A Lagrangian-trajectory study of a gradually mixed ... - Kristofer Döös
A Lagrangian-trajectory study of a gradually mixed ... - Kristofer Döös
A Lagrangian-trajectory study of a gradually mixed ... - Kristofer Döös
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Continental Shelf Research 31 (2011) 1811–1817<br />
Contents lists available at ScienceDirect<br />
Continental Shelf Research<br />
journal homepage: www.elsevier.com/locate/csr<br />
Research papers<br />
A <strong>Lagrangian</strong>-<strong>trajectory</strong> <strong>study</strong> <strong>of</strong> a <strong>gradually</strong> <strong>mixed</strong> estuary<br />
Bror F. Jönsson a, , Krist<strong>of</strong>er Döös b , Kai Myrberg c , Peter A. Lundberg b<br />
a Princeton University, Department <strong>of</strong> Geosciences, Guyot Hall, Princeton, NJ 08544, USA<br />
b Department <strong>of</strong> Meteorology/Physical Oceanography, Stockholm University, SE-10691 Stockholm, Sweden<br />
c SYKE, Mechelininkatu 34a, P.O.Box 140, FI-00251 Helsinki, Finland<br />
article info<br />
Article history:<br />
Received 21 October 2009<br />
Received in revised form<br />
11 July 2011<br />
Accepted 14 July 2011<br />
Available online 11 August 2011<br />
Keywords:<br />
Numerical modeling<br />
<strong>Lagrangian</strong> trajectories<br />
Estuarine mixing<br />
Gulf <strong>of</strong> Finland<br />
Baltic Sea<br />
abstract<br />
When modelling is used for investigating estuarine systems, a choice generally has to be made between<br />
applying simple mass-balance considerations or using a process-resolving three-dimensional (3-D)<br />
numerical circulation model. In the present investigation <strong>of</strong> the Gulf <strong>of</strong> Finland, a <strong>gradually</strong> <strong>mixed</strong><br />
estuary in the Baltic Sea, it is demonstrated how <strong>Lagrangian</strong>-<strong>trajectory</strong> analysis applied to the output<br />
from a 3-D model minimizes the disadvantages associated with both <strong>of</strong> the modelling techniques<br />
referred to above. This formalism made it possible to demonstrate that the main part <strong>of</strong> the Gulf is<br />
dominated by water originating from the Baltic proper, and that the most pronounced mixing with<br />
fresh water from the river Neva takes place over a limited zone in the inner part <strong>of</strong> the Gulf. Dynamical<br />
insights were furthermore obtained by using the <strong>Lagrangian</strong> formalism to construct overturning<br />
stream-functions for the two source waters.<br />
& 2011 Elsevier Ltd. All rights reserved.<br />
1. Introduction<br />
When undertaking oceanographic analyses, one is frequently<br />
interested in following the paths <strong>of</strong> distinct water types taking part<br />
in the circulation. In the context <strong>of</strong> numerical modeling, <strong>trajectory</strong><br />
analysis has proved to be a valuable tool for determining the origin<br />
as well as the fate <strong>of</strong> specific water masses, see, e.g. Jönsson et al.<br />
(2004). This holds true when examining processes ranging from<br />
those taking place over global scales (e.g. the thermohaline<br />
circulation) down to such local phenomena as the dispersion <strong>of</strong><br />
pollutants from a point source. The present <strong>study</strong> will focus on<br />
intermediate-scale processes, exemplified by those in the northeastern<br />
part <strong>of</strong> the Baltic (an area subject to considerable ecological<br />
threats due to its proximity to large cities and their associated<br />
wastewater disposal). The largest single freshwater inflow to the<br />
Baltic Sea, viz. the river Neva, is furthermore found here.<br />
The present <strong>study</strong> is thus focused on the Gulf <strong>of</strong> Finland,<br />
cf. Fig. 1, which is an elongated estuary (with a mean depth <strong>of</strong> only<br />
37 m) where physical processes ranging from small-scale vortices<br />
to the large-scale cyclonic circulation take place (Palmén, 1930;<br />
Andrejev et al., 2004a). The mean circulation pattern is one <strong>of</strong> an<br />
inflow to the Gulf taking place in a deep layer along the Estonian<br />
coast whereas the outflow, primarily in the form <strong>of</strong> fresher<br />
surface water, takes place slightly north <strong>of</strong> the longitudinal axis<br />
<strong>of</strong> the Gulf.<br />
Corresponding author. Tel.: þ1 617 818 1096.<br />
E-mail address: bjonsson@princeton.edu (B.F. Jönsson).<br />
The western end <strong>of</strong> the Gulf is a direct and gradual continuation<br />
<strong>of</strong> the Baltic Proper without any morphological constraints.<br />
The eastern end <strong>of</strong> the Gulf receives the discharge from river<br />
Neva, which with a mean run<strong>of</strong>f <strong>of</strong> 2700 m 3 s 1 constitutes<br />
15–20% <strong>of</strong> the total Baltic freshwater inflow. This leads not only<br />
to a salinity stratification in the vertical but also to pronounced<br />
east–west salinity gradients in which the cyclonic mean horizontal<br />
