ZENTRUM FÜR MEERES UND KLIMAFORSCHUNG

BERICHTE AUS DEM

ZENTRUM FÜR MEERES- UND KLIMAFORSCHUNG

Reihe B: Ozeanographie

Nr. 49

9th German Polish Seminar on

Dynamics of Coastal Waters and their Modelling

Ed. by J. Sündermann

__________________________________________________________ Zentrum für Meeres- und Klimaforschung der Universität Hamburg

Institut für Meereskunde 2005

Berichte aus dem Zentrum für Meeres- und Klimaforschung


Nr. 49

9th German Polish Seminar on


(Ratzeburg, 4-6 June 2003)

Proceedings

Ed. by J. Sündermann

Institut für Meereskunde

Hamburg 2005

Die "Berichte aus dem Zentrum für Meeres- und Klimaforschung" erscheinen in 6 Reihen mit folgendem Inhalt:

Reihe A: Meteorologie Reihe B: Ozeanographie Reihe C: Geophysik Reihe D: Biogeochemie und Meereschemie Reihe E: Hydrobiologie und Fischereiwissenschaft Reihe Z: Interdisziplinäre Zentrumsberichte

Alle Beiträge sind unredigiert und geben allein die Meinung des Verfassers wieder. Sie sollen in erster Linie dem sich jeweiligen Thema befassenden Personenkreis als Arbeitsunterlagen dienen und sind oft gleichzeitig Berichte für die Institutionen, die die betreffenden Arbeiten gefördert haben.

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____________________________________________________________________

ISSN 0936-949X

Zentrum für Meeres- und Klimaforschung der Universität Hamburg

9th German Polish Seminar

on


Proceedings

Ratzeburg, 4-6 June 2003

Contents

Preface .......................................................................................................................3 Paplińska-Swerpel, Paprota, Przewłócki, Sulisz:

Analyses of wave records from the Southern Baltic Sea with the emphases on large wave events .................................................................................................5

Sulisz, Wiśniewski: Prediction of the generation and propagation of wave trains.............................17

Wilde, Sobierajski: The transformation of random waves along the path of propagation ................31

Robakiewicz: Seasonal changes of hydrodynamics in the Gulf of Gdańsk – possibilities of representation by numerical model ....................................................................47

Ostrowski: Cross-shore sediment transport and sea bed evolution by a quasi phase-resolving model ..................................................................................................................60

Meyer, Krupiński: Sediment transport modelling in lower Odra River ...........................................75

Fröhle, Kohlhase: Influence of sea level changes on coastal sediment transport ............................86

Siegismund: Interannual to decadal variability in the sea level of the Baltic Sea explained by the local wind field.............................................................................................99

Programme .............................................................................................................104 Participants .............................................................................................................106

2

Preface The 9th German-Polish Seminar on Dynamics of Coastal Waters and their Modelling took place on 4-6 June, 2003, in Ratzeburg. The specific topic was “The Southern Baltic Sea Coast under Global Change”. 12 participants from the Polish side and 10 from the German attended the meeting; 8 Polish and 7 German papers were presented. On the third day there was a geographical-hydrological excursion through the Ratzeburg Lake and the river Wakenitz to the City of Lübeck . The topical emphasis was on the large-scale Baltic environment, coastal hydrodynamics and coastal morphodynamics. In the first part, “The large-scale Baltic environment”, the dominating influence of the whole Baltic region (and moreover the North Atlantic) on the German-Polish coast was documented. Frank Siegismund analyzed 40 years’ time series of meteorological forcing (NCEP data) and corresponding oceanographic model data and showed clearly the atmospheric origin of anomalies and trends in the sea. Specifically, the North Atlantic Oscillation (NAO) is present in nearly all hydrographic data. According to Ralf Weisse there is up to now no evidence for more frequent and stronger storms or higher sea waves. An exception is the Eastern North Sea with an increase of the mean wave height of 1.8 cm/year. Andreas Lehmann discussed decadal simulations of salt water intrusions from the North Sea into the Baltic (important for the deep water ventilation) and showed again a significant link with the NAO. Thomas Neumann presented long-term calculations with an aggregated ecosystem model of the Baltic Sea containing the chemical and biological variables as well as the physical state variables. It turned out that for the present eutrophication situation of the Baltic reductions of the nutrient inputs will only have a visible effect in about 20 years. In the section “Coastal hydrodynamics” Barbara Swerpel presented results of wave measurements along the Polish coast. She found that extreme waves are normally local events and do not propagate over long distances. These findings have been confirmed and explained by Wojciech Sulisz on the basis of non linear model calculations and observations in a wave tank. Piotr Wilde completed these theoretical interpretations by means of the Schrödinger equation. Malgorzata Robakiewicz presented a realistic hydrodynamical modelling of the Gdansk Bay. The main problem for operational simulations is still the lack of field data for forcing and assimilation. The concluding paper by Wolfgang Rosenthal was an epilogue on the Odra flood which was a main topic in earlier seminars. He stated that an operational protection and warning system still suffers from dispersal of responsibilities. The session on “Coastal morphodynamics” was opened by Rafael Ostrowski with a discussion of new empirical approaches for sediment transport in rivers. Zygmunt Meyer continued with investigations on the influence of salt tongues and consequent density currents. Peter Fröhle presented an empirical transport model and its application to the migration of coastal flats under rising sea level. Jacek Beldowski reported on biochemical analyses of the “fluffy layer” and the tracing of continent-born substances in the sea (e.g. from the Odra river into the Arkona basin). A second paper of Zygmunt Meyer was concerned with the transport of river sediments. He established empirical relations between water depth, discharge and grain size distribution. Finally Klaus-Dieter Pfeiffer spoke about the transfer of sediment models developed for domestic waters to other coastal regions around the world.

3

Summarizing, the German-Polish Seminar proved again to be a vital, stimulating and harmonizing platform in a current field of supra-national environmental research. Presentations and discussions both reached a high scientific level. As in the past, bilateral cooperations have been agreed upon to be realized within regional or European frameworks during the next years. In this manner the seminar has fulfilled successfully its traditional main function as communication and conceptional tool with respect to environmental issues in a specific region (which is currently becoming more interesting due to the growing EU). This is supported by the friendly human atmosphere and the continuous incorporation of new partners and young scientists. The 10th German-Polish Seminar is scheduled for 2005/2006 in Poland. Thanks are due to the German Research Foundation and to the Polish Academy of Sciences for their long-lasting support as well as to the GKSS Research Centre for the publication of the papers. I am indebted also to those authors who provided manuscripts for a proceeding volume in a time of peer reviewed journals. Hamburg, October 2004

Jürgen Sündermann

4

ANALYSES OF WAVE RECORDS FROM THE SOUTHERN BALTIC SEA WITH THE EMPHASES ON LARGE WAVE EVENTS Barbara Paplińska-Swerpel1, Maciej Paprota1, Jarosław Przewłócki2, Wojciech Sulisz1

1 Institute of Hydroengineering of the Polish Academy of Sciences 2 Technical University of Gdańsk, Poland Introduction

Increasing shipping traffic and number of accidents related with extreme waves require much better understanding of the extreme sea conditions. Wave forecasts of the Meteorological Offices usually consists of sea states parameters like significant wave height and mean wave direction. To forecast the corresponding information on the occurrence of the extreme individual waves is a difficult task. Some progress in that field can be achieved by the analysis of data sets of reliable measurements. Such a set of time series of free-surface elevation measurements, obtained by Directional Waverider buoy, is available in the Institute of Hydroengineering of Polish Academy of Sciences, Gdańsk (IBW). The database can provide some useful information on extreme waves and wave events in the southern part of the Baltic Sea.

The main subject of the presented work is to re-analyze the available wave measurements at several locations along the Polish coastline of the Baltic Sea in order to select extraordinarily large water waves, whose heights exceed by factor 2 the significant wave height. The results were used to indicate periods and areas where extreme waves occur more frequently.

Description of wave measurements

Wave measurements were performed using the Directional Waverider Buoy manufactured by Datawell. The buoy determines sea surface displacements by integrating acceleration measurements. The signals are sampled every 3.84 Hz. The measurements are stored in a form of three files, including raw data, statistical and spectral parameters. All of them were used in the analysis.

The raw data file contains 20-minutes records of free-surface elevations recorded with sampling frequency 1.28 Hz. Each record consists of 1536 samples.

The statistical file comprises maximum, average and significant wave heights, maximum, average, significant wave periods, heights of crest and trough of successive waves and theirs period, etc. A “wave” is defined by applying zero up-crossings procedure. The wave height is the difference between the crest and the trough. The Hmax is the highest wave in the record and the Hs is a significant wave height being an average of the highest 1/3 waves in a record.

The spectral file also consists of different parameters, but only mean wave directions have been farther applied. The wave direction is defined as the angle of the vector measured from geographical north clockwise – the direction where the waves are coming from.

Measurements were taken every hour, and during storms, as often as it was possible (allowed by the DWR).

5

Description of buoy localizations

The measurements of free-surface elevation were conducted in the southern part of the Baltic Sea at three buoy stations located in the vicinity of Niechorze (Pomeranian Bay), in Lubiatowo and in the Puck Bay. The locations of all measuring points are shown in Figure 1.

Fig. 1 Localization of Directional Waverider buoy stations

The periods of measurement varied from a few to several months. Some basic features of buoy stations, their locations and measurements periods are given in Table 1.

Table 1 Localization of Directional Waverider buoy stations and periods of measurements

Name of place

Geographical coordinates

Water depth (m)

Distance off-shore (km)

Period of measurements

1. Niechorze 54º 09’N; 15º 03’E 18 6 18 Apr. 97-9 May. 97

05 Sep. 97-14 Jan. 98

54º 51’N; 17º 48’E 20 5 01 Nov. 96-31 Jan. 97

54º 53’N; 17º 51E 20 5 15 Jan. 98-31Aug. 99

54º 50’N; 17º 48’E 20 5 10 Oct. 00-21 Oct. 01

2. Lubiatowo

54º 51’N; 17º 50’E 20 5 05 Sep. 02–04 Dec. 02

3. Puck Bay 54º 26’N; 18º 43’E 15 6 22 Aug. 95-4 Jan. 96

The buoy station Nr. 1, named Niechorze, was located in the eastern part of the Pomeranian Bay in vicinity of the Niechorze village. It was about 6 km off shore at the water depth 18 m.

6

The set of 2801 data records was collected during 7 months of 1997/1998. Waves are generated there on the fetch limited by Bornholm Island and the West Baltic coastline. Waves with Hs greater then 1m come mainly from W-NW and N-NE direction. The significant wave height exceeding 1.0 m occurred 37% of time and greater than 2 m-5%, respectively. The maximum measured height of an individual wave was 6.5 m and the highest significant wave height was 3.3 m (Paplińska, 2001).

The buoy stations Nr. 2 at Lubiatowo was located in the middle part of the Polish coastline near Lubiatowo village. The measurements were conducted in 1997-2002. The total data set coming from this period comprises of 19664 records. The coastline is there straight on a long section. A sea bottom is characterized by the presence of well-formed set of four longshore sand bars. Waves with Hs greater then 1 m are coming mainly from W-WNW and N-NNE directions. The biggest measured height of an individual wave was 7.6 m, the highest significant wave height was 4.0 m. The significant wave height exceeding 1.0 m occurred 29% and greater than 2.0 m-6%, respectively.

The buoy station Nr. 3 in the Puck Bay was located about 6-7 km off shore, where the water depth was approximately 15-16 m. The data come from 6 months of 1995/6. The Puck Bay is separated from the Baltic Sea by the Hel Spit. Due to this specific geographical location , wave field in the Puck Bay consists of waves coming from two different sources: waves generated on the short fetches in the Bay of Gdańsk and the waves generated on the long fetches in the Baltic Sea (Paplińska, 1995). Waves with Hs greater then 1 m are coming mainly from NNE-N directions. The significant wave height exceeding 1.0 m occurred 15% of time and over 2 m 1%, respectively. The maximum measured height of an individual wave was 6.1 m and the highest significant wave height was 3.5 m.

Data analysis

Extreme waves considered in the present study can be regarded as the wave events or large waves characterized by the extreme values of the tail of statistical probability distribution.

The whole wave data base is limited to the records fulfilling the following condition (Paprota, 2003):

1. Significant wave height is larger than Hs>1 m.

From the reduced measurement data sets, there were selected individual waves coming up the assumed, arbitrary chosen criterion (Chien, 2002), on the basis of which the extreme waves can be determined. Such waves in a case of the Baltic Sea were defined as:

2. Waves exceeding twice the significant wave height Hmax/Hs>2,

The application of this definition resulted in 339 extreme wave records. Most of records (271) come from Lubiatowo, 47 from Niechorze and 21 from the Puck Bay (Table 2).

Table 2 Number of wave records coming up assumed criterion

Number of records

1. Niechorze 2. Lubiatowo 3. Puck Bay

All measurements 1875 19276 2888

7

Hs>1 m 717 5339 319

Hs>1 m & Hmax/Hs>2 47 271 21

For each buoy localization one record, containing largest value of the maximum wave height and fulfilling extreme wave criteria, was chosen. The records are as follows:

• Niechorze 1997 October 02 Hmax =6.22 m; Hs =2.82 m; Hmax/Hs=2.20

• Lubiatowo 1998 January 31 Hmax =7.57 m; Hs =3.33 m;. Hmax/Hs=2.27

• Puck Bay 1995 August 31 Hmax =4.10 m; Hs =1.99 m; Hmax/Hs=2.06

Selected time series of the extreme storms for all buoy locations are plotted in the form of wave crest, trough and height envelopes in Figures 2 a, b and c.

The records of highest ratio of the maximum and the significant wave heights were also chosen. These records are:

• Niechorze 1997 September 17 Hmax/Hs=2.35; Hs=1.13 m; Hmax=2.66 m

• Lubiatowo 1999 June 24 Hmax/Hs=2.63; Hs=1.14 m; Hmax=3.00 m

• Puck Bay 1995 November 04 Hmax/Hs=2.32; Hs=1.33 m; Hmax=3.08 m

wave height [m]crest height [m]trough height [m]

date: 08-31-1995 time: 11:14 location: Puck BayHs=1,99m Hmax=4,10m

time [s]

-3

-2

-1

0

1

2

3

4

5

0 200 400 600 800 1000 1200 1400

Figure 2a. Envelopes of “extreme” storm record, the Puck Bay 1995.

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date: 10-02-1997 time: 05:08 location: NiechorzeHs=2,82m Hmax=6,22m

time [s]

-4

-2

0

2

4

6

8

0 200 400 600 800 1000 1200 1400 1600

Figure 2b. Envelopes of “extreme” storm record, Niechorze 1997.

Figure 2c. Envelopes of “extreme” storm record, Lubiatowo 1998.


date: 01-31-1998 time: 13:09 location: Lubiatowo Hs=3,33m Hmax=7,57m

time [s]

-6

-4

-2

0

2

4

6

8

10

0 200 400 600 800 1000 1200 1400 1600 1800

Similar plots for highest ratio of the maximum and significant wave heights are shown in Figure 3 a, b, and c, respectively.

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date: 11-04-1995 time: 14:44 location: Puck BayHs=1,33m Hmax=3,08m

time [s]

-2

-1

0

1

2

3

4

0 200 400 600 800 1000 1200

Figure 3a. Envelopes of the highest Hmax/Hs ratio of the record from the Puck Bay, 1995.


date: 09-17-1997 time: 19:06 location: NiechorzeHs=1,13m Hmax=2,66m

time [s]

-2

-1

0

1

2

3

0 200 400 600 800 1000 1200

Figure 3b. Envelopes of the highest Hmax/Hs ratio of the record from Niechorze, 1997.

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date: 06-24-1999 time: 04:00 location: Lubiatowo Hs=1,14m Hmax=3,00m

time [s]

-2

-1

0

1

2

3

4

0 200 400 600 800 1000 1200 1400

Figure 3c. Envelopes of the highest Hmax/Hs ratio of record from Lubiatowo, 1999.

Extreme waves and mean wave direction

Some efforts have been taken to assess the effect of wave direction on the formation of individual extreme waves. The analysis was conducted on the basis of data sets recorded at Lubiatowo buoy station in 2001.

The time series of a mean wave direction, significant wave height and ratio sHH max

1≥sH are

shown in Figs. 4-7 for several storm periods, indicated by a horizontal line at m. A horizontal line at 2max ≥sHH indicates the threshold that arbitrarily discriminates the height of extreme waves. Every cases of extreme waves occurrence is pointed out by black circle. Some typical extreme wave shapes (marked by letters) are shown in the lower part of these figures. The graphs contain segments (100 s each) of 20 min. records. Note the different scale on the vertical axis.

During long storms with easterly mean wave direction constantly turning northward, extreme waves occurred more frequently then during the other storms (Fig. 4). In that case, the extreme waves assume form of single waves (Figs. 4, 5). Also single extreme waves came from westerly storms turning northward (Fig. 6), but for such conditions they are not so frequent phenomenon. For westerly storms provided long duration, extreme waves are rarely phenomenon (Fig. 7)

A collection of wave groups containing extreme waves is shown in Fig. 8. The majority of such groups came from the northern direction, but also they come from all other directions. For the most of them, the ratio 04.2max ≤sHH do not exceeds two standard deviation from the mean value (Table 3).

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Table 3. Statistics of ratio sHH max=χ .The records fulfilling the condition Hs>1 m are considered.

Statistics of sHH max=χ

Mean value

χ

Standard deviation sd

sd2+χ sd3+χ

1 Niechorze 1.73 0.16 2.06 (29)1 2.22 (4)2

2 Lubiatowo 1.70 0.17 2.04 (208)1 2.21 (58)2

3 Puck Bay 1.71 0.17 2.05 (16)1 2.22 (2)2

1 Number of records coming up the criterion: sd2+χ≥χ

2 Number of records coming up the criterion: sd3+χ≥χ

It should be emphasized that the above analysis was performed on the basis of limited data set. Thus, the deduced conclusions are of limited use, and cannot be generalized at this stage. However, the proposed approach can indicate some directions for a further research in that field

Conclusions

The analysis of available wave records from the southern Baltic Sea revealed a large number of extreme wave, defined as individual waves coming up assumed criterion (Hs>1 m and Hmax/Hs>2). The highest ratio of the maximum and the significant wave heights equal 2.63, was recorded in Lubiatowo.

The formation of individual extreme waves and wave groups is effected by storm direction and duration. A preliminary analysis indicates that storms with easterly mean wave direction constantly turning northward are candidates for the occurrence of extreme waves.

12

77 78 79 80day of the year 2001 (March)

0

1

2

3Si

gnifi

cant

wav

e he

ight

[m]

and

Hm

ax / H

s

N

E

S

W

N

Mea

n w

ave

dire

ctio

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a bc

d

Significant wave height

Mean wave direction

Hmax/Hs

a. b. c. d.

420 480Time (s)

-2

-1

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Sur

face

dis

plac

emen

t (cm

)

2001-03192330.RAWHs = 2.04mHmax = 4.26mHmax/Hs = 2.09

300 360Time (s)

-200

-100

0

100

200

Surfa

ce d

ispl

acem

ent (

cm)

2001-03200130.RAW

Hs = 2.08mHmax = 4.15mHmax/Hs = 2.0

300 360Time (s)

-2

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Sur

face

dis

plac

emen

t (m

)

2001-03200400.RAW

Hs = 2.02mHmax = 4.27mHmax/Hs = 2.11

1080 1140Time (s)

-2

-1

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1

2

3

Sur

face

dis

plac

emen

t (cm

)

2001-03201200.RAWHs = 187 cmHmax = 458 cmHmax/Hs = 2.45

Fig. 4 Easterly storm turning northward. Time series of mean wave direction (points), significant wave height (solid line) and ratio sHH max (triangles). Extreme waves are pointed out by black circle. The shape of extreme wave shapes marked by letters are shown in the lower part of the graph.

109 110 111 112day of the year 2001 (April)

0

1

2

3

Sig

nific

ant w

ave

heig

ht [m

]an

d H

max

/ Hs

N

E

S

W

N

Mea

n w

ave

dire

ctio

na

b


Mean wave direction

Hmax/Hs

c

a. f. c.

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840 900Time (s)

-2

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Sur

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t (m

)


240 300Time (s)

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Surfa

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m)


600 660Time (s)

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Surfa

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m)


Fig. 5 The same as above.

47 48 49day of the year 2001 (February)

0

1

2

3

4

Sig

nific

ant w

ave

heig

ht [m

]an

d H

max

/ Hs

N

E

S

W

N

Mea

n w

ave

dire

ctio

n

aSignificant wave height

Mean wave direction

Hmax/Hs

b c

a. b. c.

540 600Time (s)

-3

-2

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0

1

2

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Surfa

ce d

ispl

acem

ent (

m)


120 180Time (s)

-4

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Surfa

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ispl

acem

ent (

m)


60 120Time (s)

-4

-2

0

2

4

Surfa

ce d

ispl

acem

ent (

m)


Fig. 6 The same as above. An example of westerly storm turning northward.

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250 251 252day of the year 2001 (September)

0

1

2

3

4S

igni

fican

t wav

e he

ight

[m]

and

Hm

ax / H

s

N

E

S

W

N

Mea

n w

ave

dire

ctio

n

a


Mean wave direction

Hmax/Hs

b

a. b.

