function along the Gulf
is shown as a function of the depth. The river discharge is seen to
progress towards the Baltic Proper predominantly in the form of a
surface flow, although note the progressively deepening streamlines
indicating an entrainment of Neva water into the depths of
the Gulf.
In Fig. 6(B), the analogous set of results for the Baltic inflow to
the Gulf across the Hanko–Hiiumaa transect is shown. The
character of these streamlines is seen to differ markedly from
that associated with the Neva discharge. In Fig. 6(B), the dominating
overturning cell is constituted by the incoming saline
deep water from the Baltic Proper, which upwells in the Gulf as it
mixes with the river Neva water and returns westwards as much
fresher water in the surface layers. This is the contribution from
the Gulf of Finland to what has been termed the Baltic haline
conveyor belt (Döös et al., 2004).
It is also possible to add these two separately determined
stream-functions to one which corresponds to the standard
Eulerian stream-function, cf. Fig. 6(C). From this diagram it is
recognised that the fluxes associated with the ‘‘Baltic Proper
circulation cell’’ are an order of magnitude larger than the
transport from the river Neva and thus dominate the vertical
circulation in the Gulf of Finland.
Instead of projecting the stream-function against depth, this
can be done versus density. This intuitively less straightforward
visualisation has the advantage of revealing more of the physical
processes underlying the circulation. Fig. 7(A)–(C) thus show the
Lagrangian overturning stream-function against density. (A projection
versus salinity proved to manifest pronounced similarities,
underlining the well-known fact that the stratification and
circulation of the Baltic are mainly determined by the salinity,
not the temperature.)
A useful feature associated with these projections against
density/salinity is that they facilitate a qualitative as well as
quantitative understanding of how the salinity of water originating
from the Baltic Proper gradually is decreased as it circulates in
the Gulf of Finland. This deep overturning cell is associated with a
flux of around 5000 m 3 s 1 , hereby exceeding the value of
4000 m 3 s 1 deduced from the depth- projected stream-functions.
This discrepancy between the results from the different
projections indicates that the transversal slope of the isopycnals/
isohalines only plays a subordinate role for the vertical circulation
in the Gulf of Finland.
In the depth-projected results there are, however, indications
of a shallow circulation cell which is absent in the corresponding
salinity- and density-projected results. This feature is most likely
due to a horizontal circulation on the transversally inclined
isopycnals, similar to what holds true for the Southern-Ocean
Deacon Cell (Döös and Webb, 1994; Döös, 1994), but which
disappears when lateral averaging is carried out over isopycnals.
(A north–south slope of the shallow part of the horizontal
circulation in the Gulf of Finland can be discerned from individual
trajectories, an example of which is given in Fig. 8.)
5. Summary and discussion
The trajectory results reported here shed some new light on
the estuarine mixing dynamics characterizing the Gulf of Finland.
It has been shown that the main part of this estuary is dominated
by water from the Baltic Proper and that the most pronounced
mixing with Neva water takes place over a rather small area in
the inner parts of the Gulf, the location of this zone tending to
fluctuate somewhat. In contrast to previously held views (Meier
and Kauker, 2003), these variations do not appear to be directly
linked to either the freshwater influx from the river Neva or the
long-term variations of the wind forcing.
Depth (m)
50
100
750
350
-350
Depth (m)
50
100
4500
1500
-1500
150
-750
150
-4500
24°E 26°E 28°E
24°E 26°E 28°E
4500
Depth (m)
50
100
1500
-1500
150
-4500
24°E 26°E 28°E
Fig. 6. Lagrangian overturning stream-functions projected with depth as the vertical coordinate in the diagrams. The upper left panel (A) shows the behaviour of the water
debouching into the Gulf of Finland from the river Neva (River Neva Meridional Stream-function). The upper right panel (B) represents the motion of the Baltic water
entering across the Hanko–Hiiumaa transect. The bottom panel (Baltic Proper Meridional Stream-function) (C) shows a combination of the results from the upper two
panels, and corresponds to the standard Eulerian stream-function (Total Meridional Stream-function).

