Numerical modeling of the semidiurnal tidal exchange through the Strait of Gibraltar

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1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 109,, doi: /2003jc002057, 2004 Numerical modeling of the semidiurnal tidal exchange through the Strait of Gibraltar G. Sannino, A. Bargagli, and V. Artale Ocean Modeling Unit, Special Project Global Climate, ENEA C. R. Casaccia, Ente per le Nuove Technologie, l Energia e l Ambiente, Rome, Italy Received 21 July 2003; revised 22 February 2004; accepted 5 March 2004; published 7 May [1] A three-dimensional sigma coordinate free surface model is used to investigate the semidiurnal tidal exchange through the Strait of Gibraltar. The model makes use of a coastal-following, curvilinear orthogonal grid that includes the Gulf of Cadiz and the Alboran Sea, with very high resolution in the strait (<500 m). A lock-exchange initial condition is used: the western part of the model domain is filled with Atlantic water, whereas the eastern part is filled with Mediterranean water. The model is forced at the open boundaries through the specification of the semidiurnal (M 2 and S 2 ) tidal surface elevation. The model is run over a spring neap cycle (fortnightly period), and the results are compared with available observed data. Simulated cotidal maps of the M 2 and S 2 tidal elevation components are in quantitative and qualitative good agreement with observed data as well as with the simulated major and minor axis of tidal ellipse. The model reproduces the generation and the subsequent propagation of internal bores both eastward and westward, showing that they are always generated during the fortnightly period. However, the principal aim of this work is to quantify the effects of tidal forcing on mean quantities, entrainment, and transport of Atlantic and Mediterranean water along the strait. Model results reveal that the contribution of the semidiurnal tidal component (M 2 )tothe transport is relevant over Camarinal Sill, whereas it is negligible at the eastern end of the strait. Model results indicate, also, that the effect of the semidiurnal tide is to increment the mean transport by about 30% both for the inflow and the outflow. INDEX TERMS: 4255 Oceanography: General: Numerical modeling; 4560 Oceanography: Physical: Surface waves and tides (1255); 4512 Oceanography: Physical: Currents; KEYWORDS: Gibraltar, tides, POM Citation: Sannino, G., A. Bargagli, and V. Artale (2004), Numerical modeling of the semidiurnal tidal exchange through the Strait of Gibraltar, J. Geophys. Res., 109,, doi: /2003jc (Roma) Horrenda late nomen in ultimas extendat oras, qua medius liquor secernit Europen ab Afro, qua tumidus rigat arva Nilus. (Rome) Feared everywhere, let her extend her name to the uttermost shores, where the midway water separates Europe from Africa, where the swollen Nile irrigates the fields. [Horatio, 8 B.C., pp ] 1. Introduction [4] As 2000 years ago so now the Strait of Gibraltar separates Europe from Africa and connects the Atlantic Ocean to the Mediterranean Sea. It is 60 km long and 20 km wide with a minimum width of less then 15 km near the contraction of Tarifa Narrows and a shallow sill located near Camarinal (west of Tarifa) with a minimum depth of less than 300 m (Figure 1). [5] The excess of evaporation over precipitation and river runoff in the Mediterranean basin represents, together with the conservation of mass and salt, the main driving force of a mean circulation through the strait. This circulation, Copyright 2004 by the American Geophysical Union /04/2003JC generally called inverse estuarine, is characterized by two counter flowing currents: in the upper layer warm and relatively fresh Atlantic water, with a salinity of 36.2 practical salinity units (psu), flows eastward, spreading into the Mediterranean Sea, and in the lower-layer cold Mediterranean water, with a salinity of 38.5 psu, flows westward toward the Atlantic Ocean [Lacombe and Richez, 1982]. As initially suggested by Bryden and Stommel [1984], the mean circulation of the strait can be described as a two-layer system hydraulically controlled at Camarinal Sill. Many other papers have subsequently dealt with the applicability of the hydraulic control theory to the Strait of Gibraltar, and the number and possible locations of such controls within the strait (see, for example, Armi and Farmer [1985], Farmer and Armi [1986], Bryden and Kinder [1991], and more recently, for a modeling study, Sannino et al. [2002] (hereinafter referred to as SBA02)). [6] Various processes, at different timescales, modify the mean flow through the Strait of Gibraltar. The mean flow shows seasonal [Garrett et al., 1990] and interannual variability, weekly modifications driven by the wind and by atmospheric pressure differences between the Atlantic and Mediterranean Sea [Candela et al., 1989; García 1of23

