sea level variability and the ‘milghuba’ seiche oscillations in the...

Physics and Chemistry of the Earth 34 (2009) 948–970

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Physics and Chemistry of the Earth

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Sea level variability and the ‘Milghuba’ seiche oscillations in the northern coastof Malta, Central Mediterranean

Aldo Drago *

Physical Oceanography Unit, IOI-Malta Operational Centre, University of Malta, Msida MSD2080, Birkirkara, Malta

a r t i c l e i n f o a b s t r a c t

Article history:Received 8 March 2009Accepted 12 October 2009

Keywords:Sea level oscillationsTidesShelf resonancesSeichePower spectraBay

1474-7065/$ - see front matter � 2009 Elsevier Ltd. Adoi:10.1016/j.pce.2009.10.002

* Tel./fax: +356 2144 0972.E-mail address: [email protected]

The characteristics of the water level variations in the northern coastal area of Malta are studied by a setof densely sampled data collected in the period 1993–1996 (43 months) at a permanent coastal sea levelinstallation in Mellieha Bay. These measurements constitute the first digitised sea level recordings in theMaltese Islands and were collected as part of a research programme that produced a long time series ofsimultaneous water level and meteorological parameters in the Central Mediterranean.

Tidal oscillations reach a maximum range of only 20.6 cm on average and are predominantly semi-diurnal. Important non-tidal signals are however found to span the whole spectral range of frequencies.Seiche oscillations in the form of large amplitude sea level fluctuations, known locally as the ‘milghuba’,carry substantial energy in the range of long wave frequencies (0.2–2 cph) and often mask completely thetidal signal. These coastal seiches are believed to be the expression of shelf scale resonances; in thenumerous embayments on the northern coastline of the Maltese Islands, these seiches are greatly ampli-fied and have associated swift alternating currents that are useful for the mixing and exchange of thewater body in the embayments with the adjoining open sea areas, but can constitute a nuisance to nav-igation especially at harbour entrances. In the synoptic and sub-synoptic time scales, variations in atmo-spheric pressure associated with mesoscale meteorological phenomena produce a predominant effect onthe sea level, but the response of the sea is non-isostatic and carries the signature of oceanographic con-ditions in the region as well as that of non-local forcing resulting from intra-basin differences. Strong sea-sonal non-eustatic fluctuations in the mean sea level are characterised by a high sea level in Decemberand is typically followed by a sharp fall to a minimum in February/March. This seasonal variability is amanifestation of the adjustments in the mass balance of the whole Mediterranean basin.

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1. Introduction

The quest to improve predictions of the residual circulation andto refine the skill of ocean forecasting numerical models in dealingwith mesoscale activity containing energy at the synoptic timescale, has in recent years accentuated the challenge to understandthe interaction of the sea with the atmosphere. This considerationis indeed crucial for the Mediterranean Sea where the tide is gen-erally weak and the tidal circulation is greatly surpassed by themeteorologically forced motions. In the particular case of theSicilian Channel, the synoptic variability and the mesoscale phe-nomena constitute very important components of the total flow(Lermusiaux and Robinson, 2001). The signatures of these phe-nomena are captured by the sea level signals which carry a high le-vel of variability at the synoptic, seasonal and interannual scales(Robinson, 1999), and as in the rest of the Mediterranean representan energetic part of the sea level spectrum.

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The Sicilian Channel is a large and dynamically active areaconnecting the eastern and western Mediterranean sub-basins.The general flow in the Channel is mainly driven by the slow(vertical) Mediterranean thermohaline basin scale circulation.The region is known to contain a number of significant hydrody-namical processes and phenomena that span the full spectrum oftemporal and spatial scales (Grancini and Michelato, 1987; Manz-ella et al., 1990; Moretti et al., 1993). The mesoscale processesare triggered by the synoptic scale atmospheric forcing. The heatand momentum fluxes at the air–sea interface represent thedominant factor in the mixing and pre-conditioning of the Med-iterranean Atlantic Water (MAW) on its way to the easternMediterranean.

The highly irregular bottom topography of the Channel takesthe form of a submarine ridge which connects in the east and westto the respective Mediterranean basins only through a system ofnarrow sills. This ridge restrains the exchange between the twoMediterranean basins and has a controlling function, in additionto that at the Strait of Gibraltar, on the adjustment of the sea levelin the Mediterranean to meteorological forcing. The ridge is

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characterised by a long-shaped NW–SE basin cutting deep into thecontinental shelf (Fig. 1). The average depth of this intermediatebasin is in the order of 500 m. Owing to the nature of its differentbehaviour with respect to the main basins, it is generally consid-ered as a third basin: the Central Mediterranean basin. In contrastto the rest of the Mediterranean, the continental margins in theChannel are rather wide and shallow. On the African coast the shelfarea covers more than a third of the areal extent of the Strait withwater depths less than 30 m in the Gulf of Gabes. Along the south-ern coast of Sicily the shelf takes the form of two disconnected andrelatively shallow (<200 m) banks. The Maltese Archipelago, con-sisting of a group of small islands aligned in a NW–SE direction,is located close to the southeastern margin of the Sicilian shelf.The islands are thus located at an oceanographically strategic posi-tion in the middle of the exchange flow through the Channel, andact like a permanent station close to the shelf break.

A set of densely sampled sea level, barometric pressure and windvector recordings were collected at two stations on the northerncoastal perimeter of Malta. These datasets constitute the basis ofthis study and were acquired through the Physical OceanographyUnit of the IOI-Malta Operational Centre of the University of Malta.On the merit of the position of the island within the Sicilian Chan-nel, the datasets are particularly important to understand the roleof the Channel in controlling the exchange between the two majorbasins of the Mediterranean Sea. With an internal Rossby radius ofjust a few tens of kilometres on the shelf areas, mesoscale phenom-ena in the Channel are impossible to detect and follow unless a de-tailed observation set is available in both time and space. Undersuch circumstances, sea level measurements become of great rele-vance as an indicator of the general dynamics of the sea (Wunsch,

Fig. 1. The Central Mediterranean showing the main isobaths and sea level stationstopography of the Siculo–Maltese continental shelf and the location of the stations deploEmpidocle = PE; Capo Passero = CP; Sfax = SX; Pantelleria = PA; Mazara del Vallo = MZ; T

1972) especially when measured, as in the case of Malta, at a loca-tion away from the continental mainland.

The overall objective of this work is to make a detailed analysisof the sea level measurements collected in Malta. The aim is toidentify and quantify the tidal and non-tidal signals composingthe sea level spectrum in the vicinity of the islands, and study theiroccurrence and variability in relation to the adjustment of theMediterranean Sea to meteorological forcing and to other pertinentoceanographic processes. Section 2 describes the datasets and thedata processing techniques used in this paper; it also gives a gen-eral overview of the sea level characteristics on the basis of the en-ergy distribution between different frequency bands obtained fromspectral analysis, and with a focus on the tidal component. This isfollowed in Section 3 by a more detailed analysis of the short per-iod and large amplitude coastal seiches. It is interesting to notethat one of the first scientific studies on seiches by Sir George Airy(1878) refers precisely to the Grand Harbour in Malta. Since thenthe seiche phenomenon in Malta remained unstudied and thiswork forms part of a renewed effort to study these high frequencysea level oscillations in this part of the Mediterranean (Drago,1999, 2001, 2007). The subsequent sections deal more specificallywith the other sea level components – meteorological and sea-sonal. The final section discusses the results and poses questionsfor further research.

2. Materials and methods

Densely sampled sea-level data (at the variable rate of 30 or 60samples per hour) have been collected at a permanent tide gauge

used for tidal constituents in Table 2. The insert in the figure shows the seabedyed for this study. Gabes = GB; Zarzos = ZA; Lampedusa = LP; JANUS SG = SG; Porto

ripoli = TR; Malta = GR.

1 The term ‘milghuba’ comes from the Maltese verb ‘laghab’ which means ‘play’.The terminology refers to the ‘play of the sea’.

950 A. Drago / Physics and Chemistry of the Earth 34 (2009) 948–970

installation positioned at the head of Mellieha Bay which is a smallembayment on the northwestern coast of Malta (indicated by MLin the insert in Fig. 1). The data set used for this study is that col-lected in the period June 1993–December 1996. An Endeco Type1029/1150 differential pressure tide gauge is used and is situatedinside a small stilling well connected to the sea. The pressuretransducer is located in a subsurface case and is awakened everysampling interval for a total of 49 s in order to filter waves. Theinstrument measures absolute pressure; atmospheric pressure iscompensated by means of a vented tube which passes throughthe topcase unit and terminates inside an environmental isolatorin the form of a small exposed PVC tube with a bladder.

Meteorological parameters are also measured by Aanderaa sen-sors at a nearby automatic weather station in Ramla tal-Bir (MT inthe insert in Fig. 1) which is situated on the coastal strip overlook-ing the South Comino Channel. The sensors are positioned in anunobstructed location at a height of 20 m from mean sea level.The data set consists of wind speed and direction, air pressureand temperature, relative humidity and net atmospheric solar radi-ation each measured at 1 or 2 min intervals, in the period April1994–December 1996. These data sets represent the first long termdigitised measurements of hydro-meteorological parameters inMalta. Air pressure data are expressed in millibars and correctedto mean sea level.

Short analogue records of hourly sea-level data in the GrandHarbour and of 3-h sampled air pressure at Luqa are provided bythe Hydrographic Office of the Malta Maritime Authority, and bythe local Meteorological Office, respectively. Upon digitisation,these data sets are used in addition to the above data sets.

Observations at the shelf scale level are provided from the ded-icated Malta Channel Experiment. The Malta Channel is the stretchof sea separating the Maltese Islands from the southern coast ofSicily (refer to the insert in Fig. 1). These observations consist ofa series of simultaneous sea level measurements at two locationsacross the Malta Channel. The data comprise simultaneous bottompressure recordings at Qawra station outside Mellieha Bay(depth = 30 m; position = 35� 59.350N; 14� 25.810E), at Pozzallo onthe southern coast of Sicily (depth = 18 m; position = 36� 42.210N;14� 50.120E) and at the Mellieha coastal station (refer to locationsQW, PZ and ML, respectively, in the insert in Fig. 1). Pozzallo sta-tion is approximately to the north of Qawra station at a separationof about 100 km across the whole shelf area. These measurementsspan a period of 20 days (8:50 GMT 10th September to 11:54 GMT30th September 1996) and were collected during the Malta ChannelExperiment carried out in co-operation with the Italian CNRInstitute at Mazara del Vallo, Sicily. Bottom pressure fluctuations(in mb) are translated into sea level variations (in cm) by directequivalence. Atmospheric pressure at MSL is added to coastalsea-level data to obtain adjusted sea levels.

