Continental Shelf Research 111 (2015) 1–8

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Continental Shelf Research

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Review Papers

Continental shelf waves on the Alborán sea

M.A. Serrano n, M. Díez-Minguito, M. Ortega-Sánchez, M.A. LosadaAndalusian Institute for Earth System Research, University of Granada, Avda. del Mediterráneo, s/n, 18006 Granada, Spain

a r t i c l e i n f o

Article history:Received 4 March 2015Received in revised form26 October 2015Accepted 7 November 2015Available online 11 November 2015

Keywords:Continental shelf wavesAlborán SeaAtmospheric pressureWind stressSpectral analysis

x.doi.org/10.1016/j.csr.2015.11.00443/& 2015 Elsevier Ltd. All rights reserved.

esponding author.ail address: mserranog@ugr.es (M.A. Serrano).

a b s t r a c t

The existence of free continental shelf waves in the Western Mediterranean Sea is studied using a sta-tistical approach. Hourly mean sea level at six locations extended along the northern and southern coastof the Alborán Sea were analyzed by standard correlation and spectral analysis techniques for twoperiods of 91 and 92 days, respectively. The mean sea level was adjusted for the effect of atmosphericpressure according to the hydrostatic approximation. The wind-coherent response of sea level was re-moved by calculating the transfer function between the input alongshore wind stress and the outputadjusted sea level. The cross-correlation analyses indicate time lags that are consistent with the hy-pothesis of the propagation of the free shelf waves predicted by the continental shelf wave theory. Thepresence of a westward-travelling signal on the NCS and a west-to-east travelling signal on the SCS of theAlborán Sea is confirmed. The cross spectra between residual sea level at the six locations show sig-nificant coherence at long periods. For these periods, the phase difference spectra show that the di-rections of propagation of the free waves are consistent with the interpretation of the cross correlationanalysis. There is evidence of a seasonal variation in the phase speed of the waves, which may be relatedto variations in the speeds of currents in the Alborán Sea. The observations suggest the presence ofcontinental shelf waves on both coasts of the Alborán Sea.

& 2015 Elsevier Ltd. All rights reserved.

Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12. The study area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23. Data sources. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24. Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35. Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

5.1. Wind transfer function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45.2. Cross correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45.3. Cross spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

6. Discussion and final remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1. Introduction

The Alborán Sea is a back-arc region located in the westernMediterranean Sea (Fig. 1). This region is connected to the AtlanticOcean through the Gibraltar Strait and opens to the east into theBalearic Basin through the Alborán Trough (Lobo et al., 2006).

Similar to many other sub-basins, the Alborán Sea is highly dy-namic, and complex interactions arise from tidal, atmospheric, andmixing processes (Renault et al., 2012). In the Alborán Sea, oceanicmotions involve a great variety of both spatial and temporal scales.The tides are mainly semidiurnal and co-oscillate with the Atlanticwaters, which propagate into the Alborán Sea through the Gi-braltar Strait (Defant, 1961). Subinertial flows induced by atmo-spheric pressure gradients and water mass distributions generatedby the wind action were investigated by Lafuente et al. (1998) and

Fig. 1. Contour map of the Alborán Sea showing the bathymetry. The dashed lineindicates the base of the continental slope. Labels NCS and SCS indicate theNorthern Continental Shelf and Southern Continental Shelf, respectively. Circles,squares and triangles indicate the stations for sea level, wind, and atmosphericpressure observations, respectively (the NCS triangles include both wind and at-mospheric pressure data). Labels S1–S6 represent the Almería (S1), Motril (S2),Málaga (S3), Estepona (S4), Ceuta (S5), and Melilla (S6) stations.

M.A. Serrano et al. / Continental Shelf Research 111 (2015) 1–82

Bárcenas et al. (2011), respectively. The influence of the windwaves near the coast, which are typically generated during thepassing of extratropical storms, has also been intensively studied(Ortega-Sánchez et al., 2008).

However, a detailed study of long-period waves propagatingalong the continental shelves of the Alborán Sea is still lacking.Among these long-period waves, continental shelf waves (CSWs)are a subclass of the so-called coastally trapped waves, which arecharacterized by a roughly exponential decrease in the amplitudeof dynamic pressure anomalies with increasing distance from thecoastline (Mysak, 1980; Brink, 1991; Yin et al., 2014). CSWs aresubinertial waves generated by large-scale weather systemsmoving along or across the continental shelf (Robinson, 1964;Mysak, 1967; Adams and Buchwald, 1969; LeBlond and Mysak,1978; Thiebaut and Vennell, 2010).

