Low-frequency sea waves generated by atmospheric ...jadran.izor.hr/~vilibic/meteotsunami/deJong_Battjes_JGR_2004_Nort… · Low-frequency sea waves generated by atmospheric convection

Low-frequency sea waves generated by

atmospheric convection cells

M. P. C. de Jong and J. A. BattjesEnvironmental Fluid Mechanics Section, Delft University of Technology, Delft, Netherlands

Received 25 April 2003; revised 30 June 2003; accepted 25 August 2003; published 13 January 2004.

[1] The atmospheric origin of low-frequency sea waves that cause seiches in the Port ofRotterdam is investigated using hydrological and meteorological observations. Theseobservations, combined with weather charts, show that all significant seiche eventscoincide with the passage of a low-pressure area and a cold front. Following these frontpassages, increased wind speed fluctuations occur with periods on the order of 1 hour.The records show that enhanced low-frequency wave energy at sea and the seicheevents in the harbor occur more or less simultaneously with these strong wind speedfluctuations. These oscillatory wind speed changes are due to convection cells that arise inan unstable lower atmosphere in the area behind a cold front, where cold air movesover the relatively warm sea surface. It is shown that the moving system of a cold frontand trailing convection cells generates forced low-frequency waves at sea that can causeseiche events inside the harbor. The occurrence of such events may be predictableoperationally on the basis of a criterion for the difference in temperature between the air inthe upper atmosphere and the water at the sea surface. INDEX TERMS: 4560

Oceanography: Physical: Surface waves and tides (1255); 4564 Oceanography: Physical: Tsunamis and storm

surges; 3339 Meteorology and Atmospheric Dynamics: Ocean/atmosphere interactions (0312, 4504); 3307

Meteorology and Atmospheric Dynamics: Boundary layer processes; 1610 Global Change: Atmosphere (0315,

0325); KEYWORDS: low-frequency sea waves, seiches, atmospheric convection cells, generation of seiches

Citation: de Jong, M. P. C., and J. A. Battjes (2004), Low-frequency sea waves generated by atmospheric convection cells,

J. Geophys. Res., 109, C01011, doi:10.1029/2003JC001931.

1. Introduction

[2] Resonant harbor oscillations (seiches) influence ship-ping activities since they can hamper ship manoeuvring orcause flooding of harbor areas. In the Port of Rotterdam,Netherlands, these waves also affect the design and man-agement of dykes and a storm surge barrier which protectthe harbor and the surrounding area against flooding.[3] The outer Rotterdam harbor area, which is seiche-

prone, and the location of the storm surge barrier are shownin Figure 1. In this harbor area, seiches occur mainly in along semiclosed basin called the Calandkanaal, with mea-sured amplitudes up to 0.9 m at the closed end nearRozenburgse Sluis (ROZ). This basin has a length ofapproximately 20 km and a depth of approximately 20 m.For almost all of the observed seiche events at ROZ, thedominant frequency is the lowest eigenfrequency (quarter-wavelength mode), equivalent to a period of 90 min(somewhat dependent on the water depth which varies withthe tide, 85–100 min). An example of a tide registrationincluding a seiche at ROZ is shown in Figure 2, togetherwith a registration at a reference location that is not seiche-prone (no closed end), near the city center further inland(south of Rotterdam Airport indicated in Figure 1). This

example event (1 and 2 January 1995) is the seiche with thelargest amplitude in our database, spanning 1995–2001. DeJong et al. [2003a] have recently presented analyses of theoccurrence of these seiche events and results of numericalsimulations.[4] A special situation occurs in case the movable storm

surge barrier is closed. In that case an additional semiclosedbasin is created, north of the Calandkanaal (Figure 1), with aneigenperiod of approximately 30 min. In extreme cases, thetrough of a seiche can cause a critical situation when the waterlevel on the sea side of the barrier drops below the level on theriver side. This could cause failure of the storm surge barriersince it is primarily designed for protection against highwaterlevels on the sea side. Therefore the water level is monitoredby the Ministry of Public Works on both sides of the barrierfor a more or less ad hoc warning method. However,presently a seiche prediction system is not available for theclosure management of the barrier.[5] For ship traffic control (e.g., assignment of tidal

windows to ships with a large draught) the RotterdamMunicipal Port Management predicts water levels insidethe harbor including the effects of tides and wind set-up. Sofar, it has not been possible to include seiche events in thisprediction.[6] The generating mechanisms of the seiches in the Port

of Rotterdam need to be known in detail in order to developa prediction system. Observations in the past have shown

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 109, C01011, doi:10.1029/2003JC001931, 2004

Copyright 2004 by the American Geophysical Union.0148-0227/04/2003JC001931$09.00

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that seiche events occur during periods of rough weather.This indicates an atmospheric origin of the seiches, but theprocess through which the generation takes place has so farnot been identified. Other known potential generatingmechanisms are either extremely rare near Rotterdam(e.g., tsunamis), or they do not generate energy near theeigenfrequencies of the harbor (e.g., surf beat). The possi-bility of meteorologically generated seiches in harbors alongthe Dutch coast has been suggested for a long time [see,e.g., Wemelsfelder, 1957]. However, most previous studiesof seiches in the Port of Rotterdam focused on the phenom-ena inside the harbor [see, e.g., De Looff and Veldman,1994]. To study the origin of the seiches in the Port of

Rotterdam, observations during two seiche events wereanalyzed in more detail by Veraart [1994]. For these cases,he found a correlation between the time of the passage of acold front over the southern North Sea and the time ofoccurrence of increased seiche activity. Numerical simula-tions of the generation of seiches on the southern North Seawere described by de Jong et al. [2003a], but these wereonly partially successful as a result of insufficient resolutionof the atmospheric forcing.[7] Meteorologically generated seiches have been ob-

served at numerous locations all around the world, fromrelatively small harbor basins and inlets to seas. Theserelatively small areas are influenced by mesoscale phenom-

Figure 1. The Port of Rotterdam. Hydro-meteo observations at Rozenburgse Sluis, Hook of Holland,and Rotterdam Airport.

Figure 2. (top) Surface elevation records at ROZ and near Rotterdam center (approximately 40 kminland) for 1 and 2 January 1995. The latter shows the tide-dominated signal, whereas the observations atROZ also show the seiche superposed on the tide signal. (bottom) Surface elevation records filtered forthe seiche frequency band (0.1–2.0 mHz).