360 420Time (s)

-2

0

2

Sur

face

dis

plac

emen

t (m

)

2001-09090300.RAW

Hs = 2.49mHmax = 5.17mHmax/Hs = 2.08

120 180Time (s)

-2

-1

0

1

2

Surfa

ce d

ispl

acem

ent (

m)


Fig. 7 The same as above. An example of long, westerly storm.

240 300Time (s)

-2

-1

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1

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3

Surfa

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ispl

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m)


600 660

Time (s)

-2

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m)

2001-04141600.RAW

Hs = 1.62mHmax = 3.26mHmax/Hs = 2.01

600 660Time (s)

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m)


1140 1200

Time (s)

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)

2001-05201200.RAW

Hs = 1.66mHmax = 3.39mHmax/Hs = 2.04

180 240Time (s)

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m)


720 780Time (s)

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m)

2001-07100700.RAW

Hs = 1.58mHmax = 3.19mHmax/Hs = 2.02

660 720

Time (s)

-1.5

-1

-0.5

0

0.5

1

1.5

Surfa

ce d

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m)


180 240Time (s)

-3

-2

-1

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1

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Surfa

ce d

ispl

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m)


Fig. 8 Examples of wave groups containing extreme waves.

15

References

Chien H., Kao C-C., K., Chuang L. Z. H. (2002), On the characteristics of observed coastal freak waves, Coastal Engineering Journal, 44, 301-319.

Paplińska B. (1995) Waves with Two-peaked Spectrum in the Gdańsk Bay, Proc. of Coastal Dynamics, Gdynia, 33-44.

Paplińska B., (2001) Specific features of sea waves in the Pomeranian Bay, Arch. of Hydro-Eng. and Env. Sci.R 48(2), 55-72

Paprota, M., Przewłócki, J., Sulisz, W. and Swerpel, B. (2003) Extreme waves and wave events in the Baltic Sea. Chapter 1.2, Proc. of the Symposium on Damaging Ocean Waves, WMO, Geneve. GKSS Printing Office, Germany.

Acknowledgments

Financial support for this study was provided by EU, contract no. EVK3-CT-2000-00026 MAXWAVE, SCSR, contract no. 5 T07A 030 22, and Institute of Hydroengineering, Polish Academy of Sciences, Gdansk, Poland.

16

PREDICTION OF THE GENERATION AND PROPAGATION OF WAVE TRAINS

W. Sulisz , M. Wiśniewski Polish Academy of Sciences, Institute of Hydroengineering Gdańsk, Poland

Abstract

A theoretical approach is applied to predict the generation of a wave train by a

wavemaker and the propagation of water waves in the train. A closed form analytical solution

was achieved. The solution was derived within the frame of linear wave theory. The solution

was applied to predict the time series of the free surface elevation along a wave flume. A

main attention is paid to the application of the derived solution to predict the formation of

large waves at pre-specified locations in the wave flume. The results show that the present

theory can be applied to predict the generation of a wave train by a wavemaker and the

propagation of water waves in a train. The derived solution can also be applied to predict the

formation of large waves at requested locations in the wave flume. Laboratory experiments

were conducted to verify theoretical results. Free surface elevation recorded by a system of

wave gauges was compared with the results provided by the analytical solution. Theoretical

results are in reasonable agreement with experimental data. A reasonable agreement between

theoretical results and experimental data is observed often even for steep waves. This indicate

that the derived theoretical solution can be applied far beyond its expected rage of

applicability.

17

Introduction

The physical modeling of the interaction of water waves with coastal or offshore

structures is still widely conducted in hydraulic laboratories worldwide. In fact, data collected

during laboratory experiments are often of primary importance at a design stage despite a

progress made in a recent decade in the numerical modeling of the interaction of water waves

with coastal and offshore structures.

The process of the physical modeling of the interaction of water waves in wave flumes

requires certain wave condition at the position of a structure in a wave flume. In consequence,

a procedure is required to predict wave field along a wave flume. Some maritime structures

such as decks of jetties or platforms, elements of balustrades, superstructures of breakwaters,

etc., are often exposed to wave attacks despite precautions at a design stage. This is caused by

the formation of extreme waves at the position of a structure and impact of such waves at the

bottom of a structure. Waves of abnormal height are often the results of focusing of waves of

deferent frequencies at the position of a structure. Thus, it is very important to be able to

predict this phenomenon in the wave flume.

The wavemaker theory was first derived by Biesel and Suquet (1953) for linear water

waves. The theory was extended to include weakly nonlinear waves within the frame of

Lagrangian approach by Fontane (1961) and within the frame of Euler approach by Hudspeth

& Sulisz (1991), Sulisz and Hudspeth (1993) and later by Schaffer (1996). The wavemaker

theories were derived in frequency domain. Although available solutions for the wavemaker

problem may be useful to predict water waves in a wave flume, time-domain solutions to

various problems related with water waves are given more and more attention. However, the

main attention has been paid up to now to numerical solutions (Grill and Svendsen 1990,

Zhang and Williams 1996) which are very time consuming and often unstable.

In this work, a theoretical approach is applied to predict the generation of a wave train

by a wavemaker and the propagation of the train in a wave flume. First, a boundary-value

problem is formulated to describe the generation and propagation of water waves in a wave

flume. Then, the boundary-value problem is solved analytically. The results are discussed with

the emphases on the prediction of the free surface elevation along a wave flume and the

formation of large waves at requested locations in the wave flume. Finally, theoretical results

are compared with experimental data and the conclusions are specified.

18

Statement of Problem

We consider the generation and propagation of a wave train in a two-dimensional wave

flume. The geometry of the problem is presented in Fig. 1. The excitation is provided by a

wavemaker located at the beginning of a flume.

Fig. 1. Definition sketch and coordinate systems.

In general, it is very difficult to find the exact solution to the nonlinear wavemaker problem.

This is because boundary conditions must be applied on a free surface which is unknown and

is a part of a final solution. In order to solve the problem of the generation and propagation of a

wave train, it is assumed that:

• The fluid is inviscid and incompressible.

• The motion is irrotational.

• The walls are impervious.

According to the assumptions the fluid motion may be described in terms of a

two-dimensional velocity potential Φ (x, z, t), such that the velocity vector V = ∇Φ . The fluid

motion is governed by the Laplace equation

.02 =Φ∇ (1a)

and the Bernoulli equation in the following form

01 =++∂Φ∂ gzPt ρ

, (1b)

where ρ is the fluid density, P is the pressure, g is the acceleration due to gravity.

19

The problem is specified if the velocity potential Φ (x, z, t) is known. The velocity

potential has to satisfy the following boundary-value problem. Namely, the Laplace equation

02 =Φ∇ (2a)

with boundary conditions

0,0 ==Φ+Φ zg ztt , (2b)

hzz −==Φ ,0 , (2c)

0, ==Φ xx χ& , (2d)

bxx ==Φ ,0 , (2e)

and initial conditions:

Φz (x, 0, 0) = 0, 0 ≤ x ≤ b, (2f)

Φt (x, 0, 0) = 0, 0 ≤ x ≤ b. (2g)

Solution

The boundary-value problem, Eq. (2), can be solved analytically. In order to obtain an

analytical solution, the velocity potential is assumed in the following form:

( ) ( ) ( )[ ] ( ) ( )∑=

++−−+=Φ

1

22 coscosh

cosh21,,

nn

n

nn x

hhz

tAbxhzb

tzx λλ

λχ& , (3a)

where

πλb

nn

1−= . (3b)

The potential function, Eq. (3), satisfies the Laplace equation, Eq. (2a), and the boundary

conditions, Eqs. (2c)-(2e). In order to satisfy the combined free-surface boundary condition a

following relation must be satisfied:

( ) 0costanh1cos21

21

11

22 =++++−− ∑∑== n

nnnnn

nn xhAgghb

xAhb

bxb

λλλχλχχ &&&&&&&&& . (4)

Equality of the coefficients of the Fourier series requires that:

( )[ ]

∫

∫

∫

∫−

−−=+ b

n

b

n

b

n

b

n

nnnn

dxxb

dxxgh

dxxb

dxxhbxhAgA

0

2

0

0

2

0

22

cos

cos

cos2

costanh

λ

λχ

λ

λχλλ &&&&&& . (5)

20

By conducting integrations in the above equation one obtains:

− for 1=n

bgh

bhbA χχ &&&&&& −

−=

2

1 21

6, (6a)

− for 1>n

χλ

ν &&&&&2

2 2

nnnn b

AA =+ , (6b)

where the coefficients nν are related with nλ by

hg nnn λλν tanh2 = . (6c)

The wavemaker position can be expressed in the following form:

( ) ( )∑=

+=1

21 sincosj

jjjj ttt ωξωξχ . (7)

where 1 and are coefficients of the Fourier series. Moreover, in order to ensure that the

wavemaker is initially at rest, the coefficients have to satisfy the following condition:

jξ jξ2

jξ2

∑=

=1

2 0j

jjωξ . (8)

By substituting Eq. (7) into Eq. (6) and assuming that ωj ≠ νn one obtains

− for n = 1

( )∑=

+−

+

−++=

1

212

321

21

11 cossin

21

61

jjjjj

jj

j

ttb

ghbhbtAAA ωξωξ

ωω

ω, (9)

− for n > 1

( ) ( )∑=

−−

++=1

21222

321 cossin2sincos

jjjjj

njn

jnnnnn tt

btAtAA ωξωξ

λωνω

νν . (10)

The coefficients 1An and 2An , n ≥ 1, may be determined from the initial conditions:

Φz (x, 0, 0) = 0, 0 ≤ x ≤ b, (11a)

Φt (x, 0, 0) = 0, 0 ≤ x ≤ b. (11b)Accordingly

( )∑ ∑∑= ==

=

−−+

+

2 1222

321

111

21

1 0costanh2

tanhn

nnnj jnn

jjn

jj

j

xhb

Ahb

ghA λλλωνλ

ωξλλξ

ω, (12a)

and

21

( )

( ) .0cos2

21

6221

2 1222

412

1

221

12

1

212

212

∑ ∑

∑∑ ∑

= =

==

=

−++

+

+

−−+−−

nn

j jnn

jjnn

jjj

jjjjj

xb

A

bgh

bhbA

bhbx

b

λωνλ

ωξν

ωξωξωξ

(12b)

Thus, for n = 1:

∑=

−=1

2

11

j j

j

bghA

ωξ

, (13a)

∑=

=1

11

2

jjb

ghA ξ , (13b)

which secures that Φ (x, z, 0) = 0, and for n > 1:

( )∑

∫

∑ ∫=

=

−+

−−=

1222

42

0

2

1 0

221

2

)(1

cos

cos21

j njn

jj

nb

nn

j

b

njj

n bxdx

xdxbxb

Aλων

ωξν

λν

λωξ, (14a)

( )∑= −

=1

222

321 2

j jnn

jjn b

Aωνλωξ

. (14b)

Finally, the potential function may be written as:

( ) ( )

( ) ( )[ ] ( ) ( )( )[ ( )] ( ) .cos

coshcoshcoscossinsin

2cossin21

cossin21

6,,

21

2 1222

2

1

2122

21

1

2

1

1

1

2

xh

hztttt

bttbxhz

b

ttb

ghbhbt

bgh

bghtzx

nn

njnnjjjjnnj

n j jnn

j

jjjjjjj

jjjjj j

jj

jj j

j

λλ

λωννωξωωννξ

ωνλω

ωωξωωξ

ωξωξω

ωξωξ

+−++−×

×−

++−−−++

++−

+

−++−=Φ

∑∑∑

∑∑∑

= ==

===

(15)Results

The solution obtained in the foregoing section has been used to predict the free surface

oscillations along a wave flume. In order to verify theoretical results laboratory experiments

were conducted in the wave flume at the Institute of Hydroengineering of Polish Academy of

Sciences, Gdańsk. The free surface elevation was recorded by a system of wave gauges

located along the flume. The comparison of theoretical results and experimental data is

conducted for regular and irregular water waves.

Regular waves

The comparisons of theoretical results and experimental data were first conducted for wave

trains with regular water waves. The calculations of a free surface elevation and the laboratory

experiments were conducted for several ratios of the wave length (L) to the water depth (h)

22

and different steepness of waves in a wave train. The different steepness was achieved by

changing the factor of the amplitude of the wavemaker motion.

0 20 40 60

-4

0

4

0 20 40 60

-4

0

4

0 20 40 60

-4

0

4

0 20 40 60

-4

0

4

0 20 40 60

-4

0

4

0 20 40 60

-4

0

4

0 20 40 60

-4

0

4

η [

cm ]

x = 4 m

x = 8 m

x = 12 m

x = 16 m

x = 20 m

x = 24 m

x = 28 m

h = 0.6 m, L/h = 4, ___ theory, --- experimental data

t [ s ]

Fig. 2a. Time series of free surface elevation for regular wave train - factor 1.

0 20 40 60

-8

0

8

0 20 40 60

-8

0

8

0 20 40 60

-8

0

8

0 20 40 60

-8

0

8

0 20 40 60

-8

0

8

0 20 40 60

-8

0

8

0 20 40 60

-8

0

8

η [

cm ]

x = 4 m

x = 8 m

x = 12 m

x = 16 m

x = 20 m

x = 24 m

x = 28 m


t [ s ]

Fig. 2b. Time series of free surface elevation for regular wave train - factor 1.5.

23

0 20 40 60

-8

0

8

0 20 40 60

-8

0

8

0 20 40 60

-8

0

8

0 20 40 60

-8

0

8

0 20 40 60

-8

0

8

0 20 40 60

-8

0

8

0 20 40 60

-8

0

8

η [

cm ]

x = 4 m

x = 8 m

x = 12 m

x = 16 m

x = 20 m

x = 24 m

x = 28 m


t [ s ]

Fig. 2c. Time series of free surface elevation for regular wave train - factor 2.

0 20 40 60 80

-2

0

2

0 20 40 60 80

-2

0

2

0 20 40 60 80

-2

0

2

0 20 40 60 80

-2

0

2

0 20 40 60 80

-2

0

2

0 20 40 60 80

-2

0

2

0 20 40 60 80

-2

0

2

η [

cm ]

x = 4 m

x = 8 m

x = 12 m

x = 16 m

x = 20 m

x = 24 m

x = 28 m


t [ s ]

Fig. 3a. Time series of free surface elevation for regular wave train - factor 1.

24

0 20 40 60 80

-4

0

4

0 20 40 60 80

-4

0

4

0 20 40 60 80

-4

0

4

0 20 40 60 80

-4

0

4

0 20 40 60 80

-4

0

4

0 20 40 60 80

-4

0

4

0 20 40 60 80

-4

0

4

η [

cm ]

x = 4 m

x = 8 m

x = 12 m

x = 16 m

x = 20 m

x = 24 m

x = 28 m


t [ s ]

Fig. 3b. Time series of free surface elevation for regular wave train - factor 2.

0 20 40 60 80

-8

0

8

0 20 40 60 80

-8

0

8

0 20 40 60 80

-8

0

8

0 20 40 60 80

-8

0

8

0 20 40 60 80

-8

0

8

0 20 40 60 80

-8

0

8

0 20 40 60 80

-8

0

8

η [

cm ]

x = 4 m

x = 8 m

x = 12 m

x = 16 m

x = 20 m

x = 24 m

x = 28 m


t [ s ]

Fig. 3c. Time series of free surface elevation for regular wave train - factor 3.

25

For waves of low steepness, the nonlinear effects associated with the interaction of

water waves in a wave train are negligible and theoretical results are in a very good agreement

with experimental data. The higher factor of the amplitude of the wavemaker motion the

stronger the nonlinear interaction between waves in a wave train. This causes some

modification of the frequency spectrum of a wave train. However, the plots presented in Figs.

2c-3c and additional analysis show that theoretical results are in reasonable agreement with

experimental data often even for fairly steep waves. This indicate that the derived theoretical

solution can be applied far beyond its expected rage of applicability.

Irregular waves

In reality regular waves do not exist in a sea (Fig. 4). The prediction of the generation and

propagation of regular waves is necessary to understand the mechanism of the generation and

propagation of irregular waves trains. This is because irregular waves can be considered, at a

first approximation, as a superposition of regular waves.

0 200 400 600 800t [ s ]

-4

-2

0

2

4

η [

m ]

Fig. 4. Record of free surface elevation (Baltic Sea).

The prediction of the propagation of irregular waves in a wave flume is indispensable in

the procedure of the physical modeling of the interaction of water waves with a structure. A

standard procedure requires certain wave condition at the position of a structure in a wave

flume. Some maritime structures such as decks of jetties or platforms, elements of balustrades,

26

superstructures of breakwaters, etc., are exposed to wave attacks despite precautions at a

design stage. This is caused by the formation of an extreme wave at the position of a structure

and impact of such a wave at the bottom of a structure - Fig. 5. Waves of abnormal height are

often the results of focusing of waves of deferent frequencies at the position of a structure and

it is very important to be able to predict this phenomenon in the wave flume.

Fig. 5a. Sketch of the model of wave impact on a structure.

18.8 19.2 19.6 20 20.4 20.8t [s]

-4

0

4

8

12

Pre

ssur

e [k

Pa]

Fig. 5b. Record of pressure induced by a wave impact at the bottom of a structure.

The plots in Fig. 6 show comparison between theoretical results and experimental data for

irregular waves. For the smallest considered factor of the amplitude of the wavemaker

motion, Fig. 6a, the theoretical results fit very well to the experimental data even at the

focusing point where the wave steepness is much higher then anywhere else in the wave train.

A similar situation has been observed for steeper waves in the train, Fig. 6b.

27

0 20 40 60

-4

0

4

0 20 40 60

-4

0

4

0 20 40 60

-4

0

4

0 20 40 60

-4

0

4

0 20 40 60

-4

0

4

0 20 40 60

-4

0

4

0 20 40 60

-4

0

4

η [

cm ]

x = 4 m

x = 8 m

x = 12 m

x = 16 m

x = 20 m

x = 24 m

x = 28 m

h = 0.6 m, ___ theory, --- experimental data

t [ s ]Fig. 6a. Time series of free surface elevation for irregular wave train - factor 2.5.

0 20 40 60

-8

0

8

0 20 40 60

-8

0

8

0 20 40 60

-8

0

8

0 20 40 60

-8

0

8

0 20 40 60

-8

0

8

0 20 40 60

-8

0

8

0 20 40 60

-8

0

8

η [

cm ]

x = 4 m

x = 8 m

x = 12 m

x = 16 m

x = 20 m

x = 24 m

x = 28 m

h = 0.6 m, ___ theory, --- experimental data

t [ s ]Fig. 6b. Time series of free surface elevation for irregular wave train - factor 4.

28

Summary

A theoretical approach is applied to predict the generation of a wave train by a

wavemaker and the propagation of water waves in the train. A closed form analytical solution

was achieved. The solution was derived within the frame of linear wave theory. The solution

was applied to predict the time series of the free surface elevation along a wave flume. A

main attention is paid to the application of the derived solution to predict the formation of

large waves at pre-specified locations in the wave flume.

The results show that the present theory can be applied to predict the generation of a

wave train by a wavemaker and the propagation of water waves in a train. The derived

solution can also be applied to predict the formation of large waves at requested locations in

the wave flume. This implies that the solution can be applied to predict wave field at the

position of a structure for physical modeling of the interaction of water waves with a

structure. In particular, the derived solution can be applied to predict the formation of a large

wave at a position of a structure for the problem of wave impact at the bottom of a deck, a

platform, etc.

Laboratory experiments were conducted to verify theoretical results. Free surface

elevation recorded by a system of wave gauges was compared with the results provided by the

analytical solution. Theoretical results are in reasonable agreement with experimental data. A

reasonable agreement between theoretical results and experimental data is observed often

even for fairly steep waves. This indicate that the derived theoretical solution can be applied

far beyond its expected rage of applicability.

Acknowledgments

Financial support for this study was provided by EU, contract no. EVK3-CT-2000-00026

MAXWAVE, SCSR, contract no. 5 T07A 030 22, and Institute of Hydroengineering, Polish

Academy of Sciences, Gdańsk, Poland.

References

Biesel, F. & Suquet, F. (1953) Laboratory wave generating aparatus. Project Report No. 39, St.

