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Article Analysis of the Water Level Variation in the Polish Part of the Vistula Lagoon (Baltic Sea) and Estimation of Water Inﬂow and Outﬂow Transport through the Strait of Baltiysk in the Years 2008–2017

Michał Szydłowski * , Wojciech Artichowicz and Piotr Zima

Faculty of Civil and Environmental Engineering, Gda´nskUniversity of Technology, Narutowicza 11/12, 80-233 Gda´nsk,Poland; [email protected] (W.A.); [email protected] (P.Z.) * Correspondence: [email protected]; Tel.: +48-58-347-1809

Abstract: The Vistula Lagoon is located in both Poland and Russia along the southern coast of the Baltic Sea. It is connected to the Baltic Sea in the Russian part by the Strait of Baltiysk. The purpose of the paper is to identify the dominant factors underlying the water level variation mechanism at Tolkmicko in the Vistula Lagoon, revealed by a statistical analysis of the measured data and a discussion on the inﬂow and outﬂow transport variation through the strait, estimated by numerical modeling. Seawater transport is exceptionally valuable in terms of the hydrological water balance in

the lagoon. Historical research on the hydrology of the lagoon shows that the water exchange in the lagoon is quite complex due to the presence of several different sources of water balance, such as

Citation: Szydłowski, M.; seawater inflow, river inflow, groundwater inflow, precipitation, and evaporation. Unfortunately, Artichowicz, W.; Zima, P. Analysis of there are no current data on seawater inflow and outflow through the Strait of Baltiysk due to the the Water Level Variation in the lack of continuous flow measurements in the strait. A novelty of the current work is an in-depth Polish Part of the Vistula Lagoon statistical analysis of the water level variation in the Polish part of the lagoon over a long time period (Baltic Sea) and Estimation of Water and an estimation of water transport through the Strait of Baltiysk by use of a numerical model. The Inflow and Outflow Transport model reproduces well the water level variation responding to variations in the sea level outside through the Strait of Baltiysk in the the lagoon and the wind action over the lagoon. The years 2008–2017 were chosen as the analysis Years 2008–2017. Water 2021, 13, 1328. period. A two-dimensional free surface shallow water numerical model of the lagoon was adapted to https://doi.org/10.3390/w13101328 simulate the water level variation in view of the wind over the lagoon and the sea level variation at one open boundary. Finally, it was concluded that the water level variation on the Polish side of the Academic Editor: Majid Mohammadian Vistula Lagoon is dominated by two factors: the water level in the Gulf of Gdańskand the wind over the lagoon. The average annual marine water inflow into the Vistula Lagoon was estimated to be 3 Received: 17 March 2021 equal to 15.87 km . Accepted: 8 May 2021 Published: 11 May 2021 Keywords: Vistula Lagoon; Tolkmicko; water stage elevation (WSE); Strait of Baltiysk; seawater transport; mathematical modeling Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- iations. 1. Introduction Coastal lagoons are defined in [1] as inland and shallow water bodies with depths not exceeding a few meters and separated from the sea by a barrier (usually sand). They are connected to the sea by at least one restricted inlet. The orientation of a lagoon may vary, Copyright: © 2021 by the authors. but being parallel to the shore is the dominant feature. According to [2], coastal lagoons Licensee MDPI, Basel, Switzerland. occur on about 12% of the length of the world’s coastline. The formation of lagoons can This article is an open access article be caused by alluvial progradation, the transgression of barriers in response to a sea level distributed under the terms and rise or they can be formed behind storm-surge barriers. Some examples of this last type of conditions of the Creative Commons formation can be found on the southern shores of the Baltic Sea. One of them is the Vistula Attribution (CC BY) license (https:// Lagoon (Figure1) as well as the neighboring Curonian Lagoon [3]. creativecommons.org/licenses/by/ 4.0/).

Water 2021, 13, 1328. https://doi.org/10.3390/w13101328 https://www.mdpi.com/journal/water WaterWater2021 2021, 13, 13, 1328, x FOR PEER REVIEW 2 ofof 1515

FigureFigure 1.1.Location Location ofof thethe VistulaVistula LagoonLagoon [ 4[4].].

TheThe waterwater movementmovement inin thethe VistulaVistula LagoonLagoon isis relatedrelated to to variations variations in in the the sea sea level level in in thethe GulfGulf ofof GdańskandGdańsk and toto thethe windwind action action on on the the water water surface surface of of the the lagoon lagoon [ 5[5].]. When When thethe waterwater stagestage ofof thethe GulfGulf ofof GdańskrisesGdańsk rises overover thethe waterwater levellevel inin thethe lagoon,lagoon, aa waterwater currentcurrent occursoccurs directeddirected intointo thethe lagoon.lagoon. AA reductionreduction inin thethe seasea levellevel inin thethe BalticBaltic SeaSea toto belowbelow the the lagoon lagoon water water level level causes causes the the lagoon lagoon water water to flowto flow into into the the sea sea creating creating a current a cur- inrent the in strait the instrait the oppositein the opposite direction. direction. The long-term The long-term damming damming of water inof thewater southern in the partsouthern of the part lagoon of the can lagoon be a cause can be of a flood cause risk of forflood the risk lowland for the areas lowland of Zuławy˙ areas of Elbl Żu ˛askieławy (FigureElbląskie1)[ (Figure4]. The 1) mechanism [4]. The mechanism underlying underlying the water level the water variation level in variation the Polish in part the ofPolish the Vistulapart of Lagoonthe Vistula is considered Lagoon is inconsidered this paper. in Itthis was paper. estimated It was in estimated the past that in the the past mean that total the 3 volumemean total of seawater volume flowingof seawater into theflowing lagoon into reaches the lagoon 15–18 kmreachesper year15–18 [6 km,7].3 Unfortunately, per year [6,7]. thereUnfortunately, are no full, there continuous, are no andfull, current continuous, data (measurements) and current data on (measurements) the seawater transport on the betweenseawater the transport Gulf of Gdańskandbetween the theGulf Vistula of Gda Lagoon.ńsk and Recently, the Vistula Russian Lagoon. researchers Recently, started Rus- tosian measure researchers the flow started through to measure the strait, the flow but sothrough far only the twostrait, series but so of far measurements only two series of approximatelyof measurements one of month approximately are available one [8 ].month The annual are available volume [8]. of marineThe annual water volume inflow isof usuallymarine citedwater and inflow estimated is usually by researchers cited and estimated like [4,5,9– by12] researchers using only datalike [4,5,9–12] from historical using sourcesonly data such from as [6historical,13]. This sources identified such knowledge as [6,13]. gapThis in identified the water knowledge balance of thegap Vistula in the Lagoonwater balance is also investigatedof the Vistula in Lagoon this paper. is also investigated in this paper. TheThe mainmain objectivesobjectives beingbeing alsoalso partsparts ofof thethe presentpresent work work are: are: • • aa statisticalstatistical analysis of of the the water water level level variation variation at atTolkmicko Tolkmicko station station to identify to identify the thefactors factors of water of water damming damming in the in thesouthern southern part part (Polish) (Polish) of the of theVistula Vistula Lagoon Lagoon (the (themeasurements measurements of the of water the water stages stages in the in lagoon the lagoon and in and the inGulf the of Gulf Gda ofńsk, Gdańsk,as as well as well as wind observations at Nowa Pasł˛ekameteorological station were used for wind observations at Nowa Pasłęka meteorological station were used for this re- this research), search), • a validation of the shallow water equations model [4] adopted to simulate the hydro- • a validation of the shallow water equations model [4] adopted to simulate the hy- dynamics of the lagoon over a long period of time (the water stage field measurements drodynamics of the lagoon over a long period of time (the water stage field meas- from Tolkmicko were used for comparison with the calculations), urements from Tolkmicko were used for comparison with the calculations), • a computational estimation of the marine water transport between the Baltic Sea • a computational estimation of the marine water transport between the Baltic Sea and and the Vistula Lagoon (a numerical simulation of the hydrodynamics of the Vistula the Vistula Lagoon (a numerical simulation of the hydrodynamics of the Vistula Lagoon was carried out for the period 2008–2017). Lagoon was carried out for the period 2008–2017). To the best of the authors’ knowledge, such a long-term study (10-year period) on To the best of the authors’ knowledge, such a long-term study (10-year period) on the hydrodynamics of the Vistula Lagoon has not been conducted so far. Moreover, the the hydrodynamics of the Vistula Lagoon has not been conducted so far. Moreover, the results of the work improve the general knowledge of the hydrological characteristics of theresults lagoon. of the It work should improve be noted the that general the waterknowle flowdge rateof the in hydrological the Strait of Baltiyskcharacteristics reaches of eventhe lagoon. 10,000 mIt 3shoulds−1 [6, 7be]. Thenoted total that inflow the water from flow all the rate rivers in the entering Strait of into Baltiysk the lagoon reaches is 3 −1 abouteven 10,000 180 m 3ms−s1 [ 6[6,7].,7], yet The this total was inflow not investigated from all the in rivers this caseentering study in dueto the to lagoon the lack is about 180 m3s−1 [6,7], yet this was not investigated in this case study due to the lack of

