The initial condition for the baroclinic flow is the lock-exchange configuration, to spontaneously reproduces the equal exchange flow with the correct hydraulic controls. The duration of the experiment is then limited by the volumes of water that can be exchanged. This method does not allow to take into account the net barotropic transport of about 0.05 Sv towards the Mediterranean to compensate the excess of evaporation . Since it represents 1.5 %–2 % of the tidal barotropic oscillations, results will not be affected by this approximation.

The tank was divided in two equal volume compartments (with 5.5 m³ effectively available for the exchange flow) by a gate positioned within the strait (cf. Fig. , red strait line). The tank was filled simultaneously in both compartments up to a total depth of 65 cm (9.6 cm above CS) and put in anticlockwise rotation with a rotation period of Tc=124.8s, as from the scaling given in Table . One reservoir was filled with saline water with density ρM, representing the Mediterranean Sea, the second with a lighter fluid of density ρ0 corresponding to the Atlantic Ocean, both at the same temperature of ≈18° so that in the experiment the density is a function of salinity only. The density difference was kept constant with Δρ0=19(±1) kg m⁻³ in accord with the scaling given in Table . Considering an emptying speed Ug and the half-section at CS, the experimental duration was estimated to be about 30 min, corresponding roughly to t/Tc∼15 rotational days.

At the start of the experiment following the gate removal, the flow through the strait undergoes through three regimes (): an initial unsteady phase in which the flow in each layer increases rapidly, a stationary regime called of maximum exchange and, finally, a sub-maximal exchange in which the flow decreases because no sufficient density difference is still present to maintain the initial hydrostatic pressure gradient. The useful phase for the experiment is the maximal exchange regime and this phase must be maintained long enough for the outflowing Mediterranean water to reach the end of the topography along the Portugal coast in geostrophic adjustment. During the maximum exchange phase, the strait is bounded by two hydraulic controls, preventing the changing boundary conditions of the two basins from influencing the flow at CS and keeping it stationary . Fig. a shows the time evolution of the flow speed (blue continuous line) in the salty layer at CS at 1.6 cm (40 m in the ocean) from the bottom, where a stationary, maximal exchange flow regime is rapidly reached, with an average velocity of 6.9 cm s⁻¹. The speed resulting from applying G2=2Ug2/(g′Hs)=1 corresponds to 6.8 cm s⁻¹. The figure also shows that the baroclinic experiment lasts at least 25 min (21.7 d for the ocean) during the stationary maximal exchange. The grey continuous line in Fig. a gives the salinity evolution from a CTD placed at the end of the topography (-7.830°,36.840°) 1 cm (25 m in the ocean) from the bottom along the Portugal continental shelf (see Fig. ). It shows that the first saline water is detected after 7 min after gate removal in the Strait (≈6 d in the ocean). This gives a stationary condition everywhere for 7<T(days)<22.

The experimental parameter that has been varied in the experiments is the strength of the barotropic forcing: without tides (purely baroclinic flow) and stationary spring and neap tides. To reproduce tidal forcing, two different plungers A1 and A2 were both placed in the Atlantic basin (cf. Fig. ), moving vertically and each displacing up to 80 L of water, at a frequency fM2=35.77s (12 h 25 min in the real ocean case), which corresponds to the frequency of the main tidal component in this region, i.e. the semi-diurnal tide M2 (). The neap tide current has been reproduced using a single plunger with a given amplitude (A), whereas the spring tide was reproduced with a second plunger moving with the same amplitude A and frequency fM2 as the first plunger. Further experiments have been performed also adding the modulation of the M2 component with the S2 component (corresponding to a 12 h period), in which case the plungers worked with a different amplitude and a different frequency (fM2 and fS2=34.56s corresponding to 12 h in the real ocean). These measurements have been only used in Sect.  in the present paper.

The tide in the Strait of Gibraltar creates a barotropic current of the order of magnitude of 1 m s⁻¹ at CS during neap tide and up to twice during spring tide (). The barotropic flow for both neap tide and spring tide conditions in the laboratory model are reported in Fig. b. The ADV (Acoustic Doppler Velocimetry, see Sect. ) placed at CS at z=-7.2cm from the free surface, measured an amplitude of 10 cm s⁻¹ for the purely barotropic velocity

The purely barotropic velocity was measured in ad-hoc experiments in which the full rotating tank was filled with homogeneous fresh water with ρ0 and only the barotropic forcing was applied.