Furthermore an ENDECO/YSI tethered-type current meter isused to measure seiche-induced subsurface currents in MelliehaBay. The instrument is deployed within the embayment at a stationlocated at 35� 58.80N, 14� 22.80E where the total depth is 28 m (re-fer to Fig. 8). Measurements consist of the vector averaged sea cur-rent and temperature sampled every 2 min at 13 m from thesurface and cover the period 16:04 GMT 25th July–07:52 GMT30th July 1994. Sea level is also measured every 2 min at the coast-al station situated at the head of the embayment. Meteorologicalparameters, including wind, surface pressure and air temperatureare measured at the nearby Ramla tal-Bir station.

Finite Impulse Response filters are used on the original datatime series in two steps and with decimation to first produce a10-min series and subsequently hourly averaged values in the sec-ond step. A cosine filter with nine weighting factors, pass-bandending at 0.156 cycles/data interval (99% gain) and stop-bandstarting at 0.326 cycles/sampling interval (1% gain) is employed

in the first step. A Doodson Xo filter (IOC-UNESCO, 1985) with 27weighting factors and a half-gain at 0.39 cycles/h is subsequentlyused to obtain the hourly values. The mean sea level (MSL) is de-rived by applying the A24A24A25/(242 � 25) tide-killing filter(Godin, 1972) to the hourly data; this filter has a half-gain at a per-iod of 2 days and thus retains only the longer period variations ofthe signal. Daily and monthly averages are calculated by takingthe simple arithmetical mean of MSL over 24 h or 1 month,respectively.

Tidal harmonic analysis is made by means of a least square pro-cedure using TIRA, a software developed at the Proudman Oceano-graphic Laboratory in Bidston, Birkenhead, UK, (Murray, 1964).TIRA allows analysis even in the presence of gaps. A few short gapsin the original data set, each smaller than 1 day, are thus interpo-lated by using predicted values. Residuals prior to and after a datagap are linearly interpolated and used to estimate and add the non-tidal component to the predicted elevation during gaps. In order toisolate the non-tidal higher frequency component in the data, thepredicted tides are subtracted from the original records and furtheranalysis is then performed on the residual series.

The sea level and meteorological data are also used to calculateenergy distributions and power spectra. 50% overlapping segmentsare taken in each case. Trend and mean are removed and a Kaiser-Bessel window (Harris, 1978) applied to each segment. The taperedsegments are then subjected to Fast Fourier Transform (FFT) anal-ysis to calculate the spectra by using the Welch method. In the caseof spectra obtained from an average over a long series, the influ-ence of transient effects is suppressed and the results thus deter-mine the general phenomenology in the region of the measuringstation.

2.1. General data analysis

Water level records from Mellieha Bay demonstrate that thetide is mainly semi-diurnal and with low amplitude. The rangefor spring tides is on average 0.206 m, and is reduced to 4.6 cmduring neap tides. Fig. 2 is a typical data series of 2-min sampledwater levels covering the period from mid-September to mid-Octo-ber 1995. The most remarkable feature in the trace is the presenceof a band of high frequency signals with periods ranging from sev-eral hours to as low as a few minutes. The long term measurementspresented in this work constitute the first digitised data set thatpermit the scientific study of these non-tidal short period sea levelfluctuations which are the expression of a coastal seiche, known bylocal fishermen as the ‘milghuba’.1 This phenomenon has now beenobserved to occur all along the northern coast of the Maltese archi-pelago and manifests itself with very short resonating periods ofthe order of 20 min in the adjacent coastal embayments. Analysisof the full data set shows that weak seiching is present uninter-rupted and appears like a background ‘noise’ on the tidal records.During random sporadic events the seiche oscillations can howeverbecome greatly enhanced and mask completely the astronomicalsignal. It is interesting to note that reference to similar sea levelvariations (known as the ‘Marrubbio’) on the southern coast ofSicily is found in the Italian ‘Portolano’ for ship navigation. Theiroccurrence is reported to be most frequent in May or June in asso-ciation to south easterly winds, and their crest-to-trough ampli-tudes can reach as high as 1.5 m. Literature on the ‘Marrubbio’ isvery scarce with the most relevant publication being that by Col-ucci and Michelato (1976) who quote typical periods of 14.6 min(main peak), 33.6 and 48 min (secondary peaks) in Porto Empido-cle where the maximum seiche amplitude is however reported toreach only 35 cm. Similar short period oscillations are known to


occur in other places including the Mediterranean where they havebeen extensively studied in Ciutadella Harbour, Menorca in theBalearic Islands (Rabinovich and Monserrat, 1996, 1998; Monserratet al., 2006; Vilibic et al., 2008).

The longer period signals are better studied from the filteredhourly values. Fig. 3 presents a representative record of observed,predicted (sum of tidal periodic variations) and non-periodic resid-

Fig. 2. Time series of sea level in Mellieha Ba

Fig. 3. Time series of (a) the observed tide and mean sea level, (b) the predicted astronMarch–10th May 1995; (d) is the inverted atmospheric pressure at MSL for the same pe

ual (observed minus predicted) water level fluctuations, togetherwith the corresponding inverted barometric pressure for the peri-od covering 18 March–10th May, 1995. The smooth curve drawnupon the observed data gives the variation of MSL and shows thepresence of long-period oscillations due to both long-period tidalconstituents and meteorological influences. These signals have aperiodicity of several days and are related to the large-scale cyclic

y (14th September–14th October 1995).

omical tide and (c) the residual sea elevation in Mellieha Bay for the period 18thriod.


atmospheric patterns in the region. The meteorological origin ofthese long-period variations is evidenced by the very consistent in-verse relation between MSL and barometric pressure (Fig. 3).

2.2. Spectral analysis and energy distribution

The characteristics of the sea level signals are best studied byspectral analytic techniques. Fig. 4a and b is a normalised powerspectral density plot of the full 2-min sampled data set from 1/6/93to 2/1/97 plotted on a linear scale. Since the spectrum is obtainedfrom an average over a long series, the influence of transient effectsis suppressed and the results thus determine the phenomenologyof the sea surface vertical movement in the region of this station.The Kaiser-Bessel spectral windows with 50% overlap are chosento be 217 records for the lower frequency range (Fig. 4a) and 214

records for the higher frequencies (Fig. 4b). The different windowsizes permit an optimal resolution for the long-period and short-period components, respectively. The linear spectral plot gives abetter visualisation of the relative distribution of energy. Thecorresponding logarithmic plots in Fig. 5a–e display the same char-acteristics, but the components with smaller energy inputs areenhanced. In Fig. 5a an additional frequency averaging is per-formed to smooth the spectral estimates at the higher frequencies.In this case the degrees of freedom are 22 and 110 for the lowerand higher frequency ends, respectively, of the spectrum.

The analysis is performed on three main frequency bands: (1)the low frequency (long-period) band (LB) in the range 0–0.8 cpd(T (h) >30); (2) the tidal frequency band (TB) in the range 0.8–4.8cpd (30 > T (h) > 5); (3) the long wave frequency (short-period)band (SB) in the range 4.8 cpd and upwards (T (h) < 5). The term‘long wave’ is here associated to sea level fluctuations with periodsintermediate between those of the long swell and of the astronom-ical tide. These spectra for these three frequency bands are drawnseparately (Fig. 5b–e) for better visualisation. The tidal band is

Fig. 4. Normalised power spectra on a linear scale calculated from 2-min sampled sea levfreedom are (Bmin = 0.52; Bmax = 2.5); the spectrum normalisation factor is 0.2048 m2/Bmax = 1.3). The spectrum normalisation factor is 0.1572 m2/cph.

plotted for the observed (Fig. 5c) and the residual (Fig. 5d) data.The sea level variations are dominated by energy inputs from thelow frequency signals and the semi-diurnal tidal components(Fig. 4), with a secondary contribution from the diurnal fluctua-tions. In the tidal frequency range, fluctuations are dominated byenergy inputs of semi-diurnal frequency with weaker contribu-tions from diurnal signals. Comparison between the two tidal en-ergy density peaks yields a ratio of amplitudes of the order of1:7 which agrees well with results from harmonic analysis ofcoastal tidal records. The peaks towards the higher frequency endsof the tidal range refer to the 1/3- and 1/4-diurnal contributions.

The energy distribution at different frequencies is expressed asa percentage of the total energy in the records (Table 1). These per-centages quantify the dominance of the low frequency inputs, pre-sumably of large scale meteorological origin, which contribute for57.3% of the total energy in Mellieha Bay. Tidal energy inputs(35.8%) mainly result from the semi-diurnal component (32.7%).The high frequency (>4.8 cpd) inputs, due to the coastal seiches,contribute only 6.6%. This figure is an average of seiche energy overthe whole time span covered by the data series and greatly under-estimates the real energy carried by the large amplitude seicheswhich are transient events lasting only for relatively short periodsof time (from a few hours to a couple of days). A full characterisa-tion of different kinds of seiches experienced in Mellieha Bay, andtheir relative power spectral signatures is given in Drago (1999).