Water 2021, 13, 1328 3 of 15

of continuous measurements. However, in order to make a full hydrological balance of the water exchange in the lagoon, each of the hydrological elements such as river inﬂow, ground inﬂow, precipitation and evaporation should be considered.

2. Materials and Methods 2.1. Study Area The Vistula Lagoon [6,7,10,14] is a classic example of a coastal lagoon in the southern Baltic Sea (Figure1). The length of the lagoon is 90.7 km and its width varies from almost 6 km to 13 km. The mean width is 8.9 km. The lagoon is a very shallow basin with a mean depth of only 2.75 m. It is separated from the Gulf of Gdańskby the Vistula Spit of a length of about 65 km. The only connection between the Vistula Lagoon and the Baltic Sea is through the Strait of Baltiysk, which is 2 km long, 440 m wide and approximately 8.8 m deep. The total area of the lagoon measures 838 km2, of which 472.5 km2 belongs to Russia, and the shoreline is 270 km long [5–7,10]. Historically, the Vistula Lagoon was formed as part of the estuary of the Vistula River [10]. The regulation and changing of the course of the Vistula River at the beginning of the 20th century, due to frequent flooding, considerably modified the hydrological regime of the lagoon. Consequently, the annual inflow of the Vistula waters into the lagoon decreased from about 8–9 km3 to 0.7 km3. Since that time, the hydrological and sedimentation regimes of the lagoon have changed. The lagoon has evolved from a freshwater plain estuary to an estuarine lagoon significantly influenced by the Baltic Sea [10]. The largest rivers flowing into the Vistula Lagoon are the Pregoła (providing 62% of the freshwater) and other small rivers in the Russian part, and the Pasł˛eka,the Elbl ˛ag,the Nogat and the Szkarpawa on the Polish side. The inflow of fresh water from the Vistula to the lagoon, as a result of the functioning of locks on the Nogat and the Szkarpawa, is very limited. The average total annual river water inflow to the lagoon is about 180 m3s−1 [6,7]. Some correction of this component of the lagoon water budget was made in [11]. Nowadays [6,11], the hydrology of the Vistula Lagoon is controlled by marine water inflow (18.13 km3 per year) and freshwater gain, which consists of catchment runoff (4.97 km3 per year), precipitation (0.55 km3 per year), and evaporation (a loss of 0.53 km3 per year). The total outflow from the Vistula Lagoon to the sea is estimated to be equal to 23.69 km3 per year. The water balance is complemented by the groundwater flow. The hydrodynamics of the Vistula Lagoon are usually influenced by changes in the sea level in the Gulf of Gdańskand by wind action on the water surface of the lagoon [6,7]. Inflow to the lagoon takes place when the water level on the open sea side of the strait is higher than on the lagoon side, and vice versa for the outflow. The occurring difference in the water levels (the water head) between the sea and the lagoon causes a strong water current in the strait. Except for the water transport through the Strait of Baltiysk, the hydrodynamic conditions of the lagoon depend on wind action. In the region of the Vistula Lagoon, SW winds with a velocity from 4 to 6 ms−1 prevail [3,6,7]. These winds cause a rise in the water level in the Gulf of Gdańskand in the Vistula Lagoon of even 0.8 m above the mean sea level. However, NE winds in particular cause a dangerous water level rise in the southern part of the Vistula Lagoon. For long periods of strong NE winds, a rise in the water level can be observed exceeding +1.0 m and in extreme conditions reaching +1.5 m above the mean sea level [4].

2.2. Hydro-Meteorological and Bathymetric Data In order to analyze the mechanism of water damming on the Polish side of the Vistula Lagoon and model the hydrodynamics of the Vistula Lagoon, as well as to ﬁnally estimate the total water transport [15] between the lagoon and the Baltic Sea, hydro-meteorological data are necessary. It was assumed that the analysis and numerical simulation of the water movement in the Vistula Lagoon and the water discharges in the Strait of Baltiysk would be analyzed in one long period from 2008 to 2017. Water 2021, 13, x FOR PEER REVIEW 4 of 15

Water 2021, 13, 1328 simulation of the water movement in the Vistula Lagoon and the water discharges 4in of the 15 Strait of Baltiysk would be analyzed in one long period from 2008 to 2017. The raw hydro-meteorological data contain the water level measurement time series at TolkmickoThe raw hydro-meteorological and Hel stations in datawhich contain the measurement the water level frequency measurement was equal time series to 10 atminutes. Tolkmicko Time and intervals Hel stations between in which 13% of the the measurement measurements frequency at Tolkmicko was equal exceed to 10 the min. 10 Timemin intervalsinterval due between to missing 13% of observations. the measurements Such ata situation Tolkmicko does exceed not theoccur 10 minin the interval meas- dueurements to missing taken observations. at Hel. The granularity Such a situation of the doeswind not speed occur and in direction the measurements measurements taken is at1 Hel.hour. The There granularity are no missing of the windobservations speed and in the direction wind measurements data is 1 h. set. There The aresta- notistical missing analysis observations of the data in the was wind performed measurements for the data set set. integrating The statistical the analysisraw water of theele- datavation was and performed wind data. for the data set integrating the raw water elevation and wind data. AsAs the the input input data data for for the the water water flow flow simulation, simulation, the the daily daily (24-h) (24-hour) averages averages of the waterof the levelwater measurements level measurements in the Gulf in the of GdańskatGulf of Gda Helńsk station at Hel were station used were along used with along the observed with the (oneobserved per hour) (one windper hour) direction wind and direction speed overand thespeed lagoon over atthe Nowa lagoon Pasł˛ekastation at Nowa Pas (Figurełęka sta- 1tion). The (Figure investigation 1). The investigation of the variability of the of thevariab waterility levels of the (water water stagelevels elevations, (water stage WSE eleva- (m a.s.l.))tions, inWSE the (m lagoon a.s.l.)) and in thethe hydrodynamic lagoon and the model hydrodynamic verification model were carriedverification out usingwere thecar- dailyried out averaged using waterthe daily level averaged measurements water atlevel Tolkmicko measurements station. at All Tolkmicko these measurements station. All arethese public measurements and are published are public online and are by publis the Polishhed online Institute by ofthe Meteorology Polish Institute and of Water Mete- Managementorology and Water [16]. TheManage examplement data[16]. forThe the example year 2008 data are for presentedthe year 2008 in the are Resultspresented and in Discussionthe Results section. and Discussion The full setsection. of measurements The full set of is measurements available in [17 is]. available in [17]. ArchivalArchival bathymetry bathymetry data data [4 ,[4,7]7] were were also also used used to buildto build the digitalthe digital model model of the of lagoon the la- bottomgoon bottom (Figure (Figure2). 2).