Figure  presents characteristics of the mean velocity and density fields around CS averaged over seven tidal cycles during the stationary maximal exchange regime for the middle transect (cf. Fig. b) under three different tidal forcing scenarios: without tide (left column), neap tide (middle column), and spring tide (right column). The same for the northern and southern transects is presented in Figs.  and , respectively, given in the Appendix . The arrows in the top panels represent the in-plane mean velocities. Note that the quiver plot is displayed with unequal vertical and horizontal scaling. Arrow angles are defined in data coordinates, so a velocity tangent to the seabed appears as an arrow tangent to the bottom. Because the figures are vertically stretched for readability, preserving geometric angles would misrepresent the actual flow direction. Arrow length is hence determined solely by the in-plane speed as commonly done in the literature. The colormap highlights the mean dimensionless ratio of density differences ρ*≡Δρ/Δρ0=(ρ‾-ρ0)/(ρM-ρ0), with ρ‾ being the averaged density over the seven considered tidal cycles. Comparison of the top panels indicates that increasing tidal forcing leads to a thickening of the mixed layer on both sides of the sill. This effect is partly due to the time-averaging procedure, as the interface oscillates more strongly with increasing tidal amplitude, and partly due to enhanced mixing associated with stronger tides, as will be shown in Sect. , where tidal averages corresponding to maximum out- and inflow are presented.

The mean density fields ρ* shown in the top panels of Fig.  reveal that the time-averaged density is overall more diluted, by approximately (0.15–0.2)ρ* east of the CS, when tides are present, particularly during spring tide, compared with the purely baroclinic case (left panel). Direct comparison of the mean dimensionless ratio of density differences ρ* at different position in the along-strait direction for the three tidal forcings is given in the Appendix . This will be also further discussed in Sect. .

Also, it appears that Mediterranean waters are more diluted (of about 0.3ρ*) west of CS in neap tide compared to spring tide conditions, an aspect which will be further discussed in Sect. . These observations are also applicable in the case of the northern (Fig. ) and southern (Fig. ) transects given in the Appendix.

In the absence of tides, the surface of maximum vertical gradient of the along-strait velocity hu closely follows the pycnocline hp on the eastern side of the sill, defined as the iso-density lines with ρ*=1/2, and illustrated by the grey line presented in the middle rows panels of Fig. . The alignment between hu and hp is disrupted when tidal forcing is introduced, suggesting that tides generate entrainment of Mediterranean water to the west of CS. This aspect is further discussed in Sect.  as well. Along-strait velocities u averaged over the tidal cycles weaken west of CS when the tide is applied, compared to the purely baroclinic case (see second row panels of Fig. ).

The abrupt downward plunge of the pycnocline and the increased thickness of the mixed layer west of CS suggests the presence of an internal hydraulic jump for all tidal conditions. This is supported by the strong vertical velocity w‾ change in the third row panels of Fig.  observed at the same location. When the tidal amplitude increases in spring tide conditions, the positive/negative maxima of the vertical velocity descend. This can be understood considering that the differences between inflow and outflow are more pronounced during spring tide as compared to neap tide: during outflow, the Mediterranean waters are advected further downstream and the internal hydraulic jump oscillates west of CS with the tidal phase and disappears even during inflow, such that the averaged velocity fields are homogenized over the entire considered field on the west of CS. During neap tide instead, the internal hydraulic jump is stably localized close to the western flank of the sill and persists longer during the tidal cycle, even during part of the inflow. This will be further discussed in the following Sect.  below.

The local composite Froude number G2 provides insights into the local criticality of the flow. To compute the composite Froude number, a two-dimensional two-layer model has been defined, with the interface defined as the pycnocline ρ*=1/2, each layer of mean thickness h‾i characterized by a constant along-strait mean velocity u‾i and a local value for g′ (see Appendix ), i.e. 9G2=u‾12g′h‾1+u‾22g′h‾2. The validity of these assumptions is still largely debated in the community, as several studies emphasize that the mixed layer plays an important role in controlling exchange-flow dynamics (e.g. ). In the Appendix , we present and discuss different ways of computing the composite Froude number including three-layers models. Since results do not present particularly relevant differences, we will use the classical two-layer model with the pycnocline ρ*=1/2 defined as interface and using a local g′ instead of the initial constant value of g0′, which are more consistent with the hypothesis under which the composite Froude number is derived.