During the summer months, in particular July and August, theseiche is very sharp, with a duration that can often be as short asa few cycles. The associated sea level fluctuations have a tsunami-like nature, starting with an abrupt and large impulse that subse-quently decays after a few oscillations. Seiches during this time ofyear are particularly strong. On several occasions, seiche heightsclose to 1 m have been recorded. One of the strongest is that on22nd August 1996 with an excursion in sea level of 1.3 m. In otherperiods of the year the seiche height does not generally reach these

els in Mellieha Bay (1/6/93–2/1/97): (a) the 95% confidence factors, for 22 degrees ofcpd, (b) the 95% confidence factors, for 176 degrees of freedom are (Bmin = 0.85;

Fig. 5. (a) Normalised power spectrum on a logarithmic scale calculated from 2-min sampled sea levels in Mellieha Bay (1/6/93–2/1/97). Up to f = 3 cpd, the spectrum iscalculated with 22 degrees of freedom and 95% confidence factors Bmin = 0.52 and Bmax = 2.5). For f > 3 cpd, the spectrum is calculated with 110 degrees of freedom and 95%confidence factors Bmin = 0.78 and Bmax = 1.35). The normalising factor is 0.2048 m2/cpd. Normalised power spectrum on a logarithmic scale for (b) the low frequency band;(c) the tidal band from observed sea levels. Normalised power spectrum on a logarithmic scale for (b) the low frequency band; (c) the tidal band from observed sea levels. Thespectrum is calculated with 22 degrees of freedom and the 95% confidence factors are (Bmin = 0.52; Bmax = 2.5); the normalising factor is 0.2048 m2/cpd. Normalised powerspectrum on a logarithmic scale for (d) the tidal band from residual levels, and (e) the short period band. In (d), the spectrum is calculated with 22 degrees of freedom and the95% confidence factors are (Bmin = 0.52; Bmax = 2.5); the normalising factor is 0.2048 m2/cpd. In (e) the spectrum is calculated with 110 degrees of freedom and the 95%confidence factors are (Bmin = 0.78; Bmax = 1.35); the normalisation factor is 0.1572 m2/cph.


extremes. The seiches are however much more persistent, lastingeven for days on some occasions. The cumulative seiche energy isthus much higher. A short buildup period generally preceeds these

seiche events. The seiche then develops as a succession of randomintensifications which follow closely one another resulting in a ser-ies of bursts of large amplitude fluctuations.

Fig. 5 (continued)


2.3. The tidal constituents

The general tidal pattern for the Mediterranean Sea as a wholegives nodal locations in the Strait of Sicily, and the magnitude of

Table 1Percentage energy distribution in Mellieha Bay.

Frequency band %

Low frequency (<0.8 cpd) 57.3Diurnal (0.8–1.2 cpd) 3.0Semi-diurnal (1.8–2.2 cpd) 32.7Quarter diurnal (3.8–4.2 cpd) 0.06High frequency (>4.2 cpd) 6.6Other 0.36

Table 2Harmonic constants for the Central Mediterranean Sea. Amplitudes in centimetres; phase

Station Latitude Longitude M2 (cm/deg) S2 (cm/deg) K1 (cm/d

Gabes 33�530 10�070 51.1 36.4 2.579 107 349

Sfax 34�440 10�460 41.6 26.7 1.876 103 4

Zarzis 33�300 11�070 21.9 15.3 2.077 103 31

Pantelleria 36�470 12�000 1.6 1.9 1.331 42 184

Lampedusa 35�300 12�300 6.6 4.2 0.945 58 3

Mazara del Vallo 37�380 12�350 4.3 1.8 3.5161 151 114

JANUS SG 36�100 12�590 4.8 3.1 0.550 57 78

Tripoli 32�540 13�120 11.1 5.4 2.060 75 13

Porto Empidocle 37�150 13�300 4.5 3.3 1.878 77 91

Malta 35�540 14�310 6.3 4.0 1.047 57 19

Capo Passero 36�410 15�090 6.7 3.5 1.962 67 52

the tide in the region is thus generally small (Defant, 1961). Thephase and amplitude of the four main tidal harmonic constituentsfor the main ports in the Central Mediterranean (Table 2) are basedupon values from Mosetti and Purga (1989), Molines (1991) andTsimplis et al. (1995). The values are in most cases derived fromshort series of tide gauge data. The constituents for Mazara delVallo, the island of Pantelleria, the island of Lampedusa and anoffshore mooring (indicated by SG in Fig. 1) situated at approxi-mately 20 nautical miles in the NNE of Linosa Island are taken from9 months of bottom pressure measurements carried out during theJANUS Experiment (Astraldi et al., 1987). The known longesthistorical sea-level data set in the region refers to the GrandHarbour (GR in Fig. 1) in Malta; these chart records are kept atthe British Hydrographic Office and cover the period 1876–1926.

s in degrees, relative to UT.

eg) O1 (cm/deg) Mean spring range Mean neap range Form number

0.5 175 29.4 0.034810.8 136.6 29.8 0.038820.9 74.4 13.2 0.0781021.4 7.0 0.6 0.771

0.7 21.6 4.8 0.148

1.6 12.2 5.0 0.836740.9 15.8 3.4 0.177

0.6 33.0 11.4 0.1581211.4 15.6 2.4 0.410760.8 20.6 4.6 0.175550.9 20.4 6.4 0.27546


In general there is a discrepancy in the phases of the constituentsquoted by various sources and calculated from one or more yearsof this data from the Grand Harbour; the values in Table 2 are thosebased on an analysis of 13 months of data from May 1990 to May1991 (Drago, 1992). Besides the tabulated values, other importantconstituents in this port are N2 (1.1 cm; 48�) and K2 (1.3 cm; 47�).

The tidal oscillations in the Strait of Sicily are dominated by thesemi-diurnal constituents with supplementary contributions fromthe diurnal constituents especially on the North African coast. M2,S2, K2, N2, K1 and O1 are the only components greater than 1 cm(Purga et al., 1979). The higher frequency components are reportedto be less than 1 mm on the Sicilian coast (Mosetti et al., 1983).Notwithstanding their small size, the tides in this region denotea behaviour of particular interest. They are greatly connected withthe hydrodynamics of the whole Mediterranean Sea and theirdevelopment is related to the influence of tidal co-oscillations.The basin morphology has also important effects and the relativelyshallow bathymetry on the African shelf results in an amplificationof the tide. The syzigial excursions in the Gulf of Gabes reach216 cm (Mosetti and Purga, 1989). The M2 tide alone has a remark-able amplitude of 51 cm in Gabes (Molines, 1991).

The diurnal tides are everywhere small and without nodes inthe Strait (Manzella et al., 1988). The amplitude of K1 is uniformwith values of 3.1 cm at Gabes and 2.1 cm at Cape Passero. Onthe other hand the influence of the rotation of the earth causesstrong transverse (North–South) oscillations that transform thenodal lines of the semi-diurnal components in the region into amp-hidromies contra solem. On the basis of semi-empiric consider-ations, Sterneck (1915) had prognosticated an M2 amphidromicpoint close to Pantelleria even before tidal coastal data could pro-vide a confirmation by means of field measurements. Numericalsimulations of the barotropic tide (Mosetti and Purga, 1989 andMolines, 1991) have recently succeeded to reproduce these tidalfeatures and to elaborate on the associated hydrodynamics.

Table 3 compares the amplitudes and phases of the harmonicconstituents (with H > 1 mm) in Mellieha Bay. The tidal analysisis performed on the hourly sea-level data (from June 1993 toDecember 1996) on the basis of 61 constituents. The main constit-

Table 3Tidal Harmonic constituents in Mellieha Bay with phases relative to GMT.

Harmonic constituent H/cm g/deg

Sa 8.85 201Ssa 1.45 152Mn 0.55 185Mf 1.22 337Msf 0.21 067Q1 0.27 056RO1 0.10 134O1 0.78 056chil 0.17 253pil 0.30 149P1 0.26 051S1 0.58 281K1 0.70 053PSI1 0.27 0572N2 0.11 082MU2 0.18 092N2 0.93 064nu2 0.19 064M2 6.04 055L2 0.30 056T2 0.15 081S2 3.77 062K2 1.15 065M3 0.10 169M4 0.15 271MS4 0.16 316

uent is M2, but the contribution of the solar radiation tidal input ishigh (S2 is 62.4% of M2), which is typical of the Mediterranean. Thetide is predominantly semi-diurnal (Form number = 0.15) and themain diurnal constituents K1, O1 and P1 are relatively weaker inthe region of the Maltese Islands compared to the Sicilian shoreto the North. These diurnal constituents cause the minor diurnalinequalities.

These results for Mellieha Bay are in good agreement with thevalues for the Grand Harbour in Table 2. The two locations are veryclose to one another and the differences, particularly in the phase,are probably more related to temporal changes. Changes in thetidal constituents between different locations can also howeveroccur due to the presence of continental shelf waves which areknown to propagate in the area. Besides producing an anomalousintensification of the tidal currents, these continental shelf wavesare responsible for the appearance of small-scale variations inthe harmonic constants (Rabinovich and Zhukov, 1984).

In the case of the Grand Harbour data, an analysis on separatemonths (May 1990–May 1991) is also performed. From 29-day ti-dal analysis of successive months, Drago and Ferraro (1996) showthat there is considerable variability in both the amplitudes andphase of the main constituents, especially N2, K1 and O1. For theseconstituents, the standard deviation is of the order of 20% in theamplitude, and 23% in the phase. For M2 and S2 the variation ismuch less but not negligible. In the case of M2, the amplitude fluc-tuates in the range 58–67 mm, and the phase in the range 42–53�.This variation can be attributed to the relative contributions of theequilibrium forcing and the tidal wave through Gibraltar in thepropagation of the tide in the Mediterranean. A high-resolution,two-dimensional model of the whole basin (Tsimplis et al., 1995)has revealed that the incoming wave through the Strait of Gibraltaris important in tuning the tides in the whole Mediterranean Sea. Inthe Strait of Sicily the forcing at Gibraltar causes a wandering of theM2 amphidrome to the eastern part of Sicily and gives rise to dou-ble amphidromes in the propagation of both K1 and O1 in the areabetween Malta and Sicily. Comparison of the constituents at differ-ent ports (refer Table 2) needs therefore to be reconsidered espe-cially in those cases where the analysis is based on data sets ofdifferent length and different year or month. From the values inTable 2 it is however clear that for all the principal constituentsthe change in phase across the eastern side of the Central Mediter-ranean region is much more gradual with respect to the west. Thisis in agreement with tidal model results. The phase differences be-tween Malta and the nearby Sicilian ports to the north suggest arelative spatial concentration of phase contours over the shallowcontinental shelf.