FigureFigure 2. 2.Simpliﬁed Simplified shape shape and and bathymetry bathymetry (m (m a.s.l.) a.s.l.) of of the the Vistula Vistula Lagoon Lagoon (in (in meters) meters) [4 [4]:]: A—Strait ofA—Strait Baltiysk, of B—Nowakowo, Baltiysk, B—Nowakowo, C—Tolkmicko, C—Tolkmicko, D—Kaliningrad, D—Kaliningrad, E—Nowy Swiat.´ E—Nowy Świat.

2.3.2.3. StatisticalStatistical MethodMethod TheThe analysis analysis was was carried carried out out using using the exploratory the exploratory data analysis data analysis approach. approach. The mutual The dependencemutual dependence of the considered of the consider eventsed was events tested was using tested the χusing2-independence the χ2-independence test [18]. The test equality[18]. The of equality the empirical of the empirical probability probability density functionsdensity functions of water of levels water waslevels tested was tested with thewith Kolmogorov–Smirnov the Kolmogorov–Smirnov test [ 19test]. [19]. The timeThe time series series comparison comparison was was performed performed using us- aing correlation a correlation analysis analysis [20]. [20]. The The statistical statistical analysis analysis was was performed performed using using the the Anaconda Anaconda distributiondistribution of of Python Python 3.8 3.8 [ 21[21,22].,22].

2.4.2.4. MathematicalMathematical ModelModel ManyMany typestypes ofof mathematicalmathematical modelsmodels ofof thethe VistulaVistula LagoonLagoon havehave beenbeen developeddeveloped over the years, including two-dimensional (2D) hydrodynamic models [4,13,14,23–26], over the years, including two-dimensional (2D) hydrodynamic models [4,13,14,23–26], two-dimensional models composed of hydrodynamics, water quality, and eutrophication two-dimensional models composed of hydrodynamics, water quality, and eutrophica- modules [27,28], and the more recently developed three-dimensional (3D) models used for tion modules [27,28], and the more recently developed three-dimensional (3D) models sediment transport and migration [9,12,29,30]. used for sediment transport and migration [9,12,29,30]. The problem of how complex a model is needed to mathematically describe the The problem of how complex a model is needed to mathematically describe the hy- hydrodynamics of the Vistula Lagoon and calculate the water transport through the Strait drodynamics of the Vistula Lagoon and calculate the water transport through the Strait of Baltiysk depends on flow characteristics and the aim of the numerical simulation. of Baltiysk depends on flow characteristics and the aim of the numerical simulation. Simplified 2D barotropic models can be applied in cases of horizontal water flow with Simplified 2D barotropic models can be applied in cases of horizontal water flow with small depth and limited vertical velocity. It seems that in general this shallow lagoon meets small depth and limited vertical velocity. It seems that in general this shallow lagoon these conditions [7], except for the Strait of Baltiysk area, where two-layer and two-stream currentsmeets these occur conditions [30]. Łazarenko [7], except and Majewski for the St [6rait] reported of Baltiysk that 74.9%area, where of inflow two-layer or outflow and aretwo-stream uniform currents,currents occur 11.7% [30]. two-layer Łazarenko currents and (inflow Majewski in the [6] bottomreported layer that and 74.9% outflow of in- at the surface), and 13.4% two-stream currents. In such a case, if a strong variation in the flow parameters is expected, fully 3D hydrodynamic models should be used. However, the present study uses a 2D barotropic model of the entire lagoon and two-layer (baroclinic) Water 2021, 13, 1328 5 of 15

water transport events cannot be resolved. This simplification means that the calculation of the water transport through the strait is treated as an estimation. Despite the significant number of applications of numerical models to the Vistula Lagoon, a multi-year water level variation analysis and a simulation of seawater transport from the Gulf of Gdańskto the Vistula Lagoon have not been carried out so far. A one-year (1994) period was analyzed in [27]. In the present study, the 2D SWE model is used to simulate the hydrodynamics of the lagoon. The classic set of shallow water equations can be written as [31]:

2 2 2 ∂U ∂U ∂U ∂h gn ∂ U ∂ U Tx + U + V + g + U|W| − νo + − = 0 (1) ∂t ∂x ∂y ∂x H4/3 ∂x2 ∂y2 H

2 2 2 ∂V ∂V ∂V ∂h gn ∂ V ∂ V Ty + U + V + g + V|W| − νo + − = 0 (2) ∂t ∂x ∂y ∂y H4/3 ∂x2 ∂y2 H ∂h ∂(UH) ∂(VH) + + = 0 (3) ∂t ∂x ∂y where: x,y—spatial coordinates; t—time; U,V—depth-averaged components of velocity in x- and y-direction, respectively; |W| = (U2 + V2)1/2—modulus of the velocity vector; h—water elevation above some plane of reference; H—water depth; g—acceleration due to gravity; n—Manning friction coefficient; ν0—coefficient of horizontal turbulent viscosity; Tx—wind stresses in x-direction; Ty—wind stresses in y-direction. The splitting technique was used to decompose the 2D mathematical model described by Equations (1)–(3) [4] and then 1D equations were integrated using the finite element method proposed in [32]. The model was verified and validated by its application to simulate the Vistula Lagoon hydrodynamics during short-time surges driven by the wind [4]. Herein, the model was applied to simulate a 10-year-long unsteady flow in the Vistula Lagoon and could be validated by a comparison of the calculated water levels to the measured ones at Tolkmicko station. In order to simulate the lagoon hydrodynamics, a structured square numerical mesh of grid dimensions ∆x = ∆y = 400 m was applied. The contour of the calculation area is shown in Figure2. The initial condition was determined by the hydrostatic state with the horizontal water table corresponding to the initial water level in the Gulf of Gdańsk and the Strait of Baltiysk. The boundary conditions were chosen in relation to the type of boundary. At the open boundary, where the connection of the Vistula Lagoon with the Gulf of Gdańskexists through the Strait of Baltiysk (composed of 3 computational cells, see Figures1 and2, point A), the water surface elevation was forced at the boundary grid cell. Such a condition represents the variation of the sea level forming the water flow through the strait. The water discharge in a normal direction to the cross-section of the strait was always calculated based on the actual value of normal velocity and the water depth. The other lagoon boundaries are located along the shore line and they were treated as closed boundaries with free-slip conditions. The inflow of the rivers entering the lagoon was neglected due to the lack of continuous flow rate measurements, as well as the groundwater flow, precipitation, and evaporation. The roughness of the lagoon bottom was assumed uniform with the Manning coefficient n = 0.024 m–1/3 s [7]. The simulation was carried out for real (observed) data for the time period from 2008 to 2017 with a 5-min time step. The example water level variation measured in the Gulf of Gdańsk(Hel station) and the wind observed at the Nowa Pasł˛ekameteorological station in 2008 (January) are shown in Figure3. A uniform wind field over the whole surface of the Vistula Lagoon was assumed. The numerical modeling of the hydrodynamics provided information about the water surface variation in the Vistula Lagoon, and the unsteady flow (discharge) through the Strait of Baltiysk. The former was used to validate the simulations and the latter allowed the water transport from the sea to the Vistula Lagoon to be calculated. Water 2021, 13, x FOR PEER REVIEW 6 of 15