C01011 DE JONG AND BATTJES: SEA WAVES GENERATED BY CONVECTION CELLS

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ena such as sharp or periodic changes in atmosphericpressure (in some cases caused by cold fronts). Examplesare: Nagasaki Bay, Japan [Hibiya and Kajiura, 1982] andMenorca, Spain [Monserrat et al., 1991; Rabinovich andMonserrat, 1998]. Larger sea areas are influenced bycomplete low-pressure areas, giving rise to the so-called‘‘inverted barometer’’ effect that causes a water levelincrease (in some cases combined with an additional windset-up) that can initiate the oscillations. Examples are: theBaltic Sea [Metzner et al., 2000] and the Adriatic Sea[Cerovecki et al., 1997; Canestrelli et al., 2001; Pasaricand Orlic, 2001].[8] However, the situation for the Port of Rotterdam differs

from the circumstances that are described in the literature forother locations, since during most seiche events in Rotterdamno sharp or distinct periodic changes in atmospheric pressureare found with magnitudes that could account for theobserved amplitudes of the long waves at sea. Moreover,the (maximum) crest height of the seiches inside the harbor isnot significantly correlated with either the (maximum) onehour average wind speed or the wind-induced set-up (seebelow). This prompted the search for another mechanismthrough which wave energy is generated at sea in the harborseiche frequency band for the Port of Rotterdam (periodsfrom approximately 30 to 100 min).[9] A total of 49 seiche events were registered at ROZ in

the time interval available for this study (1995–2001). Eachsuch event is here defined as a period of enhanced seicheactivity with amplitudes (crest heights) exceeding 0.25 m atROZ. Meteorological measurements and a visual inspectionof the weather charts show that all these events coincidedwith low-pressure systems with cold fronts crossing thesouthern North Sea toward the Dutch coast. However, notall cold fronts moving from sea toward the Dutch coastgenerated a seiche event. On average approximately onecold front passes the southern North Sea per week in therough weather season and on average only 7 seiche eventsoccur per year.[10] The registrations of the surface elevation at sea

during and just preceding seiche events showed two distinctpatterns: either a single, soliton-type wave of elevation, or aburst of oscillations. The former category, only 5 out the 49events, coincided with thunder storms during the latesummer (August–September) and a front passage fromsouthwesterly directions. The corresponding seiches wererelatively weak. The bulk of the events (44) was of thesecond category, occurring mainly in the storm season(October–April) following a cold front passage from westor northwest over the observation locations, and resulted inthe highest seiche response. Because of their preponderanceand the high response, only the events of this secondcategory are investigated further in this paper.[11] To increase the insight into these events and to

eventually arrive at a prediction system, this study startedwith the aim of identifying the main generating mecha-nism(s) at sea. Measurements of surface elevations at seaand in the harbor, precipitation, air temperature, atmosphericpressure and wind speed as well as precipitation radar,infrared satellite images and weather charts over a sevenyear interval were inspected and analyzed. The presentpaper describes the results, i.e., the mechanism throughwhich so-called low-frequency waves are generated at sea

(i.e., waves in the spectral valley between the more energetictides and sea/swell) which subsequently can cause seicheevents in the Port of Rotterdam. It was found that theepisodes of low-frequency energy at sea are generated byconvection cells that in some cases follow a cold front.Observational evidence will be presented and discussed tosupport the causative link between these convection cellsand the seiche events. The mechanisms involved andtheir theoretical background will also be described. Last, acriterion is suggested for prediction of seiche-prone situa-tions based on meteorological information.

2. Measurements

2.1. Data Acquisition

[12] The sea surface elevation measurements used in thisstudy were obtained at offshore locations Europlatform(eur), Lichteiland Goeree (goe) and Meetpost Noordwijk(mpn). Figure 3 shows the locations of these offshoreplatforms, located approximately 20 km and 40 km fromthe Rotterdam harbor mouth. From June 2000, processed(validated) surface elevation values from these sea stationsare stored continuously on an operational basis with asample rate of 1 min. Meteorological measurements (atmo-spheric pressure, wind speed and wind direction) are avail-able at these platforms as 10 min average values (centeredaround time stamp) stored every 10 min for a few events in1995, 2000, and 2001.[13] Additional measurements were obtained from more

distant platforms at the North Sea. For the time interval1995–2001, sea surface temperature measurements havebeen obtained from a platform located approximately150 km to the northwest of the harbor mouth (platformK13, 53�1300400 N, 03�1301300 E). These were available as24 hour averaged daily values. Meteorological soundings(vertical profiles of air temperature and other meteorologicalparameters from balloon measurements) were available formost of the seiche events (1995–2001) at Ekofisk platform(EKO, 56�3205000 N, 03�1200600 E, approximately 500 km

Figure 3. Overview of the Dutch coast near the harbormouth of the Port of Rotterdam and the measurementlocations.


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northwest of the harbor mouth) every 12 hours or every24 hours. For a number of events these meteorologicalsoundings at sea were not available. For these casesatmospheric soundings at De Bilt have been used (locatedinland, about 75 km east of Rotterdam), which are availableevery six hours.[14] Inside the harbor, surface elevation measurements

are available from multiple locations. Observations fromROZ (see Figures 1 and 3) will be used here to describe andanalyze the seiches. These measurements have been takencontinuously since 1995 with a floater, sampled at a60 second interval. The seiche characteristics of the harborhave not been constant in this time interval, because ofchanges that have been made to the harbor lay out thatinfluenced the eigenfrequencies of certain harbor areas. Forexample, in November 1997 a part of the Beerdam (seeFigure 1) has been removed and a new connection betweena number of harbor basins was herewith created. Wherenecessary and possible, these inconsistencies in the data setshave been taken into account in the analyses described infollowing sections.[15] Long-term meteorological measurements at Hook of

Holland (Figures 1 and 3), consisting of 10 min averagevalues, were used to analyze the meteorological situationduring all seiche events in the period 1995–2000. More-over, wind speed, wind direction, atmospheric pressure,precipitation and air temperature data are obtained contin-uously since January 2000 as 1 min averages at Rotterdam

Airport (ra, see Figures 1 and 3). These have been used tostudy more recent events in more detail. Additional mete-orological observations have been obtained for the event of1 January 1995, shown in Figure 2. These were onlyavailable at Valkenburg Airport (va; see Figure 3), with asampling interval of 10 min (average values of 10 mincentered around time stamp).