circulation pattern is reflected (cf. Figs. 1 and 2). The stratification<br />
is furthermore variable in both space and time, where the<br />
large seasonal variability <strong>of</strong> the incoming solar radiation plays a<br />
considerable role (Alenius et al., 1998). This state <strong>of</strong> affairs has, as<br />
will be seen, significant dynamic consequences. The watermasses<br />
originating from the two source regions also have markedly<br />
different characteristics, not only as regards the salinity but<br />
also the degree <strong>of</strong> anthropogenic contamination, why their<br />
ultimate fate in the Gulf is <strong>of</strong> considerable practical environmental<br />
interest. This general question has been dealt with in a number<br />
<strong>of</strong> investigations, most recently in one due to Andrejev et al.<br />
(2004b) where it was shown that the renewal time <strong>of</strong> the water<br />
masses <strong>of</strong> the entire Gulf is around 1–2 years and that the waterage<br />
distribution in the Gulf is spatially non-homogenous with the<br />
highest ages ( 2 years) found in the southeastern part <strong>of</strong> the<br />
Gulf. The scope <strong>of</strong> this <strong>study</strong>, however, had the consequence that<br />
the ultimate fate <strong>of</strong> the Baltic and Neva source waters was not<br />
fully ascertained by Andrejev and co-workers.<br />
The classical technique to resolve this question is based on an<br />
investigation <strong>of</strong> the tracer equation within the circulation-model<br />
framework. This Eulerian formalism, however, has the drawback <strong>of</strong><br />
overestimating diffusive effects, and furthermore many circulation<br />
0278-4343/$ - see front matter & 2011 Elsevier Ltd. All rights reserved.<br />
doi:10.1016/j.csr.2011.07.007
1812<br />
B.F. Jönsson et al. / Continental Shelf Research 31 (2011) 1811–1817<br />
62°N<br />
61°N<br />
60°N<br />
59°N<br />
Stockholm<br />
>6.5<br />
Hanko<br />
6.0<br />
3.0 2.5 2.0<br />
Helsinki<br />
5.0 4.5 4.0 3.5<br />
5.5<br />
Tallin<br />
2><br />
Neva<br />
St Petersburg<br />
Moshnyi<br />
Island<br />
Hiiuma<br />
58°N<br />
18°E 21°E<br />
24°E<br />
27°E 30°E 33°E<br />
Fig. 1. Map <strong>of</strong> the Gulf <strong>of</strong> Finland where the thin black line represents the model coastline. The heavy grey line indicates where the ‘‘Baltic’’ trajectories are released to the<br />
Gulf and the heavy black line where all trajectories are removed. Also included are observed surface isohalines as adapted from Jurva (1951).<br />
0.0<br />
-25.0<br />
-50.0<br />
-75.0<br />
-100.0<br />
-125.0<br />
-150.0<br />
22.0 23.0 24.0 25.0 27.0 28.0 29.0 30.0<br />
Fig. 2. Salinity sections along the central axis <strong>of</strong> the Gulf <strong>of</strong> Finland. Colour-coded contours show a long-term average <strong>of</strong> the RCO-modelled salinity field. The green lines<br />
represent observations as adapted from Jurva (1951). (For interpretation <strong>of</strong> the references to color in this figure legend, the reader is referred to the web version <strong>of</strong> this<br />
article.)<br />
models tend to be somewhat deficient in reproducing the salt<br />
dynamics <strong>of</strong> the system under consideration.<br />
The present investigation will instead approach the problem<br />
from a <strong>Lagrangian</strong> standpoint, i.e. using <strong>trajectory</strong> analysis. In<br />
next section the technical details <strong>of</strong> this procedure are outlined,<br />
whereafter the subsequent section deals with the results concerning<br />
mixing and water-mass composition obtained for the Gulf<br />
<strong>of</strong> Finland. Hereafter, a discussion is undertaken <strong>of</strong> the large-scale<br />
circulation in the Gulf based on a <strong>Lagrangian</strong> decomposition <strong>of</strong><br />
the overturning stream-function, this in order to estimate the<br />
contributions <strong>of</strong> the waters originating from the river Neva and<br />
the Baltic, respectively. The <strong>study</strong> is concluded by a review <strong>of</strong> the<br />
overall outcome <strong>of</strong> the analysis.<br />
2. Methods<br />
Classical <strong>trajectory</strong> analysis, as primarily applied to the atmosphere,<br />
mainly relied upon graphical techniques used in conjunction<br />
with meteorological good sense and experience. Modern-day<br />
<strong>trajectory</strong> methods are, however, based on numerical circulation<br />
models. In the present <strong>study</strong>, velocity fields from the Rossby Centre<br />
Ocean (RCO) model (Meier et al., 2003) serve as the basis for the<br />
further investigations. This well-proven (Meier and Kauker, 2003)<br />
finite-difference model has 41 depth levels (at intervals ranging<br />
from 3 m in the surface region to 12 m at greater depths) and a<br />
horizontal resolution <strong>of</strong> 2 4 nautical miles. It is based on<br />
an Arakawa B-grid and is capable <strong>of</strong> resolving the meso-scale