Mathematically, a condition of G2⩾1 is necessary for the development of a stationary shock such as hydraulic jumps . However, the condition G2⩾1 alone does not guarantee hydraulic control at a cross-strait section . The analysis presented here remains local and restricted to each transect and focuses on the potential for the development of localized shocks.

The composite Froude number G2, shown in Fig.  below the second row panels as a purple line, exhibits averaged values below unity everywhere and approaches unity at CS in all transects. Note that some uncertainty is present in the determination of G since velocity values close to the bottom boundary and the free surface are subject to more uncertainty and in some transects or tide conditions the upper layer velocities are sometimes missing close to the free surface. Hence, we expect our computed G2 being rather underestimated. These effects combined with the three-dimensional nature of the flow, which calls for a global criticality of the flow to be defined , can explain the failure of the computed composite Froude number to capture regions in which G2⩾1 in the average. Note also that the presented values are a total temporal average: when computing the internal Froude number during a tidal cycle, values of G2⩾1 are captured, as it will be shown further below in Sect. . The same conclusions are also valid for the northern and southern transects given in Figs.  and , respectively, that are given in the Appendix.

The bottom panels of Fig.  (and Figs.  and for the northern and southern transects given in the Appendix) report the turbulent kinetic energy (TKE=(u′2‾+w′2‾)/2), which interestingly appears to be highest close to the bottom boundaries. The intensity of the TKE also increases with increasing tidal forcing and appears to be linked to the bottom boundary-layer detachment on the west side of CS and the formation of a recirculation area. In comparison, the TKE between the two layers at the interface of Mediterranean and Atlantic waters is roughly half as strong. Moreover, since the detachment appears only when the tidal forcing is present, as shown from the second row panels for the velocity u. In the purely baroclinic case, high TKE values at the interface are observed at the onset of the descent. Further downstream, the interface becomes indistinguishable due to the development of the hydraulic jump, where TKE values increase. It should be noted that, due to the larger aspect ratio used in the experiments, the TKE is not strictly in similarity to that in the real ocean. Since the characteristic velocity scales are U in the horizontal and (H/L)U in the vertical, the relative contribution of vertical velocity fluctuations (w′) to the TKE is enhanced in the experimental configuration, potentially leading to elevated turbulence levels. Nevertheless, because the vertical contribution remains smaller in magnitude than the horizontal one, the overall order of magnitude of the TKE is preserved.

An overview of the averaged flow over four tidal cycles in the Strait at different depths (from -142.5 to -365 m) is shown in Fig. , for the considered three tidal conditions (without tide, neap tide, and spring tide from left to right). Horizontal velocities are displayed by the arrows, whereas the color fields indicate the relative vertical vorticity component ω, normalized by the Coriolis parameter f.

Overall, the flow is channeled toward the Tarifa narrow increasing the velocity while approaching CS, with similar velocity amplitudes in all depths except the deepest one at z=-365 m, for all three tidal conditions. Velocities are overall smaller in the purely baroclinic case. From the colored vorticity plots, it appears again that the highest values are reported along the coastline where strong changes of bathymetry (depth) appear, highlighted by the grey contour lines in Fig. , and in the region where the hydraulic jump appears immediately downstream of CS. When approaching the bottom, the vertical vorticity is increasing overall within the displayed field especially for the spring tide conditions (left column) because of the increasing shear due to the the bottom topography and flow detachment.

The vertically averaged two-layer flow characteristics are summarized in Fig.  for the three tidal configurations and for the three transects: the left column corresponds to the southern transect, the central column to the middle transect, and the right column to the northern transect (cf. Fig. b).