2.2. Data Analysis

[16] To visually inspect the records of the surface eleva-tions during seiche events, both at sea and in the harbor, thetidal components and the high-frequency wind waves havebeen removed from these records to retain surface eleva-tions in the seiche frequency band (0.1–2.0 mHz). Thisfiltering of the records was achieved by a forward andbackward Fourier transform. The seiche frequency bandwas well isolated from lower and higher frequencies bycosine tapering the original records over 10% both at thestart and at the end of the record.[17] A wavelet analysis based on the Morlet wavelet

[Morlet et al., 1982] has been applied to the filtered surfaceelevation data and meteorological data (details on waveletanalysis are given by, e.g., Farge [1992] and Mallat[1998]). This technique is suited to the detection of fluctua-tions that come in bursts. It has been used to identify theseiche events inside the harbor and the correspondingperiods of increased levels of low-frequency energy atsea. The results of the wavelet analysis for the seiche

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Figure 4. Characteristic examples of band-pass-filtered (0.1–2.0 mHz) observed surface elevations atROZ during seiche events. The time mark refers to the passage of a cold front at Rotterdam Airport.


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frequency band are not influenced by the filter methoddescribed previously. Following first identification fromwavelet spectra, the time intervals with seiche events werereviewed in more detail.

3. Characteristics of Seiche Events

[18] Seiche events (amplitude exceeding 0.25 m at ROZ)in the studied time interval (1995–2001) were clearlyvisible as peaks in the wavelet spectra of the surfaceelevation at ROZ (both in time and in frequency) in a bandnear the longest eigenperiod of the Calandkanaal. Thecorresponding signals from the platforms at sea showedpeaks in the wavelet spectra of the seiche frequency bandprior to all these seiche events.[19] Inspection of weather maps, infrared satellite images

and images obtained from precipitation radar showed thatall these events coincided with the passage of a cold frontacross the North Sea toward the Dutch coast. Generallyspeaking, such a front passage is followed by an increase inwind speed to moderately strong values, without reachingstorm strength. Measurements at HOH have been used todetermine the maximum sustained wind speed during theseiche events (based on hourly averaged values, available1995–2000). These had an average value of about 15 m/swith a maximum of 21 m/s. The coefficient of correlationbetween the maximum seiche crest height per event at ROZand the corresponding maximum sustained wind speed wasabout 0.14, which is not significant.[20] Some characteristic examples of seiche events at

ROZ are shown in Figure 4. The times of passage of thecold fronts (over RA) during the corresponding events areindicated by the vertical lines. The figure illustrates thatthe onset of the seiche events in the Port of Rotterdam isfound following the passage of a cold front over the area.However, the converse does not hold: not all cold frontpassages give rise to seiche events. We return to thisbelow.[21] Figure 5 shows the number of independent events

per month that have been found for 1995–2001, togetherwith the average crest-to-trough height of these events.(Multiple occurrences within 48 hours have here beencounted as one event.) The average crest-to-trough heightappears to be relatively constant. However, the distributionof the number of events clearly shows that the seiches in thePort of Rotterdam are a storm season phenomenon.[22] A correlation between wind set-up at Hook of

Holland and the seiche amplitudes had been assumed forthe design of the storm surge barrier referred to in section 1.In the present study, the coefficient of correlation betweenthe maximum seiche amplitude at ROZ for each indepen-dent seiche event and the corresponding maximum windset-up at HOH (available for 1995–2000) was calculated.For the complete data set considered (1995–2000), its valuewas 0.19. However, smaller time intervals have also beenstudied separately because of the changes that were made inthe harbor lay out, which could compromise the consistencyof the data set due to changes in harbor response. Never-theless, the correlation was very weak also for these subsets:0.31 and 0.09 for the time interval prior to the main lay outchanges (1995–1997) and for the time interval after thechanges (1998–2000), respectively. These weak correla-

tions confirm that strong winds (relatively constant indirection and magnitude) that cause a significant set-up ofthe mean water level do not have to occur simultaneouslywith the phenomena that generate the seiche events.

4. Generation of Low-Frequency Waves

4.1. Observations of Wind Speed andSurface Elevation

[23] Wavelet analyses were made of records of atmo-spheric pressure and wind speed for the episodes of in-creased seiche activity. The atmospheric pressure variationsduring most of these seiche events were very weak (on theorder of 0.1 hPa) and did not show distinct periodicities thatcould point to a causative link with the observed seiches.Moreover, numerical studies showed that the occurrence ofsignificant low-frequency energy at sea is more sensitive tothe inclusion or exclusion of wind speed in the atmosphericforcing than to the inclusion or exclusion of air pressure [deJong et al., 2003b]. Therefore wind speed observationsobtained during seiche events have been studied in moredetail.[24] Wavelet spectra have been made of wind velocity time

series at different stations and of the surface elevationmeasurements at ROZ for the selected time intervals. Asso-ciated with a cold front passage, episodes (bursts) of in-creased fluctuations in the wind speed were found to occur inthe seiche frequency band. The generation of the long wavesat sea that cause the seiches inside the Port of Rotterdam isascribed to these oscillatory wind speed changes. As anexample, Figure 6a shows the north-south component ofthe wind speed (at VA) in the seiche frequency band on 1 and2 January 1995 (positivewhendirected toward the north). Thewavelet spectrum (WS) of the wind speed signal (Figure 6b)shows fluctuations starting from the time of the frontpassage at 1308 UT over VA (about 40 km north of the

Figure 5. Black bars: number of events per month 1995–2001 (amplitude event exceeds 25 cm); gray bars: monthlymean crest-to-trough height of events (dm). Seiche eventsthat occurred within 48 hours have been identified as oneevent (maximum crest-to-trough height was used).


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harbor; see Figure 1), the most energetic period being about4000 s. Figure 6c shows the surface elevation at ROZ for theseiche frequency band on 1 and 2 January 1995. Thecorresponding wavelet spectrum, shown in Figure 6d, indi-cates that approximately 90 min after the front passage overVA the harbor responds at the lowest eigenfrequency,corresponding to a period of 5400 s (90 min).[25] Time series and corresponding wavelet spectra of the

wind velocity observations at RA and surface elevation atROZ for two seiche events that occurred in November 2001are also described here. For the first event, 8 and 9November 2001, the wavelet spectra also show distinctenergy peaks for both the wind observations at RA and

the surface elevation observations at ROZ (Figure 7). Figure8 shows the results for the event of 22–23 November 2001.The wavelet spectra in this case are less distinct, but overallthe picture is similar to that in the previous examples. Forall considered events, the time interval during which theburst of wind speed fluctuations is found approximatelycoincides with the time interval during which the seicheevent occurs, allowing for a lag of a few hours at most,depending on the alignment and speed of the front and thelocation of the observation station. In some cases the patternof wind speed fluctuations and the seiche response showeda significant similarity even in a deterministic sense, i.e., inthe somewhat irregular succession of peaks and troughs of

Figure 6. Wavelet analysis results for 1 and 2 January 1995: (a) measured wind speed in north-southdirection at VA, (b) corresponding wavelet spectrum, (c) surface elevation measurements at ROZ, and(d) corresponding wavelet spectrum.