logical station in 2008 (January) are shown in Figure 3. A uniform wind field over the whole surface of the Vistula Lagoon was assumed. The numerical modeling of the hydrodynamics provided information about the water surface variation in the Vistula La- goon, and the unsteady flow (discharge) through the Strait of Baltiysk. The former was Water 2021, 13, 1328 6 of 15 used to validate the simulations and the latter allowed the water transport from the sea to the Vistula Lagoon to be calculated.

FigureFigure 3. 3. WaterWater level level (WSE) (WSE) variation variation at at Hel Hel and and Tolkmi Tolkmickocko stations stations and and wind wind speed speed and and direction direction observedobserved at at Nowa Nowa Pas Pasł˛ekainłęka in January January of of 2008 2008 (raw (raw data; data; me measurementasurement frequency of WSEWSE isis 1010 (min), (min),wind datawind 1 data (h)). 1 (h)).

3.3. Results Results and and Discussion Discussion 3.1.3.1. Variability Variability of of the the Vistula Vistula Lagoon Lagoon Water Water Levels Levels (WSE) (WSE) at at Tolkmicko Tolkmicko InIn order order to investigateinvestigate thethe mechanism mechanism underlying underlying the the WSE WSE variation variation in thein the Polish Polish part partof the of the Vistula Vistula Lagoon, Lagoon, the the measurements measurements of theof the water water level level at at Tolkmicko Tolkmicko station station and and at atHel Hel station, station, located located in thein the lagoon lagoon and and in the in Gulfthe Gulf of Gda´nsk,respectively, of Gdańsk, respectively, were considered.were considered.Moreover, Moreover, to assess to the assess impact the ofimpact the wind of the on wind the lagoonon the surges,lagoon surges, the wind the parameters wind parameters(wind speed (wind and speed direction) and directio measuredn) measured at the Nowa at the Pasł˛ekastation Nowa Pasłęka were station analyzed. were ana- The lyzed.example The of example the collected of the time collected series datatime forseries January data offor 2008 January is displayed of 2008 inis Figuredisplayed3. in FigureThe 3. empirical probability density functions (PDFs) of the observed water levels at TolkmickoThe empirical and Hel probability are presented density in the functions forms of(PDFs) a box of plot the and observed histogram water in levels Figure at4. TolkmickoThe summarized and Hel observations are presented are in given the forms in Table of 1a. box From plot the and table histogram it can be in stated Figure that 4. on average the water level at Tolkmicko is lower than at Hel; however, the dispersion of The summarized observations are given in Table 1. From the table it can be stated that on the water levels at Tolkmicko is greater than at Hel. The minimal observed values and average the water level at Tolkmicko is lower than at Hel; however, the dispersion of the Water 2021, 13, x FOR PEER REVIEWquartiles are of similar values; however, the maxima differ: the maximum observed7 water of 15 water levels at Tolkmicko is greater than at Hel. The minimal observed values and quar- level at Tolkmicko station is signiﬁcantly greater than at Hel. tiles are of similar values; however, the maxima differ: the maximum observed water level at Tolkmicko station is significantly greater than at Hel.

Table 1. Basic statistical characteristics of the WSE (m a.s.l.) at Tolkmicko and Hel stations (raw data).

Parameter Tolkmicko Hel mean 0.088 0.095 std. dev. 0.202 0.193 minimum −0.760 −0.800 q0.25 −0.040 −0.030 median 0.090 0.090 q0.75 0.210 0.210 maximum 1.460 1.150

Figure 4. WSE (raw data) data) comparison comparison at at Tolkmicko Tolkmicko (blu (blue)e) and and Hel Hel (red) (red) stations. stations. Histograms Histograms and and box plots.

The PDFs of the observed water levels at Tolkmicko and Hel are of a similar type: unimodal bell shaped, slightly positively skewed with the same medians and slightly different means. The general populations of water levels at Hel and Tolkmicko differ significantly (Kolmogorov–Smirnov test p-value = 0.00495). Although both distributions can be considered as normal-type, they are not normal in terms of the goodness of fit test outcomes when compared with the estimated normal distributions (using the maximum likelihood method). The main factor influencing the water level variations in the southern part of the Vistula Lagoon (illustrated by Tolkmicko) is the water level variation in the coastal waters of the South-Eastern Baltic or the Gulf of Gdańsk (illustrated by Hel) (Figure 3). The statistical dependence between those variables was confirmed by the χ2-independence test (p-value ≈ 0). The lag time between events at Hel and Tolkmicko was determined by correlation coefficient maximization and is equal to 6 h 50 min with the Pearson correlation coefficient R = 0.91. However, some significant discrepancies between the water levels at the considered stations can be noticed. One such discrepancy can be observed, for example in Figure 3, between 25 and 29 January. The reason for such occurrences is the moderate or strong wind originating (on average) from one direction for about one or more days. A long-lasting wind originating from a westerly direction (W, WNW, WSW) causes the water levels at Tolkmicko to decrease. Such a situation is depicted in Figure 5. If the wind originates from a northerly direction (N, NNE, NE), the water level at Tolkmicko is raised with respect to the water levels at Hel (Figure 6).

Figure 5. Water level (WSE) variation at Tolkmicko and Hel, wind speed and direction—the case (1–8 August 2008) in which wind originates from a westerly direction for a long period of time causes the water level at Tolkmicko to decrease in comparison to the water level at Hel (raw data; measurement frequency of WSE is 10 (min), wind data 1 (h)).

Water 2021, 13, x FOR PEER REVIEW 7 of 15

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Table 1. Basic statistical characteristics of the WSE (m a.s.l.) at Tolkmicko and Hel stations (raw data).