CSWs typically exhibit amplitudes on the order of centimeters,periods on the order of one to several days, and wavelengths oftens to hundreds of kilometers, depending on the continental shelfwidth and slope (Adams and Buchwald, 1969; Schumann andBrink, 1990). Their phases travel parallel to the coast in only onedirection, similar to classical Kelvin waves (Båth, 2012). Robinson(1964) analyzed the non-equilibrium response of sea level to thebarometric factor to identify CSWs along the coastline. The surfacefriction of the alongshore wind stress component associated withweather systems acting over a variable-depth shelf is consideredto be another possible mechanism for generating CSWs (Gill andSchumann, 1974; LeBlond and Mysak, 1978; Middleton and Cun-ningham, 1984).

After the pioneering work of Hamon (1962) on the propagationof CSWs along the Australian coast, these waves have receivedample attention, and their presence on different continentalshelves around the world has been reported (e.g., Cutchin andSmith, 1973; Kaoullas and Johnson, 2010; Schulz et al., 2012).Nevertheless, to our knowledge, no studies have examined thepresence of CSWs on the continental shelf of the Alborán Sea. Thisstudy area is interesting because it features two opposite andapproximately parallel continental shelves that are relatively closeto each other (Fig. 1), which connect at the Gibraltar Strait andopen the possibility of interaction between outgoing and incomingCSWs. Within this context, the objective of this study is to identifyand characterize the CSWs on the continental margins of the Al-borán Sea using data from sea level gauges and meteorological

stations along the northern continental shelf (NCS) and thesouthern continental shelf (SCS) (Fig. 1).

This paper is organized as follows. First, the study area and thefield data and methodology used in this investigation are de-scribed in Sections 2, 3, and 4, respectively. The wind transferfunction is calculated to remove the wind-coherent response ofsea level (Section 5.1). The correlations between adjusted sea le-vels at different stations are determined at different time lags(Section 5.2), and the coherence and phase between tide gaugedata located along the continental shelf and the observed phasespeeds are calculated to compare theoretical predictions withobservations (Section 5.3). Finally, the results are discussed andsummarized (Section 6).

2. The study area

The Alborán Sea basin has been subjected to convergence be-tween Eurasia and Africa and has developed a complex physio-graphy (Comas et al., 1999). This tectonic shortening and theprogressive emergence of segments of the ancestral Alborán basinhave reduced the basin to a 400�200 km low straddling theAfrican and Eurasian plate boundary (Comas et al., 1999). Theclimate regime in the Alborán Sea provides a contrast betweenboth the Atlantic and Mediterranean weather systems (Giorgi andLionello, 2008). The Atlantic weather systems travel from west toeast. The time scale is from 2–4 days to 2 weeks, and the pre-dominant wind direction is eastward (Rodríguez, 2007). TheMediterranean weather systems are typically smaller and shal-lower than Atlantic systems and have shorter lifetimes (Ulbrichet al., 2012). They are generated between the Alborán Sea andNorth Africa; the time scale is approximately 2–4 days; and thepredominant winds blow westward (Rodríguez, 2007).

Continental shelves are zones of both generation and propa-gation of characteristic long-period waves because of their parti-cular geometries, with depths that vary according to the distancefrom the coastline (Csanady and Gacic, 1981). The study area in-cludes both the NCS and SCS of the Alborán Sea (Fig. 1). The Al-borán Basin is narrow and reaches depths of 2000 m. The averageNCS is approximately 20 km wide and widens at the far ends. Inthe southern Alborán Sea, the continental shelf is approximately20–60 km wide between Occidental and Oriental Morocco.