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B.F. Jönsson et al. / Continental Shelf Research 31 (2011) 1811–1817
0
750
0
4500
σ0 (kg -1 m 3 )
4
350
-350
σ0 (kg -1 m 3 )
4
1500
-1500
8
-750
24°E 26°E 28°E
24°E 26°E 28°E
8
-4500
σ0 (kg -1 m 3 )
0
4
4500
1500
-1500
8
24°E 26°E 28°E
-4500
Fig. 7. Lagrangian overturning stream-functions projected with density ðs 0 Þ as the vertical coordinate in the diagrams. The upper left panel (A) shows the behaviour of the
water debouching into the Gulf of Finland from the river Neva (River Neva Meridional Stream-function). The upper right panel (B) represents the motion of the Baltic water
entering across the Hanko–Hiiumaa transect (Baltic Proper Meridional Stream-function). The lower panel (C) shows a combination of the results from the upper two panels
(Total Meridional Stream-function).
62°N
61°N
60°N
59°N
58°N
18°E 21°E
24°E
27°E 30°E
33°E
Fig. 8. Example of a trajectory originating from the Baltic Proper which enters the southern part of the Gulf of Finland in the surface layer and exits the Gulf in a subsurface
layer farther north. The color scale indicates the subsurface depth of the trajectory in meters. (For interpretation of the references to color in this figure legend, the reader is
referred to the web version of this article.)
A closer examination of the vertical dynamics of the Gulf of
Finland using Lagrangian overturning stream-functions confirmed
the picture of a system dominated by a haline-driven circulation
from the Baltic Proper with a magnitude of about 5000 m 3 s 1 .
When this circulation was projected on the depth, it was also
possible to detect an additional closed shallow circulation cell,
most likely caused by a meridional slope of the sea-surface
height.
The present study has shown the feasibility of using Lagrangian
trajectories for studying complex and/or less well-defined
estuaries. When investigating systems of this type, a balance must
be struck between choosing a simple and robust model approach
as typified by mass-balance models (which, however, require a
number of specific criteria to be satisfied) and more advanced 3-D
numerical models. The latter type of model is somewhat complex
to apply, but does not, e.g. call for the system under investigation
(or parts thereof) to be more-or-less spatially homogeneous, and
furthermore yields a much better understanding of the physical
processes than mass-balance models do. 3-D modeling, however,
generates an abundance of highly resolved data in time and space.
Even if physical processes are described in a satisfactory manner,
it remains a challenge to specify the ‘‘representative state’’ of the
system, a common goal in not least estuarine studies. In situations
such as these, Lagrangian-trajectory methods can serve a useful
purpose, since these techniques are capable of providing a
coherent synthesis of the time-evolution of large data-sets, while
still including any intrinsic variability of the system. It is also
possible to define prognostic scalars characterizing the estuaries,
which facilitates a systematic comparison between different
systems.
Lagrangian approaches have classically been employed for
studies of the dispersion of pollutants, sediments, etc. In contrast,

B.F. Jönsson et al. / Continental Shelf Research 31 (2011) 1811–1817 1817
the present use of trajectories for analyzing the dynamics of an where T x i,j,k,n , Ty i,j,k,n , and Tz i,j,k,n
estuarine system can be regarded as a way of merging some of the
advantages accruing to simple mass-balance models and those
associated with more sophisticated 3-D numerical models. This is
zonal overturning stream function C LZ
accomplished by using the trajectories to synthesize the essentials
from the vast data sets generated during 3-D simulations. In C LZ
i,k C LZ
i,k 1 ¼ X X
T y i,j,k,n ,
the present study focus has been on an intermediate-scale Baltic
j n
estuary, but the utility of the method is not limited to modestsized
systems of this type. Lagrangian techniques have, for
instance, recently been used for a decomposition of the Southern-Ocean
Deacon cell (Döös et al., 2008) and are presently being
employed for a detailed analysis of the global conveyor belt, with References
particular emphasis on identifying and tracking the various water
masses contributing to this process (B. Blanke, pers. comm.).
Acknowledgments
The work herein reported has benefited from support provided
by the Bert Bolin Centre for Climate Research and the International
Meteorological Institute at Stockholm University. The
authors furthermore wish to extend their thanks to two unknown
reviewers for constructive comments.