Parameter Tolkmicko Hel mean 0.088 0.095 std. dev. 0.202 0.193 minimum −0.760 −0.800 − − q0.25 0.040 0.030 median 0.090 0.090 Figure 4. WSEq (raw0.75 data) comparison at Tolkmicko0.210 (blue) and Hel (red) stations. 0.210 Histograms and box plots. maximum 1.460 1.150

The PDFs of the observed water levels at Tolkmicko and Hel are of a similar type: The PDFs of the observed water levels at Tolkmicko and Hel are of a similar type: unimodal bell shaped, slightly positively skewed with the same medians and slightly unimodal bell shaped, slightly positively skewed with the same medians and slightly different means. The general populations of water levels at Hel and Tolkmicko differ different means. The general populations of water levels at Hel and Tolkmicko differ significantly (Kolmogorov–Smirnov test p-value = 0.00495). Although both distributions significantly (Kolmogorov–Smirnov test p-value = 0.00495). Although both distributions can be considered as normal-type, they are not normal in terms of the goodness of fit test can be considered as normal-type, they are not normal in terms of the goodness of fit test outcomes when compared with the estimated normal distributions (using the maximum outcomes when compared with the estimated normal distributions (using the maximum likelihood method). likelihood method). The main factor influencing the water level variations in the southern part of the The main factor influencing the water level variations in the southern part of the VistulaVistula LagoonLagoon (illustrated (illustrated by by Tolkmicko) Tolkmicko) is is the the water water level level variation variation in thein the coastal coastal waters wa- ofters the of South-Easternthe South-Eastern Baltic Baltic or theor the Gulf Gulf of Gdańsk(illustratedof Gdańsk (illustrated by by Hel) Hel) (Figure (Figure3). 3). The The χ2 statisticalstatistical dependencedependence betweenbetween thosethose variablesvariables waswas confirmedconfirmed byby the theχ 2-independence-independence testtest ((p-valuep-value ≈ ≈ 0). The lag time between events at Hel andand TolkmickoTolkmicko waswas determineddetermined byby correlationcorrelation coefficient coefficient maximization maximization and and is is equal equal to to 6 h6 50h 50 min min with with the the Pearson Pearson correlation correla- coefficienttion coefficientR = 0.91. R = 0.91. However,However, somesome significantsignificant discrepanciesdiscrepancies betweenbetween thethe waterwater levelslevels atat thethe consideredconsidered stationsstations cancan bebe noticed.noticed. OneOne suchsuch discrepancydiscrepancy cancan bebe observed,observed, forfor exampleexample inin FigureFigure3 3,, betweenbetween 2525 and and 29 29 January. January. The The reason reason for for such su occurrencesch occurrences is the is moderate the moderate or strong or strong wind originatingwind originating (on average) (on average) from one from direction one fordirection about onefor orabout more one days. or Amore long-lasting days. A windlong-lasting originating wind from originating a westerly from direction a westerly (W, WNW,direction WSW) (W, causesWNW, the WSW) water causes levels the at Tolkmickowater levels to at decrease. Tolkmicko Such to a decrease. situation Such is depicted a situation in Figure is depicted5. If the in wind Figure originates 5. If the fromwind aoriginates northerly from direction a northerly (N, NNE, direction NE), the (N, water NNE, level NE), at the Tolkmicko water level is raisedat Tolkmicko with respect is raised to thewith water respect levels to the at Hel water (Figure levels6). at Hel (Figure 6).

FigureFigure 5.5. Water level (WSE) variation at at Tolkmicko Tolkmicko and and Hel, Hel, wind wind speed speed and and direction—the direction—the case case ((1–81–8 AugustAugust 20082008)) inin whichwhich wind wind or originatesiginates from from a awesterly westerly direction direction for for a long a long period period of time of time causescauses thethe waterwater levellevel at Tolkmicko to decrease in comparison to to the the water water level level at at Hel Hel (raw (raw data; data; measurementmeasurement frequencyfrequency ofof WSEWSE isis 1010 (min),(min), windwind datadata 1 1 (h)). (h)).

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FigureFigure 6.6. Water level (WSE) (WSE) variation variation at at Tolkmicko Tolkmicko and and Hel, Hel, wind wind speed speed and and direction—the direction—the case case (9–16(9–16(9–16 OctoberOctober 2009) 2009) inin whichwhich windwind wind originatesoriginates originates fromfrom from aa a northerly northerly direction direction for for a along long period period of of time time causes the water level at Tolkmicko to increase in comparison to the water level at Hel (raw data; causes the water level level at at Tolkmicko Tolkmicko to to increase increase in in comparison comparison to to the the water water level level at at Hel Hel (raw (raw data; data; measurement frequency of WSE is 10 (min), wind data 1 (h)). measurement frequency ofof WSEWSE isis 1010 (min),(min), windwind datadata 11 (h)).(h)).

ToTo confirmconfirm thethe abovementionedabovementioned patternpattern inin windwind durationduration andand directiondirection influencinginfluencing thethe increaseincrease andand decreasedecrease ofof waterwater levelslevels atat Tolkmicko,Tolkmicko, anan analysisanalysis ofof suchsuch windwind eventsevents waswas undertaken.undertaken. AllAll noticeablenoticeable eventsevents inin whichwhich thethe centralcentral tendencytendency ofof thethe waterwater levellevel inin thethe southernsouthern partpart ofof thethe VistulaVistula LagoonLagoon (illustrated(illustrated byby Tolkmicko)Tolkmicko) differeddiffered significantlysignificantly fromfrom thethe centralcentral tendencytendency inin thethe South-EasternSouth-Eastern BalticBaltic (illustrated(illustrated atat Hel)Hel) werewere chosenchosen andand analyzed.analyzed. The analysis was performed with respect to thethe speedspeed andand thethe durationduration ofof thethe windwind originatingoriginating onon averageaverage fromfrom oneone dominatingdominating direction.direction. For eacheach such eventevent thethe differencedifference betweenbetween the the mean mean water water levels levels at Tolkmickoat Tolkmicko and Hel,and theHel, maximum the maximum wind speedwind andspeed the and dominating the dominating originof origin the wind of the were wind noticed. were noticed. The scatter The plot scatter displayed plot displayed in Figure in7 presentsFigure 7 thepresents mean the difference mean difference between the between water levelsthe water at Tolkmicko levels at Tolkmicko and Hel versus and Hel the windversus event the wind duration. event Additionally, duration. Additionally, the plot displays the plot the displays maximum the observed maximum wind observed speed usingwind speed point sizesusing and point the sizes dominating and the winddominating direction wind with dire labels.ction with labels.

FigureFigure 7.7. Difference between thethe average waterwater levelslevels (WSE)(WSE) atat TolkmickoTolkmicko andand HelHel versusversus thethe windwind eventevent durationduration time.time. TheThe labellabel denotesdenotes the the dominating dominatingting wind windwind direction, direction,direction, the ththe size size denotes denotes the the observed ob- maximumserved maximum wind speed. wind speed.

ItIt cancan bebe easilyeasily noticednoticed thatthat ifif thethe windwind originatesoriginatesiginates fromfromfrom oneoneone directiondirectiondirection forforfor 202020 ororor moremoremore hours,hours, aa signiﬁcantsignificant differencedifference inin waterwater levelslevels betweenbetween TolkmickoTolkmicko andand HelHel appears.appears. IfIf thethe dominatingdominating windwind direction direction is ais westerly a westerly one (W, oneone WNW, (W,(W, WNW,WNW, WSW), theWSW),WSW), water thethe level waterwater at Tolkmicko levellevel atat

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Tolkmicko decreases. If the wind originates from a northerly direction (N, NNE, NE), the water level at Tolkmicko increases with respect to the water level at Hel. Moreover, it can decreases.Tolkmicko Ifdecreases. the wind If originates the wind from originates a northerly from a direction northerly (N, direction NNE, NE), (N, theNNE, water NE), level the be noticed that a long-lasting wind originating from one dominating direction influences atwater Tolkmicko level at increasesTolkmicko with increases respect with to the respect water to level the water at Hel. level Moreover, at Hel. itMoreover, can be noticed it can only the lagoon water level variation at Tolkmicko (Figure 8a) but a similar pattern can- thatbe noticed a long-lasting that a long-lasting wind originating wind originating from one dominatingfrom one dominating direction direction inﬂuences influences only the not be clearly recognized in the Gulf of Gdańsk at Hel station (Figure 8b). lagoononly the water lagoon level water variation level atvariation Tolkmicko at Tolkmicko (Figure8a) but(Figure a similar 8a) but pattern a similar cannot pattern be clearly can- recognizednot be clearly in therecognized Gulf of Gda´nskatin the Gulf Hel of Gda stationńsk (Figureat Hel station8b). (Figure 8b).