3. Data sources

Hourly data from six sea level gauges (Si in Fig. 1, where i in-dicates the 1–6 study stations) located along the northern andsouthern coastlines of the Alborán Sea were available for thisstudy. Data at S1–S3 and S6 were provided by Puertos del Estado(Ministerio de Fomento, Spain), hereafter PdE, and the elevationdata at both S4 and S5 were provided by Instituto Hidrográfico de laMarina (Spain), hereafter IHM. The atmospheric pressure wasmeasured every six hours by the meteorological stations of theJunta de Andalucía (Consejería de Medio Ambiente, Spain), hereafterJA-CMA, located near the sea level stations. In the NCS, wind di-rection and velocity data (at three hour intervals) were obtainedfrom observations of the meteorological stations of the JA-CMA,which were located near the sea level gauges. In the SCS, winddata (at three hour intervals) were obtained from the nearesthindcasting points of the model SIMAR (also provided by PdE)(Fig. 1). The analyzed time interval (2011–2012) was limited by theavailability of simultaneous data records. Missing data for intervalsof less than 12 h in all series were completed by linear interpola-tion. Climatic data were interpolated to hourly intervals.

The data record chosen for the analysis extends from June 19 to

M.A. Serrano et al. / Continental Shelf Research 111 (2015) 1–8 3

September 19 (hereafter referred to as P1 (2209 data points)) andfrom September 19 to December 19 (hereafter referred to as P2(2185 data points)). P1 represents the summer season and P2 theautumn/winter season. These intervals were selected because theytypically exhibit different meteorological and oceanographic con-ditions, in the sense of the influence of the different storms en-tering the region, the dominant wind magnitude of the winds, andthe different surface current patterns controlled by the Atlanticwater inflow.

4. Methodology

Standard harmonic analyses were applied to the sea level data(Pawlowicz et al., 2002) to remove the tidal effects. A fifth zero-phase Butterworth low-pass filtering with a cut-off frequency of0.9 cpd was performed to remove the remaining high-frequencyoscillations. The filtered signals η were adjusted for the effect ofthe atmospheric pressure on sea level according to the hydrostaticapproximation to eliminate the assumed in-phase effect of atmo-spheric pressure variations on sea level (Cutchin and Smith, 1973).

Fig. 2. The gain of the transfer function TF (a–c, g–i) and the coherence squared Φ2 (d–f, jin the periods P1 and P2. The 95% confidence limits for the gain are shown by the shad

By calculating the spectral transfer function (hereafter referred toas TF) between the input alongshore wind stress and the outputadjusted sea level ηadj, the wind-coherent signals were removedand then, the residual sea level was examined (Ou et al., 1981;Kim, 2014).

The existence of travelling waves was revealed from correlationfunctions of the lagged cross-correlation, CC, between the residualsea levels at various stations by the existence of time lags thatcorrespond to the maximum correlation.

The relationship between the residual sea levels ηres at thedifferent stations was expressed in terms of both the coherencesquared Φ2 and the phase difference τ as a function of the fre-quency (cpd). This provided correlations and time displacementsbetween the data records. The phase shifts were interpreted as lagtimes using τ ω θ ω ω( ) = ( )/ , where θ is the phase difference ex-pressed in degrees at frequency ω. The observed phase velocity cat any given frequency was obtained from the phase shift via theequation ω τ ω ω θ ω( ) = ( ) = ( )c D D/ / , where D is the along-shoredistance between stations (km) (Fig. 1).

The 95% statistical significance level for the squared coherencewas also determined. The spectral analysis of the data was

–l) between the input alongshore wind and the output sea level at the study stationsed areas, and for Φ2 are shown by the dashed lines.

M.A. Serrano et al. / Continental Shelf Research 111 (2015) 1–84

performed by applying a Fast Fourier Transform, averaged with theWelch method, and smoothed using a Hanning window. The inputdata were used with a Hanning window of length 631 points and a50% overlap.

5. Results

5.1. Wind transfer function

In attempting to study free waves, the wind driven componentof each ηadj signal must first be removed using wind spectraltransfer functions. Wind transfer function TF is calculated as theratio of the co-spectrum of the adjusted sea level ηadj and thealong-shore wind stress to the autospectrum of along-shore windstress. The along-shore component of wind stress was calculatedas ρ= · | |−F v v3 10y a y

3 , where ρa is the air density (10�3 g/cm�3) andvy is the component of the wind speed v to the east (NCS) andwest (SCS). According to the CSW propagation direction, westwardwinds were analyzed on the NCS and eastward winds on the SCS.