Anthony Falls Hydraulic Laboratory. University of Minessota, Minneapolis, Minnesota, U.S.A

29

Fontanet, P. (1961) Theórie de la génération de la houle cylindrique par un batteur plan. – La

Houille Blanche, 16, 3-31

Hudspeth, R.T & Sulisz, W. (1991) Stokes drift in two-dimensional wave flumes. – Journal of

Fluid Mechanics, 230, 209-229

Schaffer, H.A. (1996) Second-order wavemaker theory for irregular waves. – Ocean

Engineering, 23, 47-88

Sulisz, W. & Hudspeth, R.T. (1993) Complete second-order solution for water waves generated

in the flume. – Journal of Fluids and Structures, 7, 253-268

Zhang, S. & Williams, A.N. (1996) Time-domain simulation of the generation and propagation

of second-order Stokes waves in a two-dimensional wave flume Part I: Monochromatic

wavemaker motion. − Journal of Fluids and Structures, 10, 319-335

30

THE TRANSFORMATION OF RANDOM WAVES ALONG THE PATH OF PROPAGATION P. Wilde, E. Sobierajski Institute of Hydroengineering of the Polish Academy of Sciences Gdanks, Poland

Introduction In planning the experiments described in this paper the authors had experience in the analysis of phenomena in the transformation and stability of regular deep water waves as described by P. Wilde & al (2003). In such waves the energy moves with the group velocity that is equal to one half of the phase velocity. It is a dispersive wave. If the wave height is not small the phase velocity depends upon the wave amplitude and thus the behavior can not be described within the linear wave theory. The nonlinear deep water wave is not stable and very small modifications can result in very significant changes in the surface elevations (wave profiles), as shown by Lighthill (1965). A random, narrow band, deep water wave has a dominant frequency. It may be looked at such a wave, as a regular nonlinear wave that is modified by a random process. Thus we expected that the transformations, along the path of propagation, have to be considerable. To obtain good statistical estimates we considered long random wave trains (over 300 wave length). Wilde P. & Kozakiewicz A. (1993) described a method to construct realizations of random waves based on the Brownian Motion Process. Time series of piston motion constructed by this method for narrow band, stationary random processes are Gaussian and have random amplitudes and random phases. Wilde P. and Wilde M. (2001) described in detail the generation process. A second type of time series for random piston motions was based on finite number of terms of Fourier series. The amplitudes of the terms were calculated by replacing the given spectral density by a discrete set of amplitudes of trigonometric series. In this case, all the realizations correspond exactly to trigonometric series with the same amplitudes and the randomness is due only to the choice of phases from a standard generator of random numbers with a uniform distribution in the interval -π, π. Such a series has a spectral density close to the desired one for relatively short wave trains These realizations were used to compare the changes in the Fourier series of the data measured along the flume at the installed gauges. Benjamin (1967) and Benjamin & Feir (1967) studied the instability and disintegration of periodic wave trains on deep water and showed that side bands grow along the path of propagation. Similar behavior may be seen in the transformation of random waves along the flume. Zacharov (1968) showed that the behavior due to the instability of periodic deep water waves of finite amplitude may be described by the nonlinear Schrödinger equation. Lake & al (1977a) and (1977b) described their experimental results and the comparison with the corresponding theoretical solutions. The results are summarized in a book by Yuen & Lake (1982). In our laboratory we also investigated periodic solitons that have a closed form solution. In this paper the results of measurements of the surface elevations and the envelopes of a soliton with an infinite period are presented. It may be seen that this solution of the nonlinear Schrödinger equation has some similarity with the behavior of intervals of the random waves with large amplitudes.

31

Random waves based on the Brownian Motion Process Let us base the time series of a sample function of a random function on the solution of an Itô differential equation as described in the book by P. Wilde & A. Kozakiewicz (1993). Let us look for a stationary random process for the surface elevation that has continuous accelerations; it means it is twice differentiable. We start with the analysis of the following set of Itô differential equation in time t measured in seconds (s) ( ) ( ) ( )

( ) ( )

( ) ( ),12

01

0

tAtAdtd

tAtAdtd

tdBtAdtd

ηη

ηη

αη

=

+

=

+

=+

(1)

where the parameter η has dimension 1/s, the parameter α has dimension of length (m or cm) and B(t) is the Brownian Motion Process. In a matrix notation the differential equation may be written in the form

( ) ( ) ( ),tdBttd gψAA += (2)

where . ( ) ( ) ( ) ( )[ ] [ ,0,0,,,, 210 α== gA tAtAtAtT ]

−−

−=

110011001

ηψ

The superscript T denotes a transpose of the matrix. The general solution is given by the following expression

( ) ( ) ( ) ( ) ( ),,,000 ∫+= t

t udButtttt gΦAΦA (3)

where Φ is the fundamental solution ( 0, tt )

( ) ( )[ ] ( )( ) ( )

−−−−−=

1!1/!2/01!1/001

exp,

02

02

000

tttttttttt

ηηηηΦ

It should be noted that the fundamental solution is a function of only. Thus the value of the initial time is not important and may be taken for example equal to zero without loss of generality.

0tt −

In view of the expression (3), if at the initial time the mean value of the matrix A is a zero matrix then the mean value is zero for all times t. We are looking for a solution corresponding to a stationary process with the mean value equal zero. If such solution exists its covariance matrix defined by

0t

( ) ( ) ( ) ttEt TAAP = must be constant in time. Thus the differential of P(t) has to be zero. It follows

( ) ( ) ( ) ( ) ( ) 0=++= dBdBtdtttdEtd TTT ggAAAAP . Substitution of the relation (2) and calculation of the integrals and keeping in mind that the expected value of dBdB is equal , where is the variance parameter, assumed one in our considerations, leads to the following expression for the covariance matrix

dt2σ 2σ

0ggPψψP =++ TT . (4) For the considered case of a twice differentiable function the elements of the covariance matrix are

32

=

8/38/34/18/32/12/14/12/11

2

2

ηαP .

The twice differentiable function is the function and thus its variance is . We can calculate the value of the parameter α when the variance and the

value of the parameter η are known. It is easy to calculate that for this case the correlation function of the twice differential function is

( )tA2

( ) ( ηα 16/33,3 2=P )

( )tA2

( ) ( ) ( ) ( )[ ]3/1exp3,3 222

ητητηττ ++−= PC AA , where 0tt −=τ . This expression shows that when τ goes to infinity the correlation goes to zero. How fast the correlation diminishes depends upon the value of the parameter η. The spectral density function is the Fourier transform of its auto-correlation function

( ) ( ) ( )∫ −=∞

∞−ττωτω dCiS exp (5)

where ω is the angular frequency. For the twice differentiable function the spectral density is ( )tA2

( ) ( )( )[ ]32,

/1

10316

22ηωη

ω+

= CS AA . (6)

The maximum value is for ω equal zero. The spectral density is a symmetric function and it diminishes when ω goes to infinity and minus infinity. For small values of η the stochastic process has a narrow band. Let us introduce a discrete set of times T with a regular spacing . The elements of the (n+1)x1 matrix T are

t∆

[ ]tnttT ∆∆∆= ,...,2,,0T . To construct the corresponding time series of the random wave we have to go back to the relation (3) and to substitute it into the formula for the covariance matrix P. With the notation

it follows that 0ttt −=∆

( ) ( ) ( ) ( ) duututEtt t TTT ∫ ∆∆+∆∆= ∆0 ,,0,0, ΦggΦPΦΦP .

On the left side is a symmetric matrix P. The first term on the right side is a symmetric matrix too. Thus the expected value of the integral must be a symmetric matrix. A symmetric matrix may be represented by a lower triangular matrix q multiplied by its transposeq . Thus it follows

T

( ) ( )0,0, tt TT ∆∆−= PΦΦPqq , and the corresponding expression for a realization of the value of is ( )0,t∆A

( ) ( ) ( ) qUAΦA +∆=∆ 00,tt , (7) where U is a (3x1) column matrix corresponding to a Gaussian white noise sequence. The three elements of the matrix U are chosen as independent values from the generator of random numbers with zero mean value and standard deviation equal to one. It is obvious that the formula (6) may be repeated in each step to obtain a realization of a random sequence

. ( ) nsts ...,,2,1,0, =∆A Let us now consider two independent realizations of a stationary and twice differentiable random function defined as the solutions of the set of differential equations (1) , and introduce a complex number function by the definition ( ) ( )tDtA 22 , ( )tZ2

( ) ( ) ( )[ ] ( titiDtAtZ d )ω−+= exp222 , (8)

33

where dd T/2πω =

2X

is the dominant angular frequency and T is the dominant period. The real part and the imaginary part, both have the same stochastic properties and are denoted respectively as andY . It is easy to calculate that the corresponding correlation functions are

d

( ) ( )2Re Zt = ( ) ( )22 Im Zt =

( ) ( ) ( ) ( ) ( ) ( )τωτττωττ dAAYYdAAXX CCCC sin,cos22222222

== and the corresponding spectral density is

( ) ( ) ( ) 2/2/222222 dAAdAAXX SSS ωωωωω ++−= . (9)

To study the transformation of random waves along the flume seven wave gauges were placed that measure the surface elevation at the distances: 4, 8, 16, 24, 32, 40, 48m from the piston of the generator at rest. The water was 0.60m deep, the wavelength of the dominant wave was and the corresponding wave period wasT . Thus deep water waves were studied. In the calculation of the control time series for the motion of the piston linear wave theory was used. In Fig 1 the surface elevations measured by the gauges placed at the distances 4, 8, 16 and 24m are depicted for a randomness parameter η=0.4554

mLd 20.1=

( )dT

sd 8783.0=

/4.0=η . On the graphs the surface elevation (in cm) are functions of time (in s). The phase velocity is equal to v and the group velocity s. Thus the shift in time from the first to the second gauge in is 5.85s and the shifts in the consecutive graphs are 11.713s. It should be noted that the transformations of the measured surface elevations are considerable. In general the waves become steeper along the path of propagation. To get an insight into the behavior the zoomed intervals are depicted in Fig 2. The second graph is shifted to the first one by 5s and the consecutive ones by 10s The first two graphs show random functions with slowly increasing and decreasing amplitudes forming groups including many waves. A boat floating on such waves moves gently up and down. In the case of the waves shown in the fourth graph the boat would meet an almost vertical wall of water that increases in one period from two to eight units. The measured surface elevations in the other gauges are not shown in Fig. 1 and Fig.2 , but the behavior is similar. The wave transformations continue, as measured in the succeeding gauges. Some parts of wave profiles become gentler and other steeper on the path of propagation. It has to be noted that in the experiments white caps appear accidentally on top of very high and steep waves. They dissipate energy and thus diminish the mechanical energy of the progressive wave.

smTL ddph /3663.1/ == /6832.02/ mvv phgr ==

The random waves generated by Fourier series of the piston motion The aim of this paragraph is to describe a second procedure used to obtain samples of a random function for an arbitrary prescribed spectral density. Such procedures are often used in flumes with standard generators. We apply the procedure to generate the time series corresponding to the one described in paragraph II. The starting point is the spectral density defined by the relations (9) and (6). In relation (5) the spectral density function is calculated as the Fourier transformation of the covariance function. The correlation function may be calculated as the inverse Fourier transformation according to the formula

( ) ( ) ( ) ωωτωπ

τ diSC exp21∫= ∞

∞− .

For 0=τ the value of the correlation function is equal to the variance of the stationary stochastic process. The integral is equal to the area under the spectral density curve divided by 2π. The integrals defined by the integral

34

( ) ωωπ

ωωωω dSsV s

s AAw ∫= ∆+∆−

2/2/ 222

1)( , (10)

correspond to the variances of the interval of length ω∆ attributed to the angular frequencies sω . Introduction of a new dimensionless variable prescribed by the relation ηω /=x and integration leads to the following expression for the variance of an interval:

( ) ( ) ( ) ( ) ( )[ ]lul

l

u

u

l

l

u

uw xaxaC

xx

xxC

xx

xxCsV tantan

11141432

222222−+

+

−+

+

+−

+=

πππ, (11)

where ( )[ ] ηωω /2/∆+= sxu and ( )[ ] ηωω /2/∆−= sxl . Let us introduce a set of angular frequencies ( )sω with a regular spacing ω∆ described by the following relations:

( ) nnsss ,...1,0,1,..., −−=∆= ωω . (12) If s is equal to zero the corresponding angular frequency is equal zero too. The number of discrete points is equal to and the column matrix Ω representing the corresponding set of angular frequencies has m elements. The matrix with elements corresponding to the variances of elements has the same size. We have calculated the values for the spectral density corresponding to the process without dominant frequency. It is easy to see that the same results are obtained when the spectral density (9) corresponding to the case with a dominant angular frequency

m 12 +n

dω is considered. Our aim is to calculate time series of realizations of the above defined random process with variances of the components given by the relation (11) and random phases that are statistically independent and have a uniform distribution in the interval [ ]ππ,− . The suitable standard expression for such a discrete Fourier series in complex number notation is

( ) ( ) ( )( )[∑ −∆+−+=∆=

=

ms

sd strsisAAtrZ

1exp0)( ψωω ( )] , (13)

where ( )sψ is a set of phases obtained, for a considered realization, from a standard computer program for random numbers with a uniform distribution . The real and imaginary parts correspond to two real processes

( ) ( ) ( )( )[ ]

( ) ( ) ( )( )[ ]∑ −∆+−=∆

∑ −∆+=∆

=

=

=

=ms

sd

ms

sd

strssAtrY

strssAtrX

1

1

sin

,cos

ψωω

ψωω ( )

( ). (14)

The calculation of variances of the terms in both the Fourier Series in (14) leads to the following expressions:

( ) ( ) ( ) ( ) mssAsVAV ww ,...,2,1,2/,00 22 === . (15) Thus the Fourier series coefficients may be calculated on the basis of the variancesV . When the mean value of the stochastic process is zero and the corresponding variance is zero then is zero, otherwise it must be given. It should be noted that the two real random series in (9) are complementary.

( ) ( )sAA ,0

( )0A( ) ( )sVww ,0

Up till now the surface elevation is considered. In the standard problem of wave generation it is necessary to feed the horizontal displacements of the piston into the control system of the wave generator. The piston motion introduces in the fluid a uniform horizontal velocity field. Within the linear wave theory a regular standing wave is formed in the neighborhood of the generator. The progressive wave and the motion of the piston have different horizontal amplitudes. Within the linear wave theory the horizontal displacements of the piston are given by the following expression

35

( ) ( ) ( )[ ]( )[ ] ( ) ( )( )[ strssA

hskhskhsktrW d

ms

sψωω −∆+∑

+=∆=

=sin

2sinh22sinh2

1( )]

)

, (16)

where the surface elevations correspond to the as given in the relations (14). ( trX ∆For very deep waves the coefficient tends to 1/2 and thus a good approximation is given by the following formula for the motion of the piston

( ) ( ) ( ) ( )[ strssAtrWms

sψω −∆≈∆ ∑

=

=

sin21

1

] . (17)

It means that in the case of a piston type generator the horizontal displacements of the piston are almost equal to the half of the surface elevations of the progressive wave. The waves generated in the paragraph II and III are based on the same spectral density and both correspond to Gaussian processes. The process in paragraph II based on the Brownian motion is twice differentiable, while the process based of a Fourier Series with finite number of terms has all time derivatives. It has to be remembered however that this remark is true for the control series but not for the real motion of the piston. In Fig 3 the set of amplitudes of the corresponding Fourier series approximation is depicted. In Fig 4 the surface elevations measured by seven gauges are plotted. The positions of the gauges and the parameters of the random wave are the same as in the example illustrated in Fig 1. These realizations of the random function are generated by time series based on the relation (11). Because only a finite number of terms are taken there is a large period (the generated random function has two such periods). The transformation of the wave along the path of propagation may be clearly seen. The groups move with the group velocity and the groups change shape by becoming steeper and smoother again. In Fig 5 the zoomed one period of the measured surface elevations at the gauges 2 - 5 with the corresponding amplitudes of the Fourier series terms are depicted. In the third and fourth graphs large jumps in wave height may be seen. The amplitudes of Fourier series of the piston motion are very regular and there are no terms with the double frequency. In the first graph in Fig 5 the amplitude of the dominant angular frequency is the largest one. In the third and fourth graphs the Benjamin's & Feir's (1967) side bands can be seen. Such behavior is typical for the unstable case of deep water waves with considerable heights. Thus the transformation of the wave along the path of propagation may be due to the loss of stability in deep water waves.

Periodic solitons, the physical phenomena Witham in his book (1974) discusses the various differential equations that describe nonlinear dispersive waves. One of them is the nonlinear Schrödinger equation

021

8222

2

2

2 =−∂∂−

∂∂+

∂∂ AAk

xA

kxA

ktAi ll

l

l

l

l ωωω . (18)

where the function ( ) ( ) ( )[ txitxatxA ,exp,, ]θ= is called the complex amplitude, a is the absolute value and θ is the phase of the complex number for fixed values of x and t. It has to be noted that the differential equation has complex number coefficients and the third term has a coefficient that is proportional to the square of the amplitude ( ). The Schrödinger equation describes the behavior of a propagating wave train with a dominant wave which has the carrier wave number and the carrier angular frequency

2a

llk ω which are related by the dispersion relation of the linear theory. When the complex amplitude is calculated as a solution of the differential equation (18) the surface elevation is give by the following expression

( tx, )

)]

y

( ) ( ) ( )[ ] ([ txkitxitxatxy ll ωθ −= exp,exp,Re, . (19)

36

To discuss the physics described by the Schrödinger equation let us consider the case of a regular second order Stokes wave and assume that the value of the wave number is k . For the case of a deep water wave the dispersion relation is

l

( ) ( ) ,,1,1 20

221

020

220 llllll gkakgkakgk =+=+= ωωω (20)

where 00 /2 Tπω = is the angular frequency, T0 is the period, k ll L/2π= is the wave number, is the wave length and a is the amplitude (half of the wave height) of the first term in the

formula for the Stokes wave. The second expression corresponds to the binomial expansion and the third to the corresponding angular frequency of the linear theory . The expression for the first term of the second order Stokes wave is well known and thus the surface elevation for this case, according to the relation (19), is

lL 0

( )00 →a

( ) ( )[ ] ( )[ ] ( )[ ] txkitiatxkiatxy llll ωωωω −−−=−= expexpReexpRe, 0000 . It follows that in view of the dispersion relation (20) the complex amplitude is

( ) [ ]takiatxA ll20

221

0 exp, ω−= , and it is very easy to verify that it satisfies the nonlinear Schrödinger equation (18). Following the paper by Martin D.A. and al (1980) let us introduce the following transformation of variables:

( ) ( ) 2/,2,2/12/1

0/1,

20

llll

l

ll

l AkBBk

AXT

kkxt

xt

kXT

==

−−

−=

−−

=

ωωω

. (21)

In the new dimensionless variables the differential equation (18) assumes the following form:

021 2

2

2

=+∂∂+

∂∂ BB

XB

TBi . (22)

The standard separation of variables leads to the expression ( ) ( TiXB 2exp γΦ= ) , (23)

where is a real dimensionless number, and Φ satisfies the second order ordinary nonlinear differential equation

2γ ( )X

021 32

2

2

=Φ+Φ−∂

Φ∂ γX

. (24)

This equation has a periodic solution expressed by the Jacobi elliptic function ( ) ( )[ mXXdnX ,0−=Φ ]ββ , (25)

where the parameters m,β are real numbers, , and . 10 ≤≤ m ( )[ ] 2/122/2 m−= γβ Let us go back to the initial independent variable x, t and write down the expression for the surface elevation (19) in terms of the periodic solution given by the relations (25), (23) and (21) . It follows

( ) ( )[ ] ( ) ( )[ ]

−−−= txkitimtxkdnk

txy llllll

ωωγωββ expexp,22Re, 2

The function is equal to one for and for all arguments corresponding to the multiples of the period. For other arguments its value is less then one. Thus the coefficient placed before the function dn is equal to the maximum value of the surface elevation

[ mXXdn ,0− ] 0XX =

max/ akl =β2 and thus 2/maxakl=β . It follows from the notation introduced in

the solution given by the relation (25) that 2/maxakl=γ

l

2/1 2m−

l

. Thus there are only three independent parameters a , m and k or max ω in the solution. Finally the expression for the surface elevation assumes the following form

( ) ( )[ ] ( )[ ]ψωγω ++−−= txkmtxkakdnatxy lllll2

maxmax 1cos,2/2, (26)

37

where ψ is an arbitrary phase shift. It should be noted that in the argument of the cosine function the following notations may be introduced: ( ) lωγω 2

0 1+= and 00 /2 ωπ=T .These notations correspond to the angular dominant frequency and its period, when nonlinear behavior is taken into account. To understand the physical meaning of the solutions let us first consider the very special case m = 0. It follows that γβ = and that for m = 0 the Jacobi elliptic function dn is equal to one for all values of the arguments. In this case, it follows from the relation (26), that it corresponds exactly to the first term in the second order, deep water Stokes wave. Its phase velocity is equal to lk/0ω . The surface elevation represented by the solution (26) has two periods in time for a fixed x. The first period T corresponds to the dominant frequency of the peak in the spectral density. The second period is due to the periodicity of the Jacobi elliptic function dn. It is well known that the period of this function is equal to twice the value of the complete elliptic integral K(m). Thus the number of wave periods in the group in time is

0

( ) ( )0/2 TmKN lgT βω= (27) and the number of wave lengths in the group in space is

( ) ( )llgL LkmKN 2/2 β= . (28) It is obvious that in the expression (28) is equal to 2π. In the relation (27) the following identity is true

ll Lk

ll TTT /2 00 πω = . Thus the number of wave periods in the group in time is close to twice the number of wave length in space becauseT is close to one. T/0

The second special case is when the parameter . In this special case the Jacobi elliptic function goes over to the hyperbolic secant

1=mX ( Xhdn )ββ sec=)1,( and the period goes

to infinity. Thus for a simple soliton it is sufficient to replace the Jacobi elliptic function dn by the hyperbolic secant, [ Xh ]βsec and 2/maxakl=γ .