3. Characterisation of the seiches

The broad ‘hump’ in the frequency range of 1–10 cph in thepower spectrum (Fig. 5a and e) reveals a very interesting selectiveenhancement of a band of short-period signals in Mellieha Bay thatexplains the nature of its seiche oscillations. In this amplificationprocess the response of the embayment is not restricted to theeigenperiods of its water body, but covers a wide range of frequen-cies whose energy inputs are considerably increased above thebackground values. The characteristic eigenoscillations stand outas well-pronounced peaks upon this overall amplified responseof the embayment. In particular, three sharp maxima with respec-tive periods of 25.1, 21.2 and 16.8 min (corresponding to 2.4, 2.8and 3.6 cph) feature in the rather intricate spectrum.

Higher frequency maxima with frequencies above 4 cph are anexpression of the higher order bay modes. It is important to ascer-tain that this spectral structure in not contaminated by aliased sig-nals. The energy density computed at a frequency f can have

Fig. 6. Cross-spectral analysis of 2-min sampled water level observations at Qawra station (MALTA) and Pozzallo station (SICILY in the period (08h50) 10th September to(11h54) 30th September 1996. Power spectral density in (a) is calculated over 54 degrees of freedom with a 95% confidence factor of 3.3 dB (Bmin = �0.7; Bmax = 1.5). The 95%confidence level for the coherence estimate is 0.11.


contributions from aliased frequencies f ± 2mNf with m = 1, 2, 3,etc. But the sampling interval (Nyquist frequency Nf of 30 cph) inthis case is sufficiently small to rule out effects due to aliasing.Contributions of the energy densities in the sea level at frequenciesof 60, 90, 120, . . . cph are in fact known to be too small to be of anyimportance. The measured peaks of energy are thus real.

The case of Mellieha Bay is particularly interesting. Its wide en-trance increases the number and variety of the observed seichemodes. Furthermore, movement of its water body is coupled toan adjacent embayment and the bathymetry at its wide openmouth is complicated by a shallow bank offshore. Seiches are alsoamplified by a shoaling effect on the shallow sand bank at its head.The main peak at 16.8 min is verified by numerical computation(Drago, 1999) to be related to the Helmholtz mode of the bayand corresponds to a Merian along-axis extent that exceeds the ac-tual physical dimensions of the bay. The volume of the excitedwater body is thus dictated by the bathymetry outside and in theimmediate vicinity of the bay. At times water level records in thebay take the form of quite regular oscillations, especially duringperiods when excursions are strong; at other times the seichingis irregular and is characteristic of a coastal configuration in which

fractional areas are not as fixed as in the case of a cul-de-sac orsmall bay that is suddenly terminated. The other two peaks withslightly longer periods represent oscillations involving the shelfarea immediately adjacent to the bay.

The SB band (Fig. 5e) also presents a series of sharp, well-de-fined and equally energetic peaks on its lower frequency end.The main maxima have periods of 3.7, 2.2, 1.7, 1.1 h, and 58, 46,34.3, 29.5 and 27.7 min. These peaks are probably related to shelfoscillations in the Malta Channel.

The long wave structure of sea level oscillations in the region istypically characterised by (a) forced motions, (b) free long wavespropagating along the shelf, and (c) eigenoscillations of individualcoastal areas including embayments and inlets. The bay oscilla-tions are treated elsewhere (Drago, 1999), and are found to con-cern signals with periods lower than 20 min. Long waves withlonger periods are however also persistent and rise above the back-ground levels during seiching events. Their periods normally rangefrom 20 min to a few hours. Main signals with periods 3.7, 2.2, 1.7,1.1 h, and 58, 46, 34.3, 29.5, 27.7, 25.1, 21.2 min were identifiedfrom the general spectral analysis. The bays are shallow coastalindentations with shapes that do not permit natural resonant peri-

Fig. 7. Theoretical spectrum for a straight ledge (width = 100 km; shelf depth = 100 m; c2 = 5) in the case of normal incidence.

Fig. 8. Bathymetry of Mellieha Bay and positioning of instruments.


ods to match these values. This shows that these oscillations mustnecessarily concern a larger water body.

3.1. Results of the Malta Channel experiment

This section treats the findings from the simultaneous sea levelmeasurements at two locations across the Malta Channel, thestretch of sea on the continental shelf area linking the Maltese Is-lands to the southern shores of Sicily. These measurements wereconducted to identify the shelf scale components of the seiches.

The similarity and temporal coincidence of the bottom pressurerecordings at Pozzallo (PZ) and Malta (QW) in Fig. 6 is an indicationthat the lower frequency long wave oscillations are not related tothe local topography at the respective stations, but are instead anexpression of the larger scale movements of the water body overthe continental shelf. The cross-spectral analysis between thetwo records shows that the principal signals at the two stationsare identical, with energies being on average an order of magni-tude higher in Pozzallo throughout the range of frequencies0.4 < f < 1.1 cph. The analysis shows two main sudden variationsin phase and coherence. The records are consistently in phaseand highly coherent for f < 0.7 cph. In this range of frequenciesthe energy spectrum carries a broad peak centred at 0.285 cph(T = 3.5 h) followed by sharper and stronger peaks at T = 2.2, 1.7,1.45 h. At 0.78 cph the phase changes abruptly and the signals be-come in antiphase between 0.8 and 1.1 cph. The strongest energypeak (with T = 1.1 h) at Pozzallo is registered in this frequencyrange. It is accompanied by a weaker peak at T = 58 min. At both

these two peaks, the coherence rises to high levels. The phase is re-versed back to zero at 1.16 cph. At each of these phase reversals thecoherency dips to zero. These phase reversals between 0� and 180�are indicative of the occurrence of standing waves on the shelf.

These results can be tentatively explained by considering theshelf to be very simplistically represented by a straight steep coastwith uniform bathymetry alongshore (y-axis) and a step cross-shore (x-axis) profile with constant width L, depth H1 on the shelfand H2 on the deep sea side. One possible long wave motion on thiscontinental shelf model is a system of leaky waves normal to theshore (Snodgrass et al., 1962) with an antinode at the shore anda node at the shelf edge. In the case of waves incident normallyfrom the deeper ocean with an associated spectrum So(f), the spec-trum at the shoreline (x = 0) is a2(f)So(f), where a2(f) = [1 +tan2(2pfL/c1)]/[1 + c�2tan2(2pfL/c1)] and c2 = c2/c1, c1 =

p(gH1),

c2 =p

(gH2). The spectrum at an offshore position (x = X < L) onthe shelf is a2(f)cos2(2pfX/c1)S(f). Taking a typical value for theshelf width L = 100 km and a mean depth H1 = 100 m, the theoret-ical spectra on the Sicilian coast (solid) and on the northern coastof Malta (dotted; with position of the island taken at X = 0.75 L)for normal incidence (from the south) are shown in Fig. 7 with val-ues normalised to S(f) = 1. The signals are in phase up to A (0.37cph), become in antiphase from A to B (1.11 cph), and return inphase from B to C (1.85 cph). The coherence (not shown) followsa negative delta function, falling to zero with a very narrow troughcentred on the frequencies at each phase reversal, and is equallyhigh on both sides. Successive spectral peaks on the Sicilian coasthave also the same amplitude and occur at To = 3.55 h and To/3 =1.18 h. These periods compare well to the peaks X and Y in theobserved spectra (Fig. 6). It is also possible to account for thesecond phase reversal which coincides to that observed at 1.16cph. Some remarkable differences are however noted. The ob-served fundamental mode is weaker than the first resonance mode.The observed dips in coherence are smeared into troughs, and thecoherence remains low beyond 1.2 cph. The first phase reversal isexpected at a third of this frequency (f = 0.37 cph) whereas it actu-ally occurs at a much higher frequency (f = 0.78 cph). The observedfundamental peaks in Malta and Sicily have also equal amplitudeswhich again does not agree with theory. For other angles of inci-dence (h – 0), the resonance peaks and anti-resonances shift tohigher frequencies by a factor

p(1 � c�2sin2h) and the amplifica-

tion at the coast is reduced. In the extreme case of glancing inci-

Fig. 9. Current stick plot of subsurface sea currents observed in Mellieha Bay at a depth of 11 m from the water surface.


dence (h = 0) this amounts to a shift of 12% in frequency which stilldoes not account for the differences between theory and observa-tions, especially with regard to the fundamental mode. This modethus appears to be greatly modified by the shape and limited lati-tudinal extent of the shelf. The failure of the coherence to recoverafter the second phase reversal also suggests that the presence ofmultiple stationary modes is important. These modes apparentlycarry an appreciable fraction of in-phase energy even after mostof the energy is out of phase.

The simplified model above does not thus fully apply to theshelf area between Malta and Sicily on account of the shelf’s com-plicated bathymetry and its very abrupt termination on the east. Amulti-directional distribution of incident energy from the deeperocean may indeed partly explain the smearing of the features inthe observed spectra and coherence, but the anomalous character-istics pertaining to the gravest mode should be sought in the effectproduced by the coastal configuration of the northern borderlandand the irregular shape of the shelf. A three-dimensional numericalmodelling approach would be necessary to resolve thesecharacteristics.

Other important intermediate energy peaks (at T = 2.2, 1.7,1.45 h and 58 min) are not explained by the above model. Thesesignals are apparently related to latitudinal stationary modes onthe shelf. These modes are attributed to the trapping of long waveenergy on the shelf area as a whole as well as in localised areasalong isolated features on the sea bed. One such feature is thecrescentic submarine ridge which runs close to the eastern andsouthern perimeter of the shelf. This inner shelf ridge can act asa waveguide for long waves.