2 Figure 1. Chart of the Strait of Gibraltar showing the principal geographic features referred to in the text. Locations of current meter moorings deployed during the Gibraltar Experiment (October ) and during the Canary Islands Azores Gibraltar Observations (CANIGO) observations (October 1995 April 1996) are also shown with red and blue solid circles, respectively. Lafuente et al., 2002], semidiurnal variations due to strong tides and finally, on very short timescales, modifications due to internal bores (internal wave reaching amplitudes of up to 150 m [Richez, 1994]). [7] The tidal forcing in the strait has been extensively studied and analyzed in the past. On the basis of data collected during the Gibraltar Experiment during [Bryden and Kinder, 1988]. Candela et al. [1990] (hereinafter referred to as CA90) and Bryden et al. [1994] described the structure of the barotropic M 2 tide and of the tidal transport through the strait, respectively, Bruno et al. [2000] have described the vertical structure of the semidiurnal tidal current at Camarinal Sill, while Wang [1993] used a numerical model to study tidal flows, internal tide as well as fortnightly modulation. Recently, others studies have been carried out, based on direct observations collected during the Canary Islands Azores Gibraltar Observations (CANIGO) project ( ) [Parrilla et al., 2002]: Tsimplis [2000] has described the vertical structure of tidal currents at Camarinal Sill, Tsimplis and Bryden [2000] (hereinafter referred to as TB00) have estimated the water transports through the strait, García Lafuente et al. [2000] have analyzed in detail the tide at the eastern section of the strait, and Baschek et al. [2001] (hereinafter referred to as BA01) have estimated the transport with a tidal inverse model. [8] To estimate the effect of tidal forcing on mean flow, Farmer and Armi [1986] included tides into their hydraulic theory by using a quasi-steady approximation in which the steady solution is verified at each time of a tidal cycle. However, Helfrich [1995] showed that this approach is not valid for dynamically long straits, i.e., straits having a length greater than the distance traveled by an internal wave during a tidal cycle, which is precisely the situation that occurs in the Strait of Gibraltar. Both theories assert that the exchanged flows increase with the strength of the barotropic tidal forcing, but the quasi-steady theory always predicts more flow than the time-dependent theory. [9] The purpose of this work is to implement a threedimensional (3-D) high-resolution, primitive equation, freesurface numerical model, of the circulation in the strait region and to use it to: (1) reproduce the semidiurnal tides within the Strait of Gibraltar, (2) estimate the water transports through the strait and (3) evaluate the effect of tidal forcing on the mean exchanges and entrainment. [10] The paper is organized as follows. Section 2 contains a description of the model used to simulate the tide in the strait. In section 3, model results are compared with available data of surface elevation, currents and internal bores measurements. Section 4 is devoted to the study of the tidal effect on water transports and entrainment through the strait, while summary and conclusions complete the paper. 2. Model Description [11] The numerical model used for this study was implemented in SBA02, where it was used to investigate the mean exchange through the Strait of Gibraltar. The model was only forced by the density contrast between the Alboran Sea and the Gulf of Cadiz, without any other forcing, such as tides, wind or atmospheric pressure. The main differences introduced in the present model regard the treatment of open boundary conditions, forcing and vertical resolution. In the following we only focus on the principal 2of23

3 model characteristics and on the main differences with respect to the model implemented in SBA Model Grid and Bathymetry [12] The region covered by our model includes the Strait of Gibraltar and the two adjacent subbasins connected to it: the Gulf of Cadiz and the Alboran Sea. The horizontal model domain is discretized by a curvilinear orthogonal grid made by grid points (see Figure 2 in SBA02). The resolution in the strait is much higher (500 m) than in the eastern (8 15 km) and western ends (10 20 km), so that the dynamics in the strait will be well resolved. The vertical grid is made of 32 sigma levels, logarithmically distributed at the surface and at the bottom, and uniformally distributed in the rest of the water column. The model topography has been obtained by merging the high-resolution (<1 km) topographic data set of the Strait of Gibraltar provided by the Laboratoire d Oceanographie Dynamique et de Climatologie with the relatively low-resolution (5 min) U.S. Navy Digital Bathymetric Database-5 data set (available from U.S. Naval Oceanographic Office, Bay St. Louis, Mississippi, at for the Alboran Sea and the Gulf of Cadiz. In an attempt to reduce the well-known pressure gradient error produced by sigma coordinate grids in regions of steep topography [Haney, 1991] an additional smoothing has been applied where dh/h > 0.2, as suggested by Mellor et al. [1994]. In order to estimate the residual pressure gradient error, we have integrated the model for one year without initial horizontal density gradient, i.e., with salinity and temperature fields only varying with depth, with no open boundary applied, i.e., closed domain, and without any other external forcing. In this integration the maximum intensity of erroneous currents introduced by the sigma coordinates is of about 2 cm s 1. Since the expected baroclinic velocities are up to 1 m s 1 this error seems to be tolerable. The resulting model topography in the region of the strait, with the minimum depth of the shelf set to 25 m, is shown in Figure 2. The dominant topographic features of the strait (from west to east) are clearly recognizable: Spartel Sill (Sp), Tangier basin, Camarinal Sill (Cm) with a minimum depth of 284 m and Tarifa Narrows Boundary, Initial, and Forcing Conditions [13] Near the eastern and western ends of the computational domain two open boundaries are defined, where values of velocity, temperature, and salinity must be specified. In order to minimize the contamination of the interior model solution due to wave reflection at the boundaries, an Orlanski radiation condition [Orlanski, 1976] is used for the depth-dependent velocity at both boundaries. A forced Orlanski radiation condition [Bills and Noye, 1987] is used for the surface elevation at the western and eastern boundaries: z n 1 2 z nþ1 Ti þ z Mi 2 i ¼ 1 þ Cr=2 ðcr=2þz n 1 2 i þ Crz n i 1 ; ð1þ where z n i represents the surface elevation at the i grid point of the open boundary at time step n, Cr = cdt/(2dx) isa Courant number defined in the x direction, z n 1 Ti is the forcing tide elevation at the grid point i and time step n 1, and z Mi is the time-independent mean elevation at the grid point i, which is set to about 12 cm at the western open boundary and to 0 cm at the eastern open boundary. Condition equation (1) incorporates a radiation mechanism that allows the undesired transients to pass through the open boundaries, going out of the model basin, without contaminating the desired forced solution [Arnold, 1987]. A zero gradient condition is used for the depth-integrated velocity. [14] The time-independent mean elevation values used at the open boundaries (z M ) are obtained running the model in barotropic mode. This model, as the baroclinic version, has at the eastern and western ends of the computational domain two open boundaries where values of barotropic velocity and surface elevation must be specified. For the surface elevation an Orlansky radiation condition [Orlanski, 1976] was used at the western boundary while a clamped to zero condition was used for the eastern end. For the barotropic velocity a zero gradient condition was used at both ends. In this way the barotropic model was able to freely adjust the western surface elevation, after 180 days of simulation, to about 12 cm. [15] Temperature and salinity are specified on the open boundaries by using an upwind advection scheme that allows the advection of temperatures and salinities into the model domain under inflow conditions. As in SBA02 the normal velocities are set to zero along coastal boundaries, at the bottom, adiabatic boundary conditions are applied to temperature and salinity and a quadratic bottom friction, with a prescribed drag coefficient, is applied to the momentum flux. This is calculated by combining the velocity profile with the logarithmic law of the wall: C D ¼ max 2: ; k 2 lnðdz b =z 0 Þ ; ð2þ where k is the Von Karman constant, z 0 is the roughness length, set to 1 cm, and Dz b is the distance from the bottom of the deepest velocity grid point. [16] For the initial condition we have used the same lock-exchange condition as in SBA02, i.e., we have filled the model with two water masses, horizontally uniform and vertically stratified, separated by an imaginary dam in the middle of the strait (longitude W) that is removed at the initial time. Initial temperature and salinity fields for the Alboran basin have been obtained from a horizontal average of the spring MODB data (available at modb.oce.ulg.ac.be/modb), while the spring Levitus [1982] data set has been used to set initial values over the Gulf of Cadiz (see Figure 5 in SBA02). As in SBA02 and Napolitano et al. [2003], we have used the Smolarkiewicz upstream-corrected advection scheme [Smolarkiewicz, 1984, 1990], in order to simulate correctly the free flow adjustment to the density gradient within the strait after the dam is removed. [17] The model is forced at the open boundaries through the specification of the surface tidal elevation. Candela et al. [1990] and more recently Tsimplis [2000] have found that 75% of the current variability in the strait is due to the semidiurnal tide, so we have limited our modeling study to the semidiurnal component, forcing the 3of23