( ) ( )[ ] ( )[ ]ψωω ++−−= takxktxkakhatxy llllll 4/1cos2/2sec, 2max

2maxmax . (29)

It follows that the dominant angular frequency is ( ) ll ak ωω 4/1 2max

20 +=

( )ll k2/

. It should be noted the expression in the round decreases from the case m = 0 to the value at m = 1. Thus the phase velocities change while the group velocity, equal to ω , stays constant. Experiments with periodic solitons described by the relations (26) and (29) were performed in our laboratory. Because we considered deep water waves as already mentioned the motion of the piston may be taken as one half of the functions describing the surface elevation. It has to be stressed however that with a piston type generator the production of the solitons in the flume is not perfect. In Fig 6 the values and the calculated envelopes are shown for the first graph corresponding to the data measured at the distance 19.36m from the piston at rest and the consecutive ones at spacing equal to 0.20m. The measured data were filtered by a Kalman Filter as described in the book by P. Wilde & A. Kozakiewicz (1993) to remove the small components corresponding to double frequencies and to consider the measured soliton as a realization of a random function with a dominant frequency. It may be seen that the envelopes move with the group velocity and the phase velocity is approximately twice as large so that the motion of the wave with respect to the envelope is clearly seen. As in all measurements there is a tail that follows a soliton. No smoothing is applied to the envelopes.

38

Conclusions 1. A deep water, narrow band, random wave generated in a standard flume transforms along the path of propagation. Very steep waves appear due to the transformation. 2. Waves generated on the basis of the Brownian Motion Process are represented by long random wave trains and it may be seen that big jumps between consecutive wave crests appear and disappear along the flume in time. 3. Waves were also generated by time series based on realizations of a random process represented by a Fourier series with amplitudes corresponding to a prescribed spectral density and random phases that have a uniform distribution in the interval -π, π. The amplitudes of the Fourier series change along the flume, the spectral densities become wider with side bands. 4. The two applied methods are different from the mathematical point of view, but the behavior of the realizations is similar. In the first method the realizations of the generated time series have continuous second derivatives and no higher derivatives while in the second method a finite number of terms are used and thus the time series have all derivatives. It must be however noted that the measured displacements of the piston are based on the constructed time series but depend also upon the technical solution of the control system. It is not perfect. 5. The phenomena observed in the transformations of the investigated random along the flume depend strongly on the value of the wave height. The biggest changes occur at a distance of the order of 25 wave lengths from the generator. No available, professional programs of wave transformation can reproduce the measurements along the flume. 6. The simplified description of behavior by the nonlinear Schrödinger differential equation has solutions that correspond to the phenomena observed in our experiments on random waves. The known solutions for periodic solitons were used to generate such waves in our laboratory. In the motion of deep water waves the behavior is influenced by the dispersion relation and the nonlinear effects. A perfect soliton occurs if these two effects cancel each other. In a random wave there is no equilibrium and therefore the wave are becoming smoother and locally steeper in transformations along the flume. The authors believe that the nonlinear Schrödinger differential equation may be a convenient starting point for a mathematical model that could describe the basic phenomena in the transformation of random waves along the flume.

References Benjamin T.B., (1967), Instability of periodic wavetrains in non-linear dispersive systems, Proc. R. Soc. London , Ser. A v. 299 p.59-75 Benjamin T.B., Feir J.E., (1967), The disintegration of wavetrains on deep waters. Part 1. Theory – J. Fluid Mech., v.27 p.417-430 Lake B. M., Yuen H.C., (1977a), A note on some non-linear water wave experiments and the comparison of data with theory, J. Fluid Mech., v.83 p. 75-81 Lake B.M., Yuen H. C., Rungaldier H, Ferguson W.,(1977b), Non-linear deep-water waves: theory and experiment. Part 2. Evolution of a continuous wave train, J. Fluid Mech. V. 83, p. 49-74 Lighthill M.J ,(1965), Contributions to the theory of waves in non-linear dispersive systems, J.Inst. Math. Appl., v, 1 p. 269-306 Martin D.U., Yuen H.C., Saffman P.G.,(1980) Stability of plane wave solutions of the two-space dimensional nonlinear Schrödinger equation, Wave Motion 2 p. 215-229.

39

Wilde P., Sobierajski E., Chybicki W., Sobczak Ł.,(2003) Laboratory investigations of deep water wave transformation and stability, Archives of Hydro-Engineering and Environmental Mechanics, submitted for publication. Wilde P, and Wilde M. (2001), On the generation of water waves in a flume, Archives of Hydro-Engineering and Environmental Mechanics No4, p.69-83 Witham G.B.,(1974), Linear and non-linear waves, Wiley, New York. Yuen H.C., Lake B.M., (1982), Non-linear dynamics of deep water waves, Advances in Applied Mechanics, Volume 22, p. 67-229, ACADEMIC PRESS. Zacharov V.E, (1968), Stability of periodic waves of finite amplitude on the surface of a deep fluid, Sov. Phys. J. Appl. Mech. Tech. Phys. 4, p. 86-94.

Acknowledgment The research on solitons was sponsored by KBN (the Polish Committee for Scientific Research) within the research project 5T07A03022, "The impact of a progressive water wave on a horizontal plane of an elastically supported structure".

40

Fig. 1 Measured surface elevations (in cm) at distances 4, 8, 16 and 24m from piston at rest as functions of time (in s).

41

Fig. 2 Zoomed and shifted in time the graphs of the same part of wave train registered by four gauges along the flume.

42

0,0

0,2

0,4

0,6

0,8

1,0

0 2 4 6 8 10

Fig. 3 Amplitudes of Fourier coefficients (in cm) as functions of angular frequencies (in 1/s).

43

-6-4-20246

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150

-6-4-20246

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150

-6-4-20246

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150

-6-4-20246

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150

-6-4-20246

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150

-6-4-20246

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150

-6-4-20246

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150

Fig. 4 Measured surface elevations (in cm) at distances 4, 8, 16, 24, 32, 40 and 48m from piston as function of time (in s).

44

-6,00

-4,00

-2,00

0,00

2,00

4,00

6,0038 48 58 68 78

-6,00

-4,00

-2,00

0,00

2,00

4,00

6,0048 58 68 78 88

-6,00

-4,00

-2,00

0,00

2,00

4,00

6,0058 68 78 88 98

-6,00

-4,00

-2,00

0,00

2,00

4,00

6,0028 38 48 58 68

0,00

0,20

0,40

0,60

0,80

1,00

0 5 10 15 20

0,00

0,20

0,40

0,60

0,80

1,00

0 5 10 15 20

0,00

0,20

0,40

0,60

0,80

1,00

0 5 10 15 20

0,00

0,20

0,40

0,60

0,80

1,00

0 5 10 15 20

Fig. 5 Zoomed shifted parts of wave trains at distances 8, 16, 24 and 32m and the corresponding amplitudes of the Fourier series (in cm) as functions of angular frequencies.

45

-10

-5

0

5

10

0 5 10 15 20 25

-10

-5

0

5

10

0 5 10 15 20 25

-10

-5

0

5

10

0 5 10 15 20 25

-10

-5

0

5

10

0 5 10 15 20 25

-10

-5

0

5

10

0 5 10 15 20 25

-10

-5

0

5

10

0 5 10 15 20 25

Fig. 6 Surface elevations of solitons (in cm) with an infinite period and envelopes measured at the distance 19.36m, from the piston and the consecutive ones with a spacing of 0.20m

46

SEASONAL CHANGES OF HYDRODYNAMICS IN THE GULF OF GDAŃSK – POSSIBILITIES OF REPRESENTATION BY NUMERICAL MODEL Małgorzata Robakiewicz Institute of Hydroengineering, Polish Academy of Sciences, Gdańsk, Poland Introduction Hydrodynamic conditions have considerable influence on the ecological changes in the Gulf of Gdańsk. Knowledge of the seasonal variations of water temperature, salinity, as well as magnitude and direction of flow enables better understanding of chemical and biological processes observed there. This information is interesting from the scientific point of view; it has also very important implications for the sustainable development of the area (i.e. in case of new investments located in the coastal zone). The basic information about processes taking place in nature comes from the in situ measurements. Quite often the necessary data are not available or are very incomplete; in such cases modelling approach can be an alternative. Modelling can be a very efficient tool, but before it can be applied in practice its calibration and verification for various conditions is required. This paper concentrates on possibilities of reproduction of seasonal changes of hydrodynamics in the Gulf of Gdańsk by the use of 3D model. Modelling approach General description of the area The Gulf of Gdańsk is situated in the southern part of the Baltic Sea (Fig. 1), and by convention is limited by the strait line connecting cape of Rozewie (54o50’N, 18o20’E) and cape of Taran (54o58’N, 19o59’E). The area is characterized by complex hydrodynamic conditions due to: interactions between the Baltic Sea and the Gulf through the wide connection; interactions between fresh water discharged by Vistula river and weakly saline waters of the Gulf; changeable meteorological conditions over the area, complicated bathymetry and diversified coast-line. All these features of the Gulf of Gdańsk have to be incorporated in the modelling approach to be able to reproduce hydrodynamic conditions taking place there.

Model set-up The numerical model of the Gulf of Gdańsk is based on Delft3D-FLOW package (wl|Delft Hydraulics, 1999). The area of interest is covered by curvilinear orthogonal grid in horizontal (Fig. 2) and 20 layers using sigma co-ordinates (Tab 1). The connection with the Baltic Sea is introduced by northern and eastern open boundaries, each of them divided into three sections to enable better representation of water temperature and salinity changes in time and space (see Fig. 2). To enable realistic representation of inflow of fresh water into the sea the lower most part of the Vistula river was incorporated in the model a schematic way. The bathymetry introduced into the model is given in Fig. 3. Some calibration works have been done using short term measurements; very intensive data from POLRODEX’96 experiment were in use (Robakiewicz 1997a, 1997b, 1998). To assess

47

possibilities to reproduce seasonal changes in the Gulf of Gdańsk the representation the year 1994 was chosen. Below data used to run the model and obtained results will be discussed.

Table 1 . Vertical discretization applied in the Gulf of Gdańsk model

Layer no Thickness [%] Layer no Thickness [%] 1 0.5 5-10 5.0 2 1.0 11-16 7.5 3 2.0 17-19 5.0 4 4.0 20 2.5

Initial and boundary conditions

The initial three dimensional fields of salinity and water temperature were created

based on spatial distribution of those parameters given in the layers 1 and 20 (see Figs. 4a, b; 5a, b). At the starting moment of calculations the uniform velocity 0.0 m/s in the Gulf was introduced. To reach realistic conditions for the January 1st 1994, the proposed model was run at first with average hydro-meteorological conditions for December 1993 (wind SW- 5 m/s, Vistula River – 750 m3/s, Pregoła – 115 m3/s and Niemen 550 m3/s) to reach steady state. As the next step calculation continued to reach conditions representing January 1st 1994. This was the staring point for the reproduction of the year 1994. Data characterizing the year 1994

The reproduction of the yearly changes required detailed information on the major forcing. The Gulf of Gdańsk is sensitive to local changes of hydro-meteorological conditions; as a consequence the data set including wind conditions (Hel station – Fig. 6a, b), water level variations in the area (Fig. 7a, b) and river discharges were collected (Fig 8). Additionally it was necessary to describe the open boundaries, including changes of water level, salinity and temperature. Based on the available in situ measurements on the South Baltic Sea it was possible to describe variation of salinity and temperature with the frequency three times per month. An example of the prepared data set for January 1994 is given in Table 2, where salinity in the layer above and below halocline and temperature above and below termocline are given. In January 1994 location of halocline and termocline at the northern and western boundaries were introduced at the depth of 70 meters. In the model exchange of heat between water and air was taken into account, based on collected appropriate set of data (solar radiation, humidity, air temperature). Having all those data it model was run for the chosen year 1994.

48

Table 2 Description of the open boundaries – example of January 1994

1 2 3 4 5 6 7 z-1 z-3 n-2

decade

z-2 n-1 n – 3 Salinity (psu)

7.46 7.36 7.36 7.40 7.5 7.63 6.9 ja1 7.86 12.1 12.1 8.20 9.5 9.84 7.84 7.50 7.40 7.40 7.4 7.5 7.7 7.0 ja2 8.80 11.9 11.9 8.0 9.5 10.0 8.0 7.48 7.40 7.39 7.40 7.53 7.66 6.95 ja3 8.03 12.05 12.05 8.02 10.4 10.4 7.92

Temperature (oC) 3.90 4.50 4.50 4.00 4.0 4.0 3.9 ja1 3.65 5.70 5.40 4.80 4.5 5.1 4.8 3.0 3.6 3.4 3.3 3.4 3.4 3.1 ja2 3.5 5.5 5.0 4.5 4.5 5.5 4.5 2.9 3.4 3.32 3.26 3.25 3.20 3.0 ja3 3.3 5.1 4.80 4.25 4.35 5.1 4.40

Comparison of measurements and calculations The crucial point in verification of any model is comparison between measured and reproduced values. For the year 1994 only salinity and temperature data were available; dates and locations are given in Table 3. Tab. 3 Locations and dates of available measurements

P1 P101 P110 P116 ZN2 K 19.01 20.01 20.01 19.01 20.01 20.01 8.02 7.02 8.02 8.02 8.02 8.02 3.03 2.03 3.03 3.03 2.03 3.03 21.04 19.04 20.04 21.04 20.04 20.04-1 21.05 20.05 21.05 21.05 20.05 20.04-2 5.07 4.07 10.07 5.07 4.07 21.05 13.07 22.09 12.07 12.07 12.07 11.07 7.08 23.11 3.08 4.08 3.08 22.11 6.12 23.11 6.12 22.09

5.12 22.11 Some results of calculations in comparison with measurements in locations P1, P110 and Zn2 are given in Figures 9, 10 and 11.

49

Additionally to visual comparison some simple statistical measures can be used: Average measured value: ∑ =

= n

i itpn

tp1

1 ,

Averaged modelled value: ∑ == n

i itmn

tm1

1

Relative error: [%]100*

tptmtp

et−

=

Averaged relative error: [ ]%100*1

1∑ =

−= n

ii

ii

tptmtp

neet

Root – mean square error: ( )

−= ∑ =

n

i ii tmtpn

ert1

21

where: tp - measured value (salinity or temperature) in the chosen vertical and on the given depth,

i

tmi - modelled value (salinity or temperature) in the chosen vertical and on the given depth.

Table 4 Results of statistical analysis for location P1 temperature salinity date tp tm et eet ert sp sm es ees ers oC oC % % oC psu psu % % psu 19.01.94 4.59 5.25 14.35 24.24 1.07 8.48 8.42 0.79 3.52 0.39 8.02.94 3.99 4.83 21.05 30.02 1.22 8.25 8.56 3.76 4.12 0.62 3.03.94 2.98 3.11 4.40 46.18 1.09 8.69 8.23 5.26 5.08 0.79

21.04.94 4.01 3.07 23.48 22.86 1.09 8.78 8.34 5.03 4.76 0.76 21.05.94 4.97 5.02 0.98 29.79 1.05 8.97 8.03 7.39 6.63 1.08 5.07.94 8.85 6.37 28.05 27.95 3.51 8.54 8.37 1.99 5.76 0.78

12.07.94 9.08 7.12 21.59 26.46 3.17 8.63 8.34 3.43 6.58 0.88 7.08.94 9.24 8.91 10.24 21.55 2.56 8.74 8.03 8.05 7.23 1.19 6.12.94 6.33 6.92 9.33 14.87 1.03 8.17 8.79 7.50 10.11 1.30

Table 5 Results of statistical analysis for location P110 temperature Salinity Date tp tm et eet ert sp sm es Ees ers oC oC % % oC psu psu % % psu 20.01.94 3.68 4.80 30.54 30.61 1.15 7.49 7.43 0.91 1.33 0.11 8.02.94 2.94 3.69 25.60 29.25 0.94 7.42 7.20 2.93 2.96 0.31 3.03.94 2.34 2.29 1.85 56.41 1.12 7.65 7.29 4.61 4.38 0.50

20.04.94 3.92 2.20 43.97 42.88 1.84 7.32 6.98 4.61 5.39 0.76 21.05.94 5.12 4.04 21.09 24.39 1.66 7.51 6.77 9.75 9.21 0.99 10.07.94 11.16 9.08 18.61 24.70 2.89 7.54 7.07 6.28 5.77 0.70 12.07.94 11.11 9.29 16.40 21.90 2.51 7.46 7.06 5.31 5.13 0.52 3.08.94 9.71 10.12 4.22 33.94 2.18 7.44 7.03 5.52 5.53 0.46

23.11.94 7.26 7.66 5.61 8.14 0.79 7.38 7.41 0.49 2.97 0.31 5.12.94 6.51 7.04 8.21 9.68 0.88 7.57 7.30 3.54 4.38 0.76

50

Table 6 Results of statistical analysis for location Zn2 for the surface layer temperature Salinity Date tp tm et eet ert sp sm es Ees ers oC oC % % oC psu psu % % psu

K 7.30 5.67 22.42 19.67 2.52 6.25 6.09 2.46 9.43 0.60 p101 7.84 5.85 25.36 37.67 3.36 7.06 6.23 11.72 12.10 1.08 Zn2 10.83 7.97 26.37 30.61 33.75 5.00 3.11 37.67 40.51 2.43

The number of measured data was limited, however some conclusions based on calculations carried out can be drawn: - The results obtained show relatively good agreement between measured and

calculated salinity and water temperature in the Gulf of Gdańsk, taking into account that rather general information on wind conditions and heat exchange transfer (one measuring station in each case) was available;

- The reproduction of yearly changes of hydrodynamics in the area where forcing is of random character requires detailed information on its changes in time and space;

- The representation of temperature in the upper part of the water column is better than that in the lower part.

- It is very characteristic that the calculated errors are higher for temperature than for salinity, additionally the averaged relative error is higher than the relative error. This indicates disagreement between vertical profiles: measured and calculated.

- Analysis based on simple statistical measures enables comparison of different parameters which is difficult to achieve based on visual comparison.

References: Robakiewicz M., 1997a, Time variability of water flow, temperature and salinity – comparison of experiment’s results with HIROMB model forecast, Bull. Maritime Institute, Gdańsk, 24 (1), 23-34. Robakiewicz M. 1997b, Hydrodynamic conditions in the Gulf of Gdańsk during

POLRODEX’96 experiment in comparison with numerical model, Oceanological Studies, 16 (4), 145-159.

Robakiewicz M., 1998, Modelling hydrodynamics conditions in the South Baltic Sea –

TRISULA applications, Hydroinformatics’98, Balkema, 1447-1454. Wl|Delft Hydraulics, 1999, DELFT3D-FLOW User Manual, the Netherlands

51

Fig. 1 General view on the Gulf of Gdańsk

Fig. 2 Horizontal grid for the 3D model

52

Fig.3 Bathymetry of the Gulf as applied in the 3D model

Fig. 4a Salinity distribution in the top layer – initial conditions

53

Fig. 4b Salinity distribution in the bottom layer – initial conditions

Fig. 5a Temperature distribution in the top layer – initial conditions

54

Fig. 5b Temperature distribution in the bottom layer – initial conditions

J F M A M J J A S O N D0

60

120

180

240

300

360

[deg

]

J F M A M J J A S O N D

0

2

4

6

8

10

12

[m/s

]

Fig. 6 Wind direction (left) and velocity (right) as measured in 1994 at Hel station

55


440

460

480

500

520

540

560

580

Gdansk

Fig. 7a Water level changes at the Gdańsk station as measured in 1994

01 03 05 07 09 11 13 15 17 19 21 23 25 27 29 31January 1994

480

500

520

540

560

580

GdanskGdyniaHel

Fig. 7b Comparison of water level changes at the stations: Gdańsk, Gdynia and Hel as

measured in January 1994

J F M A M J J A S O N D(months)

0

1000

2000

3000

4000

Q[m

3 /s]

Prego³aNiemenVistula River

Fig. 8 Discharge changes in rivers Vistula, Pregoła and Niemen in the year 1994

56

Salinity (psu) Temperature (oC)

surf

ace

J F M A M J J A S O N D6.4

6.8

7.2

7.6


5

10

15

20

25

20 m


7.0

7.2

7.4

7.6


5

10

15

50 m


7.2

7.6

8.0

8.4


5

10

15

70 m


7

8

9

10

11


5

10

botto

m


11

12

13

14


4

5

6

7

Figure 9 Comparison between measured (symbol) and calculated (solid line) salinity

and temperature variations in location P1 in the year 1994 on the chosen depths

57


surf

ace


6

8


5

10

15

20

25

20


7.0

7.2

7.4

7.6


4

8

12

16

30


7.0

7.2

7.4

7.6


4

8

12

16

50 m


7.2

7.6

8.0


4

8

12

Bot

tom


4

8

12


2

4

6

8

10

Figure 10 Comparison between measured (symbol) and calculated (solid line) salinity

and temperature variations in location P110 in the year 1994 on the chosen depths

58


surf

ace


2

4

6

8


510

15

20

25

botto

m


6.4

7.2

8.0


5

10

15

Figure 11 Comparison between measured (symbol) and calculated (solid line) salinity and temperature variations in location ZN2 in the year 1994 on the chosen depths

59

CROSS-SHORE SEDIMENT TRANSPORT AND SEA BED EVOLUTION BY A QUASI PHASE-RESOLVING MODEL Rafał Ostrowski Institute of Hydroengineering of the Polish Academy of Sciences – IBW PAN Gdansk, Poland Introduction Coastal changes (evolution) can be accumulative or erosive. Accumulative trends are inconvenient at harbour entrances, approach channels and water-ways, where they cause navigable problems, mostly due to silting up (sedimentation). Coastal erosive threats are obvious, especially in view of predicted climate changes resulting in accelerated sea level rise, increase of storms intensity and a number of other effects. Erosive processes occur at a lot of coastal segments of the Baltic and other seas. They are modelled theoretically and investigated in situ, for instance at the IBW PAN Coastal Research Station in Lubiatowo, Poland. Abrasion of beach and dune results from erosive changes of the entire cross-shore profile. Therefore, there is a need of thorough investigations of the cross-shore profile evolution. A lot of models have been developed to simulate multi-scale changes of the sea bed: simplified engineering models, data-driven models and more or less sophisticated process-based models. However, the results of modelling are still a long way from excellence and further studies are undertaken over and over again. The scheme of the conventional deterministic modelling system is presented in Fig. 1. Such a set of models has been developed by the IBW PAN research team of the Department of Coastal Engineering and Dynamics. The computational framework follows a classical way based on theoretical description of the physical processes occurring in a coastal zone.