Figure 8. Averaged water levels (WSE) at (a) Tolkmicko, (b) Hel versus wind event duration time. Label denotes the dominating wind direction, size denotes the observed maximum wind speed. Figure 8. Averaged water levels (WSE) at at ( (aa)) Tolkmicko, Tolkmicko, ( (bb)) Hel Hel versus versus wind wind event event duration duration time. time. LabelThe denotes conditions the dominating caused windby winds direction,direction, originating sizesize denotesdenotes from thethe a observedobserved single direction maximummaximum for windwind a long speed.speed. time are the main factors for the occurrence of events at Tolkmicko that can be considered ex- The conditionsconditions causedcaused by by winds winds originating originating from from a singlea single direction direction for for a long a long time time are treme. theare mainthe main factors factors for thefor occurrencethe occurrence of events of events at Tolkmicko at Tolkmicko that canthat be can considered be considered extreme. ex- The wind measurements performed in Nowa Pasłęka are presented in the form of a treme.The wind measurements performed in Nowa Pasł˛ekaare presented in the form of a histogram (Figure 9a) and a wind rose (Figure 9b). The maximum likelihood estimation histogramThe wind (Figure measurements9a) and a wind performed rose (Figure in Nowa9b). The Pas maximumłęka are presented likelihood in estimation the form of of a of the parameters of the Weibull distribution, thehistogram parameters (Figure of the 9a) Weibull and a wind distribution, rose (Figure 9b). The maximum likelihood estimation of the parameters of the Weibull distribution,𝑘 𝑥 (/) 𝑓(𝑥) = k−1 𝑒 , (4) k x −(x/λ)k f (x) = 𝜆 𝜆 e 𝑘 𝑥 , (4) 𝑓(𝑥) =λ λ 𝑒(/) , (4) was performed for the wind speed observations,𝜆 𝜆 resulting in 𝑘 = 1.722 and 𝜆 = 5.802. Thewas performedprobability fordensity the wind function speed of observations, the Weibull resulting distribution in k = is1.722 presentedand λ in= 5.802Figure. The 9a was performed for the wind speed observations, resulting in 𝑘 = 1.722 and 𝜆 = 5.802. withprobability a line. densityBased on function the plot, of it the can Weibull be stated distribution that the wind is presented speed at in Nowa Figure 9Pasa withłęka ais The probability density function of the Weibull distribution is presented in Figure 9a distributedline. Based onaccording the plot, to it canthe beWeibull stated thatdistribution; the wind however, speed at Nowathe null Pasł˛ekais hypothesis distributed of the with a line. Based on the plot, it can be stated that the wind speed at Nowa Pasłęka is Kolmogorov–Smirnovaccording to the Weibull goodness distribution; of fit however,test was rejected the null ( hypothesisp-value = 1.42 of the× 10 Kolmogorov–−6). Based on distributed according to the Weibull distribution; however,− the6 null hypothesis of the theSmirnov plot of goodness the wind of rose, ﬁt test it wascan rejectedbe stated (p that-value the = most 1.42 × frequent10 ). Based directions on the from plot which of the Kolmogorov–Smirnov goodness of fit test was rejected (p-value = 1.42 × 10−6). Based on thewind wind rose, originates it can be stated are SE, that then the SW most and frequent SSW. directionsThe most fromfrequent which wind the windspeed originates intervals the plot of the wind rose, it can be stated that the most frequent directions from− which1 are 0–7.1 SE, then m s SW−1 and and 7.1–14.1 SSW. The m s most−1. It can frequent be noticed wind that speed stronger intervals winds are 0–7.1(14.1–21.2 m s mand s−1) the wind originates−1 are SE, then SW and SSW. The most frequent wind−1 speed intervals more7.1–14.1 frequently m s . Itoriginate can be noticed from westerly that stronger directions winds (SW, (14.1–21.2 SSW, and m s W)) morethan from frequently other are 0–7.1 m s−1 and 7.1–14.1 m s−1. It can be noticed that stronger winds (14.1–21.2 m s−1) directions.originate from The westerlywind pattern directions resembles (SW, SSW, the andwind W) pattern than from observed other directions.in Baltiysk The by windChu- more frequently originate from westerly directions (SW, SSW, and W) than from other barenkopattern resembles et al. [3]. the wind pattern observed in Baltiysk by Chubarenko et al. [3]. directions. The wind pattern resembles the wind pattern observed in Baltiysk by Chu- barenko et al. [3].

−1 1 Figure 9. ((aa)) Wind Wind speed (m s s ) )histogram histogram and and estimated estimated Weibull Weibull distribution distribution probability probability density density functionfunction and ( b) wind rose at Nowa Pas Pasł˛ekameteorologicalłęka meteorological station (2008–2017). Figure 9. (a) Wind speed (m s−1) histogram and estimated Weibull distribution probability density function and (b) wind rose at Nowa Pasłęka meteorological station (2008–2017).

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3.2.3.2. Validation Validation of of the the Hydrodynamic Hydrodynamic Model Model for for the the 10-Year-Long 10-Year-Long SimulationSimulation Simulation PeriodPeriod Period TheThe validation validationvalidation of ofof the thethe model modelmodel was waswas preceded precededpreceded by using byby usingusing hydrodynamic hydrodynamichydrodynamic and meteorological andand meteoro-meteoro- conditionslogicallogical conditions conditions as observed. as as observed. observed. In order In In toorder order verify to to theveri veri results,fyfy the the results, results, a control a a control control point onpoint point the on Polishon the the Polish shorePolish ofshoreshore the of Vistulaof the the Vistula Vistula Lagoon, Lagoon, Lagoon, located located located near Tolkmicko near near To Tolkmickolkmicko (see Figures (see (see Figures 1Figures and2, point1 1 and and C)2, 2, waspointpoint chosen. C) C) was was Thechosen.chosen. example The The example (forexample 2008) (for (for observations 2008) 2008) observations observations and results an an ofdd results theresults numerical of of the the numerical numerical simulations simulations simulations of the water of of surfacethethe water water variation surface surface at variation variation this point at at are this this presented point point are are in pres pres Figureentedented 10 .in in It Figure canFigure be seen10. 10. It It that can can thebe be computationseen seen that that the the outcomescomputationcomputation can outcomes outcomes be considered can can be be satisfactory. considered considered sati Thesatisfactory.sfactory. full set ofThe The measurements full full set set of of measurements measurements and calculation and and resultscalculationcalculation used results results for the used used model for for validation the the model model is validation validation available inis is available [available17]. in in [17]. [17].

FigureFigure 10. 10. Observed Observed (black) (black) and and calculated calculated (blue) (blue) waterwate waterr level level (WSE) (WSE) variation variation at at Tolkmicko Tolkmicko inin in 2008.2008. 2008.