B.F. Jönsson et al. / Continental Shelf Research 31 (2011) 1811–1817 1813<br />
eddies <strong>of</strong> importance for the dynamics <strong>of</strong> the Baltic. For the present<br />
investigation with focus on the north-eastern Baltic, the remotely<br />
applied boundary conditions at the border to the North Sea do not<br />
give rise to any spurious effects. Given 0.21 0.21 gridded standard<br />
meteorological forcing (sea-level pressure, geostrophic 10 m wind<br />
components, 2 m air temperature and relative humidity, precipitation,<br />
and cloud cover) provided by the Swedish Meteorological and<br />
Hydrological Institute (SMHI) at 3-hour intervals, the circulation<br />
model yields the evolution in time <strong>of</strong> the velocity, temperature and<br />
salinity fields. As an example <strong>of</strong> what can be achieved within the<br />
RCO framework, Fig. 2 also includes a long-term average <strong>of</strong> the<br />
modeled salinity along the central axis <strong>of</strong> the Gulf <strong>of</strong> Finland. The<br />
two data sets show a high degree <strong>of</strong> resemblance to one another,<br />
with the exception <strong>of</strong> a more pronounced Neva-river plume in the<br />
observations. This discrepancy indicates that the RCO model may<br />
have too large a diffusion (which in turn affects the distribution<br />
<strong>of</strong> tracers).<br />
In the present <strong>study</strong>, a <strong>trajectory</strong> scheme based on results due<br />
to Döös (1995) as well as Blanke and Raynaud (1997) was used to<br />
compute <strong>Lagrangian</strong> paths from the three-dimensional velocity<br />
fields in the Gulf <strong>of</strong> Finland provided by the RCO model. The<br />
<strong>trajectory</strong> algorithm is based on analytical calculations using a<br />
prescribed velocity field which is five-fold more highly resolved in<br />
time than the circulation-model data, from which it is generated<br />
by linear interpolation. This permits analyses <strong>of</strong> the water-parcel<br />
motion over smaller scales than the model grid by interpolation<br />
<strong>of</strong> the zonal, meridional, and vertical velocities defined in the<br />
corners <strong>of</strong> the grid cell.<br />
It should be noted that true <strong>Lagrangian</strong> trajectories are<br />
only affected indirectly by the sub-grid parameterisation <strong>of</strong><br />
viscosity and diffusion implemented within the circulation model.<br />
(An alternative method <strong>of</strong> water-mass analysis is based on the<br />
tracer equation, which includes diffusion in contrast to what is<br />
the case for true <strong>Lagrangian</strong> trajectories. However, this diffusion is<br />
not only physical, but also numerical due to, e.g. finite-difference<br />
truncation error.) Following Döös and Engqvist (2007), sub-grid<br />
turbulence affecting the trajectories has thus been parameterized<br />
by adding a random turbulent velocity at each ‘‘high-resolution’’<br />
time step, this in order to include a measure <strong>of</strong> turbulent diffusion<br />
in the analysis <strong>of</strong> the present <strong>study</strong>.<br />
It should furthermore be underlined that the <strong>trajectory</strong> calculations<br />
can be run in an autonomous fashion with regard to the<br />
circulation model, i.e. <strong>of</strong>f-line. This makes it possible to carry<br />
through the analysis without having to take recourse to excessive<br />
computer resources.<br />
Convection has not been taken into account in the present<br />
<strong>study</strong>, but the effects <strong>of</strong> this process have been examined in the<br />
course <strong>of</strong> a previous investigation (Döös, 1995) by assigning a<br />
water parcel a random depth whenever it enters a convectively<br />
unstable water column. (Like the velocities used to calculate the<br />
trajectories, these convective events also originate from the<br />
circulation model.) This <strong>study</strong> showed, however, that the effects<br />
<strong>of</strong> convection did not affect the results to any significant degree.<br />
To examine the behavior <strong>of</strong> the system, all water parcels<br />
entering the Gulf <strong>of</strong> Finland from the Baltic and the river Neva<br />
were ‘‘tagged’’ with the aim <strong>of</strong> determining the origin <strong>of</strong> the water<br />
masses characterizing the system in ‘‘steady state’’. This refers to<br />
when the number <strong>of</strong> water parcels released into the system equals<br />
that exiting and when a ‘‘saturated’’ ratio between the number <strong>of</strong><br />
trajectories originating from the two sources is established over a<br />
meridional transect half-way into the Gulf. These criteria were<br />
found to be amply satisfied for numerical experiments <strong>of</strong> an<br />
approximately 5000-day duration, well above the estimated turnover<br />
time <strong>of</strong> the system (Andrejev et al., 2004b) being 1–2 years.<br />