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varying heights and the intervals between them. Figures 9and 10 give marked examples of this. (The wind speedrecord has been shifted forward in time by a few hours toaccount for the difference in arrival times at the twomeasurement stations, as determined from the waveletspectra.)

4.2. Generation of Low-Frequency Sea Wavesby Convection Cells

[26] Wind speed fluctuations such as those that have beenobserved following the cold front passages that coincidedwith seiche events can be generated by atmospheric con-vection cells. These convection cells are a mesoscale

atmospheric phenomenon [Atkinson and Zhang, 1996], withtypical horizontal spatial scales of 30–100 km. Theirgeneration is similar to the formation of Rayleigh-Benardcells, which form in a fluid in a gravitational field betweentwo horizontal, stress-free, perfectly conducting plates incase the fluid at the bottom plate is heated [Tritton, 1977;Getling, 1998].[27] Conditions conducive to the formation of convection

cells over the North Sea occur when cold, polar air movesover relatively warm seawater, which is likely to occur infall and winter. Especially in the area behind the cold front,the relatively warm seawater causes heating of the air in thelower layers, which results in an unstable lower atmosphere

Figure 7. Wavelet analysis results for 8 and 9 November 2001: (a) measured wind speed in north-southdirection at ra, (b) corresponding wavelet spectrum, (c) surface elevation measurements at ROZ, and(d) corresponding wavelet spectrum.


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that causes air to rise to higher altitudes in narrow plumes.The result is a more or less cellular pattern of convectioncells. If sufficient moisture is present in the air, condensa-tion of water vapor can locally cause a release of latent heat,further driving the circulation of air (see Figure 11). Narrowbands of clouds can then be formed surrounding clear skies,which can be seen on satellite images and, in case ofprecipitation, also on radar precipitation images. Becausethe system of front and convection cells moves, these cellscause more or less periodic wind speed fluctuations super-posed on the mean wind velocity in a fixed point at thesurface. These wind speed fluctuations can generate low-frequency sea waves. An example of the surface elevation

response to these wind speed changes caused by theconvection cells is sketched in Figure 11.[28] To illustrate and support the preceding statements,

we use the event of 22 and 23 November 2001. A detail ofthe weather chart of 23 November, 0000 UT is shown inFigure 12. This meteorological situation is representativefor most cases that have been observed during seicheevents. The chart shows a low-pressure area over Scandi-navia that causes a southward flow of cold polar air (a ‘polaroutbreak’) across the North Sea. This flow of cold airapproaches the Dutch coast from the northwest.[29] Figure 13 shows satellite images of 21 till 23

November 2001. These images, of a thermal infrared

Figure 8. Wavelet analysis results for 22 and 23 November 2001: (a) measured wind speed in north-south direction at RA, (b) corresponding wavelet spectrum, (c) surface elevation measurements at ROZ,and (d) corresponding wavelet spectrum.


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frequency range (10.3–11.3 mm), are obtained a few times aday as polar orbiting satellites pass over the area of interest(these images have been obtained by the NOAA 12 andNOAA 16 satellites). The top left image, of 21 November,

1324 UT, shows a dense cloud formation bordering arelatively clear area across the northern North Sea, indicat-ing the location of the cold front approximately 24 hoursprior to passing the Dutch coast near Rotterdam. Moreover,

Figure 9. Detail of surface elevation and filtered wind speed observations of 1 and 2 January 1995. Atime shift of 90 min has been applied to the wind data. Left y axis: surface elevation at ROZ; right y axis:wind speed measurements at VA.

Figure 10. Detail of surface elevation and filtered wind speed observations of 8 and 9 November 2001.A time shift of 184 min has been applied to the wind data. Left y axis: surface elevation at ROZ; righty axis: wind speed measurements at RA.


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cellular cloud formations are visible behind the cold frontindicating the presence of convection cells. The topright image, of 1617 UT on 22 November 2001, showsthe situation around the start of the seiche event (seeFigure 8). Around this time the cold front passes the Dutchcoast near Rotterdam and convection cells are visible acrossthe North Sea area from the northern region to the Dutchcoast. The bottom left image, 0117 UT on 23 November2001, shows the situation at a time about halfway throughthe event, when fewer convection cells are visible, since

most have moved over land and have dissolved. Finally, thebottom right image, 1302 UT on 23 November 2001, showsthe situation around the time the event finished (a fewremaining cells are present over the North Sea near theDutch coast; most cells have dissolved over land). The timeinterval during which the seiche event occurred (approxi-mately from 1600 UT on 22 November 2001 till 1400 UTon 23 November 2001, see Figure 8) more or less coincideswith the time interval during which the system of convec-tion cells passed the Dutch coast.

Figure 11. Sketch of convection cells in the area behind a cold front together with a theoretical surfaceelevation response for a subcritically forced situation (see section 4.3).

Figure 12. Detail of weather chart for central Europe of 23 November 2001, 0000 UT. Figure showsisobars with atmospheric pressure in hPa. Black rectangle indicates area shown in Figure 14. # Crowncopyright, Met Office. Reproduced under license number MetO/IPR/2/20030022.


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[30] The precipitation image in Figure 14 of 0130 UT,23 November 2001, shows oval patches of rain thatcorrespond to the convection cells as they pass acrossthe North Sea toward the Dutch coast from the northwest.Since the cells dissolve further inland, no precipitation

occurs in regions further away from the coast. Twoexample cells are indicated. Successive precipitationimages (available every 15 min) indicate that these cells,with a length scale of approximately 100 km, passed afixed point in approximately 100 min (between 6 and 7

Figure 13. Satellite images of infrared frequency band: (a) 21 November 2001, 1324 UT; (b) 22November 2001, 1617 UT; (c) 23 November 2001, 0127 UT; and (d) 23 November 2001, 1302 UT. Thelocation of the Port of Rotterdam is indicated by black and white dots. Source images: Dundee SatelliteReceiving Station, Dundee University, Scotland.