4 Figure 2. (top) Model bathymetry, computational grid, and transects for the presentation of model results within the Strait of Gibraltar. The gray levels indicate the water depths. The points Cm and Sp mark the points where Spartel Sill and Camarinal Sill are located, respectively. (bottom) Bathymetry along the longitudinal section E. model with only the M 2 tide, with period of hours, and the S 2 tide, with period of hours: z T ðy; tþ ¼ X2 A n ðyþcosðs n t j n ðyþþ; ð3þ n¼1 where A n (y) and j n (y) are the prescribed surface elevation amplitude and phase of the nth tidal constituent and s n is its frequency. The M 2 and S 2 surface tidal elevation amplitudes and phases have been obtained from the global tidal model of Kantha [1995] and Kantha et al. [1995]. The resulting z T (y m w, t) applied at the middle point of the western open boundary (y m w ) during the neap tide ranges from 48 to +75 cm, while during the spring tide ranges from 128 to +140 cm. Owing to the strong velocities generated by the tidal forcing short external and internal time steps of 0.1 and 6 s need be used in the simulation. 3. Model Results [18] The model was run for 360 days without tidal forcing (z T ( y, t) = 0) in order to achieve a steady two-layer exchange system. The steady exchange obtained is characterized by an inflow (toward the Mediterranean) and an outflow (toward the Atlantic Ocean) of 0.62 and 0.51 Sv 4of23

5 Table 1. Comparison Between Observed and Predicted Amplitudes A and Phases P of M 2 Tidal Elevation a Observed M 2 Predicted M 2 Difference (Pre Obs) Location Latitude North Longitude West A, cm P, deg A, cm P, deg A, cm A, % P, deg Tsimplis et al. [1995] Gibraltar b García Lafuente [1986] c Pta. Gracia ± ± Tarifa ± ± Pta. Cires ± ± Pta. Carnero ± ± Candela et al. [1990] DN DS SN SS DW TA AL CE DP a Station locations are shown in Figure 1. b Calibration. c ± indicates standard errors. at the Camarinal Sill section, and of 0.69 and 0.58 Sv at the Gibraltar-Ceuta section (1 Sv = 10 6 m 3 /s 1 ). [19] Transports were computed integrating the alongstrait velocity vertically from the bottom up to the depth where the along-strait reverts its direction for the outflow, and from this depth up to the surface for the inflow, and then meridionally, across the Camarinal Sill and Gibraltar-Ceuta sections (sections C and D in Figure 2): INðÞ¼ x OUTðÞ¼ x Z North Z 0 South hx;y ð Þ Z North Z hx;y ð Þ South bottom ux; ð y; zþdzdy ux; ð y; zþdzdy; where u is the along-strait velocity, h is the depth of the interface, and x is the longitude. The computed transports are about 15% less than the estimation carried out in ð4þ SBA02. This difference mainly depends on the value of mean elevation (z M ) used in the open boundary condition equation (1) and on the better vertical resolution implemented in this present work. [20] In order to achieve a stable time-periodic solution, the model was run for further 29 days, forced only by the two principal semidiurnal tidal components. Finally, after reaching the stable time-periodic regime, the model was run for a further fortnightly period and the least squares harmonic analysis was applied to the surface elevation and currents Tidal Elevation [21] In Tables 1 and 2 the observed [Tsimplis et al., 1995; García Lafuente, 1986; Candela et al., 1990] and simulated amplitudes (A) and phases (P) are compared, for the M 2 and S 2 tidal elevation, respectively. A good agreement between observed and predicted values is found; the maximum Table 2. Comparison Between Observed and Predicted Amplitudes A and Phases P of S 2 Tidal Elevation a Observed S 2 Predicted S 2 Difference (Pre Obs) Location Latitude North Longitude West A, cm P, deg A, cm P, deg A, cm A, % P, deg Tsimplis et al. [1995] Gibraltar ! García Lafuente [1986] b Pta. Gracia ± ± Tarifa ± ± Pta. Cires ± ± Pta. Carnero ± ± Candela et al. [1990] DN DS SN SS DW TA AL CE DP a Station locations are shown in Figure 1. b ± indicates standard errors. 5of23