Fig. 2 shows the gain of the TF (Fig. 2a–c, g–i) and the co-herence squared Φ2 (Fig. 2d–f, j–l) between the input alongshorewind stress and the output ηadj at the study stations as a functionof frequency for the periods P1 and P2. As a measure of statisticalsignificance, the 95% confidence levels are also shown, which werecomputed following Bendat and Piersol (1986). The results in-dicate that the response to wind stress at S1–6 stations is small atthe studied frequencies, suggesting the signal contained a sig-nificant amount of energy originating from a distant source. Allfrequencies where the wind stress and ηadj are significantly cor-related (at 95% level) were removed of the ηadj.

Once isolated, the residual sea level ηresi was analyzed. Severalclues indicating the existence of CSWs on the Alborán Sea wereprovided by the results shown in Fig. 3. This figure depicts thelow-pass ηresi time series from the study stations (a–b), the at-mospheric pressure patm (c), and the alongshore wind velocity data

η resi (

cm)

P atm

(mba

r)w

ind

vel

(cm

/s)

ηresi

(cm

)

08/01/12 09/01/12-10

0

10 S1 S2 S3

westward NCS08/15/12

2:00 h (08/09/12)

11:00 h (08/09/12)9:00 h (08/10/12)

07/01/12 08/01/12-10

0

10 S5 S6

SCS

07/15/12

eastward

1:00 h (07/16/12)

0:00 h (07/15/12)

08/01/12 09/01/121005

1010

1015

1020

08/15/12

4:00 h (08/09/12) 11:00 h (08/09/12)

19:00 h (08/09/12)

08/01/12 09/01/12

-200

0

200

08/15/12

2:00 h (08/09/12)

9:00 h (08/09/12)

18:00 h (08/09/12)

Fig. 3. A segment of the time series of the residual sea level data for the northernstations S1–3 (a), southern stations S5–6 (b), atmospheric pressure (c), andalongshore wind velocity data (d). Sea level data have been adjusted for the effectof the atmospheric pressure on sea level according to the isostatic approximationand the wind-coherent signal has been removed. The squares, which are depictedto guide the eye, indicate the dates on which a maximum peak is attained at thevarious stations. The propagation direction of the signal is also indicated.

(d). The lags between the maximum sea level at the S1–3 stationsare 9 h (between S1–S2) and 31 h (between S1–S3) (Fig. 3a plot)for the NCS. For the SCS, the time lag between the maximum sealevel at the S5–S6 stations is 23 h (Fig. 3b plot). Similar differencesare not observed in patm (Fig. 3c plot) nor in the wind velocityrecords (Fig. 3d plot). The time lags between the maximum patmlevel and wind velocity are different than the time lags foundbetween the study stations. These observations indicate that thespeed with which weather systems move is different than thepropagation speed of the residuals analyzed. Moreover, residualsea level fluctuations have no apparent association with weathersystems, suggesting that the residuals analyzed correspond to freewaves. The results suggest the presence of waves that travelwestward and eastward on the NCS and SCS, respectively. Tosupport this hypothesis and to gain insight into the propagationprocess, cross-correlations and cross spectrum analysis were per-formed on the data series.

Fig. 4 shows the autospectral density functions R of hourly sealevel (both η and ηresi) at S1–6 stations for the P1 (upper plots) andP2 (lower plots) periods using the Welch method. In this figure thefrequency was plotted against the logarithm of the energy spectraldensity. Dotted lines show the 95% confidence limits of the spec-tra, which were computed according to the method of Bendat andPiersol (1986). In general, the amplitudes of the η spectra exceedthose of the ηresi. This is a consequence of removing the effects ofatmospheric forcing. The most energetic peaks appear at thelowest frequencies for all stations in both periods. These peaks arefound in η, and ηresi data. In the NCS, the most common energy inthe S1–4 spectra is found in the frequency band 0–0.14 (0–0.28)cpd in the period P1 (P2). In the SCS, the most of the energycommon in the S5–6 stations is found in the frequency band 0–0.14 cpd in both the periods P1 and P2, and there is a second peakwith less energy in the 0.42–0.51 cpd in the period P1. The resultsindicate that the response to atmospheric forcing at the studystations is small at low frequencies, suggesting that the residualsignal contains a significant amount of energy originating from adistant source.

For the study region, theory predicts that CSWs should travelwestward on the NCS; in other words, the adjusted sea level at S1should lead that at the S4 station. In contrast, on the SCS, this typeof wave should travel eastward, and S6 should lag that at S5station.

The following sections explain more results to support thesebehaviors.