The first row of the figure presents the pycnocline depth hp defined as ρ*=1/2. The pycnocline depth hp shows minimal variation with changes in tidal forcing. In the northern plane, it remains essentially unchanged regardless of the tidal conditions. In the central plane, the tides cause a slight elevation of the pycnocline on the eastern side of CS, likely due to the thickening of the mixed layer, as predicted by hydraulic theory. This depth increase has been also reported in the numerical simulations of and in observations . On the western side, the downward plunge of the pycnocline is less pronounced for spring tide conditions, likely as a result of the detachment of the Mediterranean waters from the bottom boundary during outflow. The surface of maximum vertical gradient of the along-strait velocity hu exhibits limited sensitivity to tidal forcing as well, except near the sill where it becomes slightly shallower for spring tide, again likely linked to the boundary-layer detachment occurring during outflow. The third row panels of Fig.  shows the vertically averaged horizontal velocities in both the Atlantic (dashed lines) and in the Mediterranean (dotted lines) layers. Very little difference in these averaged values is reported for the different tidal conditions, except of lower values for the purely baroclinic case, as already reported in the literature and observed in the previous Fig. . In the Atlantic layer, velocities are increasing when moving toward the East due to the decreasing pycnocline depth and are nearby constant in all three sections for x>≈30 cm (≈7 km west of CS). In the Mediterranean layer velocities are increasing when moving to the West, reaching the highest values at CS, and then decreasing to a nearby constant value after the flow re-adjustemnt with the internal hydraulic jump for x>≈30 cm (≈7 km west of CS).

The local composite Froude number G2 computed from Eq. () and shown in the following row panels, remains subcritical throughout the sections as already mentioned above, but exhibits a noticeable increase approaching the sill location. The maximum value of G2 is up to 50 % higher during neap tide compared to the spring tide condition with exception of the northern transect where values remain similar for all tidal conditions. A local increase in G2 is also reported in correspondence of other topographical features at x=-30 and x=-45cm (corresponding to -7.5 and -11.25 km in the real ocean) West of CS, where other hydraulic controls have been suggested by previous authors . The criticality of the flow varying with the tidal phase is further discussed in the following section.

Figure  displays the in-plane horizontal velocities averaged over the four measured tidal cycles at the given phase with superposed relative vorticity field (colormap). We see that during inflow, the flow is arrested and slightly reversed within the Strait at this depth (z=-190 m) and is clearly reversed during spring tide. The mean flow is directed following the continental shelf and influenced by the Coriolis force, resulting in canalizing the flow toward the North and the Spanish coast. We also see that very high values of relative vorticity (5 to 10 times the Coriolis frequency f) are concentrated along the coasts in proximity of the continental shelf characterized by strong depth gradients. Generally, during outflow and high water slack (the top row panels) the northern coastline is characterized by negative relative vorticity values, whereas the southern coastline is characterized by positive relative vorticity. During inflow and low water slack (latter two row panels) positive relative vorticity dominates overall. Also, during low water slack, bands of positive and negative vorticity are present just east of CS, possibly suggesting the presence of internal waves.

Similar to the panels in Fig.  presented in the previous section, Fig.  shows the results from simultaneous velocity and density measurements, averaged over the seven measured tidal cycles at maximum inflow (first and second rows) and maximum outflow (third and fourth rows) for both neap tide and spring tide conditions. Here, we focus on the middle transect only.

The density fields displayed in the left panels show that during both maximum inflow and outflow, dilution is enhanced under neap tide conditions on both sides of the sill, consistent with the total averaged fields in Fig. , except during the spring tide maximum outflow, where the strong barotropic tide produces slightly greater dilution ≈0.1ρ* east of the sill.

A closer inspection, presented in Fig. , demonstrates that tidal forcing is capable of advecting the full water column down to 500 m of mixed waters east of CS during spring tide, with oscillations of the density ratio as a function of the tidal phase of the order of (0.15-0.2)ρ*. Since in the experiment we do not have different density components in the Mediterranean and in the Atlantic waters, in the experiment the whole Mediterranean layer is well mixed from the bottom to the interface with clearer differences between inflow and outflow. A direct comparison between our measurements at the MO5 mooring and the in situ observations from the PROTEVS GIB20 field experiment confirms the presence of the same tide-phase-dependent oscillations and the full-depth density ratio oscillation east of the sill. A pronounced thickening of the pycnocline is also evident when tidal forcing is applied, compared with the purely baroclinic case in Fig. .

During maximum neap tide inflow, examination of both the density (left panels) and velocity fields (further right panels) shows that the Mediterranean flow can still overflow the sill, forming a thin layer that is spilling down the western flank of the sill. In contrast, under maximum spring tide inflow, the Mediterranean Outflow is blocked. The internal hydraulic jump disappears for both neap tide and spring tide maximum inflow conditions, as indicated by the purple line representing the composite Froude number G2 below the along-strait velocity field. Note that now G2 is computed taking into account the contribution from the mean and the tidal flow, i.e.: 10G2=(u‾1+ũ1)2g′(h‾1+h̃1)+(u‾2+ũ2)2g′(h‾2+h̃2).