6 Figure 3. Cotidal charts of the (a) M 2 and (b) S 2 surface tides. Solid lines are phase contours in degrees; dashed lines are amplitude contours in cm. differences do not exceed 6.2 cm in amplitude (with a maximum error of about 15%) and 13 in phase. The maximum differences are confined to coastal points as Ceuta (CE), Algesiras (AL), Tarifa and Pta. Cires, since our model grid is not coastal fitted. [22] In Figure 3 are also shown the computed cotidal charts for the strait region, for the simulated M 2 and S 2 surface tidal waves. The M 2 chart is in good qualitative agreement with the empirical cotidal chart presented by CA90. The only difference is in the Camarinal Sill area, where the cotidal lines (lines of constant phase) undergo a deviation toward North. The principal features to be noted on this chart are the reduction (more than 50%) of the amplitude in the along-strait direction, the invariability of the amplitude in the cross-strait direction (except for the eastern part of Tarifa narrow), and the southwestward phase propagation, more evident east of Camarinal Sill as far as the eastern entrance of the strait. The same features are also present on the S 2 cotidal chart even if the cotidal lines exhibit a greater deviation toward North over the Camarinal Sill. In agreement with CA90, the ratios and phase differences between the M 2 and S 2 components remain quite constant throughout the strait; the amplitude ratio is confined between 2.6 and 2.8 and the phase difference decreases from west to east of only 2 degrees between 24 to Tidal Currents [23] A direct comparison between the predicted fields of major and minor axes of tidal ellipse and data are difficult because of the lack of data in most part of the strait, with the exception of Camarinal Sill (see CA90) and of the eastern 6of23

7 Figure 4. Comparison between observed and simulated semimajor axis components of tidal ellipses. Observed data M1, M2, M3, M7, M8, M9, and F3 are from Candela et al. [1990], and N, C, and S are from García Lafuente et al. [2000]. entrance [see García Lafuente et al., 2000]. Thus in order to quantitatively compare the model results with observed data, a linear regression between predicted and observed semimajor axis, in only ten different locations, was performed (Figure 4). The mean errors and the root mean square errors are shown in Table 3. The errors are limited to 4.0 cm s 1 and 7.5 cm s 1 for the S 2 and 5.9 cm s 1 and 7.9 cm s 1 for the M 2, except for the stations M3 and F3 where the mean error reaches the value of 24.7 cm s 1 and the root mean square reaches 31.9 cm s 1. These differences are mainly due to an overestimation of the simulated lower-layer currents. [24] Figures 5 and 6 show a complete semidiurnal tidal cycle simulated by the model during spring tide at the Gibraltar-Ceuta and Camarinal Sill sections, respectively. It is apparent from Figure 5 that the lower-layer flow, at the eastern section D, is periodically reversed by tidal currents toward the Mediterranean Sea (also during neap tide, not showed). The typical currents range from 60 to 30 cm s 1 during spring tide and from 40 to 30 cm s 1 during neap tide. On the contrary, the upper layer is always directed toward the Mediterranean Sea, indicating a clear weakness of the tidal amplitude in comparison with the mean upper layer 7of23