WIND

OFFSHORE (DEEP-WATER) WAVES

WAVE TRANSFORMATION

WAVE - DRIVEN CURRENTS

BED SHEAR STRESS

SEDIMENT TRANSPORT

SEA BED CHANGES

Moveable bedroughness

PRO

CES

SB

ASE

D M

OD

ELLI

NG

-

Fig. 1. Deterministic modelling framework for coastal morphodynamics First, processes of wave transformation and breaking are determined, as well as wave-induced currents, quantitatively much dependent on the features of the cross-shore profile (these stages

60

of hydrodynamic modelling are not presented herein). Then the bed shear stresses are calculated, interactively dependent on the sea bed roughness, after which sediment transport is found, dependent on both shear stresses and the roughness. Finally, sea bed changes are modelled conventionally as a result of spatial variability of the net sediment transport rates. Bed boundary layer and roughness of moveable sea bed Let us imagine that we know the velocity profile in the bed boundary layer. For instance, let it be a logarithmic profile, as shown in Fig. 2. Further, let us assume that we know the velocity of the water-soil mixture caused by the upper flow. The velocity distribution of water flow is dependent on the friction velocity uf, bed roughness ke, von Karman constant κ and ordinate z in accordance to the following formula (note that at z0=ke/30 the flow velocity equals zero):

=30

lne

f

kzu

κu (1)

As theoretically deducted by Kaczmarek & Ostrowski (1996), the water flow overlaps with the moveable bed layer (δn thick) in the following way:

5.0=n

sx

δδ

(2)

u

c

ub

c0

A

z’’

cms

z = k /0 e 30

0δsx

δb δn

z’

granular-fluidshear region

turbulent fluidshear region

theoretical bed level

Fig. 2. Definition sketch of moveable sea bed boundary layer

The friction velocity uf depends on ke. In turn, the parameters of the bedload layer (δn, δb and ub) depend on uf. The water flow layer is solved interactively with the moveable bed layer to find the roughness ke. Within the iterative procedure, a matching point A is found (see Fig. 2) at which the velocity profiles cross each other, yielding a slip velocity ub at the theoretical bed level. The computations run for a number of wave conditions and two diameters of sand grains yield the following approximate formulas, represented graphically in Fig. 3:

61

658.05.2 47 −= θ

dke (regular sinusoidal waves) (3)

71.05.2 64.26 −= θ

dke (irregular waves) (4)

in which, after Nielsen (1992):

gdsa

ff m

)1()(

21

21 2

15.215.25.2 −

==ωψθ (5)

−

= 977.55.2213.5exp

194.0

15.2

madf (6)

where g is the acceleration of gravity, d – grain diameter, a1m=U1m/ω (amplitude of water motion), U1m – amplitude of oscillatory free stream velocity (for sinusoidal waves U(ωt)=U1msin(ωt)).

0.01 0.10 1.00 10.00θ2.5

10

100

1000

k e/d

large scale conditions, d=0.0002 mlarge scale conditions, d=0.0004 msmall scale conditions, d=0.0002 msmall scale conditions, d=0.0004 mpower approximation, Eq. (3)power approximation, Eq. (4)

Fig. 3. Calculated roughness of moveable sea bed

Bed shear stress

Equation of motion in the bed boundary layer reads:

( )z

Uut

∂∂=−

∂∂ τρ (7)

where U is a free stream velocity which for different wave regimes can come from the sinusoidal wave theory, Stokes approximations or cnoidal theory.

Making use of the logarithmic velocity profile described by Eq. (1), Fredsøe (1984) integrated Eq. (7) over the bed boundary layer thickness δ :

( ) b

z

z

z

z

dzz

dzuUt

ττρδδ

−=∂∂=−

∂∂− ∫∫

++ 0

0

0

0

(8)

Assuming that the bed shear stress is defined as:

62

2fb uρττ =≈ (9)

and introducing a dimensionless variable:

κfu

Uz =1 (10)

the following differential equation is obtained:

( ) ( )( )( ) ( )td

dUUze

zezzek

Utd

dzz

z

ze

1111

1130

1

11

1

21

1

1

1 ωωκ

ω +−−−−

+−= (11)

For combined co-linear wave-current flow, with the friction velocity related to a steady current at the top of the bed boundary layer uf0, analogous considerations are as follows:

( ) ( )∫+

−∂∂−=−

30/

30/

N

N

k

k

b dzuUt

δ

ρτ

ρδτ (12)

( ) 20fuρδτ = (13)

κ0

1ff uu

Uzm

= (14)

( )( )( ) ( )

( )( ) ]11[

30111

1)(

1

01

1

11111

1

+−+

++−

−+=

zeuzU

ktddU

Uzeezz

tdzd

zf

ez

z

ωκκ

ωω (15)

From the solution of Eqs. (11) and (10) or Eqs. (15) and (14), the friction velocity uf0 can be calculated, as well as the boundary layer thickness:

( 130

1 −= ze ekδ ) (16)

The results of the bed shear stress calculations are used in the sediment transport model. Sediment transport The presented theoretical sediment transport system consists of three layers, as proposed by Kaczmarek & Ostrowski (1998), namely the bedload layer, the contact load layer (transitional layer) and the outer flow region, see Fig. 4. The collision-dominated bedload layer granular-fluid region stretches below the theoretical bed level while the turbulent fluid region extends above it constituting the contact load layer and the outer flow layer. The outer region of pure suspension is characterised by very small concentrations, where the process of sediment distribution may be considered as a convective and (or) diffusive process. In contrast, the granular-fluid region below the nominal bed is characterised by very high concentrations, where the inter-granular resistance is predominant. The velocity profile in the contact load layer is assumed to be continuous. Its intersection with the nominal seabed is the apparent slip velocity ub, identified as a characteristic velocity of sediment moving in the form of bedload. The downward extension of the velocity distribution in the outer zone of the main flow yields a fictitious slip velocity u0 of the fluid at the nominal static bed level. Clearly, the fluid velocity u0 is greater than the sediment velocity ub.

63

Fig. 4. Three-layer sediment transport model

The mathematical model of the bedload transport is based on the water-soil mixture approach presented by Kaczmarek (1991). In this approach, the bedload sediment transport model is based on a collision-dominated drag concept and uses the effective roughness height, ke from Eq. (3), of the above-bed wave-induced flow. In the sub-bed flow region, the sediment concentration is high and chaotic collisions of grains are the predominant mechanism. Particle interactions are assumed to produce two distinct types of behaviour. The Coulomb friction between particles give rise to rate-independent stresses (of the plastic type) and the particle collisions bring about stresses that are rate-dependent (of the viscous type). The use of the mathematical description for determination of the stress tensor and the balance of linear momentum according to Kaczmarek (1991) leads to the following equations

22

100

'2sinsin f

m

uzu

cccc ρ

∂∂µψϕα =

+

−

− (17)

( ) ( )

( ) ∫−+

+

=

++−

−

−

=

'

0

2

1

20

2

2000

'

2sinsin1

0

z

sf

cc

m

cdzgu

zcu

cccc

ρρρµ

µµ

∂∂µµψϕα

(18)

where: c0, cm - solid concentrations corresponding to fluidity and the closest packing,

respectively; µ0, µ1, µ2 - functions of the solid concentration c, after Sayed & Savage (1983):

5.121

)(03.0ccd ms −

=ρµ (19)

75.1220

)(02.0ccd ms −

=+ρ

µµ (20)

ϕ - angle of internal friction ψ - angle between the major principal stress and the horizontal axis, Sayed & Savage

(1983):

24ϕπψ −= (21)

64

Further, the following assumptions were made by Kaczmarek (1991):

10

=gdsρ

α cm = 0.53 c0 = 0.32 ϕ = 24.4° (22)

In the contact load layer, the sediment velocity and concentration is modelled using the following equations, proposed by Deigaard (1993):

( ) 22

2222

'32

23

fMD

M

s

udzdulcscd

ccs

dzdu

wd =

++

β+

+α (23)

cwdzdc

dzdulc

dzdud

ccs

dzdu

wd

sD

M

s

−=

+

β+

+α 22

2

323 (24)

where ρuf’2(ωt) - “skin friction”, calculated from the integral momentum method (see the previous section) with the roughness ke’ = 2.5d (effective skin roughness, Nielsen (1992)). ws - settling velocity of grains cM, cD - added mass and drag coefficients, respectively α, β - coefficients, Deigaard (1993) l - mixing length (l = κz) Boundary conditions for Eqs. (23) and (24) are the instantaneous sediment velocity u (equal to ub) and concentration c (equal to c0) at the upper limit of the bedload layer. Besides, it is assumed that (s+cM)=3.0 and cD=1.0. The sediment transport rates in the bedload layer and in the contact load layer are calculated as the phase-resolving (instantaneous) quantities, i.e.:

∫∫ ⋅+⋅=c

e

b

k

dztzctzudztzctzutqδδ

ω30/'0

),(),('),'(),'()( (25)

For the outer flow, there are a lot of difficulties in proper determination of time-dependent concentrations. At higher levels, the measured time series of concentration show rather complicated structure and compliance in phase between theoretical models and the data is lost, as reported e.g. by Davies et al. (1997). Therefore, the net sediment transport rate in the outer flow has to be determined using the following simplified formula:

∫ ⋅=h

y

c

dzzczuqδ

)()( (26)

where the time-averaged concentration is obtained from a conventional relationship, e.g. by Ribberink & Al-Salem (1994):

1

)()(αδδ

==

zzczc c

c (27)

in which the concentration decay exponent α1 is an unknown value, which has to be determined, e.g. from experiments. The sediment transport modelling system has been thoroughly tested versus available laboratory and field data. Some results are depicted in Figures 5 and 6.

65

10 100computed net sediment transport *10-6 [m2/s]

10

100m

easu

red

net s

edim

ent t

rans

port

*10-

6 [m

2 /s]

bedload + contact load,Katopodi et al. (1994)total load (incl. outer region),Katopodi et al. (1994)bedload + contact load,Ribberink & Al-Salem (1994, 1995)total load, field tracer experiment,Balouin & Howa (2000)perfect compliancedisagreement factor 1.5

Fig. 5. Net sediment transport rates: present model results versus laboratory data of Katopodi et al. (1994), Ribberink & Al-Salem (1994, 1995)

and field tracer data of Balouin & Howa (2000)

10 100computed net sediment transport *10-6 [m2/s]

10

100

mea

sure

d ne

t sed

imen

t tra

nspo

rt *1

0-6

[m2 /

s]

Dohmen-Janssen (1999)perfect compliancedisagreement factor 1.5d=0.21 mm; bedload+ contact loadd=0.21 mm; total loadd=0.32 mm; bedload+ contact loadd=0.32 mm; total load

Fig. 6. Net sediment transport rate: present model results versus laboratory data by Dohmen-Janssen (1999)

Sea bed changes

66

Coastal hydrodynamics (waves and currents) is a driving force for sediment transport and sea bed evolution. In the cross-shore domain, interaction of wave motion and return flow (undertow) is of key importance. It should be noted that the balance or imbalance between wave asymmetry and undertow can lead to various types of resultant flow (and sediment flux), as depicted in Fig. 7, in which the scheme on the left-hand side is typical for the surf zone, while the scheme on the right-hand side represents the situation at a location far offshore, where no wave-driven currents occur.

z z z z

bed

boun

dary

laye

r

returnflow

returnflow

returnflow

direction ofwave propagation

bedload bedload bedloadtheoreticalbed level

outer flow (beyond bed boundary layer)

bedload

contact load contact load

Fig. 7. Schemes of wave-current interaction in the nearshore zone

For the wave-current situations of Fig. 7, the net sediment transport rates are calculated along the entire cross-shore profile. Consequently, from these net transport quantities, the sea bed profile evolution can be modelled. Following conventional approaches, the evolution of the sea bed profile is determined on the basis of spatial variability of net sediment transport rates, from the following continuity equation for sediment in the direction perpendicular to the shore:

( ) ( )xnt ∂−∂ 1

txqtxh ∂=∂ ,1,

( )

(28)

where: q – net sediment transport rate [m2/s] in the cross-shore direction per unit width n – the soil porosity while deposited x – cross-shore coordinate t – time Solution of Eq. (28) can be easily obtained, e.g. by a finite difference scheme. To control excessive and non-physical slope growth, Rakha et al. (1997) introduced an additional diffusive term into Eq. (28), which yielded the following differential equation:

21 xxhnt ∂∂∂−∂

2 ,,,1, txhKtxhtxqtxh ∂=∂∂−∂ (29) ( ) ( ) ( )

in which, following Watanabe et al. (1982), K is a diffusion coefficient assumed to be proportional to q, namely:

qK ε= (30) where ε is an empirical coefficient. To solve Eq. (29) explicitly, stability criteria should be satisfied with respect to the spatial step ∆x and the time step ∆t. Secondly, the solution requires an arbitrary assumption with respect to the coefficient ε in Eq. (30).

67

Within an alternative approach, a so-called modified Lax scheme can be used, with a dissipative interface, as given by Rakha et al. (1997), yielding the following finite-difference equation:

xqq

nthh j

ij

ij

ij

i

∆−

−=

∆− −+

+

211 11*

1

ji 1−

(31)

( ) ji

ji

ji hhhh 1* 21+ +−+= ααα

in which: (32)

In the above equations, used in the present study, the counters i and j refer to the spatial grid and time, respectively, while αL is a coefficient assumed as 0.25. According to Rakha et al. (1997), the value αL≤0.5 provides the stable solution. Practically, application of Eq. (32) denotes a kind of smoothing of the sea bed profile at the time j. This smoothing neutralises inaccuracies of sediment transport calculations, resulting in unrealistic sea bed changes, which have been assumed as model instabilities by Rakha et al. (1997). It is very convenient to start the computations from the offshore location, where the sediment transport does not exist as the waves are deep water waves and do not affect the sea bed. Furthermore, there are no wave-driven currents at this location (except for the wave drift between wave crest and trough, which does not cause any sand motion). While going onshore with the solution of Eq. (31), at smaller water depths, the net sediment transport appears and increases. Simultaneously, the compensative wave-driven return current starts to play more and more important role. This current, called the undertow in the surf zone, can be a predominating factor, locally causing the offshore sand transport. This is all accounted for in determination of q(x,t), which is used in Eq. (31), yielding the change of water depth h(x,t), as shown in Fig. 8.

erosion∆x

∆x

∆x

accumulation

erosion

qi+1 qi

qi+1 qi

qi+1 qi

X. .

. ..

.

.

.. .

.

. . .

..

..

..

....

...

.. .

.

.. .

.

.

. ... .

..

. .

..

.

.

. ..

...

.

..

.

.

...

. ..

... ...

... .

..

.. ..

.

h.

.

.

..

.

.. .

..

..

.. . . . . . ...

...

.

. .

.

..

.

.

..

.

.

.

.

.

.

..

.

.

..

.. . .. .

.

.. .

.

surf zone

wavebreakinglocation

region beyond surf zone.

Fig. 8. Definition sketch for calculations of the cross-shore profile changes

The model have been first run with the following sea bed soil parameters: median grain diameter d50=0.21 mm, settling velocity ws=0.026 m/s and relative density ρs/ρ=2.65. The sea bed porosity has been assumed as n=0.4. One should note that n=0.5 corresponds to loosely packed grains while n=0.25 is a minimum porosity for non-graded spherical grains.

68

The initial test has been made for the sea bed having a uniform slope of the cross-shore profile. The computations have been carried out for the deep water wave height Hrms=1.5 m and the period Tp=6.5 s. The results after the first time step, comprising the wave height, undertow nearbed time-averaged velocity and net sediment transport rates, are presented in Fig. 9. Aside from the total net sediment transport rate qtotal, all its components determined within the three-layer sand transport model are distinguished in Fig. 9, namely the bedload qb, the contact load qc (denoting the sediment suspended in a thin nearbed layer) and the suspended load qs (sediment particles suspended in the water column high above the sea bed). It can be seen from Fig. 9 that the flow velocity increases very slowly landwards, attaining only a few cm per second before wave breaks. Simultaneously, wave asymmetry causes a distinct increase of all the sediment transport components. The suspended load transport far from the bed qs relatively increases as well, although it is small in comparison to the other components. The nearbed flow grows up rapidly at the wave breaking point and the proper undertow starts to affect the bed boundary layer. This causes local great variabilities (also qualitative) in the net sediment transport rates. The wave breaking points appears to be a location at which the sediment fluxes converge. Further landwards, the wave motion restores after breaking and simultaneously becomes more and more asymmetrical due to decreasing depth. This asymmetry effect predominates over the undertow and the resultant sand transport is directed onshore. Close to the shoreline, waves collapse and the return flow increases which results in offshore directed sediment flux. The spatial variabilities in the net transport yield the greatest sea bed changes at the location of wave breaking, which seems to be realistic.

0 200 400 600 800 1000 1200distance offshore [m]

-10

-8

-6

-4

-2

0

sea

bed

ordi

nate

[m]

0

0.4

0.8

1.2

1.6

H [m

]

0

0.2

0.4

0.6

0.8

Um

ean [

m/s

]


-1x10-5

0x100

1x10-5

2x10-5

3x10-5

4x10-5

net t

rans

port

rate

[m3 /s

/m]

wave height Hnearbed undertowvelocity Umean

qb

qc

qs

qtotal

Cross-shore profileinitialafter 1 dayafter 2 daysafter 5 days

440 460 480 500 520 540 560[m]

-5.2

-4.8

-4.4

-4

-3.6

[m]

Fig. 9. Modelled hydrodynamics, net sediment transport rates and short-term evolution of uniformly sloped cross-shore profile

69

The second test model run has been conducted for a natural cross-shore transect measured at the CRS Lubiatowo. Here, again, the offshore wave parameters have been assumed as Hrms=1.5 m and Tp=6.5 s. The results of these calculations, depicted in Fig. 10, show a more complicated system of hydrodynamics and lithodynamics. This system, however, is consistent with the idealised situation of the uniformly sloped sea bed profile of Fig. 9. In particular, it can be seen from Fig. 10 that the return flow velocity increases at the wave breaking point, then showing the variability with the maxima over the bars where the wave has its additional minor breakings. The variability of net sediment transport rates follows the cross-shore changes of hydrodynamics, revealing a great jump nearby the first wave breaking location, similarly as in Fig. 9.


-10

-8

-6

-4

-2

0

sea

bed

ordi

nate

[m] 200 400 600 800 1000 1200

0

0.4

0.8

1.2

1.6

H [m

]0

0.2

0.4

0.6

0.8

Um

ean [

m/s

]


-1x10-5

0x100

1x10-5

2x10-5

3x10-5

net t

4x10-5

rans

port

rate

[m3 /s

/m]

wave height Hnearbed undertowvelocity Umean

qb

qc

qs

qtotal

Fig. 10. Modelled hydrodynamics and net sediment transport rates on a natural multi-bar cross-shore profile

An attempt has also been made to reproduce the hydrodynamics and lithodynamics together with short-term changes of the cross-shore profile registered at CRS Lubiatowo in 2001. The natural wave conditions (irregular waves actually) were represented in the model by the root-mean-square wave height Hrms and the peak period Tp. The input wave parameters were found from the offshore measurements carried out by the directional waverider buoy. The records of representative wave parameters, calculated from the raw waverider data series, are plotted in Fig. 11.

70

0.0

0.5

1.0

1.5

2.0

Hrm

s [m

]

0

2

4

6

8

10

T p [s

]

16.0

9.01

17.0

9.01

18.0

9.01

19.0

9.01

20.0

9.01

21.0

9.01

22.0

9.01

23.0

9.01

24.0

9.01

25.0

9.01

26.0

9.01

27.0

9.01

28.0

9.01

29.0

9.01

30.0

9.01

01.1

0.01

02.1

0.01

03.1

0.01

04.1

0.01

05.1

0.01

06.1

0.01

07.1

0.01

08.1

0.01

09.1

0.01

10.1

0.01

11.1

0.01

12.1

0.01

13.1

0.01

14.1

0.01

15.1

0.01

16.1

0.01

Hrms

Tp

-200

-100

0

100de

ep w

ater

wav

e ap

proa

ch a

ngle

[deg

]

200

16.0

9.01

17.0

9.01

18.0

9.01

19.0

9.01

20.0

9.01

21.0

9.01

22.0

9.01

23.0

9.01

24.0

9.01

25.0

9.01

26.0

9.01

27.0

9.01

28.0

9.01

29.0

9.01

30.0

9.01

01.1

0.01

02.1

0.01

03.1

0.01

04.1

0.01

05.1

0.01

06.1

0.01

07.1

0.01

08.1

0.01

09.1

0.01

10.1

0.01

11.1

0.01

12.1

0.01

13.1

0.01

14.1

0.01

15.1

0.01

16.1

0.01

Fig. 11. Offshore wave conditions at CRS Lubiatowo during field campaign in 2001; signs (+) and (–) in the record of wave angle stand for wave approach

westwards and eastwards from the cross-shore profile, respectively After having analysed the wave record, the period 16-27 September was found to be particularly useful because of two distinct short storms, which could affect the cross-shore profile substantially. The wave directions during these storms were approximately perpendicular to the shoreline and therefore one could suppose the cross-shore sediment transport to be predominant over the longshore transport. The hydrodynamic deep water wave input is given in the table below.