The hydraulic control that forms above CS during the outflow phase is distinct from the control that develops during inflow to the east of the sill, in the direction permitted for internal wave propagation. The control at CS prevents the eastward propagation of internal waves, whereas the control located to the east blocks westward propagation. When the control at CS is subsequently relaxed, the internal hydraulic jump propagates eastward, potentially contributing to the generation of internal solitary waves. This behavior is more clearly illustrated in the slack-water dynamics presented below and in the Hovmöller diagram of G2 shown in Fig. . A detailed discussion of this process is provided in a separate study focusing on internal solitary wave generation .

TKE values presented in the rightmost panels of Fig.  are markedly reduced during maximum inflow for both neap tide and spring tide conditions compared with those in the total averages of Fig. . However, under neap tide conditions, since Mediterranean waters still pass over the sill, the strong shear in the thin Mediterranean vein close to the bottom boundary continue to generate localized moderate TKE.

During maximum outflow, very high TKE values are observed, concentrated in the bottom boundary layers west of the sill, with approximately half those values occurring along the sheared layer between the Mediterranean and Atlantic waters. The second-column panels suggest an offset between the pycnocline and the region of maximum velocity shear west of CS, particularly during spring tide. This will be further discussed below with Fig.  and in the Appendix .

The hydraulic jump west of CS is characterized by an abrupt reversal in vertical velocity, transitioning from strongly negative to strongly positive values over a short distance. Consequently, the vertical velocity field provides a clear indicator of the location and intensity of the internal hydraulic jump. The internal hydraulic jump is evident at the sill for both neap tide and spring tide maximum outflow conditions, along with two additional control points farther downstream (to the west), more pronounced during spring tide. Our observations show that during neap tides, the vertical velocity attains higher maximum and minimum values within a confined region west of the sill. In contrast, during spring tides, the vertical velocity extremes are lower in magnitude but spread over a broader area. This behavior indicates that the hydraulic jump is displaced farther west during spring tides, whereas it remains confined to a more restricted region closer to the western flank of CS during neap tides.

A particularly noteworthy feature, previously visible in the total averaged along-strait velocity fields of Fig.  but now more clearly defined, is the region of weak or even positive velocities on the western flank of the sill, especially during spring tide outflow conditions. This pattern indicates that the Mediterranean vein detaches from the bottom boundary, which may be due to an adverse pressure gradient generated by the barotropic flow during outflow in the divergent region west of the CS. This divergent pressure field can become sufficiently strong to overcome the opposing baroclinic pressure gradient of the gravity current. This mechanism, which accounts for the elevated TKE values observed in the bottom boundary layers during outflow (see right column panels), is further examined in the following Sect. .

Figure  presents the dynamics at high-water slack (top panels) and low-water slack (bottom panels) for both neap tide and spring tide conditions. Overall, it can be seen that even at these tidal phases, neap tide conditions produce stronger dilution of Mediterranean waters west of the CS compared with spring tide, with a difference of roughly 0.2ρ*. East of the sill, the high-water slack exhibits about 0.1ρ* greater dilution overall than the low-water slack, with mean dilution values of approximately 0.8ρ* and 0.9ρ*, respectively.

During high-water slack, a thin layer of Mediterranean water continues to spill down the western flank of the sill (see along-strait velocity fields), while part of this flow forms a recirculation cell particularly evident under neap tide conditions in the first panel of Fig.  and in the along-strait velocity fields of the second-column panels, where a circular region of positive velocity appears. Similar behavior has been observed in situ in Figs. 9b and 12 of , who reported large overturns along the slope, suggesting a reversal of the Mediterranean outflow acting as a strong gravity current, in addition to the shear instabilities in the interfacial mixed layer previously described.

An internal hydraulic control, indicated by the composite Froude number (G2>1), is present at the sill for both neap tide and spring tide conditions during high-water slack, with an internal hydraulic jump farther west, as shown by the strong positive and negative vertical velocities in the third-column panels. The thin layer flowing along the western boundary of the sill is responsible for the high TKE values observed there (see first panels of the right column).