8 Table 3. Mean and Root-Mean-Square Error of the Simulated Semimajor Axis M 2 S 2 Station Mean Error RMS Error Mean Error RMS Error Candela et al. [1990] M M M M M M F García Lafuente [1986] N C S flow, that is too strong to be reversed. Here the upper layer, the currents range from 80 to 140 cm s 1 during spring tide and from 60 to 110 cm s 1 during neap tide. These results are in good agreement with BA01, who showed very similar results for the M 2 component, computed with an inverse model at the eastern entrance of the strait. [25] At Camarinal Sill, the tidal signal is so strong to always reverse the currents, both in the upper and lower layers, for a part of each semidiurnal tidal cycle, except for the neap tide where the Mediterranean layer is not reversed completely (for the spring tidal cycle see Figure 6). To discriminate between upper and lower-layer velocities we superimposed to the velocity contours the depth of the isohaline, that, as suggested by SBA02, can be considered as an interface between the two layers. Using this method, it is possible to see that velocity in the upper layer ranges from 130 to 200 cm s 1 during spring tide and from 100 to 130 cm s 1 during neap tide. For the lower layer, velocity ranges from 230 to 150 cm s 1 during spring tide and from 190 to 70 cm s 1 during neap tide. [26] Figures 7 10 show the simulated M 2 and S 2 tidal amplitude and phase of the along-strait velocity at Camarinal Sill and Gibraltar-Ceuta cross-strait sections. Looking at Figures 7a and 8a it is clear that there is a drastic decrease in the M 2 amplitude (more then 70%) going from Camarinal Sill to the eastern entrance of the strait. At Camarinal Sill the amplitude constantly increases from 100 cm s 1 at the surface up to 140 cm s 1 at a depth of about 220 m and then decreases in the vicinity of the bottom due to the influence of friction. On the other hand, in good agreement with BA01, at the eastern entrance of the strait the amplitude increases from 8 cm s 1 at the surface to 42 cm s 1 in the lower layer. The main increase is in the upper layer: amplitude reaches the value of 34 cm s 1 in the first 200 m, and remains rather constant in the rest of the water column. It is also evident a meridional variation of the amplitude from the southern part (40 cm s 1 ) to the northern part (18 cm s 1 ) of the strait. Another point to highlight is that the phase at Camarinal Sill (Figure 8b) is quite constant from the upper layer to the lower layer; there is only a difference of 20, i.e., a difference of 40 min between the appearing of the maximum velocity in the upper layer and the appearing of the maximum velocity in the lower layer. This difference goes up to 60 (2 h )atthe eastern entrance (Figure 7b), where the phase decreases from about 210 in the upper layer to 150 in the lower layer. [27] The S 2 tidal current amplitude also decreases of more than 70% from Camarinal Sill to the eastern entrance (Figures 10a and 9a, respectively). At the eastern entrance the amplitude increases with depth, from the surface to about 250 m, of only 2 cm s 1, remaining constant at 11 cm s 1 as far as the bottom on the southern side. S 2 tidal current phase (Figure 9b) decreases from 170 to 130 in the first 200 m and increasing up to 150 at about 350 m, remaining constant below 350 m to the bottom. At Camarinal Sill the S 2 tidal current amplitude increases from surface to 90 m of about 14 cm s 1, with an increment that is not uniform along the cross section (maximum values of about 42 cm s 1 are concentrated on the south and north sides), while below 150 m the amplitude decreases going toward the bottom. Phase (Figure 10b) is constant (150 ) from the surface to the bottom for nearly the whole section Internal Bore [28] One of the most important features of the dynamics in the strait is the presence of internal bores which are generated over Camarinal Sill and propagate both eastward and westward [Armi and Farmer, 1988; Farmer and Armi, 1988]. In Figure 11 we present six sequential snapshots of longitudinal salinity sections which cover the overall spring tidal period. Here one can see that, in good agreement with the twodimensional, two-layer, hydrostatic model of Izquierdo et al. [2001], the generation of the eastward propagating internal bore begins with the formation of an interfacial depression over the western edge of Camarinal Sill, approximately 1.5 hours before high tide at Tarifa, i.e., as soon as the westward barotropic forcing over Camarinal Sill starts weakening and the interface located upstream of Camarinal Sill is not sustained any more. Subsequently, about 30 min before high tide at Tarifa, the internal bore is released from Camarinal Sill and starts to travel eastward. The bore is released when the upper layer starts to move toward east while the lower layer continues to move westward. Its initial length scale, in the along-strait direction, is about 3 km and its travel times from Camarinal Sill to Tarifa, Pta. Cires and Gibraltar sections are 2, 4 and 6 hours, respectively. It follows that, always in agreement with the two dimensional model of Izquierdo et al. [2001], the speed of the bore is about 1.7 m s 1 between Camarinal Sill and Tarifa sections, 2.5 m s 1 between Tarifa and Pta. Cires sections, and 1.5 m s 1 between Pta. Cires and Gibraltar sections. [29] In agreement with Armi and Farmer [1988], a much weaker westward propagating internal bore is also released from Camarinal Sill, just 30 min before the eastward propagating bore reaches Gibraltar-Ceuta section, i.e., 40 min before the low tide at Tarifa. The amplitude of the eastward Figure 5. (a f ) Simulated sections of the along-strait current (cm s 1 ) showing several phases of a semidiurnal (M 2 + S 2 ) tidal cycle during spring tide at the Gibraltar-Ceuta section. The time difference between the single sections is 2 hours. (g) Time moments referred to the surface elevation at Tarifa. The contour interval is 10 cm s 1. Red and blue shadows highlight outflow and inflow currents, respectively. Yellow lines represents the depth of the 38.1 isohaline. 8of23

9 Figure 5 9of23

10 Figure 6. Same as Figure 5, but for the Camarinal Sill section. Yellow lines represent the depth of the isohaline. 10 of 23

11 Figure 7. M 2 tidal constituent of the along-strait velocity at the eastern section D. (a) Amplitude in cm s 1 ; the contour interval is 2.0 cm s 1. (b) Phase relative to the Moon transit at Greenwich in degrees; the contour interval is 10. propagating bore diminishes progressively from about 100 m on the western edge of Camarinal Sill to about 50 m at the Gibraltar section. Initially the bore is characterized by two large and steep internal waves that during the eastward propagation seem to be subject to an amplitude dispersion. What happen actually is that the bore, during its eastward propagation, disintegrates into a train of internal solitary waves [Artale and Levi, 1990; Artale et al., 1990; Brandt et al., 1996]. The model is not able to reproduce these internal solitary waves since nonhydrostatic effects are neglected and the horizontal model resolution is lower in the eastern part of the domain; however the final effect is the same, since the bore is in any case dispersed. The model shows also that the bores are always released from Camarinal Sill in the course of the fortnight period, even during neap tides. 4. Transport 4.1. Effect of Tidal Forcing on Transport and Mean Quantities [30] Recent estimates of transport based on direct measurements over Camarinal Sill have been carried out by TB00. They considered two methods of estimating the mean transport across the sill: in the first one they used the timeaveraged along-strait velocities, fixing the interface depth at 147 m, while in the second method they produced 30 min time series of transport by finding the depth of the interface for each measurement. In the first case the inflow transport was estimated to be 0.46 Sv, while in the second case the average over the time series gave an estimated transport of 0.78 Sv. [31] At the eastern entrance of the strait other direct measurements have been carried out by BA01. They report an inflow of 0.81 Sv, estimated by using an inverse model to predict every instant the interface displacement and the along-strait velocities, while using an interface at constant mean depth they estimated a transport higher than 7% respect to the nonstationary interface case. [32] As initially argued by Bryden at al. [1994] and more recently by TB00 the contribution of the fluctuating terms in velocity and interface depth represents the main difference between the two methods of computation. To better explore the effect of these fluctuating terms on the mean flow along the strait, we analyze numerical results of both experiment Figure 8. M 2 tidal constituent of the along-strait velocity at the Camarinal Sill section B. (a) Amplitude in cm s 1 ; the contour interval is 10.0 cm s 1. (b) Phase relative to the Moon transit at Greenwich in degrees; the contour interval is of 23