However,However, to toto assess assessassess in inin detail detaildetail the thethe compliance compliancomplian ofcece the ofof calculationsthethe calculationscalculations with with thewith measurements, thethe measure-measure- morements,ments, detailed more more detailed detailed description description description is necessary. is is necessar necessar Therey.y. There There are two are are state-of-the-arttwotwo state-of-the-art state-of-the-art approaches approaches approaches for theforfor the estimationthe estimation estimation of theof of the the model model model accuracy: accuracy: accuracy: a a correlationa co correlationrrelation analysis analysis analysis of of of thethe the predictionspredictions predictions versusversus versus observationsobservations and andand the thethe analysis analysis of of an an an error error error [33,34]. [33,34]. [33,34 ].The The The most most most widely widely widely used used used estimator estimator estimator for for the forthe theoveralloverall overall error error error of of the the of model themodel model is is the the is mean mean the mean square square square error error error(MSE). (MSE). (MSE). However, However, However, in in the the incase case the of of case phys- phys- of physicalicalical sciences, sciences, sciences, the the square thesquare square root root root of of MSE ofMSE MSE can can can be be beused, used, used, which which which provides provides provides better better better interpretability. interpretability. Therefore,Therefore, itit was was used used in in this this work work as as aa a syntheticsynthetic synthetic measuremeasure measure of of thethe the modelmodel model accuracy.accuracy. accuracy. InIn In thethe the consideredconsidered casecase case ofof of thethe the VistulaVistula Vistula Lagoon Lagoon Lagoon simulation, simu simulation,lation, the the the estimated estimated estimated linear linear linear regression regression regression model model model is y = 0.916x – 0.0443 (Figure 11, red line). The regression line suggests that on average the isis y y = = 0.916 0.916xx – – 0.0443 0.0443 (Figure (Figure 11, 11, red red line). line). The The regres regressionsion line line suggests suggests that that on on average average the the water level values are slightly underestimated. The Pearson correlation coefficient is equal waterwater levellevel valuesvalues areare slightlyslightly underestimatunderestimated.ed. TheThe PearsonPearson correlationcorrelation coefficientcoefficient isis to R = 0.892 and is significant (p-value ≈ 0); the determination coefficient R2 = 0.796. equalequal to to R R = = 0.892 0.892 and and is is significant significant ( p(p-value-value ≈ ≈ 0); 0); the the determination determination coefficient coefficient R R2 2= = 0.796. 0.796.

FigureFigure 11. 11. Observed Observed and and predicted predicted water water level level at at Tolkmicko Tolkmicko station.station. station.

TheThe rootroot meanmean squaresquare errorerror ofof thethethe observations observobservationsations andand predictionspredictionspredictions isis givengivengiven withwithwith the thethe followingfollowing expression: expression: s n 1 2 RMSE = ∑(xi − ci) (5) n i=1

Water 2021, 13, x FOR PEER REVIEW 11 of 15

Water 2021, 13, 1328 11 of 15 1 𝑅𝑀𝑆𝐸 = (𝑥 −𝑐) (5) 𝑛

where xi are the observed water levels and ci are the outcomes of the computation where xi are the observed water levels and ci are the outcomes of the computation (i = (i = 1,...,n) with n denoting the number of values. The value of RMSE is equal to 0.104 m. 1,...,n) with n denoting the number of values. The value of RMSE is equal to 0.104 m. The absolute error can be deﬁned as The absolute error can be defined as

εi = xi − ci (6) 𝜀 =𝑥 −𝑐 (6) − ForFor such aa deﬁneddefined error error the the maximum maximum negative negative value value (overestimation) (overestimation) is εmax is= 𝜀−0.667= + −m,0.667 whereas m, whereas the maximum the maximum positive positive error value error (underestimation)value (underestimation) is εmax is= 0.620𝜀 = m. 0.620 The m.mean The value mean of value the error of the expresses error expresses the bias ofthe the bias model of the and model is equal and to isε =equal0.049 tom, 𝜀̅ = whereas 0.049 m,the whereas standard the deviation standard of deviation the error of is the equal error to iss =equal 0.0914. to s The= 0.0914. histogram The histogram of the absolute of the absoluteerror is displayederror is displayed in Figure in 12 Figure. 12.

FigureFigure 12. 12. HistogramHistogram of of the the simula simulationtion absolute absolute error. error.

TheThe results results displayed displayed in in Figures Figures 1111 andand 1212 showshow that that the the calculated calculated elevations elevations of of the the waterwater table table are are usually usually slightly slightly underestimated underestimated in in relation relation to to the the measurements measurements (Table (Table 2).2 ). ForFor the the years years 2008–2017, 2008–2017, the the average average underestimation underestimation of of the the results results of of water water levels levels in in the the reservoirreservoir is is about about 0.05 0.05 m, m, which which is is confirmed confirmed by by the the determined bias bias value. The The reason reason forfor this this is is the the fact fact that that the the external external (catchment, (catchment, river, river, ground) ground) inflow inflow to the to lagoon the lagoon is not is includednot included in the in calculations. the calculations. This assumption This assumption ignores ignores the resistance the resistance of the oflagoon the lagoon to ac- ceptto accept seawater seawater when whenthe water the waterlevel in level the inlag theoon lagoon is partly is partly controlled controlled by intensive by intensive river dischargeriver discharge (during (during spring springand autumn and autumn maximums maximums of river of discharge). river discharge). It would It wouldbe neces- be sarynecessary to examine to examine seasonal seasonal variations variations of the of difference the difference between between the model the model calculated calculated and observedand observed water water levels levels in order in order to identify to identify the the influence influence of ofriver river discharge, discharge, but but this this is is currentlycurrently beyond beyond the the scope scope of of this this work. work.

TableTable 2. TheThe basic statistical characteristicscharacteristics of of the the observed observed daily daily averaged averaged WSE WSE (m (m a.s.l.) a.s.l.) at Tolkmickoat Tolkmickoand the simulation and the simulation outcome. outcome.

Parameter Observations ComputationsComputations Error mean 0.059 0.010 0.049 mean 0.059 0.010 0.049 std. dev. 0.198 0.203 0.092 std. dev. 0.198 0.203 0.092 minimum −0.580 −0.600 −0.667 minimum −0.580 −0.600 −0.667 q0.25 −0.060 −0.120 0.000 q −0.060 −0.120 0.000 median0.25 0.060 0.010 0.045 median 0.060 0.010 0.045 q0.75 0.179 0.130 0.096 q 0.179 0.130 0.096 maximum0.75 1.319 1.250 0.620 maximum 1.319 1.250 0.620 Despite the recurring, slight understatement of the calculations relative to the observations,Despite other the recurring,statistical measures, slight understatement such as the ofcorrelation the calculations coefficient, relative mean to theabsolute obser- error,vations, or root other mean statistical square measures, deviation, such testify as the to correlationthe good quality coefﬁcient, of the mean simulation absolute results. error, or root mean square deviation, testify to the good quality of the simulation results. The comparison between the observed data and the results of the computations obtained for the control point near Tolkmicko conﬁrms that the proposed mathematical model provides reliable simulation results of the hydrodynamics of the Vistula Lagoon. Water 2021, 13, x FOR PEER REVIEW 12 of 15

The comparison between the observed data and the results of the computations obtained Water 2021, 13, 1328 12 of 15 for the control point near Tolkmicko confirms that the proposed mathematical model provides reliable simulation results of the hydrodynamics of the Vistula Lagoon.