During high-water slack, a decorrelation between the pycnocline and the region of maximum velocity shear is visible west of CS and up to x=20 cm (5 km in the real ocean) in the second-column panels of Fig. , for both spring tide and neap tide conditions, even if less marked in this latter case (see also Fig. ). The same holds true during low-water slack.

As already reported previously for the maximum inflow and outflow dynamics, tidal forcing induces a strong decorrelation between velocity shear and vertical stratification. Figure  shows vertical profiles of shear (S, continuous lines) and Brunt–Väisälä frequency (N, dashed lines) at the CS summit (see Appendix  for spatial correlation maps of S and N), corresponding to the position of the MO2 mooring, during four tidal phases: maximum outflow (first column), low-water slack (second column), maximum inflow (third column), and high-water slack (fourth column), under both neap tide (green lines) and spring tide (black lines) forcing conditions. The shear, S, is computed by accounting for all measured components of the velocity gradient, i.e., 11S=[∂x(u¯+ũ)]2+[∂z(u¯+ũ)]2+[∂x(w¯+w̃)]2+[∂z(w¯+w̃)]2. As predicted by scaling arguments, the shear is primarily governed, to first order, by the vertical gradient of the along-strait velocity, ∂z(u¯+ũ). Since the tidally averaged density profiles are statically stable, the Brunt–Väisälä frequency, N, is computed directly from the absolute value of the vertical density gradient as: 12N=-gρ0|∂z(ρ¯+ρ̃)|. Under neap tide conditions, during most tidal phases, the depth of maximum vertical density gradient is nearby co-located with the depth of maximum velocity shear. In spring tide conditions, instead, there is a permanent shift between the maximum velocity shear and the pycnocline, as highlighted by the difference between the black continuous and dashed lines in Fig. . During outflow, strong entrainment of Atlantic water shifts the shear profile upward, positioning the maximum velocity shear within the Atlantic layer. As the tide transitions from outflow to inflow, the direction of entrainment reverses, and the Mediterranean layer is advected eastward. Consequently, the region of maximum velocity shear is displaced downward, below the pycnocline, within the Mediterranean layer.

The frequent decorrelation between the region of maximum vertical density gradient N and the shear maximum S, implies that shear alone does not effectively mix the two water masses, since it is located within a layer of almost homogeneous water with a constant density. This explains why even with a stronger tidal amplitude, spring tide conditions are less effective in diluting water westward of CS compared to neap tide conditions.

As during outflow, a clear detachment of the Mediterranean vein from the bottom boundary, driven by an adverse pressure gradient in the divergent region on the western flank of the sill, is evident during low-water slack for both neap tide and spring tide conditions. This detachment again produces high TKE values in the bottom boundary layer. High values of TKE are also present in the interfacial region between Atlantic and Mediterranean waters west of the sill (x<0). Under neap tide low-water slack, a hydraulic control remains at the sill, accompanied by a weaker internal hydraulic jump farther west. In contrast, under spring tide conditions, this control propagates eastward as a bore, with the G2 line reaching criticality at approximately x≈25 cm, corresponding to a sign change in the vertical velocity at the same position. This eastward-propagating bore is not observed under neap tide conditions, consistent with the findings of , , and .

To better understand the interesting observation that the Mediterranean vein detaches from the bottom boundary, we investigate closer the conditions for the boundary-layer detachment and consider the variations with the tidal phase.

Figure  presents contours of zero-horizontal velocity u‾+ũ colored as a function of the tidal phase, highlighting the presence of a recirculation area along the western bottom slope of CS during outflow in neap tide and spring tide conditions. The tidal phase is expressed in terms of the vertical average of horizontal tidal velocity ub above CS (top panels). Since the reversal of the tide is not synchronized in both layers, this average gives slightly lower values than expected and previously measured. The detachment is characterized by contours of zero-horizontal velocities at a given phase of the tidal cycle and this is shown in the bottom panels of Fig. . During outflow (negative values of ub), detachment of the Mediterranean layer occurs, generating a reverse flow (eastward) on the western flank of CS. The detachment appears earlier in the tidal phase, persists over a longer time, and has a larger amplitude in spring tide compared to neap tide, consequently delivering stronger bottom turbulence.