12 Figure 9. S 2 tidal constituent of the along-strait velocity at the eastern section D. (a) Amplitude in cm s 1 ; the contour interval is 2.0 cm s 1. (b) Phase relative to the Moon transit at Greenwich in degrees; the contour interval is 10. (with and without tidal forcing (hereinafter TE and NTE)), in a two-dimensional two-layer formulation; in particular we have integrated the model results in the cross-strait direction choosing, as in SBA02, the psu as interface isohaline between the two layers. In this case the momentum and continuity equations for the upper layer can be u 2 þ r 0 ð h h eþ ¼ 2 @t ðhuþ ¼ where u is the velocity, h is the thickness of the layer, h is the surface elevation, h e is the equilibrium potential, r 0 is the density of surface water, and r 1 is the mean density of the layer. Moreover we decompose each model variable (c)ina mean term, plus the two semidiurnal components (M 2 and ð5þ ð6þ S 2 ) and plus a residual component that also includes the internal bore: cðx; tþ ¼ c ðþþ~c x M2 x h ðþcos x w S2 t þ j c S 2 ðþcos w M2 t þ j c M 2 ðþ x þ ~cs2 i ðþ x þ ^c ðx; tþ; ð7þ where c represents the mean component, ~c M2, ~c S2 are the amplitudes of the semidiurnal components, w M2, w S2 and j c M2, j c S2 are the frequencies and phases of the semidiurnal components, respectively, and ^c represents the residual component, which includes the internal bore. Time-averaging the continuity equation (6), we obtain the following transport equation for the u h þ ~u M 2 ~ hm2 2 cos j u S 2 j h S 2 þ ^u^h cos j u j h þ ~u ~ S hs2 2 M2 M2 2! ¼ 0: ð8þ Figure 10. S 2 tidal constituent of the along-strait velocity at the Camarinal Sill section B. (a) Amplitude in cm s 1 ; the contour interval is 10.0 cm s 1. (b) Phase relative to the Moon transit at Greenwich in degrees; the contour interval is of 23

13 Figure 11. Evolution of salinity perturbations during a tidal period. Contours are shown with an interval of 0.5 psu. The snapshots are plotted at an interval of 2 hours. The time moments are referred to the surface elevation at Tarifa (insets). 13 of 23

14 Figure 12. Along-strait total upper layer transport in the case with (A) and without (B) tidal forcing, and single component of upper layer transport in the case of tidal forcing: mean component (C), M 2 and S 2 components (D, E), and residual component (F). In Figure 12 all terms of equation (8) are plotted: the mean transport (C), the transport due to the M 2 and S 2 components (D, E) and the residual transport (F). Also plotted are the total upper layer transport (A) and the transport computed for NTE (B). This figure reveals that the contribution of the semidiurnal tidal component M 2 ((~u M2 ~ u hm2 cos(j M2 j h M2 ))/2) is relevant over Camarinal Sill whereas it is negligible at the eastern end of the strait. In practice, while in the eastern region of the strait the mean current nearly determines the whole transport, it only contributes about 60% of the total transport near Camarinal Sill, in agreement with the results of TB00 and BA01. Contributions of the S 2 component and of the bore are less than 4%, but, whereas the S 2 component has its maximum effect near Camarinal Sill, the bore is more effective in the eastern region. Deviations from the conservation relation equation (8) are mainly due to the large entrainment of Mediterranean water near the hydraulic jump, just west of Camarinal Sill, whereas in the eastern part diffusion of salinity moves the interface toward 38.1 psu (see BA01, see also section 4.2 of this paper), implying an underestimation of transport in this zone when the upper layer is limited to psu. [33] In order to investigate the effect of the tidal forcing on the mean currents of the layers we have plotted in Figure 13 (for the upper layer only) the mean currents u (A), half of the M 2 current amplitude (1/2 ~u M2 ) (C), half of the S 2 current p ffiffiffiffi amplitude (1/2 ~u S2 ) (D), the mean quadratic residual ( ^u 2 ) (E), and the current for NTE (B). It is evident that the mean current (A) shows a local minimum in the region where the amplitude of the M 2 current (C) has its maximum value, whereas in the case without tidal forcing the current (B) shows a local maximum in the same region. Residual (E) and S 2 (D) terms appear negligible. [34] Always from Figure 13 one can note that in most part of the strait, and in particular in the eastern part, the mean current is higher for TE (A) respect to NTE (B). Most of tidal energy is dissipated toward smaller scales, but we suppose that part of this energy can be transferred also to the mean flow. This supposition is based on the fact that the only difference between the two experiments is the tidal forcing, and so the difference between the mean currents can only be caused by this forcing. It is plausible that everywhere within the strait, with exception at Camarinal Sill and surroundings, tidal fluxes interact with mean motion enhancing it. At Camarinal Sill, dissipation processes (bottom friction, mixing) and energy transfer to internal bore generation drop energy probably from both tidal and mean motion. [35] The effects of tide on the interface depth are shown in Figure 14. Plotted are the mean depth of interface (A), its range of variation due to the M 2 component (B min, B max ), its range of variation due to the residual term (C min,c max ), and the interface depth for NTE (D). The variation due to the residual term is mainly due to the bore and it is of the same order of that associated with the M 2 component. In the presence of tidal forcing the mean depth of the interface rises of about 20 m just west of Camarinal Sill up to about 40 m over the sill respect to the depth of interface of NTE. The minimum difference of about 13 m is limited at Tarifa Narrow. This interface rising is probably related to the increased mixing between upper and lower layer introduced by tidal forcing. However, in spite of this reduction of the upper layer thickness the transport increases for TE, indicating that the effect of a stronger 14 of 23