3.3. Water Water Transport throughthrough the StraitStrait ofof BaltiyskBaltiysk The application of the verified verified and validated 2D hydrodynamic model of the Vistula Lagoon enabled calculations of the flowflow through the Strait of Baltiysk. Calculations of the time-varying water discharge we werere carried out on the basis of the unsteady flowflow velocity and the waterwater depthdepth in in the the strait strait (in (in one one calculation calculation cell cell near near point point A, Figure A, Figure2). The 2). example The ex- ampleresults results of the calculatedof the calculated day averaged day averaged flow rates flow for rates 2008 for are 2008 presented are presented in Figure in 13 Figure. The 13.positive The positive and negative and negative discharge discharge values values depend depend on the on flow the directionflow direction in the in strait the strait and andrepresent represent the waterthe water outflow outflow and inflowand inflow to the to lagoon, the lagoon, respectively. respectively.

Figure 13. Calculated water flow ﬂow discharge throughthrough the Strait of Baltiysk in 2008.

In order to estimate the annual seawaterseawater transport between the Gulf of GdańskandGdańsk and the VistulaVistula Lagoon,Lagoon, the the flow flow discharges discharges were were integrated integrated in one-yearin one-year time time periods. periods. Inflows In- flowsand outflows and outflows through through the strait the were strait added were separately,added separately, which made which it possiblemade it topossible calculate to calculatethe accumulative the accumulative volume ofvolume water of entering water entering and leaving and leaving the lagoon the lagoon from 1 from January 1 Jan- to uary31 December to 31 December each year. each The year. water The accumulation water accumulation was calculated was duringcalculated a simulation during a withsimu- a lation5-min with time a step. 5-minute The calculated time step. results The calculated are collected results inTable are collected3. in Table 3.

Table 3. Calculated water transport throughthrough thethe StraitStrait ofof Baltiysk.Baltiysk.

Value/Year Value/Year2008 2009 2010 20082011 20092012 2010 2013 2011 20122014 20132015 2014 20152016 20162017 2017 AverageAverage Annual accumu- Annual accumulative inflow (km3) 17.78 15.40 15.76 16.23 16.35 15.42 14.31 16.50 15.18 15.73 15.87 lative inflow 17.78 15.40 15.76 16.23 16.35 15.42 14.31 16.50 15.18 15.73 15.87 Annual accumulative outflow (km3) 17.60 15.66 15.68 15.95 15.97 15.57 14.43 16.75 15.38 15.94 15.89 (km3) Annual accumulative outflow 17.60 15.66 15.68Data regarding15.95 the15.97 water exchange15.57 between14.43 the16.75 Vistula Lagoon15.38 and15.94 the Baltic Sea15.89are (km3) not readily available, which was described in detail in [11]. Early research has shown that the total volume of seawater inflow into the lagoon reaches 15–18 km3 per year [3,6,7]. The Data regarding the water exchange between the Vistula Lagoon and the Baltic Sea total outflow from the lagoon was estimated at about 24 km3 [5]. However, researchers areworking not readily today available, on hydrological which was processes described of the in Vistuladetail in Lagoon [11]. Early use research historical has data shown only. 3 thatThe mostthe total recent volume source of data seawater cited by inflow researchers into the may lagoon be found reaches in [ 615–18] and [km13]. Noper moreyear 3 [3,6,7].up-to-date The studiestotal outflow were identified from the that lagoon would was provide estimated scientific at about data 24 for km this [5]. research However, area. researchersThe annual working average today values on hydrological of seawaterinflow processes and of outflow, the Vistula presented Lagoon in [ 6use], are historical17 km3 dataand 20.5only. km The3, respectively.most recent Thesource numerical data cited simulations by researchers presented may inbe [found13] demonstrated in [6] and [13]. the Nodynamics more up-to-date of the water studies exchange were in identified the Vistula that Lagoon would through provide the scientific Strait of Baltiyskdata forfor this a researchone-year area. period (1994). The calculation was done using the MIKE21 hydrodynamic model. The accumulativeThe annual average annual inflowvalues ofof marineseawater water inflow into and the Vistulaoutflow, Lagoon presented in 1994 in amounted[6], are 17 kmto about3 and 18.120.5 km km3;3, the respectively. annual outflow The (includingnumerical thesimulations river outflow) presented toward in the[13] Baltic demon- Sea stratedin the same the dynamics year was aboutof the 22.5water km exchange3. in the Vistula Lagoon through the Strait of BaltiyskIn the for discussion a one-year of period the calculations (1994). The and calculation historical data, was onlydone the using total the seawater MIKE21 inflow hy- drodynamicvolume can bemodel. compared. The accumulative The calculated annual outflow inflow from of the marine lagoon water to the into Gulf the of GdańskVistula (average 15.89 km3 in the period 2008–2017, Table3) is devoid of other hydrological balance components and cannot be compared with the source data (20.5 and 22.5 km3), where additional hydrological processes (e.g., river flow) were incorporated, forming the

total outﬂow. The total inﬂow to the Vistula Lagoon, calculated for the period 2008–2017 Water 2021, 13, 1328 13 of 15

(Table3) , is in the range of 14.31 km3 to 17.78 km3. The average accumulative marine inﬂow amounts to 15.87 km3. The results obtained are in a similar range to those described in [7], and moreover, close to the annual average presented in [6] and slightly lower than those estimated for 1994 [13]. This makes it possible to assess the seawater supply to the Vistula Lagoon obtained for the period 2008–2017 as reliable. These data may constitute the necessary current information to perform a full hydrological balance of the lagoon.

4. Conclusions The results of the statistical analysis confirmed that the main mechanism underlying the water level variation in the Vistula Lagoon is driven by the water level in the Gulf of Gdańsk.However, when it comes to events that can be considered as causing an extreme increase or decrease in the water level, the main factor is strong wind originating from one dominating direction over a long period of time. Thus, the two main factors determining the water stage variation on the Polish side of the Vistula Lagoon are the water level in the gulf and the wind. This result supports the previously formulated understanding of the Vistula Lagoon hydrodynamics [6,7,10] and, additionally, brings more detailed information about the wind parameters and duration needed to produce extremes. The results of the two-dimensional mathematical modeling of the hydrodynamics of the Vistula Lagoon and the simulations of the seawater transport into and from the lagoon in the period 2008–2017 allow the following to be concluded: • The Vistula Lagoon 2D hydrodynamic model was verified and the simulation results were validated using water stage elevation measurements at Tolkmicko control point. Although the results obtained for the years 2008–2017 are slightly understated in relation to the observations, the calculations should be considered reliable, which was confirmed by statistical measures. • The application of the validated hydrodynamic model enabled calculations of the flow through the Strait of Baltiysk. The calculated discharges in the strait provided the data to estimate the total accumulative inflow and outflow of seawater entering and leaving the lagoon, respectively. • The average annual marine water inflow into the Vistula Lagoon between 2008 and 2017 was calculated to be equal to 15.87 km3. The minimum and maximum total inflows were estimated as 14.31 km3 and 17.78 km3, respectively. These values, although different, are consistent with the data given in historical sources and can be used to update the hydrological balance of the waters of the Vistula Lagoon.

Author Contributions: Conceptualization: M.S.; Data curation: P.Z.; Formal analysis: M.S., W.A.; Investigation: M.S., W.A.; Methodology: M.S., W.A.; Resources: P.Z.; Software: M.S.; Validation: M.S.; Visualization: W.A.; Writing—original draft: M.S., W.A.; Writing—review and editing: M.S. and P.Z. All authors have read and agreed to the published version of the manuscript. Funding: This research received no external funding. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: The data presented in this study are openly available in repository Most Wiedzy at doi:10.34808/5wgr-5r25, reference number [17]. Conﬂicts of Interest: The authors declare no conﬂict of interest.

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