Every 6 h trajectories were seeded over the exit from the<br />
river Neva as well as over the entire breadth and depth <strong>of</strong> the<br />
Hanko–Hiumaa transect delimiting the Gulf <strong>of</strong> Finland from the<br />
Baltic proper. Each <strong>trajectory</strong> was specified as representing a<br />
volume flux <strong>of</strong> 100 m 3 s 1 , which, based on a transport <strong>of</strong><br />
7–8000 m 3 s 1 entering the Gulf, yielded a sufficient number <strong>of</strong><br />
trajectories, in this case around 80 each 6th hour. The sensitivity<br />
was tested by stepwise increasing the volume flux associated<br />
with each <strong>trajectory</strong>, corresponding to a decrease <strong>of</strong> the <strong>trajectory</strong><br />
‘‘density’’. The threshold when a further decrease <strong>of</strong> this density<br />
had a degrading effect on the outcome was found to be around 50,<br />
i.e. considerably below the value <strong>of</strong> 80 trajectories ‘‘seeded’’ every<br />
6 h during our numerical simulations.<br />
All trajectories leaving the system were removed from the<br />
model run at a meridional transect located 5 grid-points west <strong>of</strong><br />
the Baltic Proper boundary where source water originally was<br />
tagged (cf. Fig. 1). The rationale behind this procedure was to<br />
reduce the computational costs and to, as far as possible, prevent<br />
trajectories from recirculating, which would have entailed the<br />
risk <strong>of</strong> multiple tagging <strong>of</strong> water parcels. When the <strong>trajectory</strong><br />
integrations had attained a steady state in the sense described<br />
above, the results were analyzed using techniques to be outlined<br />
in what follows.<br />
3. Mixing and water mass composition<br />
Before proceeding with a discussion <strong>of</strong> the overall results from<br />
the numerical experiments, we examine some specific features in<br />
order to judge whether the model can be regarded as performing<br />
in an adequate fashion. The underlying rationale is that even if a<br />
circulation model yields more-or-less correct overall results,<br />
considerable local irregularities may arise, in particular adjacent<br />
to the boundaries <strong>of</strong> the system. In the larger Baltic perspective,<br />
the Gulf <strong>of</strong> Finland can be looked upon as a marginal area, and<br />
thus the modelers in the course <strong>of</strong> developing the RCO formalism<br />
did not focus specifically on this region, even though the discharge<br />
from the river Neva potentially could create anomalies<br />
here. The model results were thus verified by collating the timeaveraged<br />
RCO-generated velocity fields with the results from<br />
earlier studies. One interesting area for comparisons <strong>of</strong> this type<br />
is located close to Moshnyi Island in the inner part <strong>of</strong> the Gulf,<br />
where Andrejev et al. (2004b) reported exceptionally long residence<br />
times <strong>of</strong> the water. This state <strong>of</strong> affairs also manifests itself<br />
in the present circulation-model results, assuming the form <strong>of</strong> a<br />
prevalence <strong>of</strong> stable, highly persistent eddies in the area. In<br />
general, the time-averaged RCO velocity fields employed for<br />
calculating the trajectories proved to be in agreement with<br />
well-known transport patterns (Palmén, 1930) characterizing<br />
the Gulf <strong>of</strong> Finland. Thus high-saline water from the Baltic Proper<br />
enters the Gulf as a deep boundary current adjacent to the<br />
Estonian coast, whereas transports associated with the river Neva<br />
discharge tends to take place on the Finnish side <strong>of</strong> the Gulf.<br />
Once each experiment had been concluded, analysis <strong>of</strong> the<br />
<strong>trajectory</strong> behavior was undertaken by calculating a mixing ratio<br />
R defined as the number <strong>of</strong> trajectories from Neva divided by the<br />
total number <strong>of</strong> trajectories within each grid-box. This was done<br />
for each cell in the Gulf <strong>of</strong> Finland modeling domain, the value<br />
R¼1 representing a situation where all water originated from the<br />
river Neva, the value R¼0 indicating the presence <strong>of</strong> only Baltic<br />
water. To investigate the relative water-mass composition along<br />
the main axis <strong>of</strong> the Gulf <strong>of</strong> Finland, this distribution was<br />
averaged across the Gulf. By also carrying through a vertical<br />
integration, it proved feasible to construct a Hovmøller diagram<br />
representing the evolution <strong>of</strong> this water-mass distribution along a<br />
section in the Gulf <strong>of</strong> Finland as a function <strong>of</strong> time, cf. Fig. 3. From<br />
this diagram it can be concluded that a saturated ratio between<br />
the number <strong>of</strong> trajectories originating from the two sources was
1814<br />
B.F. Jönsson et al. / Continental Shelf Research 31 (2011) 1811–1817<br />