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image time steps), which is near the timescale of thewind speed fluctuations that have been observed. More-over, the images indicate that the convection cells areadvected at a relatively constant velocity, in this caseapproximately 17 m/s, which is near a characteristic longwave phase velocity at the southern North Sea (17.2 m/sfor a depth of 30 m, which is typical for this region).This near resonance occurs for many of the observations.Thus it so happens that the range of the advectionvelocities of the convection cells coincides with the rangeof the phase velocities of shallow water waves at thesouthern North Sea. This implies enhanced, near-resonantresponse. Moreover, the resulting energetic frequenciesexcited by the convection cells in a fixed point happen tobe comparable to the lowest eigenfrequency of theCalandkanaal! (What here appears to be a double coinci-dence may to some extent reflect the result of conditionalsampling, inasmuch as we have so far only consideredatmospheric disturbances that are associated with enhancedseiche activity at ROZ.)[31] Despite variations in detail from one event to

another, the pattern shown here has been observed inall the 44 cases considered here. Therefore, on the basisof the observations described, it is justified to concludethat these episodic occurrences of wave energy at sea

with periods of the order of 1 hour, possibly causingharbor seiching, are due to variations in local wind speed(thus wind stress) associated with the passage of a frontalsystem containing mesoscale convection cells.[32] To the authors’ knowledge, only Wemelsfelder

[1957] has referred to the possible role of wind speedfluctuations in the generation of long-period (25–50 min)waves at sea, together with a suggested cause of theseperiodic wind speed changes (which at that time he couldnot verify). However, he apparently suggests a Kelvin-Helmholtz instability at higher elevations as the origin ofthe wind speed changes, instead of a static instability inthe air column from the sea surface upward, as is the casewith convection cells described in the present work.[33] The above described characteristics indicate that a

specific situation is required for convection cells togenerate sea waves with significant amplitudes. Thisraises the question whether the mechanism of generationof low-frequency sea waves described here for the south-ern North Sea area can be responsible for atmosphericallygenerated seiches in other areas. Agee [1987] gives ageneral overview of the main areas in the world whereconvection cells have been observed (based on satelliteimages). It appears that, besides the North Sea area, noneof these areas comply with the criteria described above.

Figure 14. Precipitation radar image of 23 November 2001, 0130 UT. Precipitation intensity indicatedfrom white (mild) to black (intense). The ovals indicate examples of convection cells; the arrow indicatesthe approximate direction of movement of the cells. Source original image: Royal NetherlandsMeteorological Institute.


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Most of these areas are characterized by a steep coastalbathymetry (e.g., the west coast of the USA and Canada),with relatively large depths even close to the shore,which is not expected to result in a resonant generationof low-frequency sea waves by passing convection cells.At other locations (e.g., the East China Sea), the convec-tion cells arise from cold air flows that originate abovethe mainland and move offshore across a relatively warmsea, obviously resulting in convection cells that areadvected away from the coast. These could generate longwaves at sea, which however would not generate seichesin harbors at that same coast.

4.3. Generation of Forced Long Waves

[34] In the previous sections, we have presented evidencefor the generation of low-frequency waves at sea by windspeed fluctuations due to a system of moving atmosphericconvection cells over a shallow sea. We now address thequestion of the dynamics of this process. Before going intodetails, we point out two observations that are relevant inthis context. First, as stated in section 4.2, typical values ofthe speed of advection of the system of convection cells areof the order of magnitude of the long wave speed in much ofthe relatively shallow southern North Sea. This suggests thatthe hydrodynamic response there is near-resonant and for

that reason expected to be dominant over contributions fromdeeper water further to the north. A second relevant obser-vation is that the arrival times of the low-frequency dis-turbances more or less coincide with the passage of theassociated cold fronts over the same locality. Forerunnerswith significant energy have not been observed; these wouldbe expected if free waves with significant energy weregenerated in the northern parts (where the long-wave speedexceeds the advection speed of the convection cells).[35] In order to gain insight in the dynamics of the

generation process over the given bottom topography, andto investigate the validity of the ideas given above, we usea simplified model based on the forced, linearized one-dimensional barotropic shallow water equation with neglectof friction and geostrophic effects:

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twr

� �: ð1Þ

Here z is the surface elevation, g the gravitationalacceleration, h the local depth, tw is the wind stress and xa horizontal coordinate in the direction of wave propagation.The linearization and neglect of friction are justified in viewof the very low amplitudes (up to O(10�1) m at sea, indepths of tens of meters). The wind stress forcing is

Figure 15. (a) Overview of the southern North Sea. The solid line indicates the transect that has beenused to derive depth values; the location of the Port of Rotterdam is indicated by the asterisk. (b) Depthalong transect. (c) Amplitude of the forced long waves along transect for U = 17 m/s and L = 80 km. Solidlines in F igures 15b and 15c correspond to the original depth profile; dashed lines correspond to thedepth profile with the Dogger Bank removed.


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assumed to be harmonic with a constant amplitude (s),wave number (k) and speed of advection (U):

twðx; tÞr

¼ <seiðkðx�UtÞÞ ¼ <sðxÞe�iwt ð2Þ

in which s(x) = s exp(ikx) and w = kU is the frequencyobserved in a fixed point. The response to this forcing istaken as:

zðx; tÞ ¼ <hðxÞe�iwt ð3Þ

in which h is the complex amplitude of the forced waves,whose magnitude equals the surface elevation amplitude z.[36] Using equations (2) and (3), the evolution of the

forced low-frequency waves based on equation (1) obeys

w2hþ gdh

dx

dhdx

þ ghd2hdx2

¼ dsdx

: ð4Þ

For a uniform depth and a constant s, the (equilibrium)solution to this equation is

h ¼ iks

w2 � k2gh¼ i

s=kU2 � gh

ð5Þ

which indicates that the forced long waves in theseconditions will be p/2 out of phase with the forcing bythe oscillatory wind stress. For subcritical forcing (U2 < gh),the long wave response is p/2 ahead of the forcing (thissituation is sketched in Figure 11), whereas for a super-critical forcing (U2 > gh) a phase lag of p/2 occurs.[37] For a given value of U and going shoreward, starting