The depth profile normal to the ridge axis can be taken to beparabolic according to H = Ho

p(1 + x2/a2), where Ho is the mini-

mum water depth at the centre of the ridge (x = 0); at a distancex = a from the ridge axis the depth is Ho/

p2. The lower frequency

mode has a period given by T1 = 6.95(a/p

gHo) (Defant, 1961, page235) and consists of anti-phasic oscillations on opposite sides ofthe ridge, with a node over the ridge. On each separate side ofthe ridge, the oscillations remain in phase with the amplitudereaching a maximum at a distance x � 2a. The next mode is short-er, with a period given by T2 = 3.24(a/

pgHo), and consists of sym-

metrical oscillations with an antinode over the ridge, nodes atx � ±a and antinodes at x � ±2a. Taking Ho = 50 m and a = 25 kmas typical values for the inner shelf ridge, the corresponding modeshave periods T1 = 2.179 h and T2 = 1.016 h which compare well

with two of the observed spectral peaks. Pozzallo and Qawra sta-tions are on the same side of the ridge, but their longitudinal coor-dinates differ by 24.310 which amounts to an east–west separationof about 36 km. This explains the phase relationship of the oscilla-tions at the two stations which are in phase for mode T1, but inantiphase for mode T2.

The shelf itself can also in its totality be considered as a sub-merged ridge extending normal to the borderland. In this case ifwe take Ho = 100 m and a = 60 km, we obtain a second mode peri-od of 1.724 h which again compares well with the observed peak atT = 1.7 h. The signal at T = 1.45 h is probably related to a co-oscilla-tion of the plateau (averaging a depth of 150 m) on the westernflank of the shelf with the Central Mediterranean basin.

3.2. Seiche-induced currents in Mellieha Bay

This section deals with the strong oscillating currents that canbe triggered by seiches especially in semi-enclosed basins like har-bours and coastal embayments. High frequency, large amplitudesea level fluctuations can have associated horizontal motions thatcan be a damage potential on moored vessels. Ship loading/unload-ing operations can be delayed or even potentially hazardous. Con-stricted flow at harbour entrances also results in strong reversiblecurrents that can be detrimental to navigation.

Taking the example of the Grand Harbour, with a narrow open-ing to the adjacent sea, we can estimate the magnitude of thisseiche-induced flow. If the profile of the uninodal standing waveoscillation is taken to be sinusoidal with wavelength k and swingheight H, then the volume of water that must flow in half a periodacross a vertical line through the nodal point at the mouth of theharbour is Hk/2p. The time average horizontal velocity <V> is ob-tained on dividing by the time T/2 of one half period and the aver-age cross-sectional area, namely the depth d. On using the shallowwater wave propagation relation, the maximum velocity at the nodalpoint is then given by Vmax = p < V > /2 = Hk/(2Td) = (H/2)

p(g/d),

where d is the average depth of the basin.Taking H = 0.25 m, d = 18 m as typical values for the Grand Har-

bour, and assuming the constriction at the harbour mouth to behalf the extent of the average harbour width, then the actual valueof Vmax is expected to be 0.37 m s�1.

The horizontal particle excursion at the harbour mouth is givenby <V> T/2 = Hk/2pd = (HT/2p)

p(g/d) � 41 m. By proportion the

excursion at the middle of the harbour is expected to be 20 m. Thus


although the maximum velocity is not excessively high, the largehorizontal motion could create difficulties.

3.2.1. Observation of seiche currentsDirect observations of seiche-induced subsurface currents have

been conducted in Mellieha Bay by means of an ENDECO/YSI teth-ered-type current meter positioned inside the embayment (referto Fig. 8) such as to mainly pick the influence from the main modeof bay oscillation. The current meter measures the total currentwhich consists of a background current upon which the seiche-in-duced currents are superimposed. The background current is pre-dominantly established by wind-induced effects as well as by theinteraction of the bay circulation with the open sea. In general thisbackground circulation predominates and completely masks theseiche currents. During strong seiching events the associated move-ment of the oscillating water body in the bay can however producesignificant currents. In the case of Mellieha Bay these seiche currentshave the same frequency of the sea level oscillations (with a periodof around 20 min) and can usually be easily detected as very rapidfluctuations in the current stick plots. The stick plot in Fig. 9 showsan example when the seiche currents become important and thecurrent switches direction very rapidly in a matter of a few minutes.These currents are aligned in parallel to the bay axis along a NE–SWdirection and are therefore an expression of the sloshing watermovement in the bay. The current vectors in one half-cycle are notparallel to those in the next half cycle since a background currentacross the bay is superimposed during this particular interval.

The incidence of these high frequency currents can be followedfrom the along- and cross-bay axis components of the currents.These components are obtained by resolving the current vectorsalong orthogonal directions N65�E and S25�E parallel and perpen-

Fig. 10. Time series plot of water elevation in Mellieha Bay, and of su

dicular to the bay axis. The current component plots in Fig. 10show clearly the onset of the current fluctuations in coincidencewith the sea level oscillations on the 26th July. The seiche currentsare predominantly rectilinear and orientated along the bay axis.Their component across the bay is less important although it isnot completely negligible.

In order to better study the seiche currents it is important toisolate them from the background component. This is done by highpass filtering each of the resolved current component time series. AVercelli filter with 48 weighting coefficients and a cut-off period of34 min is used. The recombination of the high pass filtered compo-nents returns the seiche current vectors.

3.2.2. Estimation of seiche-generated currents from sea-level dataThe 1D model described in Appendix 2 is applied to the obser-

vations in the period 12:00 GMT–15:00 GMT 26th July 1994. To isobtained from an inspection of the average period of the water le-vel oscillations during this period. The value To = 16.8 min isadopted. The average depth is taken to be 20 m while the totaldepth at the current meter station is taken as D = 28 m. The sea le-

vel time series is used to obtain 2-min values of � To

ffiffiffiffigdp

2pDo

dgðtÞdt

� �o.

These values are compared with the filtered along-axis currentobservations. The correlation estimated is found to be very high(r = 0.865) which shows the efficiency of the model. A regression

analysis between U and � To

ffiffiffiffigdp

2pDo

dgðtÞdt

� �o

gives the optimal value of

sinðpao=2Þ as 1/1.96. This implies that ao = 0.34 and hence thatL = xo/0.34. This result confirms that the seiche oscillations arenot confined to the interior of Mellieha Bay, but that they actuallyinvolve a much larger water body which extends well beyond thepromontaries of the bay. This water body is about three times the

bsurface current components resolved along/across the bay axis.

Fig. 11. Comparison of observed and calculated along-axis current components in Mellieha Bay.


size of the bay and includes the deeper basin between the mouth ofthe bay and the White Bank to the northeast of the bay.

Fig. 11 compares the depth-averaged along-axis seiche currentcomponent derived by the above model (with ao set equal to0.34) to the observed high pass filtered currents. The correspon-dence confirms the validity of the model which can thus be usedto predict seiche currents from observations of sea level on thecoast. In particular it is interesting to note that moderate seicheamplitudes of just 15 cm can generate quite strong seiche currentswhich peak up to 10 cm s�1. This shows the potential hazard ofthese seiche-induced currents even in the case of a wide mouthedembayment such as Mellieha Bay.

4. Other oscillations of non-tidal origin

Spectral analysis of the residual sea-level data allows the sepa-rate study of the non-tidal oscillations. The analysis is performedby two alternative computations of energy spectra for (1) thewhole time series (Fig. 5d), and (2) 27 successive 50% overlapped3-month periods in order to reveal any seasonal variability. Themost important result is a strong diurnal residual energy that cov-ers a relatively broad band of spectral frequencies centred at 1 cpd.This signal is probably related to the signature of baroclinic mo-tions on the continental shelf. Measurements made during a phys-ical oceanographic survey carried out in the NW coastal area ofMalta in summer 1992 have in fact revealed the presence of diur-nal subsurface flows in the vicinity of the islands. These diurnalbaroclinic currents are believed to be the expression of a topo-graphically trapped wave that takes the form of an internal Kel-vin-like waveform in the deeper sea away from the shelf breakand is accompanied by shelf wave modes propagating over the

continental platform. An associated vertical oscillation of the ther-mocline in the form of an internal tide has been quantified to havea crest-to-trough amplitude of the order of 8 m (Drago, 1997).From CTD casts taken during the same survey, the density profilesshow that the seasonal pycnocline has a sharp gradient between 10and 30 m depth and acts as a clear interface between the surfacemixed layer and the deeper layer. Taking the mean relative densitydifference between these upper and lower layers to be 0.3%(q1ðupperÞ � 1024:5Kg m�3;q2ðlowerÞ � 1027:5kg m�3), the freesurface displacement accompanying this internal tide is estimatedto have a semi-amplitude of the order of 1.2 cm.

The variability of the diurnal residual energy over the wholeperiod of measurements (June 1993–December 1996) is studiedfrom a series of spectra calculated in each of successive 3-monthhalf-overlapping periods. The energy carried in the frequencyrange of 0.978–1.022 cpd is calculated for each spectral estimate.The relevant plot against time (Fig. 12) shows a consistent patternwith peak energy during the periods spanning mid-October–mid-January and mid-April–mid-July, respectively. This intra-seasonalvariability of the diurnal signal is probably linked to dynamicalprocesses that have a correlated temporal repeatability; there isas yet no sufficiently long hydrodynamical datasets to confirm thisrelationship. The variability of the semi-diurnal residual energycalculated over the frequency range 0.48–0.52 cpd is less pro-nounced. In particular, the semi-diurnal residual can predominateover the diurnal residual during late winter and early spring.

4.1. Variability in the low frequency range and relation to atmosphericforcing

The subtidal sea level signals are well resolved in the LB spectralband (Fig. 5b) by utilising a long window size of 65,536 records

Fig. 13. Time series of (a) residual sea level in Mellieha Bay; (b) inverted barometric pressure fluctuations at MSL; (c) and (d) along/cross-shore components of wind at Ramlastation for the period mid-November 1995–mid-July 1996 (Residual sea level is measured relative to MSL; the positive along-shore wind component corresponds to a windfrom S55�E; the positive cross-shore wind component corresponds to a wind from N35�E.).

Fig. 12. Variability of the (a) diurnal (solid) and (b) semi-diurnal (dotted) residual energy calculated for successive 3-month half-overlapping.



(182 days:01 h:04 min). The sea-level is found to have significantvariability in the 1.2–15 days period (periods from tidal to severalweeks). This is very typical of the whole Mediterranean basinwhere the sea level variability in the low frequency range repre-sents an energetic part of the sea level spectrum.