1990<br />
50<br />
Year<br />
1985<br />
Depth (m)<br />
100<br />
1981<br />
150<br />
0% 50%<br />
Particles orginating from Neva<br />
100%<br />
24°E 26°E<br />
28°E<br />
24°E 26°E<br />
28°E<br />
0% 50%<br />
Particles orginating from Neva<br />
100%<br />
Fig. 3. Hovmøller diagram showing the time-evolution <strong>of</strong> the mixing between the<br />
water masses from the river Neva (red) and those originating from the Baltic<br />
Proper (blue) following a longitudinal transect through the Gulf <strong>of</strong> Finland. (For<br />
interpretation <strong>of</strong> the references to color in this figure legend, the reader is referred<br />
to the web version <strong>of</strong> this article.)<br />
Mixingfront Location<br />
River Discharge m 3 /s<br />
28˚E<br />
27˚E<br />
3000<br />
2000<br />
0<br />
1 2 3 4 5 6 7 8 9 10 11 12 13<br />
Time (Years)<br />
Fig. 4. Diagram showing the evolution in time <strong>of</strong> the location <strong>of</strong> the zone <strong>of</strong><br />
maximum mixing in the Gulf <strong>of</strong> Finland (heavy black line), the Neva freshwater<br />
discharge (dotted line), and the observed wind at the representative Landsort<br />
meteorological station south <strong>of</strong> Stockholm in Sweden (thin black line).<br />
established after an initial adjustment period <strong>of</strong> approximately<br />
one year.<br />
The most pronounced mixing between the water-masses from<br />
the Baltic Proper and the river Neva can be expected to take place<br />
in a zone encompassing the maximal gradient between the two<br />
water-masses. From Fig. 3, this zone is seen to vary somewhat in<br />
position as well as extent, although no systematic tendencies are<br />
visible. The two most likely ‘‘suspects’’ when attempting to assign<br />
responsibility for these irregularities are the variations <strong>of</strong> the<br />
freshwater input from the Neva and the longer-term properties <strong>of</strong><br />
the overall wind conditions characterizing the Baltic region. Fig. 4<br />
thus shows the evolution in time <strong>of</strong> the position <strong>of</strong> the mixing<br />
zone (taken to coincide with the maximum gradient in Fig. 3), the<br />
Neva freshwater discharge, and the observed wind at the Landsort<br />
meteorological station south <strong>of</strong> Stockholm in Sweden (known to<br />
be representative <strong>of</strong> the larger-scale conditions affecting the<br />
10<br />
8<br />
6<br />
Wind m/s<br />
Fig. 5. Diagram showing the time-average (1980–1994) <strong>of</strong> the mixing between<br />
waters from the river Neva (red) and those from the Baltic Proper (blue) following<br />
a transect along the Gulf <strong>of</strong> Finland. (For interpretation <strong>of</strong> the references to color in<br />
this figure legend, the reader is referred to the web version <strong>of</strong> this article.)<br />
Baltic). From this diagram it is appears as though, in contrast to<br />
commonly held prejudices, the river discharge does not play a<br />
major role. Consistent with results due to Elken et al. (2003), the<br />
wind regime, however, tends to demonstrate the same temporal<br />
scales <strong>of</strong> variability as does the location <strong>of</strong> the mixing zone.<br />
Although the present results do not show any pronounced<br />
correlations, it is relevant to underline that a previous <strong>study</strong> has<br />
suggested that the Baltic system may be quite sensitive to longterm<br />
changes <strong>of</strong> the westerly winds (Meier and Kauker, 2003).<br />
These questions, nevertheless, remain an unresolved issue and<br />
further studies are needed.<br />
Even if the mixing zone tends to wander somewhat, this<br />
feature appears to be comparatively well-localized, with the<br />
water masses originating from the Neva and the Baltic Proper<br />
mainly mixing in a rather narrow zone, most frequently located<br />
somewhere between 271E and 291E. The fact that this mixing<br />
takes place comparatively close to the exit <strong>of</strong> the Neva also<br />
indicates the predominance <strong>of</strong> Baltic water in the largest part <strong>of</strong><br />
the Gulf, a state <strong>of</strong> affairs that is easily verified on the basis <strong>of</strong><br />
salinity records from the area (Jurva, 1951).<br />
In Fig. 5, the laterally averaged distribution ranging from 0 to<br />
1 dealt with above is instead represented as a time-mean over the<br />
entire period from 1980 to 1994. This diagram confirms the<br />
picture <strong>of</strong> a stable sloping mixing zone located in the inner part<br />
<strong>of</strong> the Gulf. To gain an appreciation <strong>of</strong> one <strong>of</strong> the advantages <strong>of</strong> the<br />
<strong>Lagrangian</strong> formalism, it is <strong>of</strong> considerable interest to compare<br />
the results in Fig. 5 with the salinity section in Fig. 2, where the<br />
isohaline configuration only gives weak indications <strong>of</strong> where the<br />