in relatively deep water, where gh > U2, the effectiveness ofthe wind speed fluctuations to generate forced long wavesincreases with decreasing depth until a resonance situationoccurs where gh = U2. However, if the depth decreasesfurther (gh < U2), the effectiveness of the wind speedfluctuations to generate forced long waves will againdecrease. Thus significant amplification of the amplitudesof the forced long waves generated by convection cells islimited to areas where gh � U2.[38] To investigate this further, equation (4) has been

integrated numerically for a depth profile that is realisticfor the area of study. To this end, a transect has been chosenalong the North Sea, more or less in the main path of theconvection cells. This transect and the corresponding depthprofile, smoothed with a running average over a lengthof 30 km to remove irrelevant variations, are shown inFigure 15. The shallow area approximately 300 km fromthe Dutch coast represents the Dogger Bank, which separatesthe relatively shallow southern North Sea from the deepernorthern part.[39] Utilizing the fact that the wind speed variations are

weak compared to the mean wind speed, the amplitude ofthe wind stress changes can be approximated as tw =Cdra2u0u, in which Cd is the drag coefficient (a value of0.0015 has been used), ra is the density of air, u0 is the meanwind velocity (which in this case, as an approximation, hasbeen taken equal to U) and u is the amplitude of the windspeed fluctuations. The following values have been used,which are typical for the observed values: U = 17 m/s, u =3 m/s and the wavelength 2p/k = 80 km. At the sea side

boundary of the model, values for h and dhdx

were imposedaccording to the forced equilibrium solution (equation (5))for the local depth, to prevent free waves from entering thecomputational domain.[40] Figure 15c shows the variation of the computed

surface elevation amplitude along the chosen transect, upto a depth of 20 m. The results confirm that the generationtakes place mainly in the shallow, southern region. The lowresponse in the more remote, deeper parts of the North Seaprecludes a possibility of a seiche warning system based onmonitoring of the sea surface elevation in those regions.Near the Dutch coast, in a depth of 20 m, comparable tothose of the offshore measurement stations (Figure 3),computed amplitudes are in the order of 10 cm, which iscomparable to the observed amplitudes at these locations[de Jong et al., 2003a]. Including the influence of theatmospheric pressure gradients that are caused by theconvection cells resulted in an increase of this amplitudeof approximately 20%. The amplification factor that hasbeen derived for the seiches at ROZ compared to theincoming amplitude is about 7 (average value based onmeasurements [see De Looff and Veldman, 1994]). Thisindicates that the computed amplitudes at sea can result inseiche crest heights in the order of about 0.7 m, whichlikewise is of the same order of magnitude as observed.[41] The amplitude that is found at the boundary of the

model near the coast (h = 20 m) has been calculated fordifferent values of U, keeping the values of the otherindependent parameters the same. Figure 16 shows that theresponse is nonmonotonic, with a maximum at U �18 m/s(in the range of observed values). This nonmonotonicpattern helps explain the weak correlation between thesimultaneous occurrence of significant seiche activity andhigh wind-induced setup of the mean water level found inSection 3.

Figure 16. Amplitude found at nearshore boundary of themodel as function of the advection velocity of the system ofconvection cells. Depth at nearshore boundary is compar-able to the depth at the observation locations near the Dutchcoast (h � 20 m). Solid line represents results based on theoriginal depth profile; the dashed line represents resultsbased on the depth profile with the Dogger Bank removed.


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[42] The nonmonotonic variation of the nearshore re-sponse with the speed of advection in the range below thatof the maximum response reflects the nonmonotonic depthprofile, in particular the existence of the Dogger Bank,which gives rise to multiple regions of possible resonantresponses which shift and may even vanish or appear withvariations of advection speed. Indeed, removing the DoggerBank from the model bathymetry results in a smooth,monotonic variation of the response with the advectionspeed on both sides of the maximum (see Figure 16).[43] Changes in the angle of the chosen transect did not

result in large differences in the depth profile, in particularfor the shallow part of the southern North Sea, south of theDogger Bank. Since the generation of low-frequency wavesmainly takes place in this shallow area, the calculatedresults at the nearshore boundary of the model did notchange significantly with changes in the direction of thetransect.[44] The preceding exploratory computations were based

on the steady state (purely harmonic) solution for a one-dimensional model. They illustrate the essential features ofthe dynamics of the generation of these low-frequency seawaves. For more exact numerical evaluations, more realis-tic, transient atmospheric forcing should be used in con-junction with a two-dimensional model.

5. Conditions for the Occurrence ofConvection Cells

[45] Convection cells of sufficient strength arise only if thelower atmosphere is sufficiently unstable. A number ofstorm season months has been analyzed in order to investi-

gate the conditions for the occurrence of an unstable atmo-sphere with convection cells such that low-frequency wavesare generated at sea inducing seiches in Rotterdam harbor.[46] The criterion for an unstable lower atmosphere is

based on the adiabatic lapse rate for dry air and a pseudoa-diabatic lapse rate of moist air [see Holton, 1992, pp. 53 and290]. It can be expressed as a critical value of the verticaltemperature gradient. In case the temperature lapse rate (rateof decrease in temperature from the sea surface upward) issmaller than 5.5 to 7.5�C/km, the lower atmosphere is stable;we will use 6.5�C/km as a representative critical value.Between approximately 10�C/km and (say) 6.5�C/km it isconditionally unstable (depending on the moisture content),and a temperature lapse rate larger than 10�C/km results in anunstable atmosphere (see Holton [1992] for details on thederivation of these values). For a given reference level thesetemperature gradient criteria can be converted to criteria fortemperature differences between the surface seawater and theair at the reference level. In meteorology, standard referencelevels are used in the boundary layer at 925 hPa (at an altitudeof about 700 m) or 850 hPa (at an altitude of about 1500 m).For the 850 hPa level these criteria become:�T < 10�C for astable atmosphere, 10�C < �T < 15�C for a conditionallyunstable atmosphere and �T > 15�C for an unconditionallyunstable atmosphere.[47] Sea surface water temperature values measured at the

offshore platform K13 have been obtained for 1995–2001(available as daily values). Atmospheric soundings have beenused from platform EKO or, if these were not available, fromDe Bilt (about 75 km inland, east of Rotterdam).[48] For the reference level at 850 hPa, 90% of the

observed temperature differences during a total of 42 seiche

Figure 17. (top) Temperature difference between the surface seawater at K13 and the air at 850 hPalevel at Ekofisk for December 2001. (bottom) Same but restricted to days of occurrence of a cold frontthat moved from the North Sea toward the Dutch coast (see main text). The asterisk denotes days onwhich a seiche event occurred.