Wave record at CRS Lubiatowo in the period 16-27 Sept. 2001

Hrms [m] 0.5 1.0 0.9 0.7 0.5 0.4 0.3 0.2 0.1 0.2 0.5 0.8 1.3 1.5 1.3 1.1 0.9 0.6 0.2

Tp [s] 5.7 5.6 6.6 6.6 6.4 6.1 5.9 5.6 5.0 3.9 4.4 6.1 6.7 7.7 7.6 7.2 6.8 6.0 4.0

duration [hours] 4 8 8 4 4 8 4 8 12 92 12 4 4 8 4 8 4 8 24

71

The initial cross-shore profile of 16th September was entered into the model with the spatial resolution (step) ∆x=10 m. The time step ∆t of 4 hours has been assumed, after which the sea bed changes have been calculated from Eqs. (31) and (32), as well as the wave-current transformation over the new cross-shore profile has been updated. The net sediment transport rates for the storm moment with Hrms=1.5 m and Tp=7.7 s, as well as the final sea bed profile (after the last time step) are shown in Fig. 12.

0 100 200 300 400 500 600 700 800 900distance offshore [m]

-8

-7

-6

-5

-4

-3

-2

-1

0

sea

bed

ordi

nate

[m]

initial sea bed measured on 16.09.01 final sea bed measured on 27.09.01 final sea bed calculated for 27.09.01

100 200 300 400 500 600 700 800 900

0x100

4x10-5

8x10-5

1x10-4

net t

rans

port

rate

[m3 /s

/m] Hrms=1.5 m, Tp=7.7 s

qb

qc

qs

qtotal

Fig. 12. Change of cross-shore profile at CRS Lubiatowo during field campaign in 2001: model results versus field data; plotted sediment transport rates correspond to

the storm peak The comparison presented in Fig. 12 shows that the model produces distinct sea bed changes at the second bar only, while the field data reveal the evolution of the entire cross-shore profile. At the second bar, however, the compliance between the theoretical and experimental results appears to be good. It should be noted the model results for high waves (cf. also Fig. 10) show significant changes in net sand transport mostly at the third bar. However, due to short duration of such conditions, these changes cause very small changes of the sea bed. The predominating sea bed evolution at the second bar is probably related to mean long-lasting conditions, with highly random (irregular) waves. Such conditions can not be considered by

72

the present model. Furthermore, it can be supposed that the observed evolution of the sea bed has partly resulted from coastal morphodynamics in the longshore domain. Final remarks and conclusions On the basis of presented considerations and results, the following conclusions can be drawn: • Sediment transport is very sensitive to hydrodynamics. The relationship between

these two processes is highly non-linear and ought to be investigated thoroughly. Even a small change of proportions between the wave asymmetry effects and the undertow can result in a great modification of net sediment transport rate, not only quantitative but qualitative as well.

• Randomness of coastal hydrodynamics is represented in the model only by the representative wave height (Hrms) and period (Tp). In the study of Kaczmarek & Ostrowski (1996) it was deducted that the present sediment transport approach yields proper results for natural (irregular) wave conditions if Hrms and Tp are assumed as the representative input. Certainly, this can not be guaranteed for all types of wave spectra. Thus, the prediction of coastal morphodynamics does not account for all components in random wave series.

• The present approach is restricted to one cross-shore profile and is an approximation of actual hydrodynamic and lithodynamic situations, during which the wave forcing can be directed obliquely to the shoreline. In such conditions, coastal morphodynamic processes (shoreline displacement and sea bed changes) are much influenced by longshore wave-driven currents and longshore sediment transport. However, the present achievements can be applied in a combined approach, enabling the solution of longshore coastal changes with cross-shore modifications and corrections. Such models (e.g. the Swedish-American software GENESIS and the Dutch package UNIBEST of Delft Hydraulics), based on the so-called one-line theory and supplemented by the cross-shore transport amendments, are also formulated and presented in literature, see e.g. Hanson et al. (1997). Till now, unfortunately, insufficient precision in determination of cross-shore sediment transport distributions has forced the modellers to “guess” the cross-shore transport values, thus making these models less effective.

Acknowledgements The study was sponsored by KBN, Poland, under programme 2 of IBW PAN, which is hereby gratefully acknowledged. References Balouin, Y. & H. Howa (2000). Correspondence contact with University of Bordeaux. Davies, A.G., J.S. Ribberink, A. Temperville & J.A. Zyserman (1997). Comparisons between sediment transport models and observations made in wave and current flows above plane beds, Coastal Engineering 31, 163-198. Deigaard, R. (1993). Modelling of sheet flow: dispersion stresses vs. the diffusion concept, Prog. Rep. 74, Inst. Hydrodyn. and Hydraulic Eng., Tech. Univ. Denmark, 65-81. Dohmen-Janssen, M. (1999). Grain size influence on sediment transport in oscillatory flow, FEBODRUK BV, ISBN 90-9012929-4, Enschede, The Netherlands. Fredsøe, J. (1984). Turbulent boundary layer in combined wave-current motion, J. Hydraulic Eng., ASCE, Vol. 110, No. HY8, 1103-1120. Hanson, H., M. Larson, N.C. Kraus & Michele Capobianco (1997). Modelling of seasonal variations by cross-shore transport using one-line compatible methods, Proc. Coastal Dynamics ’97, ASCE, Reston VA, 893-902.

73

Kaczmarek, L.M. (1991). Mathematical model for oscillating sheet flow, Proc. Euromech 262 Colloquium on Sand Transport in Rivers, Estuaries and the Sea, A.A. Balkema / Rotterdam / Brookfield, 197-202. Kaczmarek, L.M. & R. Ostrowski (1996). Asymmetric and Irregular Wave Effects on Bedload: Theory versus Laboratory and Field Experiments, Proc. 25th ICCE, ASCE, New York, 3467-3480. Kaczmarek, L.M. & R. Ostrowski (1998). Modelling of a three-layer sediment transport system in oscillatory flow, Proc. 26th ICCE, ASCE, Reston VA, 2559-2572. Katopodi, I., J.S. Ribberink, P. Ruol & C. Lodahl (1994). Sediment Transport Measurements in Combined Wave-Current Flows, Proc. Coastal Dynamics ‘94, ASCE, New York, 837-851. Nielsen, P. (1992). Coastal bottom boundary layers and sediment transport, Advanced Series on Ocean Engineering, Vol. 4, World Scientific, Singapore. Rakha, K.A., R. Deigaard & I. Brøker (1997). A phase-resolving cross shore sediment transport model for beach profile evolution, Coastal Engineering, Elsevier Science B.V., 31, 231-261. Ribberink, J.S. & A. Al-Salem (1994). Sediment transport in oscillatory boundary layers in cases of rippled beds and sheet flow, Journal Geoph. Res., Vol. 99, No. C6, 12707-12727. Ribberink, J.S. & A. Al-Salem (1995). Sheet flow and suspension of sand in oscillatory boundary layers, Coastal Engineering 25, 205-225. Sayed, M. & S.B. Savage (1983). Rapid gravity flow of cohesionless granular materials down inclined chutes, J. Applied Mathematics and Physics (ZAMP), Vol. 34, 84-100. Watanabe, A. (1982). Numerical models of nearshore currents and beach deformation, Coastal Engineering in Japan, Vol. 25, JSCE, 147-161.

74

SEDIMENT TRANSPORT MODELLING IN LOVER ODRA RIVER

Zygmunt Meyer, Adam Krupiński

Technical University of Szczecin, Department of Geotechnical Engineering, Poland

Abstract

Analysis of sediment transport in river junction has been presented. Based at the mathematical

model simulation of sediment mixing in river junction of Odra and Warta river at Kostrzyń

has been proceeded. The mathematical model, which is based upon the Ackers-White’s

formula and was checked and verified for the conditions of the Lower Oder River, was

applied to the calculations. Chezy’s and Manning’s formulae as well as the modified

Strickler’s formula were used to define relationships between flow intensity, roughness

coefficient and parameters of transported sediment. The analysis of results showed a big

influence of sediment composition upon the formation of water levels and flows in the river

junction. The verified model was used for mixing of sediment transport analysis in case of

varying sediment composition. Two different diameters of sediment grains were taken and the

fractial content of each of them was varying. Depending at this changes the resulting sediment

composition after the junction was calculated, and furthermore additional hydraulic

parameters were also evaluated. The results indicate the Warta river sediment stream has very

little significance on the resulting sediment stream after the junction.

Introduction During investigating the problems of sediment transport it was noticed that in some special

places (such as river junctions and estuaries) there was a phenomenon of sediment sorting,

consisting in joining both water streams and sediment transport rates, which are mixed

together and sorted. The mechanisms ruling over the sediment sorting are strictly connected

with the hydraulic parameters of the junctions. Such junction is usually a place of shallowings

and deepenings, which is very important to determine the stability of water structures and

navigation.

The main element influencing the shipping in the navigable rivers is to maintain its depth at

the higher level than the minimal navigation depth. The river water flow velocity, which

affects the vessel manoeuvring, is the other element. It is commonly believed that the flow

increase makes the river deeper, on the other hand some places can be observed in the

75

navigable river that get shallow in spite of flow intensity increase. The analysis of this

phenomenon indicates that the relationship between the flow intensity and the depth depends

strictly on the composition of the transported sediment. The field observations have shown

that the river depth after the junction is strongly dependent on the sediment composition – its

small change brings about considerable changes in the river bottom junction.

Also during the researches that have been carried out so far concerning the sediment transport

in the area of the Lower Oder River, the measurements were made in the Oder and Varta

Rivers’ junction in the area of Kostrzyń. The Varta River is one of the biggest Oder River

inflows, and the Kostrzyń junction is one of the most important junctions in the water way

connecting the industrialised Silesia with the Szczecin-Świnoujście port. The measurement

results allowed verifying the mathematical model of sediment stream mixing in the Oder and

Varta Rivers’ junction, which was constructed in the Department of Geotechnical Engineering

at the Technical University in Szczecin. Applying this mathematical model and evaluative

algorithm, one analysed the changes of river bottom after the river junction, when the

geometrical dimensions of riverbed, flow intensity and bottom material composition in

inflows are known and the riverbed width and flow in outflow are known as the one resulting

from the flow balance in the junction. In the paper it was proved also that the changes of

bottom material composition in the inflow influence significantly the process of sediment

mixing, which consequently affects the bottom position after the junction. In the other

elaboration there have been presented some researches, which allowed formulating the

conclusion that when the sediment is stable in the inflow, not big change of water flow has a

big influence upon the bottom position after the rivers’ junction. In the above elaborations it

was also stated that the Varta River affects the obtained results very little, even if water flow

intensity corresponding to 60% of flow intensity in the Oder River before the junction is

assumed to the calculation. In all papers the source data resulted directly from the field

measurements and that’s why the entire grain curve of bottom material was applied in the

evaluations. Thus, the questions arose: What happens to the single fractions during this

process? How can they influence the mixing of other fractions? Is it possible to manage the

process of sediment mixing artificially and in this way to generate the bottom position

changes after the junction e.g. by artificial sediment supply? In the present paper the authors

also analysed how the changes of sediment composition, which was represented by two

extreme natural values of bottom material diameters (0,1mm and 1,0mm), influenced the

sediment mixing and hydraulic parameters after the junction.

76

Mathematical model of phenomenon The Oder and Warta River junctions in so-called Kostrzyński junction was assumed the

natural model for the evaluations. The following data were defined on the basis of the field

measurements:

- Riverbed geometry both in inflows and outflows, by the help of echo-probe,

- Composition of material creating the bottom for each evaluative profile in each riverbed,

- Flow values on the grounds of the limnigraphic read-outs in the Gozdowice and Słubice

cross-sections.

All the calculations concerning the sediment transport rates were carried out applying the

Ackers-White’s method, which had been checked and verified for the conditions occurring in

the estuarial sections of the Oder River (Coufal, 1993). According to this method the sediment

that is transported by the flow Q is defined by the following relationship:

QgX ⋅⋅⋅= ρω (1)

However if the granulometric composition of the river sediment includes different fractions,

evaluations should be carried out within the entire sieving curve. Therefore the total sediment

transport rate amounts to:

∑=

⋅=N

iii DpD

1)()(ωω (2)

On the basis of the prior papers (Meyer, Coufal, Kotiasz, Roszak) it can be assumed that the

sorting in the junction is defined so that the sediment continuity within particular fractions is

preserved. Therefore the continuity equation of sediment transport rate will take the following

form for the river junctions:

pO(Di)*O(Di) + pW(Di)*W(Di) = pP(Di)*P(Di) (3)

where the lower indices denote:

o– the Oder River parameter before the junction,

w – the Warta River parameter,

p – the Oder River parameter after the junction.

Both in the previously presented papers and the present elaboration some necessary

conditions were introduced into the optimising procedure:

- Preserving the flow equilibrium:

QP = QO + QW (4)

- Preserving the continuity of sediment transport rates’ equilibrium:

PWO ωωω =+ (5)

77

- Preserving the sum of percentage shares in all the fractions, which were obtained after having calculated the sediment sorting after the junction:

11

=∑=

n

iip (6)

Conserving the condition, which is described by the formula 4, a number of flow divisions in

the junction was defined for different assumed roughness coefficients in the inflows for two

measurement series. The evaluations of sediment sorting were carried out for each case,

defining – by the help of iteration – such value of roughness coefficient for the Oder River

after the junction that the conditions 5 and 6 were fulfilled. The composition of sorted

sediment and optimal value of roughness coefficient were calculated for each case. The

changeability of sediment composition in the Oder Riverbed after the junction is presented for

different flow divisions in inflows in the following figure.

0,00

0,05

0,10

0,15

0,20

0,25

0,30

0,35

0,40

0,45

0,50

0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0

Di [mm]

Pimeasurment

Qo 340

Qo 360

Qo 380

Qo 400

Qo 420

Qo 440

Qo 460

Qo 480

Fig.1. Changeability of calculated sediment composition for different cases of flow division.

For the case Qp=540m3/s, nso=0,030, nsw=0,030.

As it can be noticed there is always such flow in river inflows that for the assumed flow after

the junction and for the assumed roughness coefficient the sieving curve is obtained for the

sediment after the junction, which is equal with the measured one in the discussed case.

Analysing the results of calculations for the succeeding flow values for different assumed

roughness in inflows and for both measurement series it has been noticed that there is a linear

relationship between the assumed roughness coefficients in inflows and the slope in the

riverbed after the junction. It has also been noticed that the calculated water slope shows a

78

small changeability in the function of flow after the junction for each analysed series

nS0=const.. The slope relationship was assumed as a linear function:

- For the first measurement series in July 1999:

Jp = 0,0174 nSO – 0,0003 (7)

- For the second measurement series in August 1999:

Jp = 0,0038 nSO – 0,000034 (8)

The next stage was to include the modified Strickler’s formula, which enabled calculating the

value of the roughness coefficient for the Manning’s formula, in the evaluations:

61

=

HD

Mn zs (9)

where Dz denotes the substitutive diameter according to (Meyer, Coufal, Kotiasz, 1998):

∏=

=

n

i

P

i

M

MZ i

DD

DD

1

(10)

while M expresses the constant value for any analysed river section.

On the basis of statistical methods the constant value M was specified as equal to M=0,139.

The relationship describing the change of water slope due to the depth and substitutive

diameter of sediment grains on the grounds of formulae 7, 8 and 9:

+

−

⋅

−

= 004505,001944,0139,01649,0 ,72360

61

61

61

HDz

HDz

HDzJ P (11)

Knowing the function of roughness coefficient changes (according to the formula 9) as well

as the function of water slope changes (formula 11), it was possible to calculate the flow in

the riverbed after the junction on the basis of the Chezy’s and Manning’s formulae:

JHHn

HBQs

1 61

= (12)

after having introduced the formula 9 the following relationship was obtained:

356

1

0,139

1 HDzHJBQ

= (13)

where the slope J is defined by the formula 11.

79

On such grounds it is possible to construct a nomogramm describing the flow curves for

different bottom-creating material, which is expressed by the substitutive grain diameter. This

function has the following form:

( DzHfQ ,= ) (14)

The nomogramm describing the grain size of the sorted sediment for any measured depth and

an adequate flow has been created on the basis of the relationships 13 and 11. This diagram is

presented in the following figure:

Analysing the figure 2 thoroughly it can be noticed that size of sediment grains influences the

formation of water levels in the Oder Riverbed after the junction very considerably. The

circumstances, which occurred during the measurements on 28th August 1999, were taken into

consideration as the example. According to observations of limnigraphic read-outs in

Gozdowice the depths that corresponded to the succeeding substitutive grain diameters were

selected from the figure 2 for the flow value Q=280 [m3/s]. Then the figure 3 was made

illustrating the changes of bottom position due to changes of sorted sediment’s grain size,

which is expressed by the substitutive diameter.

2,0

2,2

2,4

2,6

2,8

3,0

3,2

3,4

3,6

3,8

4,0

150 200 250 300 350 400 450 500 550 600 650 Q [m3/s]

H [m]

Dz=0,3mm

Dz=0,4mmDz=0,45mm

Dz=0,5mm

Dz=0,55mmDz=0,6mm

Dz=0,7mmDz=0,8mm

Fig.2. Modified flow curve for the Oder Riverbed in the Kostrzyński junction,

including the sediment sorting

The dashed line denotes the most probable substitutive sediment diameters for the succeeding

flows.

80

In the nature the range of the sediment composition’s observed changes corresponds to the

substitutive diameter from 0,40mm to 0,50mm for such flow. It means that the water level

may oscillate from 2,44m to 3,08m for such flow value. This situation is presented

schematically in the below figure:

H = 2,02mO

H = 2,70mW

H =2,88m, measurments D =0,46mmZ

D =0,50mmZ

H =3,08m, forP D =0,4mmZ

Q =166o s3

Q =114W s3

Q =280P s3

RIV

ER

JU

NC

TIO

ND =0,47mmZ

D =0,34mmZ

Fig.3. Scheme of bottom position changes after the Oder and Warta River junctions

for flow Qp=280 m3/s.

As it is seen in the figure small fluctuations of sediment grain sizes cause significant changes

of water depth after the junction (up to 50cm). This confirms the Authors’ presumptions that

the water depth in the riverbed after the junction is very “sensible” to changes of sediment

composition. The fact resulting from the above figures that as the substitutive diameter

increases, the depth decreases, can be explained by the considerable increment of unwashing-

out velocities for thicker sediment fractions, which can make the velocities increase in the

Oder River cross-section after the junction.

The data, which correspond to the field measurements on 28th August 1999, were taken for

the calculations. Several cases of sediment composition corresponding to the combinations of

the below tables were analysed.

81

Table 1

Percentage sediment compositions in inflows

Fraction content For the Oder River

Fraction content for the Varta River

for D1=1,0mm

for D2=0,1mm

for D1=1,0mm

for D2=0,1mm

80% 20% 10% 90% 60% 40% 20% 80% 40% 60% 40% 60% 20% 80% 60% 40% 10% 90% 80% 20%

The iteration of the roughness coefficient in the Oder Riverbed below the junction was carried

out for every assumed case of the percentage sediment composition in the inflows and for the

assumed section range of the analysed depth. The optimal depth value and sediment

composition, which follow the nomogram in the fig. 1, were searched for as the result. The

composition of mixed sediment calculated on the basis of the formula 3 was obtained also for

the obtained optimal values. The fig. 2 shows an example results of sediment mixing, in

which the entire range of possible sediment compositions was analysed for the Varta Riverbed

assuming the uniform sediment composition in the Oder River above the junction. In the

figure the particular cases of the analysed percentage content of the given fraction were

marked with number from 1 to 5 for the Varta River and the Oder River below the junction as

the mixing result with the equal sediment composition in the Oder River for each case

82

Fig.4. Example results of sediment mixing in the Kostrzyń junction

The above figure indicates that the sediment composition in the Varta River does not

influences mixing significantly as effect it does not change the sediment composition after the

junction.

Similarly evaluations for the remaining sediment compositions in the Oder River before the

junction, according to the table 1 were carried out afterwards one described relations between

the assumed sediment compositions and the calculated hydraulic parameters in the form of the

diagram. The calculation results have been presented in the following collective figures.

2,0

2,2

2,4

2,6

2,8

3,0

3,2

3,4

0,0 0,2 0,4 0,6 0,8 1,0p 1 odry

Hp [m]

P1w=10%

P1w=20%

P1w=40%

P1w=60%

P1w=80%

Fig.5. Changeability of optimal depth after the junction as the function of analysed sediment

Odr

aW

arta

Nr1

War

ta N

r2W

arta

Nr3

War

ta N

r4W

arta

Nr5

mix

ed N

r1m

ixed

Nr2

mix

ed N

r3m

ixed

Nr4

mix

ed N

r5

Odr

a War

ta N

r1W

arta

Nr2

War

ta N

r3W

arta

Nr4

War

ta N

r5m

ixed

Nr1

mix

ed N

r2m

ixed

Nr3

mix

ed N

r4m

ixed

Nr5

0,0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1,0 0,1

p i

D i [mm]

composition in the Oder River

83

Fig.6. Changeability of optimal roughness coefficient after the junction, as the function of

analysed sediment composition in the Oder River

In the fig. 5 the analysed case of sediment composition has been expressed as the value of

fraction percentage share, which is marked as „1” corresponding to the diameter D1=1,0mm

(according to the table 1). As it can be noticed in the fig. 5, the depth is a function only of

sediment composition in the Oder River before the junction, irrespective of the assumed

sediment characterisation in the Varta River. The following diagram summing up the obtained

roughness coefficient values for the analysed sediment compositions leads to the similar

conclusions, Fig.6.