5.2. Cross correlation

The propagation direction of the CSW and the time lags can bebetter determined from correlation functions of the lagged CCbetween the residual sea level at different stations (Hamon, 1966;Candela et al., 2014). Fig. 5 shows the CC between the residual sealevel data from the study stations for both periods P1 and P2. Thepositive or negative lags at the maximum CC indicate the propa-gation direction of the wave. Larger distances between stations arecorrelated with larger observed time lags (see Fig. 4).

The maximum CC between S3 and S1 stations is located at atime lag of approximately 1.1 days and 0.4 days for S3 and S2,respectively, whereas the maximum CC between S5 and S6 ap-pears at �0.8 days for the period P1 (see Fig. 5). Therefore, thepropagation directions from the correlation functions between theresidual sea level at adjacent stations were consistent with thoseof a travelling CSW in the study region. These results point to thepresence of a westward-travelling signal on the NCS and a west-to-east-travelling signal on the SCS of Alborán Sea. A seasonaldependence in the correlation functions appeared to exist, with amaximum time lag occurring in the period P1.

η η

S1

P1

S3

P1

S4

P1

S5

P1

S6

P1

S1

P2

S2

P2

S3

P2

0 0.5 1Frequency (cpd)

S4

P2

0 0.5 1Frequency (cpd)

S5

P2

0 0.5 1Frequency (cpd)

S6

P2

resi

S2

P1

R (c

m2 c

pd-1

)R

(cm

2 cpd

-1)

R (c

m2 c

pd-1

)R

(cm

2 cpd

-1)

10-2

100

102

10-2

100

102

10-2

100

102

10-2

100

102

Fig. 4. Spectral density (cm2/cpd) at stations S1–6 of both the observed and residual sea level during P1 (upper plots) and P2 (lower plots) for frequencies lower than 1 cpd.The error bounds (dotted lines) in the spectra include the 95% confidence intervals. Shaded areas indicate the most energetic frequencies.

M.A. Serrano et al. / Continental Shelf Research 111 (2015) 1–8 5

5.3. Cross spectrum

The coherence squared Φ2, which is analogous to the square ofa correlation coefficient, and the phase difference τ between theresidual sea level are determined to estimate the relationship andthe time displacement between two records as a function of thefrequency (Mysak and Hamon, 1969; Candela et al., 2014). Thisprovides additional clues on the existence and propagation ofCSWs at the study site. Fig. 6 showsΦ2 and τ between the residualsea levels as a function of the frequency. As a measure of statisticalsignificance, confidence limits are also shown, which were com-puted following Koopmans (1995). The phase differences are quitereliable when Φ2 amplitudes are below the 90–95% confidenceinterval. Phase shifts τ are shown when Φ2 has a significant value.Coherence squared Φ2 has significant peaks, within the 95% con-fidence level, at frequencies 0–0.32 cpd for all pairs of stations onthe NCS of the Alborán Sea for the period P1 (S1–4, left plots). Forthe pair of stations on the SCS, Φ2 shows a peak within the fre-quency range 0.45–0.61 cpd (S5–S6, left plot) for the period P1. Forthe period P2, Φ2 is significant at the frequency range 0–0.28 cpdon the northern shelf (S1–4, right plots) and at 0.14 cpd on thesouthern shelf (S5–S6, right plot).

The positive or negative sign of τ indicates the direction of

propagation of the wave at the significant frequencies. A negative τvalue at a frequency of 0–0.32 cpd for both P1 (S1–4 left plots) andP2 (S1–4 right plots) periods on the NCS indicates a westward-travelling signal, whereas the positive τ values at 0.45–0.61 cpd forthe P1 period (S5–S6 left plot) and 0.14 cpd for P2 (S5–S6 rightplot) on the SCS indicate an eastward-travelling signal in the Al-borán Sea. These results are consistent with both the interpreta-tion of the CC analysis and the continental shelf waves hypothesispreviously described.