Besides the effect of the tides and oceanographic factors such aswater density and currents (both geostrophic and ageostrophic),the influence of meteorology determines a great part of the synop-tic variability in the sea level. In the Mediterranean, sea level vari-ations at time scales from 1 to 10 days have been shown to beprimarily due to surface pressure changes related to synopticatmospheric pressure disturbances (Kasumovic, 1958; Mosetti,1971; Papa, 1978; Godin and Trotti, 1975). Sea level variations attime scales from 10 days to several weeks have been explainedas being due to atmospheric planetary waves (Orlic, 1983; Lascara-tos and Gacic, 1990). The contribution of the wind is also impor-tant, both with regard to its dynamic effects as well as to itseffect on rate of evaporations and the difference in air–seatemperature.

The comparative plot of the residual sea level and meteorolog-ical time series (Fig. 13) presents some very interesting first handinformation in this part of the Mediterranean. The residual sea le-

Fig. 14. General spectra for (a) residual sea elevation in Mellieha Bay, and (b) MSL atmospfactor, for 24 degrees of freedom is 5.0 dB (Bmin = 0.6; Bmax = 1.9). Data used consists of

vel displays the presence of variability at different time scales. Sealevel variations at a time scale of 1 month have the largest ampli-tudes and can reach peak-to-trough values of up to 0.35 m. Higherfrequency oscillations at time scales of several days (synoptic var-iability) appear on both the sea level and pressure time series, andare predominant during the winter months. These oscillations arerelated to natural periods of occurrence of cyclones in the region.Variations in the atmospheric pressure can reach up to 18 mbarat high frequencies, and are accompanied by equivalent variationsin sea surface height. Both high and low frequency sea level oscil-lations in fact bear a distinct visual correlation with the invertedbarometric pressure and extreme values of sea level are very wellassociated with extreme values of inverted pressure. It is also clearthat the sea level and atmospheric pressure variance differs sea-sonally. The sea level signal shows however a greater variabilitythan that implied by a simple barometric effect.

The two wind components are taken to correspond to a windvector from directions 35� (positive cross-shore component) and125� (positive along-shore component) with respect to north.These components are roughly coincident to a wind directed alongthe axis of Mellieha Bay and parallel to the northern foreshorecoastline of the island, respectively. The mean wind magnitude is

heric pressure and (c and d) wind components at Ramla tal-Bir. The 95% confidencehourly values from 18/11/95–11/8/96.

Fig. 15. Comparison spectra with semi-logarithmic scales for (a) residual seaelevation in Mellieha Bay (solid), and (b) MSL atmospheric pressure at Ramla tal-Bir(dotted). The 95% confidence factor, for 24 degrees of freedom is 5.0 dB (Bmin = 0.6;Bmax = 1.9). Data used consists of hourly values from 18/11/95 to 11/8/96.


1.15 m s�1 from the west (N87�W). The wind data are character-ised by frequent clockwise rotations of the wind direction whichis indicative of the passage of fronts over the area. During theoccurrence of these pressure lows, the wind vector can typicallyrotate by 90� and subsequently attenuate very rapidly, or even un-dergo a complete reversal without losing in strength.

The comparison is further investigated by means of auto- andcross-spectra of the data series which were computed from hourlyvalues, after linear detrending and mean removal, by means of 1350%-overlapping consecutive segments of 1024 points each. All thespectra (Fig. 14) show the ‘red’ shift of energy that is characteristicof geophysical processes. The prominent peak in the atmosphericpressure spectrum at the frequency of 2 cpd is due to the semi-diurnal atmospheric tides. The diurnal maximum is less pro-nounced. At the higher frequencies, the atmospheric spectrum fol-lows an x�2.2 power law that is in good agreement withobservations by Rabinovich and Monserrat (1996) and Kovalevet al. (1991) (x�2.3), and only slightly steeper than that describedby Herron et al. (1969) (x�2.0). The spectrum has however a dis-tinctive steeper gradient in the higher synoptic frequency range(0.2–1 cpd) where it follows an x�3.2 power law decay. The pres-sure and wind spectra, especially the NS component, exhibit a dis-tinct flattening at periods greater than about 5 days. The sea levelspectrum has no such bounding limit on power at the lower fre-quencies and the spectrum has an almost constant slope for the fullrange of frequencies. As a matter of fact, the prevalence of the sealevel spectrum over that of atmospheric pressure results in thepower in sea level rising much more rapidly than that of pressureat periods longer than about 12 days, and in an increased diver-gence between the two spectra at the higher frequencies. At plan-etary time scales (several weeks), the sea level variance is thusroughly seven times higher than that of pressure. As expectedthe seasonal signal of the atmospheric pressure is less than thatof the sea level since the latter carries other forcings especiallyfrom steric effects. At the synoptic time scale (0.05–0.5 cpd) thevariances are equal at the lower frequency end, with variance inpressure become more important with increasing frequency. Thecomparative plot of spectra on a semi-logarithmic scale (Fig. 15)shows that signals with frequencies higher than 0.5 cpd representonly a negligible fraction of the total variance in both residual sealevel and atmospheric pressure.

The wind spectra exhibit a predominance of the EW componentover the NS component in the synoptic frequency range (3:1) andespecially for the lower frequencies (12:1).

4.2. Barometer factor from spectrum analysis

The significant dependence of sea level variability on atmo-spheric pressure in the synoptic and planetary wave scales is evi-denced by the fairly high coherence between the residual sealevel and atmospheric pressure fluctuations at these frequencies(Fig. 16a and b). Coherence levels have an average of 0.7 for the re-sponse at frequencies lower than 0.5 cpd (T > 2 days). Coherencehas a value of 0.6 for the lower frequencies, drops to 0.4 for fre-quencies centred on 0.14 cpd (T approx. = 7 days), and rises againto values close to 0.8 in the frequency range of 0.3–0.5 cpd(2 < T < 3 cpd) on average. The gain (barometer factor) and phaserelationship bear a very similar dependence on frequency so thatthe response can be broadly classified into three bands offrequency.

In the lower frequency planetary wave time scale (f < 0.05 cpd)the response is over-isostatic with a gain of about 1.5, and a lag ofatmospheric pressure with respect to sea level a few tens of de-grees less than 180�; This phase relationship is equivalent to thatof an atmospheric pressure leading the inverted sea level by afew tens of degrees (i.e. an approximate phase difference of 1

day). This behaviour has been reported in other studies based oncoastal data (Palumbo and Mazzarella, 1982; Pasaric and Orlic,1992; Tsimplis and Vlakhis, 1994) and is probably due to signals,such as from steric effects and wind, which can be correlated withatmospheric pressure. For periods higher than 25 days, Le Traonand Gauzelin (1997) have also found an over-isostatic responseof the satellite-derived mean sea level to the ECMWF model anal-ysis mean atmospheric pressure calculated over the wholeMediterranean.

In the lower synoptic frequency range (0.05 < f < 0.3 cpd) thepressure variance is higher than that of the residual sea level.The phase relationship remains less than 180� and a maximum de-lay of 70� in the response of the sea level is registered at a fre-quency slightly higher than 0.1 cpd. The response is thus wellunder-isostatic, and the equivalent sea level fluctuations are onlyabout half the value expected for a perfect inverse barometer re-sponse. On the other hand the response is very close to isostaticin the upper synoptic frequency range (0.3 < f < 0.5 cpd). For thisband of frequencies the sea level is highly correlated to pressure,and responds in antiphase to pressure with an average gain of 0.7.

For further higher frequencies (f > 0.5 cpd) the coherence is ingeneral low except for a number of discrete frequencies. In partic-ular, the sea level signals close to the diurnal frequency are uncor-related to barometric pressure, which indicates that their originhas to be sought from other oceanographic influences.


4.3. Seasonal changes in mean sea level

The seasonal variation in the mean sea level in Mellieha Bay isstudied by 43 monthly averages of sea-level data covering the per-iod June 1993–December 1996. The seasonal signal in the sea levelis seen to be quite strong (Fig. 17) and is actually larger than thedaily variation. The magnitude of these sea level seasonal changesis in the order of tens of centimetres and therefore greatly exceedsin size other effects on the sea level such as of climate change..Such seasonality is typical of the whole Mediterranean Sea where,on the basis observations from sea level stations, seasonal signalsare found to account on average for approximately 20% of the totalsea level variance (Marcos and Tsimplis, 2007). Besides the spatialvariability across the Mediterranean (Emery et al., 1988), the sea-sonal signal is characterised by a strong temporal variability withchanges in amplitude and phase from year to year, and this ismainly attributed to changes in water temperature (Marcos andTsimplis, 2007). The range, pattern and regularity of the seasonalsignal varies widely for different locations in the Mediterranean

Fig. 16. (a) and (b) Coherence and phase for residual sea level and atmospheric pressure acovers the period mid-November 1995–mid-September 1996. The phase indicates the leawhich the coherence is significant at the 95% confidence level. With 24 degrees of freed

and is often influenced by local effects such as of temperatureand spells of strong winds whose incidence can vary from year toyear (Goldsmith, 1990). Indeed this renders eustatic sea leveldeterminations more difficult to detect. In general, seasonality inthe west is less regular, and is not as sharp as in the southeasternMediterranean. Satellite altimetric missions such as from the ERS-1and Topex/Poseidon platforms, have since 1992 provided a newtechnology to monitor sea level change, and permit variations tobe followed not only in time but also in space by providing a 2Dview of the variability that has revealed considerable geographicaldifferences worldwide and in the Mediterranean. Analysis of sealevels derived from TOPEX/POSEIDON satellite altimetry data dur-ing 1993–1994 show that the mean level variations in the westernand eastern Mediterranean basins are about the same in magni-tude, but have a phase lag which varies with time (Larnicol et al.,1995; Jorge del Rio et al., 2007).

In the case of Mellieha Bay the records (Figs. 17 and 18a) showthat a sea level maximum generally occurs in October while a min-imum occurs in March. The sharp lowering in the mean sea level

t MSL; (c) amplitude of the barometer factor as a function of frequency. The data setd of sea level with respect to the pressure. The dashed line indicates the level aboveom this is equal to 0.22.