mixing actually takes place. Analogously Fig. 5 does not do justice<br />
to the salinity distribution shown in Fig. 2. The Baltic surface<br />
waters entering the Gulf over the Hanko–Hiumaa transect have<br />
comparatively low salinities and in fact give rise to a haline<br />
stratification very similar to that visible in the RCO-modeled<br />
salinity section shown in Fig. 2.<br />
4. Large-scale circulation<br />
In a broader oceanographical context, the overall circulation in<br />
the Gulf <strong>of</strong> Finland is also <strong>of</strong> interest. A convenient way (Blanke<br />
et al., 1999) <strong>of</strong> representing the long-term circulation <strong>of</strong> an<br />
estuary is to use a <strong>Lagrangian</strong> stream-function, which is calculated<br />
by summing over selected trajectories describing water<br />
pathways <strong>of</strong> particular interest. Hereby one can isolate the
B.F. Jönsson et al. / Continental Shelf Research 31 (2011) 1811–1817 1815<br />
behaviour <strong>of</strong> specific water masses by following sets <strong>of</strong> trajectories<br />
from one or more initial sections (in the present case the<br />
mouth <strong>of</strong> the river Neva and the Hanko–Hiiumaa transect). A brief<br />
general outline <strong>of</strong> the procedure is given in the appendix.<br />
Applying this formalism to the previously described velocity<br />
fields from the RCO-model, the river Neva inflow to the Gulf <strong>of</strong><br />
Finland is first considered. In Fig. 6(A), the resulting, laterally<br />
averaged, <strong>Lagrangian</strong> overturning stream-function along the Gulf<br />
is shown as a function <strong>of</strong> the depth. The river discharge is seen to<br />
progress towards the Baltic Proper predominantly in the form <strong>of</strong> a<br />
surface flow, although note the progressively deepening streamlines<br />
indicating an entrainment <strong>of</strong> Neva water into the depths <strong>of</strong><br />
the Gulf.<br />
In Fig. 6(B), the analogous set <strong>of</strong> results for the Baltic inflow to<br />
the Gulf across the Hanko–Hiiumaa transect is shown. The<br />
character <strong>of</strong> these streamlines is seen to differ markedly from<br />
that associated with the Neva discharge. In Fig. 6(B), the dominating<br />
overturning cell is constituted by the incoming saline<br />
deep water from the Baltic Proper, which upwells in the Gulf as it<br />
mixes with the river Neva water and returns westwards as much<br />
fresher water in the surface layers. This is the contribution from<br />
the Gulf <strong>of</strong> Finland to what has been termed the Baltic haline<br />
conveyor belt (Döös et al., 2004).<br />
It is also possible to add these two separately determined<br />
stream-functions to one which corresponds to the standard<br />
Eulerian stream-function, cf. Fig. 6(C). From this diagram it is<br />
recognised that the fluxes associated with the ‘‘Baltic Proper<br />
circulation cell’’ are an order <strong>of</strong> magnitude larger than the<br />
transport from the river Neva and thus dominate the vertical<br />
circulation in the Gulf <strong>of</strong> Finland.<br />
Instead <strong>of</strong> projecting the stream-function against depth, this<br />
can be done versus density. This intuitively less straightforward<br />
visualisation has the advantage <strong>of</strong> revealing more <strong>of</strong> the physical<br />
processes underlying the circulation. Fig. 7(A)–(C) thus show the<br />
<strong>Lagrangian</strong> overturning stream-function against density. (A projection<br />
versus salinity proved to manifest pronounced similarities,<br />
underlining the well-known fact that the stratification and<br />
circulation <strong>of</strong> the Baltic are mainly determined by the salinity,<br />
not the temperature.)<br />
A useful feature associated with these projections against<br />
density/salinity is that they facilitate a qualitative as well as<br />
quantitative understanding <strong>of</strong> how the salinity <strong>of</strong> water originating<br />
from the Baltic Proper <strong>gradually</strong> is decreased as it circulates in<br />
the Gulf <strong>of</strong> Finland. This deep overturning cell is associated with a<br />
flux <strong>of</strong> around 5000 m 3 s 1 , hereby exceeding the value <strong>of</strong><br />
4000 m 3 s 1 deduced from the depth- projected stream-functions.<br />
This discrepancy between the results from the different<br />
projections indicates that the transversal slope <strong>of</strong> the isopycnals/<br />
isohalines only plays a subordinate role for the vertical circulation<br />
in the Gulf <strong>of</strong> Finland.<br />
In the depth-projected results there are, however, indications<br />
<strong>of</strong> a shallow circulation cell which is absent in the corresponding<br />
salinity- and density-projected results. This feature is most likely<br />
due to a horizontal circulation on the transversally inclined<br />
isopycnals, similar to what holds true for the Southern-Ocean<br />