Conclusions

The paper presents the optimisation results of river junction’s hydraulic parameters on the

basis of the sediment transport’s analysis.

The practical evaluations were carried out on the grounds of field measurements in the Oder

and Warta River junction in Kostrzyń.

The carried-out analyses indicate that the junction optimisation can be based upon the

evaluations of sediment transport.

Analysing the optimisation results two linear relationships were specified connecting both

hydraulic parameters and parameters of transported bottom material.

0,020

0,025

0,030

0,035

0,040

0,045

0,050

0,0 0,2 0,4 0,6 0,8 1,0 p1 odry

n sp

P1w=10%

P1w=20%

P1w=40%

P1w=60%

P1w=80%

The presented modified flow curve allows estimating the flow in the Oder River in the area of

junction taking the measurements of riverbed geometries and granulometric composition of

the bottom material into consideration.

The presented calculation results concerning sediment transport rates’ mixing in the junction

show that the Varta River sediment composition does not affects significantly upon either the

84

results of sediment mixing or evaluated hydraulic parameters of the Oder River after the

junction. This conclusion is important, because in the presented examples there was analysed

a case of flow divisions, for which the water flow intensity of the Varta River was bigger than

50% of the flow intensity in the Oder River before the junction. This should suggest the Varta

River’s big share in the sediment shaping after the junction.

The sediment composition in the inflows influences the result of both sediment transport

rates’ mixing after the junction. These results were obtained by applying the mathematical

model of sediment mixing procedure in the river junction. This model was verified on the

basis of field measurements. During measuring the sediment composition changed itself

within a certain range. The basic assumption of the present elaboration is that the sediment

outside this range can be analysed by the help of this model. The obtained results agree with

the observed natural tendencies as far as the sorted sediment composition and the achieved

hydraulic parameters are concerned, which indicates a certain universality of this method. The

further field measurements, which would expand the change ability range of the sediment

composition, are necessary to draw more precise conclusions

Literature

Ackers P. & White W.R. (1975) Sediment transport, new approach and analysis. Journal of

the Hydraulic Division ASCE, Vol.99, No HY 11

Coufal R. (1997) Bed changes and sediment transport at river mouth. IBW PAN, Biblioteka

Naukowa Hydrotechnika, Nr 23

Coufal R., Meyer Z., Kotiasz W. (1998) Influence of the sediment transport on the steady

water flow in the river. 3rd International Conference Hydro-Science and Engineering, ICH'’ 98

Meyer Z., Krupiński A. (2000) Sediment sortation at alluvial river junction. 4th International

Conference on Hydroscience and Engineering, Seoul

Meyer Z., Krupiński A. (2000) (Mathematical modelling of sediment mixing in river

junction). XX Ogólnopolska Szkoła Hydrauliki, Krakow-Ustroń, Poland

85

INFLUENCE OF SEA LEVEL CHANGES ON COASTAL SEDIMENT TRANSPORT

Peter Fröhle, Sören Kohlhase Institute for Hydraulic and Coastal Engineering, University of Rostock, Germany Introduction In science and coastal engineering practice in Germany a mean sea level rise of approx. 12cm/centrury to 25cm/century was expected and included in the dimensioning of constructions since more than 50 years. Nowadays It is anticipated that global warming will bring accelerated sea level rise. IPCC (1996) and IPCC (2001) predicted a sea level rise of up to 90cm/century. Consequently, the question arises how these possible changes of the mean water level will influence the sediment transport and the morphological development of a coastal stretch. Basic Considerations on Sediment Transport and Beach Reaction Geomorphological concepts are helpful for understanding the physics and the physical reactions of a sandy beach. General questions related to coastal morphology and especially coast erosion must therefore be answered on the basis of the hydrodynamic input. Interrelations between oceanongraphic influences and the corresponding morphological reactions are schematically illustrated in fig. 1.

Fig. 1: Hydrodynamic input and coastal reactions (schematic, after Kohlhase, 1991)

86

The main input to form a beach and to influence the morphological development of a coastal stretch are waves and currents and their interactions as a result of wind and the local tidal conditions. In order to assess changes of the beach morphology and topography a detailed quantitative evaluation of the local sediment transport and the sediment transport in neighbouring areas are required. General and physically correct solutions for the calculation of sediment transport do not exist, but there are many more or less empirical formulae for the calculation of sediment transport capacity induced by waves and also many empirical formulations for the sediment transport rates induced by currents. Since the real near-shore current velocity field is very complex (fig. 2), most of the theoretical approaches need simplifying assumptions.

Fig. 2: Near-shore current velocities (after CERC, 1984)

For practical engineering purposes the overall coastal sediment transport is subdivided in long-shore sediment transport and cross-shore sediment transport. Hence, two more or less one-dimensional processes are the basis for the assessment of the morphological development of coastal areas. Long-Shore Sediment Transport Long-shore sediment transport is primarily associated with wave induced long-shore currents and with tidal currents. Various formulae for calculating the long-shore sediment transport are available from the literature. These formulae are using the local currents and the local current profile or the wave energy input into the area for the calculation of the sediment transport. The current oriented theoretical formulations are in general more or less of the form:

( ) ( ) ( )∫ ⋅== dzzvzcStS where: S(t) = Sediment transport

c(z) = sediment concentration v(z) = velocity

87

The energy oriented formulations are of the form: ( ) lsPctQ ⋅=

where: Q(t) = Sediment transport capacity

c = empirical constant Pls = long-shore energy flux

Some often used formulations are given in tab. 1. Since all concepts for calculating sediment transport need individual simplifications, all formulae have their limitations. The CERC-formulation is often used in coastal engineering practice and gives also reasonable results in many cases (Schoones & Theron, 1996). Approach Formulation Galvin (1972) (outdated)

2b

6 H10646,1Q ××=

CERC (1984) ( ) ( ) l

Sl P

p1gKQ ×

−××ρ−ρ=

K=0.39 Bailard (1981 / 1984)

( ) ( ) lS

l Pp1g

KQ ×−××ρ−ρ

= (CERC)

( )f

mbb

2

wu007,02sin6,205,0K +α+=

Kamphuis (1986) bb

27bS cossinH

Dm37,83Q αα=

Bijker (1976) lslbl QQQ +=

Λ×τ×ρ

ρ−ρ×ρ−××××=

C

S50

50lb l

D27,0expg

CVD5Q

+××= 21lbls I

rh33lnIQ83,1Q

Tab. 1: Selected sediment transport formulations Despite all limitations of the theoretical formulations, long-shore sediment transport formulae give the opportunity, to explain and assess erosion and accretion tendencies of a beach in the medium term and in the long term.

88

The morphological development of a coastline is only characterized by the long-shore gradient of the sediment transport within a control section (fig. 3). If Sx1=Sx2 (∆Sx=0), then the coastline is expected to be stable. If the gradient ∆Sx is positive, accretion has to be expected; if ∆Sx is negative, erosion is probable.

Fig. 3: Long-shore sediment transport, sediment budget, control section

(schematic, after Kohlhase (1991)) Cross-Shore Sediment Transport Cross-Shore sediment transport is mainly driven by the orbital motion of waves and the wave-induced on-/offshore currents. Depending on the wave characteristics and the sediment, a sorting of sediments and a bar or a system of bars can be found in the cross-shore profile of a coast. Since the moveable bed tries to adjust to the acting wave and current forces, a dynamically stable profile is formed in time which represents so-called equilibrium conditions. This process depends on both, the hydrodynamic input and the local sediment characteristics. Equilibrium profiles can be observed mainly in hydraulic models when the model is operated with constant hydrodynamic conditions. A huge number of investigations have been performed to explain the mechanisms of sediment motion within the profile under regular and irregular wave conditions. Under real conditions natural changes of the water level dependent on tides and wind have to be considered. In engineering practice the term equilibrium profile is often used as a basis to describe the changes within the profile. Anyhow, it can be stated that even if a wave approach normal to the coast is assumed, a stable equilibrium profile of the coast dos not exist. Nevertheless, with respect to erosion and accretion of a sandy beach, and considering cross-shore transport only, erosion and accretion are balanced within a profile (fig. 4). Sand can not leave the control section and remains

89

within the cross-shore profile. Therefore, a sandy beach will recover naturally and adapt to a dynamically stable equilibrium profile even after severe damages to the beaches and the dunes.

Fig. 4: Cross-shore sediment transport, sediment budget, control section

(schematic, after Kohlhase (1991)) Formulations for the calculation of cross-shore sediment transport are mainly part of numerical models. Import models are the SBEACH – model (Larson, 1989) and the formulations used in ACES (CERC 1992). An example result is given in fig. 5.

Profilentwicklung Projekt Bug

-6

-5

-4

-3

-2

-1

0

1

2

3

4

0 50 100 150 200 250 300 350 400 450 500

x [m]

z [m

]

Fig. 5: Simulation of cross-shore sediment transport with the numerical model SBEACH

and development of the beach profile

90

Influence of sea level change on sediment transport and morphological development As shown before, the sediment transport is often separated in cross-shore transport and long-shore transport. Hence, the influences are described separately. Long-Shore Sediment Transport For the following comparison calculations, a long-shore sediment transport formulation based on the recommended CERC (1984) approach was used. This formulation is based on the long-shore component of the local energy flux dissipation. Input parameters are wind and wave climate as well as the local water depth. Information on the input parameters are necessary for the complete simulated period with constant resolution in time. Therefore, data gaps were closed using wind-wave correlations (Kohlhase et al. 1994) and a complete and closed time series of homogeneous wave and water level information covering a period of more than 20 years was created. The result of the energy flux calculations and the corresponding sediment transport over the profile and averaged over approx. 20 years for one selected area (profile 0, Westerland, Sylt) is given in fig. 6. It can be seen, that the sediment transport capacity is directly connected to the wave energy flux dissipation. Furthermore, it can be seen that the main area of the sediment transport are the offshore bar (reef area approx. 350 - 500m off the coast-line) and the beach (inshore area).

Inshore- Area

Reef- Area

Fig. 5: Long-shore energy flux and respective sediment transport along the selected cross- shore profile (example area, Sylt, (Witte et al. 2002))

91

The influence of changing water levels on the long-shore sediment transport is shown in fig. 6. The inshore area was defined as landward from the offshore bar to the beach from x=–100m to approx. x=+350m. The reef area was defined from x=+350m to x=+500m. The results are given in relation to the results shown in fig. 5, where a relation factor of 1 means that no changes to the results in fig. 5 occur, a relation factor >1 that the sediment transport rate increases and a relation factor < 1 that the sediment transport rate decreases. The sediment transport rates in the inshore area are increasing with increasing water level, and, hence, the sediment transport rates are decreasing with increasing water level. The overall sediment transport rates integrated over the profile are constant and independent from the water level increase. For the calculations which are the basis for the results shown in fig. 5, it was assumed, that the cross-shore profile of the coast is not adapting to the changed water level and constant over the time. This scenario (no reaction) may be possible, but is not very likely. Therefore, comparison calculations have been performed to assess possible reactions of the coast. Two scenarios have been compared to the no reaction scenario. The complete adaptation scenario, where the profile completely adapts to the increased water level, and, the bar adaptation scenario, where only the offshore bar adapts to the increased water level with a fixed change of +1m. The profiles of these three scenarios are shown in fig. 7.

Fig. 6: Effect of water level rise on the sediment transport capacity

for the inshore and reef area (Witte et al. 2002)

92

The calculations with the complete adaptation scenario give the same result as shown in fig. 5, since the changes in the input wave energy which are included in the wind-wave correlation calculations are neglectable.

Fig. 7: Reference profile and modified profiles Profile 1 = no reaction,

Profile 2 = complete adaptation, Profile 3 = bar adaptation +1m The results of the calculations with bar adaptation+1m scenario are shown in fig. 8. The sediment transport capacity in the inshore area is always lower (ration < 1) compared to the no reaction scenario and the sediment transport capacity in the reef area is always higher compared to the no reaction scenario. Also in these two scenarios the overall sediment transport rates integrated over the profile are constant and independent from the water level increase.

93

Fig. 8: Sediment transport capacities of adaptation bar+1m scenario related

to the results of the no reaction scenario (Witte et al. 2002) These results show that regarding the assessment of long-term changes of a shoreline, the response of the bottom must be included in the calculations. Cross-Shore Sediment Transport For the assessment of the influence of a possible sea level rise on the cross-shore sediment transport and, hence, on the morphological development of the cross-shore profile two basic scenarios have been investigated. The first scenario is the complete adaptation scenario, which means that the cross-shore coastal profile adapts entirely to the sea level rise and that therefore the cross section of the coast is rising by the amount of the sea level rise. In fig. 9 an example for a possible adaptation to a sea level rise of ∆SWL=1,0m – which is exceeding the IPCC projections of global sea level (IPCC 1996, IPCC 2001) and only exemplarily chosen – is given. The adaptation of the profile was performed by balancing the mass of the sediment in the cross-shore profile (balanced mass profile), where the, in comparison to the original profile, eroded and accreted areas are balanced. If the adaptation of the cross-shore profile to a sea level rise will happen in accordance to the balanced mass profile, then, for the selected example cross-shore profile, a retreat of the coast-line of approx. 200m must be expected corresponding to an assumed sea level rise of ∆SWL=1,0m. Mimura and Kawaguchi (1996) used an approach based on the "Dean equilibrium profile" (Dean 1991) and concluded that nearly 95% of all beaches with an average width of 35m in Japan will disappear if a sea level rise of ∆SWL=1,0m will occur.

94

-10

-8

-6

-4

-2

0

2

4

6

1500 2000 2500 3000 3500 4000 4500 5000 5500

[m]

[m]

original

water level 1m

approx. 200m

Fig. 9: Adaptation of a selected example cross-shore profile calculated under the assumption of balanced sediment masses to an hypothetic (free assumption) sea level rise of ∆SWL=1,0m Selected simulations of the development of cross-shore profiles with dunes are used to generally describe the influence of extreme events to the coastal sediment transport under changed mean sea level conditions. The numerical model SBEACH (Larson 1989) was used for the numerical simulations. The simulations were performed with storm water levels based on existing MSL (mean sea level) and with storm water levels related to an assumed increased water level which is MSL+1m for comparison. In Fig. 10 and Fig. 11 the results of the comparison calculations with the numerical model SBEACH are shown. The differences in the development of the cross-shore profile are evident. It can be seen in fig. 10 that additional dune erosion of up to 50m occur under extreme conditions. Furthermore, the results shown in fig. 11 indicate, that the dimensions and safety of flood protection dunes have to be re-assessed, if a substantial sea level rise – as it is expected for the future by IPCC – shall be included in the dimensioning of flood protection dunes.

95

Fig. 10: Example simulations of a storm event with SBEACH with an existing MSL and under the hypothesis of a mean water level of MSL+1m

Fig. 11: Example simulations of a storm event with SBEACH with an existing MSL and under the hypothesis of a mean water level of MSL+1m

Conclusions The investigations and calculations described in the paper show, that long-term changes in the mean sea level will influence the coastal sediment transport. Both kinds of sediment transport, the long-shore and the cross-shore sediment transport, are – in general – influenced. The influence mainly results from the changes in the hydrodynamic conditions caused by changes of the mean sea level. It can also be expected that the coastal profile will respond to the changed sediment transport. The rate of adaptation to the sediment transport is not yet predictable. Under the assumption of a complete adaptation of the cross-shore profile to the changed conditions, it can be

96

expected that no continuous changes of the long-shore and cross-shore sediment transport will occur, but that a strong reaction (retreat) of the shore line and, hence, a strong reaction of the complete cross-shore profile has to be expected. The comparison calculations of long-shore sediment transport showed that the long-shore sediment transport integrated over the cross-shore profile is not significantly influenced by changed water levels. Nevertheless, the area of the main long-shore sediment transport changes significantly to the inshore area. Literature

Bijker, E.W.: Sand Transport by Waves. Delft University of Technology, Delft, 1976

CERC: U.S. Army Corps of Engineers (Hrsg.): Shore Protection Manuel Vol. I and Vol. II. Coastal Engineering Research Center, Washington D.C., U.S.A, 1984

CERC, Automated Coastal Engineering System, User's Guide, Leenknecht, Szuwalski & Sherlock, Coastal Engineering Research Center, Waterways Experiment Station, Vicksburg, 1992

Dean, R.G.: Equilibrium beach profiles: characteristics and applications, Journal of Coastal Research, 7(1), pp. 53-84, 1991

Horikawa, K.: Nearshore Dynamics and Coastal Processes. University of Tokyo Press, Tokyo, Japan, 1993

IPCC (International Panel of Climate Change): Climate Change 1995 - The Science of Climate Change, Cambridge University Press, 1996

IPCC (International Panel of Climate Change): Climate Change 2001 - The Science of Climate Change, Cambridge University Press, 2001, http://www.grida.no/climate/ipcc_tar/wg1/index.htm

Kohlhase, S.: Ozeanographisch-seebauliche Grundlagen der Hafenplanung. Sonderdruck aus Heft 57 der Mitteilungen des Franzius-Instituts für Wasserbau und Küsteningenieurwesen der Universität Hannover, Hannover, 1983

Kohlhase, S. (1991a): The Concept of Sediment Budget in the Nearshore Area, Proc. Seminar on Causes of Coastal Erosion in Sri Lanka, Colombo, Sri Lanka (1991)

Kohlhase, S. (1991b): The Need to Monitor the Coastal Response to Structural Interventions, Proc. Seminar on Causes of Coastal Erosion in Sri Lanka, Colombo, Sri Lanka (1991)

Kohlhase, S., Strotmann, Th. Fröhle, P.: BMFT-Forschungsvorhaben "Untersuchungen zur Optimierung des Küstenschutzes auf Sylt", Phase II, Abschlußbericht Phase II, Hannover 1994 (unveröffentlicht)

Larson, M.: SBEACH: Numerical Model for Simulating Storm-Induced Beach Change, US Army Corps of Engineers, Washington DC, 1989

97

Moutzouris, C.I.: Longshore Sediment Transport Computations: The Wave Power Equation and the Bijker Formula. Mitteilungen des Franzius-Instituts für Wasserbau und Küsteningenieurwesen der Universität Hannover, Heft 66, Hannover, 1988

Miura, N. & Kawaguchi, E. (1996): Responses of Coastal Topography to Sea Level Rise, Proceedings of the Twenty-fifth International Conference on Coastal Engineering, Orlando, Florida, September 2-6, 1996

Schoones, Theron: Improvement of the Most Accurate Longshore Sediment Transport Formula. Stellenbosch, South Afrika, 1994

Witte, J.O., Kohlhase, S., Fröhle, P. & Radomski, J.: Strategien und Optionen der Küstenschutzplanung Sylt, in: Daschkeit & Schottes: Klimafolgen für Mensch und Küste am Beispiel der Nordseeinsel Sylt, Springer Verlag, Berlin Heidelberg, 2002

98

INTERANNUAL TO DECADAL VARIABILITY IN THE SEA LEVEL OF THE BALTIC SEA EXPLAINED BY THE LOCAL WIND FIELD Frank Siegismund Institute of Oceanography, University of Hamburg, Germany Abstract Within the period from 1958 to 1997 strong trends in the winterly wind climate of the North Sea/ Baltic Sea region can be found. In the same period cyclonic circulation within the North Sea in January and March appeared as well a strong increase in winterly mean sea level of the Baltic Sea. Using simple regression methods it can be shown that the predominent part of the trend in these hydrodynamic phenomena is a respond to changed wind climate. Introduction Wind climate changes over the North Sea/ Baltic Sea area 1958 - 1997. Are there related changes in hydrodynamics? Investigation of the wind fields, taken from NCEP-reanalysis (Kalnay et al., 1996), reveals strong changes in the wind field for the North Sea area. Mean wind speed raised about 10%. When analysing the annual cycle this trend is shown to be confined to the winter period (JFM) with strong enhancement of wind speed and frequency for winds coming from southwestern and western directions (Siegismund and Schrum, 2001). To investigate variability in the hydrodynamics results from a 'state of the art' hydrodynamic model for the North Sea/ Baltic Sea region were analysed (Schrum and Siegismund 2002). Earlier versions of the model have been successfully applied to several other regions (e.g. Backhaus, 1985; Backhaus and Hainbucher, 1987; Stronach et al., 1993; Daji, 1995; Pohlmann, 1996; Schrum, 1997). In this case the model was run with a spatial resolution of about 10km and forced by the NCEP Reanalysis as atmospheric boundary conditions. A full description can be found in Schrum and Siegismund (2002). The model results reveal strong enhancement of the cyclonic circulation within the North Sea for January and March. For these two months in- and outflow at the northern entrance increased by about 0.5 Sv (1 Sv=106m /s) and transports in the southern North Sea along the continental coastline have almost doubled (Siegismund 2004).