Fig. 17. Monthly mean sea level in Mellieha Bay as a function of time (June 1993–December 1996).


after October is interrupted by a second maximum in December/January while the increase after March is temporarily halted by asecondary minimum in late spring. The maximum range between

Fig. 18. (a) Monthly mean sea level in Mellieha Bay as a function of time with (dotted) apressure at Ramla station (period covered is December 1995–December 1996).

the extreme levels is in the order of tens of centimetres. The threeyears are however not fully alike. Both the size and phase of thefluctuations as well as the occurrence of fast variations (such asthe sharp rise in sea level in May 1996) are indicative of consider-able interannual variability. A comparison with 13 months of sea-level data covering the period May 1990–May 1991 and obtainedat a sea level station in the Grand Harbour (only within a few kilo-metres of distance from Mellieha Bay) further confirms that theseasonality is not very regular (Fig. 19a). During 1990/1991 the riseto maximum sea level in October is more gradual, while the min-imum in 1991 occurs in January rather than March. There seemsthus to be a sensible phase shift in the seasonal signal with respectto the period 1995/1996. This implies that in the case of local sealevel variations with the longer time scales and with cycles inthe order of tens of years, studies need to be based on longer timeseries of data so as to enable inferences on statistical averages thatare better representative of the predominant seasonal signals.

The seasonal signal is also present, though less energetic, in theatmospheric pressure (Figs. 18b and 19b). After applying the IBcorrection the mean sea level still retains a large part of its variabil-ity and hence other factors besides the barotropic effect of baro-metric pressure are responsible for the sea level variations at thisscale. On the basis of 44 years of model data from the HIPPOCASproject Gomis et al. (2008) attribute only a small contribution(2 cm amplitude) to the observed sea level seasonal cycle fromthe atmospheric pressure forcing; the temporal offset with respect

nd without (solid) the inverse barometer correction; (b) monthly MSL atmospheric


to the steric cycle results in an overall reduction in the amplitudeof the annual cycle. Seasonal winds and the piling of water onshoreas a result of storm surges can greatly contribute to the seasonalsea level variations (Goldsmith and Gilboa, 1987). Other processesinfluencing sea levels include currents (Pirazzoli, 1987; Thompsonand Pugh, 1986) and shelf waves (Huthnance, 1986). Palumbo andMazzarella (1982) have determined on various time scales themeteorological as well as the hydrographic and oceanic factors thatcan explain the seasonal cycle in the mean sea level. Sea watertemperature and baroclinic phenomena such as steric effectswhich are related to the volume dilation and contraction of thesea surface layer in response to changes in heat fluxes are alsoimportant factors contributing to the seasonal oscillation. Usingan ECMWF climatology of net surface heat fluxes for the Mediter-ranean, it is found that steric effects produce changes in the meansea level that have the right phase compared to observations, butwhich are only about half in size to the actual variations (Larnicolet al., 1995). On the seasonal timescale, the Mediterranean is thusprobably not in mass balance. In order to account for this discrep-ancy it seems necessary to hypothesize variations in the inflow andoutflow at the Strait of Gibraltar. Ovchinnikov (1974) suggests thatseasonal fluctuations in the inflow at the Strait could well be inantiphase to the fluctuations in the outflow, thus resulting in anet barotropic flow and an associated expression in terms of meansea level variations. On the basis of measurements made in the Bayof Naples, Palumbo and Mazzarella (1985) explain the seasonal

Fig. 19. (a) Monthly mean sea level in the Grand Harbour as a function of time withatmospheric pressure at Luqa (period covered is May 1990–May 1991).

cycle of mean sea level in terms of a mass balance that takes intoaccount local evaporation and precipitation rates and Atlanticwater inflow. In their modal analysis of the contribution of atmo-spheric pressure forcing to sea level variability in the Mediterra-nean Sea, Gomis et al. (2008) relate a basin-wide EOF (explaining66% of the variance) to the existence of a related flow exchangeat the Strait of Gibraltar.

The response of the Mediterranean mean sea level to variationsin the difference between evaporation and precipitation (E–P) inthe basin thus appears to be linked to the exchange at the seasonalscale at the straits. In the case of a barotropic adjustment to (E–P)variations, the basin has a very rapid response (gravity waves takeless than a day to cross the whole Mediterranean) and the sea levelis thus practically unaffected. On the other hand, when variationsin (E–P) give rise to baroclinic processes, such as during the forma-tion of deep waters and Levantine Intermediate Water that aretriggered by strong winter cooling and evaporation conditions,the associated vigorous vertical mixing processes are slower andan associated signature in the mean sea level is expected. These ba-sin properties affect the nature of the exchange at the Strait ofGibraltar between the Mediterranean Sea and the Atlantic Ocean.On the basis of existing evidence, Garrett et al. (1990a) concludethat the hydraulically controlled flow in the Strait of Gibraltar isvery close to the state of maximal exchange, but do not excludea switching to a marginally sub-maximal state during the secondhalf of the year. Evidence supporting this seasonal pattern also

(dotted) and without (solid) the inverse barometer correction; (b) monthly MSL


comes from the drop in the mean sea level along the strait from Ca-diz to Malaga, which is found to change during the year becomingmore enhanced during sub-maximal exchange conditions (Garrettet al., 1990b).

During sub-maximal exchange the density difference betweenthe inflow and outflow is large; the budgets of the sea require onlya slow exchange and the outflow is likely to be restricted to a thinlayer above the bottom of the sill. If the sea becomes more mixed,the hydraulically controlled outflow becomes less dense; budgetswill require greater flow rates and the outflow will occupy a thick-er bottom layer in the strait; the exchange becomes maximal. Theseasonal flips between the two states could thus be a consequenceof wintertime replenishment of dense Mediterranean water fol-lowed by summertime draining of this water by the outflow. Dur-ing sub-maximal exchange rapid baroclinic adjustment throughthe strait to basin (E–P) variations is possible and no effect onthe mean sea level is expected. On the other hand, the suppressedadjustment during maximal flow produces a direct response of themean sea level to changes in (E–P).

Unfortunately knowledge on the seasonal variations in (E–P) inthe Mediterranean is too scarce to confirm these suppositions.Whether the flow at Gibraltar is maximal or sub-maximal is stillan open question. It is however certain that this simplified pictureconsidering the Mediterranean as a single basin will have to bemodified in order to include the more complex intra-basin interac-tion between the eastern and western Mediterranean through theStrait of Sicily. The mean sea level variations observed in Malta arefound to be practically unrelated to the wind climate. The winddoes not in fact have any significant seasonal character, being dom-inated by the westerly winds practically throughout the year. Agreat part of the seasonality is thus believed to be non-local in nat-ure and to predominantly carry the signals deriving from differ-ences of meteo-marine parameters in the two basins. It is theintra-basin changes that in fact dictate the seasonality in the flowthrough the Strait of Sicily.

Taking the example of the atmospheric pressure field, very dif-ferent behaviours are observed between the two basins. From datareported by the Koninklijk Netherlands Meteorological Institute(1957) and by the Meteorological Office in London (1964), the wes-tern basin is characterised by a minimum in April. This is followedby a rise until July, with the pressure remaining practically con-stant until January when it starts to decline again until April. Therange is on average 6 m bars. In the eastern basin a deeper mini-mum occurs at mid-July and is followed by a sharp rise untilNovember. The range is about 10 mbar. The pressure remainsapproximately at the same levels during winter, with a maximumin January, after which it begins to fall until mid-July. With respectto these two characterisations, the observations in Malta (Figs. 18band 19b) show that the atmospheric pressure field has a markedinterannual variability and follows a mixed behaviour as the twopressure regimes are, respectively, pushed eastwards or westwardsover the Central Mediterranean. This will have a local effect on thesea level, but most importantly the difference in pressure over thetwo basins will govern the flow through the Strait of Sicily and thusproduce other indirect non-local effects on the sea level. These con-siderations render the measurements in Malta particularly impor-tant. Simultaneous observations at a number of locations in thispart of the Mediterranean can certainly reveal key aspects on theexchange between the two basins.

A hydraulic control may also pertain to the flow in the Strait ofSicily. This can indeed be dictated by discrepancies in the mass bal-ance of the respective basins and may not necessarily have thesame state of flow as that in the Strait of Gibraltar. It can be envis-aged that the exchange condition through the two straits may infact be different. Different factors that have a control on the straitexchange may moreover be expected to not simply play a discon-

nected role but to be indeed dynamically and mutually linked toone another. For example, the sea level drop across the straitsmay not simply be consequential of the state of flow, but may actu-ally act as a forcing agent that dictates, possibly together with anumber of other factors, the transition between marginally sub-maximal and maximal exchange strait flows.

5. Discussion and conclusion

The sea level and its variability in the Sicilian Channel is farfrom fully studied. Long data sets are generally lacking especiallyon the African coast. Storm surges and sea level seasonal fluctua-tions in this region of the Mediterranean have not been previouslystudied. The work presented in this paper has served to improvethe knowledge on the full spectrum of sea level signals in the re-gion of the Maltese Islands, a key location in the CentralMediterranean.

The analysis of densely sampled meteo-marine observationscollected in the region of Mellieha Bay on the northwestern areaof Malta demonstrates that the tidal amplitude in the vicinity ofthe Maltese Islands is small. The mean spring tidal range is20.6 cm and is reduced to 4.6 cm during neap tide. Water levelvariations are dominated by energy inputs from long-period oscil-lations of non-tidal origin. The annual Sa (365.3 days) and semi-an-nual Ssa (182.6 days) components are rather strong. Althoughvariations in atmospheric pressure associated with mesoscalemeteorological phenomena produce a predominant effect on thesea level in the synoptic and sub-synoptic time scales, the responseof the sea is non-isostatic. The response of the sea level on theatmospheric pressure is thus a complex one being determined byboth local and non-local effects. The smaller extent of the seaand the closer proximity of the land in the Central Mediterraneanarea give rise to distinctive weather patterns that pertain to the re-gion and which directly affect the sea elevation. But the main influ-ence is determined by the general synoptic situation over thewhole Mediterranean basin, with geostrophic gradients being pro-duced in the Sicilian Channel by different pressure regimes be-tween the western and eastern Mediterranean basins. Besidesmeteorlogical forcing other factors are expected to contribute toits variability mainly depending on effects derived from the influ-ence by the general circulation and from mesoscale eddies propa-gating on the Sicilian shelf.