Deacon Cell (Döös and Webb, 1994; Döös, 1994), but which<br />
disappears when lateral averaging is carried out over isopycnals.<br />
(A north–south slope <strong>of</strong> the shallow part <strong>of</strong> the horizontal<br />
circulation in the Gulf <strong>of</strong> Finland can be discerned from individual<br />
trajectories, an example <strong>of</strong> which is given in Fig. 8.)<br />
5. Summary and discussion<br />
The <strong>trajectory</strong> results reported here shed some new light on<br />
the estuarine mixing dynamics characterizing the Gulf <strong>of</strong> Finland.<br />
It has been shown that the main part <strong>of</strong> this estuary is dominated<br />
by water from the Baltic Proper and that the most pronounced<br />
mixing with Neva water takes place over a rather small area in<br />
the inner parts <strong>of</strong> the Gulf, the location <strong>of</strong> this zone tending to<br />
fluctuate somewhat. In contrast to previously held views (Meier<br />
and Kauker, 2003), these variations do not appear to be directly<br />
linked to either the freshwater influx from the river Neva or the<br />
long-term variations <strong>of</strong> the wind forcing.<br />
Depth (m)<br />
50<br />
100<br />
750<br />
350<br />
-350<br />
Depth (m)<br />
50<br />
100<br />
4500<br />
1500<br />
-1500<br />
150<br />
-750<br />
150<br />
-4500<br />
24°E 26°E 28°E<br />
24°E 26°E 28°E<br />
4500<br />
Depth (m)<br />
50<br />
100<br />
1500<br />
-1500<br />
150<br />
-4500<br />
24°E 26°E 28°E<br />
Fig. 6. <strong>Lagrangian</strong> overturning stream-functions projected with depth as the vertical coordinate in the diagrams. The upper left panel (A) shows the behaviour <strong>of</strong> the water<br />
debouching into the Gulf <strong>of</strong> Finland from the river Neva (River Neva Meridional Stream-function). The upper right panel (B) represents the motion <strong>of</strong> the Baltic water<br />
entering across the Hanko–Hiiumaa transect. The bottom panel (Baltic Proper Meridional Stream-function) (C) shows a combination <strong>of</strong> the results from the upper two<br />
panels, and corresponds to the standard Eulerian stream-function (Total Meridional Stream-function).
1816<br />
B.F. Jönsson et al. / Continental Shelf Research 31 (2011) 1811–1817<br />
0<br />
750<br />
0<br />
4500<br />
σ0 (kg -1 m 3 )<br />
4<br />
350<br />
-350<br />
σ0 (kg -1 m 3 )<br />
4<br />
1500<br />
-1500<br />
8<br />
-750<br />
24°E 26°E 28°E<br />
24°E 26°E 28°E<br />
8<br />
-4500<br />
σ0 (kg -1 m 3 )<br />
0<br />
4<br />
4500<br />
1500<br />
-1500<br />
8<br />
24°E 26°E 28°E<br />
-4500<br />
Fig. 7. <strong>Lagrangian</strong> overturning stream-functions projected with density ðs 0 Þ as the vertical coordinate in the diagrams. The upper left panel (A) shows the behaviour <strong>of</strong> the<br />
water debouching into the Gulf <strong>of</strong> Finland from the river Neva (River Neva Meridional Stream-function). The upper right panel (B) represents the motion <strong>of</strong> the Baltic water<br />
entering across the Hanko–Hiiumaa transect (Baltic Proper Meridional Stream-function). The lower panel (C) shows a combination <strong>of</strong> the results from the upper two panels<br />
(Total Meridional Stream-function).<br />
62°N<br />
61°N<br />
60°N<br />
59°N<br />
58°N<br />
18°E 21°E<br />
24°E<br />
27°E 30°E<br />
33°E<br />
Fig. 8. Example <strong>of</strong> a <strong>trajectory</strong> originating from the Baltic Proper which enters the southern part <strong>of</strong> the Gulf <strong>of</strong> Finland in the surface layer and exits the Gulf in a subsurface<br />
layer farther north. The color scale indicates the subsurface depth <strong>of</strong> the <strong>trajectory</strong> in meters. (For interpretation <strong>of</strong> the references to color in this figure legend, the reader is<br />
referred to the web version <strong>of</strong> this article.)<br />
A closer examination <strong>of</strong> the vertical dynamics <strong>of</strong> the Gulf <strong>of</strong><br />
Finland using <strong>Lagrangian</strong> overturning stream-functions confirmed<br />
the picture <strong>of</strong> a system dominated by a haline-driven circulation<br />
from the Baltic Proper with a magnitude <strong>of</strong> about 5000 m 3 s 1 .<br />
When this circulation was projected on the depth, it was also<br />
possible to detect an additional closed shallow circulation cell,<br />
most likely caused by a meridional slope <strong>of</strong> the sea-surface<br />
height.<br />
The present <strong>study</strong> has shown the feasibility <strong>of</strong> using <strong>Lagrangian</strong><br />
trajectories for <strong>study</strong>ing complex and/or less well-defined<br />
estuaries. When investigating systems <strong>of</strong> this type, a balance must<br />
be struck between choosing a simple and robust model approach<br />
as typified by mass-balance models (which, however, require a<br />
number <strong>of</strong> specific criteria to be satisfied) and more advanced 3-D<br />