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events complied with the criterion for conditional atmo-spheric instability (44 events were studied in total, no datawas available for two events). The criterion for an uncon-ditionally unstable atmosphere was met in 57% of the cases.The results that were found with corresponding criteria forthe 925 hPa reference level showed an even strongercorrelation: 95% and 81%, respectively. For the latterreference level, all noncompliance cases were based onatmospheric soundings at the land based station De Bilt. Itis expected that the number of events that comply with thecriteria will increase in case soundings are available at seafor all cases.[49] The results described above, for given situations with

seiche events, show a strong correlation between the occur-rence of a nominally unstable lower atmosphere and seicheevents. Studying the stability of the lower atmosphere whenno seiche events occur could help explain why only alimited number of the cold front passages over the southernNorth Sea result in a seiche event. Possibly, knowledge ofthe circumstances needed for convection cells to arise couldeventually even be used to predict the occurrence, and alsothe absence, of seiche events.[50] In order to determine the predictive power of this

approach, archived sea surface and atmospheric temperaturedata on consecutive days during a number of storm seasonmonths have been analyzed in a hindcast situation. For eachday atmospheric soundings of 0000 UT have been used(1200 UT if these were not available). Results for twomonths are described here: October and December 2001.These two storm season months have been chosen becausean approximately equal number of cold fronts passed theNorth Sea during each of these months, whereas during one

of these months no seiche events occurred (October) andduring the other month a relatively large number of eventsoccurred (December). The year 2001 was selected becausefor recent years archived weather charts are readily available(daily, at 0000 UT). These have been used, together withprecipitation images and infrared satellite images, to inspectthe meteorological situation during all considered days.[51] Figure 17 (top) shows the difference between the

surface seawater temperature at K13 and the air temperatureat 850 hpa level at EKO for each day of December 2001.The boundaries of the temperature difference indicating aconditionally or unconditionally unstable atmosphere arealso shown in this figure.[52] Weather maps have been used to identify the days

on which cold fronts have passed the North Sea thatresulted in a flow toward the Dutch coast (includingpossible successive days during which this flow directioncontinued). This is obviously needed for seiche events tobe generated in the Port of Rotterdam. Figure 17 (bottom)again shows the temperature differences given in the toppanel, now for only those days that comply with thissecond criterion. The days on which seiche events haveoccurred have also been indicated. (Two consecutive dayswith seiche events indicated represent a single event thatstarted during the first day and ended on the second dayindicated.) All seiche events that occurred during thismonth not only coincided with a cold front that caused aflow toward the Dutch coast (for one or more days), theyalso coincided with a lower atmosphere that was uncon-ditionally unstable (on 28 December 2001, the atmospherebecame unconditionally unstable during the course of theday, following the cold front passage).

Figure 18. (top) Temperature difference between the surface seawater at K13 and the air at 850 hPalevel at Ekofisk for October 2001. (bottom) Same but restricted to days of occurrence of a cold front thatmoved from the North Sea toward the Dutch coast (see main text).


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[53] Figure 18 shows a similar overview as in Figure 17,now for October 2001. Again, Figure 18 (bottom) shows thetemperature differences for the days during which relevantcold front passages occurred and successive days duringwhich the resulting flow toward the Dutch coast continued.Figure 18 (bottom) shows that the criterion for instability ofthe lower atmosphere was not met during these potentiallyseiche-prone cold front situations. This means that noconvection cells would be expected, which agrees withthe fact that no seiche events occurred during this month.[54] The overall quality of this method of prediction

appears to be surprisingly good, considering the limitednumber of criteria that is used. This is ascribed to therestricted variation in the relevant meteorological conditionsover the North Sea region during the occurrence of convec-tion cells. In this area, the cold fronts that result intemperature differences that are large enough for convectioncells to arise have front velocities in a rather narrow rangeof about 15–20 m/s. Likewise, the size of the convectioncells that are found over the southern North Sea appearsto have typical values in the restricted range of about70–100 km (which is relatively large compared to otherlocations [see, e.g., Atkinson and Zhang, 1996]). Thecombination of advection velocity and cell size frequentlyresults in low-frequency sea waves that are generated nearthe lowest eigenfrequency of the studied basin during thepredicted cases described above.[55] So far, only one situation has been found where the

temperature difference indicated an unconditionally unsta-ble lower atmosphere (� T � 20�C) and no seiche eventwas registered at the Calandkanaal. However, during thissituation, relatively small-scale convection cells were foundon the satellite images and a seiche event did occur, on thebasis of a local amplitude criterion, in a shorter basin in thePort of Rotterdam, with a higher eigenfrequency.[56] The above described evidence indicates that condi-

tions for the existence of convection cells together withinformation regarding the passage and direction of coldfronts can be used to identify and eventually predict seiche-prone situations. The analysis of measurements from otherbasins in the Port of Rotterdam (with higher eigenfrequen-cies) and the refinement of the suggested prediction methodare part of ongoing work. For example, if not only temper-ature is taken into account but moisture content as well, afurther improvement of the predictive capability isexpected. Both parameters are routinely computed in nu-merical weather forecasting models. This may open thepossibility of operational forecasting of seiche-prone con-ditions, which would be useful for port authorities and forthe operation of the movable storm surge barrier.[57] As described in Section 3, storm situations have not

been observed during seiche events in the studied timeinterval. So far, the question whether convection cells canalso arise and generate low-frequency sea waves in moreextreme storm situations has not been discussed. The resultsof the 1-D model described in par. 4.3 (Figure 16) indicatean inability of convection cells to generate low-frequencysea waves of significant amplitudes at high advectionvelocities (corresponding to severe storm situations). How-ever, those calculations were made with the assumption thatthe characteristics of convection cells remained constant(only the advection velocity was changed). Observations

described by, for example, Atkinson and Zhang [1996]indicate that the coherent patterns of convection cells canonly occur when the vertical wind shear is relatively small.Therefore convection cells are not expected to arise over theNorth Sea in extreme storm situations because then the cellsare diminished by the high turbulence levels and by thelarge vertical mean wind shear. Storm events need to bestudied to verify this. If it is true, this would preclude thesimultaneous occurrence of high wind-induced setup of themean water level and significant seiche activity. (Even ifconvection cells would occur in these cases, the highadvection speed would prevent them from generating low-frequency waves with significant amplitudes, as mentionedabove.) This is of practical importance in establishingdesign conditions for dikes and for the operation of thestorm surge barrier.