The phase differences, τ, between the study stations at thefrequency of the maximum Φ2 values are interpreted as lag times.The apparent wave phase speed c was evaluated and comparedwith the theoretical phase speed cth. Different authors have de-veloped some theoretical models to describe the CSW's, such asRobinson (1964), Buchwald and Adams (1968), Mysak (1980), andKaoullas and Johnson (2010). Because of its simplicity, the theo-retical phase speed cth was calculated following Robinson's model(see Table 1). Robinson proposed a simple theoretical model forbarotropic waves travelling in one direction only on a sloping fi-nite width shelf (barotropic waves involve a lower Burger number,that is, a Burger number ⪡1 (LeBlond and Mysak, 1981). The the-oretical phase velocity, c¼w/k (where w is the frequency and k is

>0

<0

>0

>0

>0

P1

P1

P1

P1

P1 P2

P2

P2

P2

P2>0

<0

>0

>0

>0

S1S3

S2S3

S2S4

S1S2

S6S5

Fig. 5. Correlation functions of CC between the adjusted sea level at different stations (S1–S6) on the coast of the Alborán Sea for 76 day time lags. The error bars (dashedareas) include the 95% confidence interval in cross correlation between study stations. Positive lags indicate that the western station of each pair lags relative to the easternstation.

M.A. Serrano et al. / Continental Shelf Research 111 (2015) 1–86

the wavenumber), is γ=c fL/ j2 (where γj2 (j¼1,2,…) is defined as a

part of the Bessel function of zero order J0(2γj)). The first zero of J0lies at γj¼1.2. For the fundamental mode of a CSW is =c fL/1.44,where L is the shelf width and f is the Coriolis parameter. As-suming typical shelf widths of the Alborán Sea ≈ –L 22 40 km and a

Coriolis parameter of = −f d7.51 1, the phase velocity for the

0 0.2 0.4 0.6 0.8 1Frequency (cpd)

0

0.5

1

Φ2

P1

0

0.5

1

Φ2

P1

0

0.5

1

Φ2

P1

0

0.5

1

Φ2

P1

P

P

P

0P

S2 S1

S3 S2

S4 S3

S5 S6

Fig. 6. Coherence squared Φ (solid) and phase τ (dots) versus frequency ω between residshifts are only shown when Φ2 has a significant value. The horizontal dashed line reconfidence range for phase estimates when the coherence exceeded the 95% significancethe eastern station.

fundamental mode ranges between 1.32 and 2.5 m/s.The phase differences, τ, indicate, for example, that the S1 sea

level leads S2 by approximately 32 at approximately 0.14 cpd. Thisphase shift between these two stations, 90 km apart, implies a c of1.63 m/s. This c value can be compared with the theoretical cthvalue. For the period P1, the wave velocity at the frequency of themaximum coherence squared Φ2 value appears to be on the same

-180

0

180Ƭ

(º)

2

-180

0

180

Ƭ (º

)

2

-180

0

180

Ƭ (º

)

2

0.2 0.4 0.6 0.8 1Frequency (cpd)

-180

0

180

Ƭ (º

)

2

ual sea levels at the study stations during the periods P1 (left) and P2 (right). Phasepresents the 95% confidence limit for the coherence. Error bars indicate the 95%level. A negative phase means that the western station of each pair lags relative to

Table 1Comparison of the theoretical and observed phase relationships, τth and τ, re-spectively, for pairs of stations, during P1 and P2. The observed wave velocity c isevaluated at the frequencies of 0.14 (0.46) cpd during P1 and 0.28 (0.14) cpd duringP2 on the NCS (SCS).

Stationpairs

cth (m/s)

c (m/s) Time lagth(day)

Time lag (day) τ (deg)

P1 P2 P1 P2 P1 P2

S1–S2 1.32 1.63 2.2 0.8 0.63 0.48 32 47S2–S3 1.43 1.2 3 0.56 0.67 0.27 35 27S3–S4 1.52 1.67 2.5 0.57 0.52 0.35 26 35S5–S6 2.5 8 2.7 0.97 0.31 0.92 41 46

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order as the cth value (1.32–1.52 m/s) of a CSW travelling along thenorthern coast of the study region (see Table 1). However, thewaves appear to travel considerably faster than predicted by thetheory along the southern shelf. For the period P2, the wave ve-locity between stations S1–S4 on the northern shelf is faster thanthat predicted by the theory (observed values of approximately2.2–3 m/s) (see Table 1).