numerical models. The latter type <strong>of</strong> model is somewhat complex<br />
to apply, but does not, e.g. call for the system under investigation<br />
(or parts there<strong>of</strong>) to be more-or-less spatially homogeneous, and<br />
furthermore yields a much better understanding <strong>of</strong> the physical<br />
processes than mass-balance models do. 3-D modeling, however,<br />
generates an abundance <strong>of</strong> highly resolved data in time and space.<br />
Even if physical processes are described in a satisfactory manner,<br />
it remains a challenge to specify the ‘‘representative state’’ <strong>of</strong> the<br />
system, a common goal in not least estuarine studies. In situations<br />
such as these, <strong>Lagrangian</strong>-<strong>trajectory</strong> methods can serve a useful<br />
purpose, since these techniques are capable <strong>of</strong> providing a<br />
coherent synthesis <strong>of</strong> the time-evolution <strong>of</strong> large data-sets, while<br />
still including any intrinsic variability <strong>of</strong> the system. It is also<br />
possible to define prognostic scalars characterizing the estuaries,<br />
which facilitates a systematic comparison between different<br />
systems.<br />
<strong>Lagrangian</strong> approaches have classically been employed for<br />
studies <strong>of</strong> the dispersion <strong>of</strong> pollutants, sediments, etc. In contrast,
B.F. Jönsson et al. / Continental Shelf Research 31 (2011) 1811–1817 1817<br />
the present use <strong>of</strong> trajectories for analyzing the dynamics <strong>of</strong> an where T x i,j,k,n , Ty i,j,k,n , and Tz i,j,k,n<br />
estuarine system can be regarded as a way <strong>of</strong> merging some <strong>of</strong> the<br />
advantages accruing to simple mass-balance models and those<br />
associated with more sophisticated 3-D numerical models. This is<br />
zonal overturning stream function C LZ<br />
accomplished by using the trajectories to synthesize the essentials<br />
from the vast data sets generated during 3-D simulations. In C LZ<br />
i,k C LZ<br />
i,k 1 ¼ X X<br />
T y i,j,k,n ,<br />
the present <strong>study</strong> focus has been on an intermediate-scale Baltic<br />
j n<br />
estuary, but the utility <strong>of</strong> the method is not limited to modestsized<br />
systems <strong>of</strong> this type. <strong>Lagrangian</strong> techniques have, for<br />
instance, recently been used for a decomposition <strong>of</strong> the Southern-Ocean<br />
Deacon cell (Döös et al., 2008) and are presently being<br />
employed for a detailed analysis <strong>of</strong> the global conveyor belt, with References<br />
particular emphasis on identifying and tracking the various water<br />
masses contributing to this process (B. Blanke, pers. comm.).<br />
Acknowledgments<br />
The work herein reported has benefited from support provided<br />
by the Bert Bolin Centre for Climate Research and the International<br />
Meteorological Institute at Stockholm University. The<br />
authors furthermore wish to extend their thanks to two unknown<br />
reviewers for constructive comments.<br />
Appendix A. The <strong>Lagrangian</strong> stream function<br />
A <strong>Lagrangian</strong> stream function can be calculated by summing<br />
over trajectories representing the desired path. A particular water<br />
mass can be isolated by following a set <strong>of</strong> trajectories between<br />
specific initial and final sections. Each <strong>trajectory</strong>, indexed by n, is<br />
associated with a volume transport T n given by the velocity, initial<br />
area, and number <strong>of</strong> trajectories released. During transit from the<br />
initial to the final section the volume transport remains<br />
unchanged; the transport/velocity field is non-divergent, permitting<br />
stream-function representations. The volume transport<br />
linked to each <strong>trajectory</strong> is inversely proportional to the number<br />
<strong>of</strong> trajectories released, viz. the <strong>Lagrangian</strong> resolution (which<br />
should be sufficiently high to ensure that the stream function<br />
does not change when the number <strong>of</strong> trajectories is further<br />
increased). A non-divergent 3-D volume-transport field is<br />
obtained by recording every instance <strong>of</strong> a <strong>trajectory</strong> passing a<br />
grid-box wall. Every <strong>trajectory</strong> entering a grid-box also exits, and<br />
hence this field exactly satisfies<br />
T x i,j,k,n<br />
T x i 1,j,k,n þTy T y i,j,k,n i,j 1,k,n þTz i,j,k,n<br />
T z i,j,k 1,n ¼ 0,<br />
are the <strong>trajectory</strong>-derived volume<br />
transports in the zonal (i), meridional (j), and vertical (k) directions,<br />
respectively. A meridional integration yields the <strong>Lagrangian</strong><br />
where the ‘‘vertical’’ index k can correspond to either the depth or<br />
the density.<br />
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ocean. Journal <strong>of</strong> Physical Oceanography 24, 1281–1293.<br />
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