Appendix A. The Lagrangian stream function
A Lagrangian stream function can be calculated by summing
over trajectories representing the desired path. A particular water
mass can be isolated by following a set of trajectories between
specific initial and final sections. Each trajectory, indexed by n, is
associated with a volume transport T n given by the velocity, initial
area, and number of trajectories released. During transit from the
initial to the final section the volume transport remains
unchanged; the transport/velocity field is non-divergent, permitting
stream-function representations. The volume transport
linked to each trajectory is inversely proportional to the number
of trajectories released, viz. the Lagrangian resolution (which
should be sufficiently high to ensure that the stream function
does not change when the number of trajectories is further
increased). A non-divergent 3-D volume-transport field is
obtained by recording every instance of a trajectory passing a
grid-box wall. Every trajectory entering a grid-box also exits, and
hence this field exactly satisfies
T x i,j,k,n
T x i 1,j,k,n þTy T y i,j,k,n i,j 1,k,n þTz i,j,k,n
T z i,j,k 1,n ¼ 0,
are the trajectory-derived volume
transports in the zonal (i), meridional (j), and vertical (k) directions,
respectively. A meridional integration yields the Lagrangian
where the ‘‘vertical’’ index k can correspond to either the depth or
the density.
Andrejev, O., Myrberg, K., Alenius, P., Lundberg, P.A., 2004a. Mean circulation and
water exchange in the Gulf of Finland—a study based on three-dimensional
modeling. Boreal Environmental Research 9, 1–16.
Andrejev, O., Myrberg, K., Lundberg, P., 2004b. Age and renewal time of water
masses in a semi-enclosed basin—application to the Gulf of Finland. Tellus
56A, 548–558.
Alenius, P., Myrberg, K., Nekrasov, A., 1998. The physical oceanography of the Gulf
of Finland: a review Boreal. Environmental Research 3 (2), 97–125.
Blanke, B., Arhan, M., Madec, G., Roche, S., 1999. Warm water paths in the
equatorial Atlantic as diagnosed with a general circulation model. Journal of
Physical Oceanography 29, 2753–2768.
Blanke, B., Raynaud, S., 1997. Kinematics of the pacific equatorial undercurrent: a
Eulerian and Lagrangian approach from GCM results. Journal of Physical
Oceanography 27, 1038–1053.
Döös, K., 1994. Semi-analytical simulation of the meridional cells in the southern
ocean. Journal of Physical Oceanography 24, 1281–1293.
Döös, K., 1995. Inter-ocean exchange of water masses. Journal of Geophysical
Research 100 (C7), 13499–13514.
Döös, K., Engqvist, A., 2007. Assessment of water exchange between a discharge
region and the open sea—a comparison of different methodological concepts.
Estuarine, Coastal, and Shelf Science 74 (4), 709–721.
Döös, K., Meier, H.E.M., Döscher, R., 2004. The Baltic Haline Conveyor Belt or the
overturning circulation and mixing in the Baltic. Ambio 33 (4–5), 261–266.
Döös, K., Nycander, J., Coward, A., 2008. Lagrangian decomposition of the Deacon
Cell. Journal of Geophysical Research 113, C07028.
Döös, K., Webb, D.J., 1994. The deacon cell and the other meridional cells in the
Southern Ocean. Journal of Physical Oceanography 24, 429–442.
Elken, J., Raudsepp, U., Lips, U., 2003. On the estuarine transport reversal in deep
layers of the Gulf of Finland. Journal of Sea Research 49, 267–274.
Jönsson, B., Döös, K., Lundberg, P., 2004. Baltic sub-basin turnover times examined
using the Rossby Centre Ocean Model. Ambio 33 (4–5), 257–260.
Jurva, R., 1951. Ympäröivät meret. Suomen maantieteellinen käsikirja, Suomen
maantieteellinen seura, Helsinki, pp. 121–144.
Meier, H.E.M., Döscher, R., Faxén, T., 2003. A multiprocessor coupled ice-ocean
model for the Baltic Sea: application to salt inflow. Journal of Geophysical
Research 108 (C8), 3273. doi:10.1029/2000JC000521 ((C8), 3273–3290).
Meier, H.E.M., Kauker, F., 2003. Modeling decadal variability of the Baltic Sea: 2.
role of freshwater inflow and large-scale. Journal of Geophysical Research 108
(C11), 3368. doi:10.1029/2003JC001799.
Palmén, E., 1930. Untersuchungen über die Strömungen in den Finnland umgebenden
Meeren. Commentationes Physico-Mathematicae, Societas Scientarium
Fennica 12.
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