3

For the Baltic Sea an increase in sea level for the winter period (JFM) is found, accounting to about 10cm in the Kattegat and 20cm in the eastern part, while in summer interannual to decadal variability in sea level is low and no significant trend is found for the investigated 4 decades.

99

Circulation in the North Sea To investigate the driving forces for the variability of the circulation in the North Sea area, a simple regression model is introduced, using input time series describing the temporal variability of the wind field over the North Sea. 9 time series were extracted from the NCEP-reanalysis to describe the variability of the wind field in monthly temporal resolution against climatological values. 8 time series are describing the area mean anomal wind stress densities in dependency of the wind direction. The wind stress densities are defined thereby as the product of the frequency and the mean squared wind speeds for a chosen sector. The 9th time series is describing the anomal wind stress curl over the North Sea area. Given a chosen section linear combination of the 9 time series regressed to results from the hydrodynamic model give an estimate of the anomal transport through this section. The comparison of the estimated transports from the regression model with results from hydrodynamic modelling shows reproduction of the variability to a high accuracy for the central and southern North Sea. The strong enhancement of cyclonic circulation in January and March for the period from 1958 to 1993 comes out to be forced by positive trends in wind stress densities for southwestern and western directions, partly reduced by decreasing wind stress curl for the same period. For the in- and outflow in the deeper parts of the northern entrance of the North Sea variability of the transport cannot be reproduced by the regression model suggesting other forcing than the wind field to be dominant for driving the anomalies. For the central and southern North Sea it could be shown that the anomaly of the circulation from climatology on monthly to decadal time scales can almost completely be explained by the wind climate. Strong enhancement of winterly cyclonic circulation is found, forced by the found changes in southwestern and western winds, damped partly by the wind stress curl, that shows a decreasing trend for the investigated period. At the northern entrance of the North Sea variability of the inflowing Atlantic water cannot be explained by the regression model, indicating that other forcing than wind climate is important there to drive the variability. A detailed description of the regression model and application to the North Sea circulation can be found in Siegismund (2004). Sea level of the Baltic Sea The variability of the sea level of the Baltic Sea on a monthly time scale can to a high accuracy be explained by the first pattern and Principal Component (PC) of an EOF-analysis. 95% of the global variance is explained. The pattern shows in rough approximation a plane with inclination from west to east with values in the easternmost parts twice as high as in the Kattegat (no fig.). The first PC (pc1) is used to describe the temporal development of the sea level of the Baltic. The idea behind the regression model introduced here, is, that the variability in the sea level, and thus the first PC, is explainable by the according variability in the wind stress in a specific region of the Baltic or North Sea, projected to a special direction. Let V be the (absolute) wind speed from the NCEP Reanalysis surface wind field for grid point i and time step t and and the zonal and meridional components respectively. Then the wind stress projected to direction γ is given by

)(ti

)(tui )(tvi

100

γγγϕ sin)(cos)()(),( ⋅⋅⋅⋅= tvtutVt iiii , assuming proportionality to squared wind speed and omitting the constant. Averaging to monthly mean values gives

∑=+−

=Φm

m

h

ljji

mmi t

lhm ),(

11),( γϕγ

)(tiΦr

(

, with [ t ] the time steps within the month m. mm hl t,...,

)γmi

))(,()( 1 γγ ii pccorrr Φ=r

With the vector containing ,Φ for all month m in the period from 1958 to 1997

gives the correlation between the projected wind stress and the first PC. Determination of that angle γ that maximises the correlation can be obtained by using the Newtonian method searching for the roots of the first derivative. Results of the procedure are found in fig.1 and fig.2. The method proves applicability. High correlation is found in a region from west of Jutland to the western Baltic Proper with values exceeding 0.8. The highest correlation is found for about the Arcona basin with a value of 0.86 (fig. 1). For that region γ is about 10 degrees, that means westerly winds with a small southern component (fig. 2).

Figure 1: Maximum correlation of the first Principal Component of the monthly sea level of the Baltic with the wind stress projected to variable direction γ. γ is given in fig. 2. See text for detailed description.

101

Figure 2: Angle γ for projection of wind stress to reveal maximum correlation with the First Principal Component of the sea level. Western direction is defined as 0 counting anticlockwise. See text for detailed description. So in summary we can conclude that the monthly mean sea level of the Baltic is in high correlation to the wind stress in the western Baltic projected to western or west-southwestern directions. The strong enhancement of southwestern and western wind speeds in the winter time for the period from the late fifties to the mid nineties together with the gain in the frequency of winds blowing from that direction can explain a considerable amount of the found sea level rise in the Baltic Sea. Acknowledgement The model simulations were performed within the frame of the Collaborative Research Centre 512 founded by the German Science Foundation. The NCEP Reanalysis data is provided by the NOAA-CIRES Climate Diagnostics Centre, Boulder, Colorado, freely available from their Web site at http://www.cdc.noaa.gov/.

102

References Backhaus, J.O. (1985): A three-dimensional model for the simulation of shelf sea dynamics. Dt. Hydrogr. Z., 38(4), 165-187. Backhaus, J.O. and Hainbucher, D. (1987): A finite-difference general circulation model for shelf seas and its application to low frequency variability on the North European Shelf. In: Three-Dimensional Models of Marine and Estuarine Dynamics, 221-244, Ed.: J.C.J. Nihoul und B.M. Jamart, Elsevier Oceanography Series, 45. Daji, H. (1995): Modelling Studies of Barotropic and Baroclinic Dynamics in the Bohai Sea. Berichte aus dem Zentrum f\"ur Meeres- und Klimaforschung, Reihe B, 17, 126 pp., University of Hamburg. Kalnay, E.; Kanamitsu, M.; Kistler, R.; Collins, W.; Deaven, D.;Gandin, L.; Iredell, M.; Saha, S.; White, G.; Woollen, J.; Zhu, Y.; Chelliah, M.; Ebisuzaki, W.; Higgins, W.; Janowiak, J.; Mo, K.C.; Ropelewski, C.; Wang, J.; Leetmaa, A.; Reynolds, R.; Jenne, R. and Joseph, D. (1996): The NCEP/NCAR 40-Year Reanalysis Project. Bulletin of the American Meteorological Society, 77(3), 437-471. Pohlmann, T. (1996): Predicting the thermocline in a circulation model of the North Sea. Part I: Model description, calibration and verification. Cont. Shelf Res., 16(2), 131-146. Schrum, C. (1997): Thermohaline stratification and instabilities at tidal mixing fronts: results of an eddy resolving model for the German Bight. Continental Shelf Research, 17(6), 689-716. Schrum, C. and Siegismund, F. (2002): Modellkonfiguration des Nordsee/Ostseemodells, 40-Jahresintegration. Berichte aus dem Zentrum f\"ur Meeres- und Klimaforschung, Reihe B, 44, 126 pp., University of Hamburg. Siegismund, F. (2004): Interannuale und dekadische Variabilität in Zirkulation und Salzgehalt der Nordsee 1958 bis 1997 – Einführung und Anwendung von Boxmodellen -. To be submitted. Siegismund, F. and Schrum, C. (2001): Decadal changes in the wind forcing over the North Sea. Clim Res, 18, 39-45. Stronach, J.A.; Backhaus, J.O. and Murty, T.S. (1993): An update on the numerical simulation of oceanographic processes in the waters between Vancouver Island and the Mainland: the GF8 model. Oceanogr. Mar. Biol. Annu. Rev., 31, 1-87, UCL Press.

103

The Southern Baltic Seacoast under Global Change

German-Polish Seminar 3 – 7 June 2003

Programme Tuesday, 3 June

Arrival in Hamburg, Check-in of Polish participants at Motel Hamburg (Hoheluftchaussee 119, Tel. 420 4141)

18.00 h: Evening Icebreaker (at green ZMK Pavillon across the street from

Bundesstraße 55; accessible from Motel Hamburg by Bus No. 5 to Hallerstr. or in about 20 min. by foot)

Wednesday, 4 June Drive to Ratzeburg 11.00 h: Prof. Dr. Jürgen Sündermann, IfM Hamburg (Introduction) 11.15-12.45: “The large-scale Baltic environment”

Dipl. Ozeanogr. Frank Siegismund, IfM Hamburg (Decadal variability of the Baltic and North Sea hydrography in relation to changes in the wind forcing) Dr. Thomas Neumann, IOW Rostock-Warnemünde (Simulation of load reduction scenarios for the Baltic Sea)

12.45-14.00: Lunch

14.00-15.30: Dr. Andreas Lehmann, IfM Kiel (Influence of global climate change on prevailing winds and resulting effects on the circulation of saline, oxygenated water in the southern Baltic Sea) Dr. Ralf Weisse, GKSS (Status of Surface Wave Statistics from numerical hindcasts for the Baltic)

15.30-16.00: Coffee 16.00-18.15 “Coastal hydrodynamics”

Dr. Barbara Swerpel (Paplinska) IBW PAN (Extreme waves and wave events in the South Baltic Sea) Dr. hab. Wojciech Sulisz IBW PAN (Prediction of wave propagation and transformation in a wave train) Prof. Piotr Wilde IBW PAN (Transformation of nonlinear random waves along the path of propagation (Co-author: Eugeniusz Sobierajski))

104

Thursday, 5 June 9.00-10.30: Dr. Małgorzata Robakiewicz IBW PAN (Seasonal changes of

hydrodynamics in Gulf of Gdańsk - possibilities of representation by numerical model) Dr. Wolfgang Rosenthal, GKSS (Technical and administrative tasks for implementing operational forecasts of river floods)

10.30-11.00: Coffee 11.00-12.30: „Coastal Morphodynamics“

Dr. Rafał Ostrowski IBW PAN (Cross-shore sediment transport and sea bed evolution by a quasi phase-resolving model) Prof. Zygmunt Meyer TU-Szczecin (Sediment transport at the Odra mouth in case of density stratification (Co-authors: Ryszard Coufal, Małgorzata Pluta))

12.30-14.45: Lunch

14.45-15.30: Dr.-Ing. Peter Fröhle, IWR Rostock (Influence of sea level changes on coastal sediment transport)

15.30-16.00: Coffee 16.00-18.15: MSc. Jacek Beldowski IO PAN (Fluffy layer suspended matter

transport and properties in the Odra River/Arkona Deep System (Authors: Pempkowiak J., Beldowski J., Emeis K.))

Prof. Zygmunt Meyer TU-Szczecin (Sediment transport modelling in the lower Odra River (Co-author: Adam Krupiuski)) Dipl. Ozeanogr. Klaus-Dieter Pfeiffer HYDROMOD Wedel (Morphodynamics of a tidal estuary (Co-authors: Kurt Duwe, Johannes Post, Jürgen Sündermann, Ingeborg Nöhren))

19.00: Dinner Friday, 6 June ca. 9.30 – 17 h: Boat excursion to Lübeck (boat departs 9.45 h from

“Lüneburger Damm” in Ratzeburg) Cathedral Drive to Hamburg

Saturday, 7 June and Sunday, 8 June

Departure

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Participants Poland:

1. Prof. Piotr Wilde, Institute of Hydroengineering Gdansk 2. Prof. Zbigniew Pruszak, Institute of Hydroengineering Gdansk 3. Dr. hab. Ewa Jasinska, Institute of Hydroengineering Gdansk 4. Dr. hab. Wojciech Sulisz, Institute of Hydroengineering Gdansk 5. Maciej Paprota, Institute of Hydroengineering Gdansk 6. Marcin Wisniewski, Institute of Hydroengineering Gdansk 7. Dr. Rafał Ostrowski, Institute of Hydroengineering Gdansk 8. Dr. Barbara Swerpel (Paplinska), Institute of Hydroengineering Gdansk 9. Dr. Małgorzata Robakiewicz, Institute of Hydroengineering Gdansk 10. MSc. Jacek Beldowski, Institute of Oceanology, Sopot 11. Prof. Zygmunt Meyer, Technical University Szczecin 12. Dr. hab. Ryszard Coufal, Technical University Szczecin

Germany:

1. Dr. Wolfgang Rosenthal, GKSS Research Center Geesthacht 2. Dipl. Ozeanogr. Frank Siegismund, Institute of Oceanography, University of

Hamburg 3. Dr. Ralf Weisse, GKSS Research Center Geesthacht 4. Dr. Andreas Lehmann, Institute of Oceanography, University of Kiel 5. Dr. Thomas Neumann, Institute of Baltic Research Rostock-Warnemünde 6. Dr.-Ing. Peter Fröhle, Institute for Hydraulic and Coastal Engineering, University

of Rostock 7. Dipl. Ozeanogr. Klaus-Dieter Pfeiffer, HYDROMOD Wedel 8. Dipl. Ozeanogr. Ingeborg Nöhren, HYDROMOD Wedel 9. Prof. Dr. Jürgen Sündermann, Institute of Oceanography, University of Hamburg 10. Dipl. Biol. Susan Beddig, Institute of Oceanography, University of Hamburg

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The participants with the Ratzeburg Cathedral in the background. From left: Ralf Weisse, Wolfgang Rosenthal, Ryszard Coufal, Piotr Wilde, Zygmunt Meyer, Jacek Beldowski, Ewa Jasinska, Maciej Paprota, Frank Siegismund, Jürgen Sündermann, Marcin Wisniewski, Barbara Swerpel, Susan Beddig, Rafał Ostrowski, Peter Fröhle, Wojciech Sulisz, Zbigniew Pruszak Photo by Małgorzata Robakiewicz

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Berichte aus dem Zentrum für Meeres- und Klimaforschung


Soweit noch Exemplare vorhanden sind werden Hefte dieser Reihe auf Anforderung zugeschickt: Zentrum für Marine und Atmosphärische Wissenschaften (ZMAW) Bibliotheks- und Informationsservice Bundesstr. 53 D-20146 Hamburg 1. Bartsch, J.: Zur Simulation der Verdriftung von Heringslarven in der Nordsee. - 1992. 2. Zou, J.: Analysis of sea level variations in the Tropical Pacific using GEOSAT radar altimeter

data. - 1993. 3. Carbajal, N.: Modelling the circulation in the Gulf of California. - 1993 4. Gurgel, K.-W.: Flächenhafte Messung der Oberflächenströmung vom fahrenden Schiff aus:

eine neue Anwendung des Hochfrequenz-Radarverfahrens am Beispiel der Arktisfront. - 1993. 5. Rizal, S.: Numerical study on the Malacca Strait ( Southeast Asia) with a three-dimensional

hydrodynamical model. - 1994. 6. Frey, H.: Ein diabatisches, wirbelauflösendes Schelfmeer-Zirkulationsmodell mit Anwendung

auf die Deutsche Bucht. - 1994. 7. Harms, I.: Numerische Modellstudie zur winterlichen Wassermassenformation in der

Barentssee. - 1994. 8. Frische, A.: Über den indo-Pazifischen Einstrom und die Zirkulation im östlichen Indischen

Ozean. - 1994. 9. Rubino, A.: Anregung und Ausbreitung von Tsunami-Wellen, die durch untermeerische

Erdrutsche angeregt wurden - 1994. 10. Jungclaus, J.: Ein numerisches Modell zur Simulation dichter Bodenströmungen im Ozean

mit Anwendung auf den "Overflow" durch die Dänemarkstraße. - 1994. 11. Karcher, M.: Dynamische Auswirkungen veränderlicher Quellen auf die Tiefenzirkulation des

Nordatlantik. - 1994. 12. Gekeler, J.: Datenassimilation in ein Randmeergezeitenmodell mit Anwendung in der Irischen

und Keltischen See. - 1994. 13. Eigenheer, A. : Simulationen der Plankton-Dynamik in der nördlichen Nordsee während

FLEX'76 mit dem Modell von FASHAM, DUCKOW & McKELVIE (1990). - 1994. 14. Kühn, W.: Ein eindimensionales physikalisch-biologisches Modell des pelagischen

Stickstoffkreislaufs. Anwendungsfall: Frühjahrsblüte in der Nördlichen Nordsee (FLEX'76). - 1994.

15. Schrum, C.: Numerische Simulation thermodynamisther Prozesse in der Deutschen Bucht. - 1994.

16. Pätsch, J.: MOCADOB a model generating synthetical time series of solar radiation for the North Sea. - 1994.

17. Huang, D.: Modelling studies of barotropic and baroclinic dynamics in the Bohai Sea. - 1995. 18. Dippner, J.W.: Untersuchung transienter Wirbelstrukturen in der Deutschen Bucht. - 1995. 19. Moll, A.: Regionale Differenzierung der Primärprodukäon in der Nordsee: Untersuchungen mit

einem drei-dimensionalen Modell. - 1995. 20. Radach, G.; Pätsch, J.; Gekeler, J.; Herbig, K.: Annual cycles of nutrients and chlorophyll in

the North Sea. - 1995. 21. Langenberg, H.: Zur Dynamik haliner Fronten und ihre lnstabilitäten in der Nordsee. - 1996 22. Lenhart, H.J. ; Pätsch, J., Radach, G.: Daily nutrient loads of the European continental rivers

for the years 1977-1983. - 1996. 23. Kämpf, J.: Thermohaline Konvektion in Polynien arktischer Schelfmeere - numerische

Prozeßstudien - 1996. 24. Segschneider, J.: Zur Ausbreitung geochemischer Spurenstoffe in der Tiefsee - Numerische

Simulationen. - 1996. 25. Rolinski, S.: Zur Schwebstoffdynamik der Tide-Elbe - Numerische Simulationen mit einem

Lagrangeschen Verfahren. - 1996. 26. Pätsch, J.: Auswirkungen anhaltender Eutrophierung der Nordsee: Langzeituntersuchung mit

dem Ökosystemmodell ERSEM. - 1997. 27. Radach, G.; Gekeler, J.: Gridding of the NOWESP data sets: Annual cycles of horizontal

distributions of temperature and salinity, and of concentrafions of nutrients, suspended particulate matter and chlorophyll of the North-West European shelf. - 1997.

28. Damm, P.E.: Die saisonale Salzgehalts- und Frischwasserverteilung in der Nordsee und ihre

Bilanzierung. - 1997. 29. Mirbach, K.: Schwebstofftransport in Küstengewässem mit Frischwassereinfluß. - 1997 30. Eigenheer, A.: Über den Einfluß von niederfrequenten Wellen auf die Monsunzirkulation im

Indischen Ozean. - 1997 31. Metzner, M.: Prozeßstudien regionaler Phänomene im Auftriebsgebiet von Nordwestafrika

unter Einbeziehung von Fernerkundungsdaten. - 1998. 32. Junge, M.M.: Stochastisch angeregte großräumige Variationen der windgetriebenen

Zirkulation - Untersuchungen mit einem isopyknischen Zirkulationsmodell. - 1998. 33. Nitz, Thomas: Untersuchung der Sensitivität eines wattenauflösenden hydrodynamischen

Modells bezüglich der iterativen Kopplung mit einem mesoskaligen meteorologischen Modell für den Bereich der Deutschen Bucht. - 1998.

34. Jiang, Wensheng: A three dimensional modeling of suspended particulate matter dynamics in the Bohai Sea. - 1999. 35. Lenhart, Hermann-Josef: Eutrophierung im kontinentalen Küstenbereich der Nordsee:

Reduktionsszenarien der Flusseinträge von Nährstoffen mit dem Ökosystem-Modell ERSEM. - 1999.

36. Festschrift zum sechzigsten Geburtstag von Prof. Dr. Jürgen Sündermann / hrsg. von Peter Müller und Thomas Pohlmann. - 2000

37. Schrum, C.; F. Janssen; U. Hübner: Recent climate modelling in North Sea and Baltic Sea.

Part A: Model description and validation. - 2000. 38. Wehde, Henning: Phytokonvektion im offenen Ozean: Feldexperimente und numerische

Prozessstudien. - 2001. 39. Schrum, C.; J. Staneva; E. Stanev: Black Sea surface climatological data for the period

1979-1993: a study based on the ECMWF atmospheric re-analysis. - 2001. 40. Lenhart, H.J.; J. Pätsch: Daily nutrient loads of the European continental rivers for the years

1977-1998. - 2001 41. Schrum, Corinna: A coupled atmosphere/ice/ocean model for the North Sea and the Baltic

Sea. - 2001. 42. Lopez, Benjamin Martinez: A study of the influence of surface forcing on the variability of the

oceanic circulation in the Gulf of Mexico using an isopycnic cordinate model. - 2002. 43. Lenz, Walter: Die treibenden Kräfte in der Ozeanographie seit der Gründung des Deutschen

Reiches. - 2002. 44. Schrum, C.; F. Siegismund: Modellkonfiguration des Nordsee / Ostseemodells: 40 Jahre

NCEP Integration. - 2002. 45. Schrum, C., B. Martinez-Lopez; F. Siegismund: Modellierte Klimatologie von Wärmeinhalt und Salzgehalt in Nordsee und Ostsee. - 2002 46. Pohlmann, Thomas: Eine Bewertung der hydro-thermodynamischen Nordseemodellierung. - 2003. 47. Taguchi, Eifu: Inverse Modellierung nichtlinearer Flachwassergezeiten und ihre Anwendung auf ein Randmeer. - 2004. 48. Pätsch, J., H.-J. Lenhart: Daily loads of nutrients, total alkalinity, dissolved inorganic carbon and dissolved organic Carbon of the European continental rivers for the years 1977-2002. – 2004. 49. Dynamics of Coastal Waters and their Modelling: Proceedings of the 9th German Polish Seminar – Ratzeburg, 4-6 June 2003 / ed. by J. Sündermann. – 2005

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