It is not easy to identify the physical processes responsible forsuch a response. It would certainly be necessary to obtain simulta-neous measurements at other sea level stations, particularly in theCentral Mediterranean area, in order to assess the dependence ofthe response on the geographical position. Only then can one iden-tify the extent to which discrepancies from the inverse barometereffect can be related to local effects as compared to the dependenceon the larger scale dynamics of the Strait of Sicily acting as a con-nection between the two major basins of the Mediterranean Sea.

Short period oscillations with periodicities that are in agree-ment to the theoretical natural periods of Mellieha Bay are also ob-served. They are attributed to coastal seiche motions that aretriggered in cascade manner by open sea modes in the nearshoreshelf areas, and on a larger spatial scale by resonances over theSicilian continental shelf. The large amplitude sea level oscillationsobserved on the northern coast of Malta in the long wave fre-quency band contain substantial energy in the range of frequencies0.2–2 cph. The lower frequency signals are associated to longitudi-nal, latitudinal and mixed stationary modes that develop on thehighly irregular shaped continental shelf. The presence of thesemodes suggests that the Sicilian coast is a good reflector to theselong waves. Their wavelengths are comparable to the shelf extentwhich thus modifies their characteristics from a simple quarter-


wave resonant effect. It is inferred that the observed waves are notonly ones that cross the shelf from the deep sea, but that compara-ble energy is presented in trapped waves associated to bathymetricfeatures on the shelf.

The higher frequency coastal seiches are characterised not onlyby eigenmodes pertaining to inlets and bays on the coastal perim-eter, but also by long period modes in the adjacent open sea areasoutside the embayments. This is particularly evident in the pres-ence of coastal shallows, reefs or banks such as in the case of Mel-lieha Bay. The bays are in double resonance with the adjacentnearshore areas; the open coastal sea is on its own count in doubleresonance with the offshore deeper shelf area. It is thus inferredthat the observed long period wave field in coastal areas is not sim-ply restricted to those oscillations directly related to the deepershelf, but that comparable energy is also present in the form of sta-tionary coastal waves that are excited in the nearshore and innershore areas either directly by local atmospheric disturbances orindirectly through the forcing by deeper sea waves. Both mecha-nisms involve a cascading effect from larger to smaller horizontalscales.

Seasonal changes in the mean sea level show a major minimumin March and a major maximum towards the last months of theyear. Besides the usual steric and direct meteorological effects, thisvariability is attributed to adjustments in the mass balance of thewhole Mediterranean basin.

Acknowledgements

The authors wish to thank Mr. A. Xuereb and Mr. J. Bianco of theHydrographic Office at the Malta Maritime Authority, and Major J.Mifsud and Capt. A. Gauci, ex-Chief Meteorologists at the Meteoro-logical Office in Luqa, for kindly making available the sea level datain the Grand Harbour and the atmospheric pressure records inLuqa, respectively. Thanks also go to Dr. Alexander Rabinovich ofthe Russian Academy of Sciences for his numerous suggestionsand advice.

Appendix 1. General description of long waves on a continentalshelf

The typical lengths of these lower frequency long waves are inthe order of tens of kilometres and are compatible to the character-istic widths of continental shelves. The constructive interferencebetween incident waves of appropriate dimensions from the dee-per ocean with waves reflected from the coast can thus lead tostanding wave patterns corresponding to the well-known leakymodes and trapped modes (also called edge waves) of waveguidetheory (Munk et al., 1964). These oscillations can develop on thecontinental shelf as well as in more restricted coastal areas pro-vided that appropriate bottom slope conditions apply.

Leaky modes re-radiate energy into the deep sea. Along a coastwith uniform alongshore bathymetry their alongshore wave-lengths are usually long in comparison with the offshore wave-lengths. For a particular frequency they have a continuousspectrum of alongshore wavelengths. Any angle of approach is pos-sible, that for normal incidence corresponding to a standing wavesystem normal to the shore. An antinode is required at the shoreand a node at the shelf edge. In the case of a shelf with constantalongshore characteristics, width L and a monotonic depth profiled(x), where x is the distance normal to the coast, the fundamentalperiod is given by T ¼ 4

R L0

dxffiffiffiffigdp . Other possible modes are given by

the odd harmonics T/3, T/5, etc. For other angles of incidence theresonant frequencies become slightly higher and the amplificationis less. Leaky waves do not however necessarily require a shelfbreak and can occur over smaller areas and close to shore. In the

case of a plane coast with a linear sloping bathymetry d(x) = ax,the leaky waves normal to the coast are described by the expres-sion (Lamb, 1932):

gðx; xÞ ¼ AJoðvÞ ¼ AJo

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4x2x=ga

q� �

where x ¼ 2p=T is the angular frequency, A is the amplitude at thecoast, Jo is the zeroth order Bessel function of the first kind and g isthe acceleration due to gravity. The function Jo has zeros when itsargument vK = 2.405, 5.520, 8.654, etc. and maxima at vM = 3.832,7.016, 10.174, etc. At any offshore position, distant X from the coast,standing waves with frequencies xK = vK

p(ga/4X) thus exist for

which Jo = 0 represents a nodal line parallel to the coastline. At fre-quencies xM = vM

p(ga/4X) antinodes are obtained. The theoretical

spectral energy distribution for this linear slope model (propor-tional to J2

o thus consists of peaks at frequencies xM and troughsat xK.

In the case of trapped modes the energy is channelled by thebathymetry and remains on the shelf. The waves propagate in adirection parallel to the coast and energy is totally internally re-flected at the continental slope. For a straight continental shelf ofuniform width and constant depth falling vertically at the shelfedge to a flat bottomed ocean, amplitudes decrease exponentiallyseawards. Only certain angles of incidence are permissible. Atglancing incidence, a lower cut-off frequency occurs in each modecorresponding approximately to periods of infinity, T/2, T/4, T/6,etc. where T = 4L/

p(gh), and the waves are thus non-dispersive.

Their offshore wavelengths have the same scale as their offshorewavelengths. For a given frequency and bottom slope the along-shore wavelengths have a discrete spectrum, so that only certainwavelengths are permitted. For a wave of a given frequency on acontinental shelf with no longshore variations, trapped modes al-ways have shorter alongshore wavelengths compared to those ofleaky modes. This may not however apply in the case of a morecomplicated bathymetry with both cross-shore and alongshorevariations. Moreover if the depth at a distance from the coast tendsto a constant value, the trapped modes leak some energy to infin-ity, although the consequent rate of decay may be exceedinglyslow (Longuet-Higgins, 1967). With a real bathymetry the transi-tion between leaky and trapped modes is thus much more tenuousthan the idealised case.

The initial energy of these long period shelf oscillations may beabsorbed by radiation of the same frequency incident from theadjacent deeper sea areas. It may alternatively be derived from asharp pulse such as due to a travelling pressure disturbance. Inthe case of the Maltese shelf area, there is evidence (Drago,1999) in favour of the dependence of long waves in the sea on pres-sure fluctuations in the atmosphere. The passage of a pressure dis-turbance is responsible for the resonance generation of shelfmodes (Kulikov and Shevchenko, 1992); it can also generate awhole range of more localised sea level signals in the mediumrange of frequencies. The periods of these shelf oscillations aredependent on both the period range in the pressure wave spectrumas well as on the bottom relief.

Appendix 2. 1D model to estimate seiche-generated currentsfrom sea-level data

Seiche-induced currents in a rectangular open ended embay-ment can be related to the sea level oscillations inside the bay bymeans of a simple 1D model. Suppose that the sea level g ismeasured by a coastal gauge at the head of the bay (origin O inFig. 20). These sea level oscillations follow the movement of thesloshing water in and out of the embayment. The proposed one-dimensional model estimates the along-axis barotropic currents


associated to this movement. Currents in the along-axis directionare calculated at X at distance xo from O. The bay is assumed tohave constant width. The distance x from the origin O is measuredalong the ‘talweg’; the depth D is taken to be the crossectionalaverage orthogonal to the ‘talweg’.

Suppose that the water body is oscillating at its gravest modewith period To. The position of the displacement node which marksthe boundary between the water body in the bay and the open seais not known a priori. The position of X with respect to this bound-ary is thus described by a non-dimensional distance ao = xo/L,where L is the hypothetical distance of the open boundary from O.

The elevation along the bay axis is also assumed to follow asinusoidal profile with a maximum amplitude at the head and zerodisplacement at the open boundary. The elevation g at O is anexpression of the instantaneous displacement of the vertical oscil-lation at O. If the free surface along the bay is assumed to retain asinusoidal profile in time, the instantaneous displacement at anyposition a (=x/L) will be given by y(a, t) = g(t)cos(pa/2).

The excess volume of water V(ao, t) at time t which has accumu-lated behind point X is given in dimensional form and for unitcross-bay width by:

Vðao; tÞ ¼ LZ a

0 oyda ¼ 2gL

p½sinðpa=2Þ��ao

0 ¼2gLp

sinðpao=2Þ:

ðA2:1aÞ

Now if d is the average depth of the embayment along the ‘tal-weg’, then

pðgdÞ ¼ 4L=To ) L ¼ ðTo=4ÞpðgdÞ;

hence

Vðao; tÞ ¼gðtÞTo

ffiffiffiffiffiffigd

p2p

sinðpao=2Þ ðA2:1bÞ

The instantaneous flow of water across unit bay width at X is

dVðao; tÞdt

¼ To


p2p sinðpao=2Þ dgðtÞ

dt

� �o

ðA2:2aÞ

where the subscript in dgðtÞdt

� �o

is included to indicate that the rate ofsea level change is measured at the origin. If Do is the depth at a = ao,the barotropic current U (+ve in the +ve x-direction) is given by

� dVdt¼ UDo: ðA2:2bÞ

Hence

U ¼ � To


p2pDo

sinðpao=2Þ dgðtÞdt

� �o

: ðA2:3Þ

If the elevation g is measured as a discrete signal with samplingfrequency 1/DT, dg/dt can be expressed in finite

Fig. 20. Schematic diagram for 1D model calculations.

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sea level variability and the ‘milghuba’ seiche oscillations in the...

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