15 Figure 13. Along-strait current amplitudes for the mean component u (A), the Mp 2 component ffiffiffiffi (1/2 ~u M2 ) (C), the S 2 component (1/2 ~u S2 ) (D), the mean quadratic residual component ( ^u 2 ) (E), and for the experiment without tidal forcing (B). mean current, together with that of tidal transport prevail on the effect of depth reduction. [36] Mean surface elevation is another quantity that shows an unexpected change due to tidal forcing. It is well known that there is a gradient of elevation between Atlantic Ocean and Mediterranean Sea that compensates for the different densities of the two seas. In the presence of tide, there is a strong gradient of elevation just west of Camarinal Sill with an extra 1.9 cm of gradient between the two seas. The equation for the mean elevation is analogous to equation (8); in the presence of tide there are terms like 1/2 fua M2 ~h M2 cos(j UA M2 j h M2 ) (where UA represents the barotropic current), that act to modify the mean elevation with respect to the case without tide. This quantity has its maximum value near Camarinal Sill, in coincidence with the maximum value of the tidal amplitude of the barotropic current fua M2. [37] The mean salinity field within the strait is also modified by tidal forcing. This appears particularly clear in Figure 15, where are shown the along-strait (section E) difference between the salinity field obtained for NTE and a fortnightly average of the tidally forced salinity field. This figure shows strong differences in the mean profiles of salinity: the yellow spot (+0.6 psu) east of Camarinal Sill is an evidence of the increased entrainment of Mediterranean water in the upper layer due to the effect of tide; while the long blue patch ( 0.3 psu) is due to an increment of entrainment of the Atlantic water in the denser layer. The hydraulic jump is characterized by strong mixing also in the case without tidal forcing, and for this reason in the region, west of Camarinal Sill, the effect of tide is less evident Effect of Tidal Forcing on Entrainment [38] In order to estimate entrainment and detrainment fluxes, we have analyzed model results in a two-dimensional four layers framework. This was accomplished integrating model results in the across-strait direction and then choosing the following three separating isohalines: 36.8, 37.5 and 38.2 psu. Figure 16 shows the thickness of the four layers and the depth of the three interface isohalines for NTE and TE; here the layers are numbered, starting from the upper one, from 1 to 4 (hereinafter L1, L2, L3 and L4), while arrows represent volume fluxes between adjacent layers, i.e., entrainment and detrainment. [39] Including to volume and salt conservation equations terms representing intrusion of volume flux from one layer into an adjacent one, it is possible to calculate entrainment and detrainment fluxes: B ð kh k Þ þr H ðb k h k u k Þ ¼ Fup kþ1 Fdw k Fup k þ Fdw k ð9þ ð B kh k S k þr H ðb k h k u k S k Þ ¼ S kþ1 Fup kþ1 S k Fdw k S k Fup k þ S k 1 Fdw k 1 ; ð10þ where k indicates the number of the layer, h k and B k are the thickness and the width of the kth layer, Dx is the longitudinal distance between two adjacent grid point (600 m), and Fup k and Fdw k represent upward and downward volume flux of the kth layer through the B k Dx surface, respectively. [40] The resulting time averaged (on a fortnight period) upward and downward fluxes, for NTE and TE, are shown in Figure 17. For NTE entrainment increases just 15 of 23

16 Figure 14. Along-strait mean depth of interface (37.25 psu) (A), its range of variation due to the M 2 (B min,b max ), its range of variation due to the residual term (C min,c max ), and interface depth for the case without tidal forcing (D). west of Camarinal Sill (Figure 17a), i.e., in the same location of the stationary hydraulic jump. Here water mass is exchanged prevalently between L3 and L4: 0.06 Sv of L4 water intrudes into L3, while 0.04 Sv of water flows from L2 to L3. Weaker entrainment is also evident at the western entrance of Tarifa Narrow. Here, water mass is exchanged prevalently between L1 and L2: Sv of L1 water intrudes into L2, while 0.01 Sv of Figure 15. Along-strait (section E) salinity difference between the field obtained in the case without tidal forcing and the field obtained by averaging the tidally forced salinity field on 15 days. 16 of 23

17 Figure of 23

18 Figure 17. Along-strait time-averaged (on a fortnight period) entrained and detrained volume fluxes between layers (the same as Figure 16) for the case (a) without and (b) with tidal forcing. Positive values (solid lines) indicate upward volume flux, while negative value (dashed lines) represent downward volume fluxes. Positive red, blue, and green lines represent entrainment from L2 to L1, from L3 to L2, and from L4 to L3, respectively. Negative red, blue, and green lines represent entrainment from L1 to L2, from L2 to L3, and from L3 to L4, respectively. water flows from L2 to L Sv of water are also exchanged from L2 to L3. [41] TE shows an increased entrainment along whole the strait respect to NTE (Figure 17b). Strong exchange between the first and second layer as well as the second and third layer are located at Camarinal Sill and within whole Tarifa Narrow, while exchange between the fourth and third layer seems to show a behavior similar to NTE. West of Camarinal Sill the most active layers are the second, third and fourth: Sv of L4 water are entrained into L3 and 0.08 Sv of L3 water into L2, while 0.07 Sv of L2 water are detrained downward L3 (see peaks number 1 in Figure 17b). East of Camarinal Sill the most active layers are the first and second: 0.12 Sv are exchanged from L1 to L2 and a slightly more is exchanged from L2 to L1 (peaks n. 2), while 0.08 are exchanged from L3 to L2. Entrainment between layers decreases from Camarinal Sill as far as Tarifa, from here to the eastern entrance of the strait entrainment increments again showing two relative maximum (peaks n. 4 and n. 5). Within Tarifa Narrow the most active layers are the first, second and third: in the western maximum, 0.08 Sv of water are exchanged between L1 and L2 in both direction, 0.05 Sv of water flows from L3 to L2, Sv are exchanged from L2 to L3, while only 0.01 Sv flows from L4 to L3. The second maximum shows a behavior similar to the first one, except for the amplitudes that are reduced of about 25% for layers 1, 2 and 3. [42] Results shown for TE are representative of a complete fortnight period, however for a complete understanding of the tidal entrainment along the strait it is necessary to investigate also the single ebb (toward Gulf of Cadiz) and flood (toward Alboran Sea) tidal periods both for spring and neap tide. [43] During ebb tide (not showed) only peaks n. 1 are present, that is entrainment is located only west of Camarinal Sill where 0.18 Sv of water is exchanged from L4 to L3 and from L3 to L2 for spring tide and 0.12 Sv for neap tide, while 0.15 Sv are exchanged between L3 and L4 for spring tide and 0.09 for neap tide. [44] During flood tide (not showed) peaks n. 1 are not present and entrainment is located only at east of Camarinal Sill and within Tarifa Narrow. East of Camarinal Sill, during spring tide, the water exchanges principally from L1 to L2 and from L2 to L1 at a rate of 0.4 Sv, and from L3 to L2 and from L2 to L3 at a rate of 0.21 Sv and 0.11 Sv, respectively. During neap tide the active layers are always L1, L2 and L3 with weaker values for peaks n. 3, 4, and 5 while peaks n. 2 are totally absent. The only possible cause of a such behavior is that the dynamical mechanisms generating peaks n. 2 are activated only when intensity of tidal currents exceed a threshold value. Observing that at Camarinal Sill only during neap tide the entire water column it is not reversed completely, we can assume that peaks n. 2 appear only when both the upper Atlantic layer and the lower Mediterranean layer flow simultaneously in the same direction Transport Estimates [45] The simple and intuitive method of computation of inflow and outflow volume transport introduced in section 3 is strictly related to the existence of an internal surface of zero along-strait velocity, used as an interface between Atlantic and Mediterranean water. However, this method cannot be used to determine the volume transport when tidal forcing is included, since, as described in section 3.2, the semidiurnal tidal signal is so strong to reverse the inflow or the outflow during part of each tidal cycle, obscuring the two-layer character of the mean flow. Another way of Figure 16. Along-strait time-averaged (on a fortnight period) thickness of four layers (L1, L2, L3, and L4) separated by three isohalines (36.8, 37.5, and 38.2 psu) for the experiment (a) without and (b) with tidal forcing. Arrows represent volume fluxes (Fup and Fdw) between adjacent layers, i.e., entrainment and detrainment. 18 of 23