6. Conclusions

[58] Analyses of meteorological data in the North Searegion, in combination with sea surface elevation recordsoffshore and inside a Rotterdam Harbor basin, have shownthat moving mesoscale atmospheric convection cells, follow-ing a cold front passing over relatively warm seawater, inducesurface wind speed fluctuations which in turn generate low-frequency (period O(1 hour)) waves at sea. The sea surfaceresponse is amplified when the system of atmospheric con-vection cells moves at a speed near that of free (long) gravitywater waves, which happens to be the case in the southernNorth Sea for typical weather situations in that region, givingrise to O(10�1)m amplitudes in the coastal zone.[59] The wind-induced low-frequency waves generated

offshore excite oscillations in coastal harbors, which can beamplified in case of resonance, giving rise to so-calledseiches. This occurs, among others, for one of the Rotter-dam harbor basins, where amplitudes up to 0.9 m haveoccasionally been observed at the closed end (near Rozen-burgse Sluis, ROZ). To the authors’ knowledge this gener-ation mechanism of low-frequency waves at sea and seichesin harbor basins as described here has not been reportedbefore in the scientific literature.[60] Analysis of surface elevation records at ROZ in the

period 1995–2001 showed a total of 49 seiche events (crestheight exceeding 0.25 m at ROZ). Most seiche events occurin the storm season and all seiche events occurred followinga cold front passage over the harbor area, approaching fromthe sea. Of these events, 44 coincided with episodes of low-frequency waves at sea. The origin of this type of seicheevent has been studied. Application of a conventionalcriterion for the occurrence of convection cells, expressedin terms of a minimum temperature difference between thecold upper atmosphere and the warmer sea surface water,has shown the feasibility of prediction of seiche-pronesituations, which may be beneficial to harbor authorities.Because the low-frequency wave response remains veryweak in the deeper parts of the northern North Sea, anadvance seiche warning system based on monitoring of thesea surface elevation there is not possible.[61] The timescales of the convection cells derived from

the ranges of the observed length scales [Atkinson andZhang, 1996] and advection velocities indicate that intheory these cells can cause wind speed fluctuations with


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periods between 20 to 160 min. This range encompasses notonly the lowest eigenfrequency of the studied basin, it alsoencompasses eigenfrequencies of other basins in the Port ofRotterdam and elsewhere, including a basin that is tempo-rarily formed when a storm surge barrier in the RotterdamWaterway is closed. The effectiveness of convection cells togenerate low-frequency sea waves at these other eigenfre-quencies is part of ongoing research.[62] It would appear that the low-frequency waves at sea

described here could in principle be detected with remotesensing techniques such as satellite-borne altimeters, be-cause in the shallow-water areas their amplitudes exceed thenoise level in such observations. If these data do in factcontain a useful signal, they could be used to further studythese low-frequency sea waves.

[63] Acknowledgments. The first author (MPCdeJ) acknowledges thefinancial support from the Dr. Ir. Cornelis Lely Foundation. Furthermore, theauthors thank the Directorate-General for Public Works and Water Manage-ment and the Rotterdam Municipal Port Management for providing themeasurement data at sea and in the harbor. The authors also thank the RoyalNetherlands Meteorological Institute for the meteorological measurements,weather charts, and precipitation images. Moreover, the authors thank A. J.van Delden of Utrecht University, Netherlands, for his extensive informationon convection cells. Atmospheric sounding data have been obtained fromthe Department of Atmospheric Science, University of Wyoming, USA.Wavelet software was provided by Torrence and Compo [1998] (availablethrough URL: http://paos.colorado.edu/research/wavelets/).

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Atkinson, B. W., and J. W. Zhang (1996), Mesoscale shallow convection inthe atmosphere, Rev. Geophys., 34, 403–431.

Canestrelli, P., M. Mandich, P. A. Pirazzoli, and A. Tomasin (2001), Wind,depression and seiches: Tidal perturbations in Venice (1951–2000), Tech.Rep. 2001, Centro Previvioni e Segnalazioni Meree, Venice.

Cerovecki, I., M. Orlic, and M. C. Hendershott (1997), Adriatic seichedecay and energy loss to the Mediterranean, Deep Sea Res., Part I, 44,2007–2029.

de Jong, M. P. C., L. H. Holthuijsen, and J. A. Battjes (2003a), Generationof seiches by cold fronts, over the southern North Sea, J. Geophys. Res.,108(C4), 3117, doi:10.1029/2002JC001422.

de Jong, M. P. C., L. H. Holthuijsen, and J. A. Battjes (2003b), Generationof seiches by cold fronts, Coastal Engineering 2002: Solving CoastalConundrums, edited by J. McKee, J. Smith, and Smith McKee, pp. 1106–1117, World Sci., River Edge, N. J.

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Hibiya, T., and K. Kajiura (1982), Origin of the Abiki phenomenon (a kindof seiche) in Nagasaki Bay, J. Oceanogr. Soc. Jpn., 38, 172–182.

Holton, J. R. (1992), An Introduction to Dynamic Meteorology, 3rd ed.,Academic, San Diego, Calif.

Mallat, S. (1998), A Wavelet Tour of Signal Processing, Academic, SanDiego, Calif.

Metzner, M., M. Gade, I. Hennings, and A. B. Rabinovich (2000), Theobservation of seiches in the Baltic Sea using a multi data set of waterlevels, J. Mar. Syst., 24, 67–84.

Monserrat, S., A. Ibbetson, and A. J. Thorpe (1991), Atmospheric gravitywaves and the ‘Rissaga’ phenomenon, Q. J. R. Meteorol. Soc., 117, 553–570.

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Torrence, C., and G. P. Compo (1998), A practical guide to wavelet anal-ysis, Bull. Am. Meteorol. Soc., 79, 61–78.

Tritton, D. J. (1977), Physical Fluid Dynamics, Van Nostrand Reinhold,New York.

Veraart, C. (1994), Events of long waves in the North Sea, Tech. Rep. RIKZ-94.034, Rijkswaterstaat, The Hague, Netherlands.

Wemelsfelder, P. J. (1957), Origin and effects of long period waves in ports,paper presented at Congres International des Navigation, Int. Navig.Assoc., London.

��J. A. Battjes and M. P. C. de Jong, Environmental Fluid Mechanics

Section, Delft University of Technology, P.O. Box 5048, 2600 GA Delft,Netherlands. ([email protected]; [email protected])


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