6. Discussion and final remarks

Our analyses and results suggest that travelling free CSWs existon both the NCS and SCS of the Alborán Sea. The results of thespectral analysis R show energy peaks in the frequency bands 0–0.14 and 0–0.28 cpd (0–0.14 and 0.42–0.51 cpd) on the NCS (SCS)for ηresi, after removing both the astronomical and meteorologicalinfluence. These results indicate the existence of a non-local sealevel response to atmospheric influence for the studied fre-quencies. The cross-correlation between the residual sea level atdifferent stations and the coherence and phase difference spectraindicate that a travelling westward (eastward) oscillation exists onthe NCS (SCS), which is consistent with the propagation directionof the CSW. Similar analyses by Cutchin and Smith (1973) andSchulz et al. (2012) also showed low-frequency oscillations on theOregon continental shelf and in the Mid-Atlantic Bight, respec-tively, which were attributed to the existence of CSWs.

The coherence squared Φ2 and phase τ spectra indicate a CSWtravelling with a phase velocity that appears to be in agreementwith the CSW theory on the NCS (SCS) for the P1 (P2) period. Incontrast, the free wave on the NCS (SCS) in the P2 (P1) periodtravels at a higher phase speed than the calculated theoreticalwave speed. Discrepancies between the theory and our observa-tions may be likely the result of the effects of local generation,density stratification, and irregular topography, which have notbeen considered. The effects of deep-sea stratification on the freewaves speed were studied by Mysak (1967) and Brink (1991). Itwas found that stratification increases frequency (hence wavespeed) for a given wave mode (Mysak and Hamon, 1969). The flowin the Alborán Sea is predominantly barotropic (Gomis et al.,2006) and then, the stratification may not explain this higherphase speed. One of the factors driving the variability of the Al-borán Sea is the temporal variability of the Atlantic inflow throughthe Gibraltar Strait (i.e., the Atlantic Jet) (Renault et al., 2012),which fluctuates in a north–south direction (Sarhan et al., 2000).These displacements could explain the wave velocities anomaliesobserved in this study. Therefore, a possible cause of the higherwave speeds on the NCS during the P2 period could be the ex-istence of cyclonic circulation forced by the Atlantic Jet along thenorthwestern shelf of the Alborán Sea because westerly windsstrengthen the jet and trigger coastal upwelling (Sarhan et al.,2000; Ruiz et al., 2013). During the P1 period, the Atlantic inflow ishigher (Renault et al., 2012), and part of the jet flows to the west

and is incorporated into an anticyclonic gyre. The remainder flowssoutheast along the African coast forming the southern peripheryof a cyclonic circulation (Nihoul, 2000), which could be the causeof the higher wave velocities observed on the SCS.

The observed time lags of the cross-correlation analysis in theperiod P2 are very small. The observed seasonal dependence onthe time lag may be due to the seasonal variations in the surfacespeed of the currents in the Alborán Sea, but this hypothesis hasnot been corroborated. Simultaneous continuous records of cur-rent and sea level might be required to further explore thishypothesis.

Another point of interest is the study of possible interactionbetween CSWs travelling in opposite directions, because of themorphology of the back-arc study area. The presence of CSWstravelling in opposite directions on the northern and southernshelves of the Alborán Sea opens the possibility of interactions andthe existence of amphidromic points associated with this type ofwaves. Since the amplitude decreases roughly exponentially withincreasing distance from the coast and the NCS and SCS are rela-tively close to each other, the CSWs travelling on the two shelvesmight interact near the Gibraltar Strait. Simultaneous continuousrecords of current and sea level along transverse sections might berequired to further explore this aspect. A possible leakage of aportion of the energy of a CSW on the NCS might occur throughthe Gibraltar Strait, but another portion might occasionally turnaround and propagate from the Gibraltar Strait to the Mediterra-nean Sea.

Finally, it is worthy mentioning that a same weather systemmight generate winds in opposite directions and might thereforegenerate simultaneous CSW travelling in both continental shelves.Thus, space scales of the weather system could be defining windvariations in two directions in both shelves. It should be necessarythat the spatial scale of the weather system was the same as thatof the study site. A detailed and broad study will be necessary totest this assumption.

Acknowledgments

This research was partially supported by the project CTM2012-32439 (Secretaría de Estado de IþDþ i, Spain) and the researchgroup TEP-209 (Junta de Andalucía). The authors thank Puertos delEstado (Ministerio de Fomento, Spain) and Instituto Hidrográfico dela Marina (Spain) for providing the sea level records. The me-teorological data were obtained from the Junta de Andalucía(Consejería de Medio Ambiente y Ordenación del Territorio, Spain)and Puertos del Estado (Ministerio de Fomento, Spain).

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