19 Figure 18. layer. Internal surface salinity interface between the upper Atlantic layer and lower Mediterranean defining the interface between upper and lower layer is by using an isohaline. For example, Bryden et al. [1994] and Candela et al. [1989] used the 37.0 and 37.5 isohalines, respectively, to define the exchange interface over Camarinal Sill, while BA01 used the 38.1 isohaline at the eastern entrance of the strait. The choice of different values for the separating isohaline has to be ascribed, as argued in the previous section, to the strong entrainment developing along the strait: in particular, along Tarifa Narrow the inflowing Atlantic water entrains denser water and west of Camarinal Sill the outflowing Mediterranean water entrains part of the inflowing Atlantic water [Bray et al., 1995; SBA02]. [46] Thus it emerges that it is incorrect to use a single isohaline as an interface for the whole strait. For this reason, an alternative definition is used in this paper. We define as interface the fortnightly averaged internal salinity surface associated with the internal surface where fortnightly averaged along-strait velocity zero occurs. The internal salinity surface obtained is shown in Figure 18; here it is possible to note that the salinity contrast between upper- and lower-layer changes from psu at Camarinal Sill up to 38.1 at the east entrance of the strait. This salinity surface is then used to find the time-dependent depth of the internal surface interface between the two layers. Now we are able to calculate the instantaneous upper (ULT) and lower-layer transport (LLT) in the whole strait by using the following equations: Z north ULTðx; tþ ¼ LLTðx; tþ ¼ Z 0 south hx;y;t ð Þ Z north Z hx;y;t ð Þ south bottom ux; ð y; z; tþdzdy ux; ð y; z; tþdzdy; ð11þ where u is the along-strait velocity and h is the timedependent depth of the interface. In Figure 19, the computed upper and lower-layer transports are shown for a complete fortnight cycle at three different cross-strait section over Camarinal Sill (section B) (Figure 19a), at Tarifa (section C) (Figure 19b), and at the east entrance of the strait (section D) (Figure 19c). In agreement with CA90 the largest amplitude of the instantaneous transport occurs in the upper layer at the sill, and in the lower layer at the eastern section. The behavior of tidal currents noted in 3.2 is apparent in the transports: it is clear that the upper currents have decreasing amplitudes going eastward and reverse their directions only as far as Tarifa, while the lower currents increase eastward and reverse their direction everywhere in the strait. [47] Red lines in Figure 20 show the mean along-strait transports, obtained averaging over the fortnight period the ULT and LLT. From west to east the upper layer transport ranges from 0.68 Sv to 0.9 Sv, while lower-layer transport ranges between 0.5 Sv to 0.75 Sv. At Camarinal Sill the transports are 0.85 Sv and 0.70 Sv for the upper and lower layer, respectively, while at the east entrance they are 0.9 Sv and 0.75 Sv. [48] At Camarinal Sill the most accurate estimates of transports from direct measurements are the ones given by Bryden et al. [1994] and, more recently, by TB00. In their computation they considered the vertical movement of the interface and determined the transport of the upper layer to be 0.72 ± 0.16 Sv and 0.78 Sv, respectively, and the transport of the lower layer to be 0.68 ± 0.15 Sv and 0.67 Sv, respectively. At the eastern entrance of the strait the last most accurate estimates of transports are from BA01. They calculated the transports using an inverse model to predict for every instant the depth of the isohaline 38.1 obtaining an upper layer transport of 0.81 ± 0.07 Sv and a lower-layer transport of 0.76 ± 0.07 Sv. The results of the present study are in reasonable agreement with all these transport estimates since they lie within the error bars. [49] Also plotted in Figure 20 (blue lines) are the transports computed for the experiment without tidal forcing (equation (4)). Comparison with the results with tidal forcing 19 of 23