According to Prandtl's boundary-layer theory, flow detachment occurs when an adverse pressure gradient is present. We consider a simplified two-dimensional model of the flow, consisting of two immiscible fluid layers separated by a pycnocline at a depth hp. Detachment occurs when ∂xp>0, where x is the along slope coordinate. The total pressure gradient ∂xp can be decomposed into three components: 13∂xp=∂xpb+ρ0g′∂xhp+∂xpnh, where ∂xpb=ρ0g∂xη represents the hydrostatic barotropic pressure gradient driven by tidal forcing, ρ0g′∂xhp is the hydrostatic baroclinic pressure gradient due to the tilt of the pycnocline hp, which is always positive for a downslope gravity current, and ∂xpnh denotes the non-hydrostatic pressure gradient. Assuming that viscous effects on the tidal flow are negligible, the barotropic pressure gradient can be approximated using Bernoulli's principle: 14∂xpb=-ρ0∂xub22.

This formulation clearly shows that deceleration of the tidal flow, such as that caused by a bathymetric divergence during outflow, leads to an adverse pressure gradient. Conversely, in regions where the tidal flow accelerates, the barotropic pressure gradient becomes negative, driving the motion of the flow. The sign of the non-hydrostatic pressure gradient cannot be determined from this simplified analysis, although its magnitude can be estimated. Considering L being the typical horizontal length scale, H the vertical length scale, Ug the baroclinic speed scale, Ub the tidal speed scale and g′ the reduced gravity, the non-dimensional momentum equation in the bottom layer becomes: 15dudt=UbUg2ub∂xub-g′HUg2∂xhp+HL2∂xpnh. In our experimental setup, the barotropic pressure gradient is of order one, by design of the tidal forcing. Similarly, the baroclinic pressure gradient scales as the inverse of the squared internal Froude number, which is also of order one. Therefore, the barotropic and baroclinic pressure gradients are expected to be dynamically similar with the real ocean case, as the ratio Ub/Ug and the internal Froude number are preserved by the experimental design. The non-hydrostatic pressure gradient is two orders of magnitude smaller than the hydrostatic terms. Due to the tenfold increase in the aspect ratio in the laboratory compared to oceanic conditions, non-hydrostatic effects are expected to be more pronounced in the experiment. Nevertheless, they only influence the flow characteristics at second order and hence differences can be neglected.

inferred strong turbulence near the summit of CS, and observed large overturns on its western flank further downstream during neap tide (compare their Figs. 8 and 11 in transects S2 and S3). These overturns extended down to depths of approximately ≈70 m, significantly larger than those typically produced by shear instabilities, which were estimated to be one order of magnitude smaller. They attributed these large overturns to the detachment of the Mediterranean vein from the western slope of CS during outflow conditions. Their explanation was based on the experiments of , who argued that on the western slope of CS, the gravity-driven flow behaves more like a turbulent plume than a smooth gravity current, resulting in substantial entrainment, rapidly mixing the current with the surrounding water, and causing it to intrude at an intermediate depth rather than to continue following the bottom topography. According to and high turbulent dissipation rates during neap tide outflows were reported, comparable to those measured under spring tide conditions at the interface between the two layers. They attributed these elevated rates to the Mediterranean intrusion phenomenon proposed by .

Our results clearly indicate that the detachment of the Mediterranean vein is primarily driven by the adverse pressure gradient induced by the barotropic flow, rather than by intense mixing and subsequent intrusion of the Mediterranean vein, a condition present during both spring tide and neap tide phases. This barotropic component of the pressure gradient exceeds the gravitational forcing that drives the dense current downslope, ultimately causing the current to detach from the boundary. Under spring tide conditions, the strength of the barotropic flow leads to reattachment and deceleration of the Mediterranean vein further downstream promoting the transport (advection) of waters mixed through shear instabilities and the internal hydraulic jump further downstream as well. In contrast, during neap tide outflows, it seems that the shear turbulence at the interface, interacts with the boundary-layer turbulence induced by the detachment of the Mediterranean vein, as argued already by , as it can be observed from the TKE distribution and the combined velocity/density fields given in Fig.  (but also in Figs.  and ). This effect, combined with a more co-localized positioning of maximum velocity shear and maximum vertical density gradient shown in Fig. , results in more localized (less advected) and intense dilution during neap tide conditions as compared to spring tide conditions.

These results therefore suggest the importance of boundary layers (bottom and coastal) in producing TKE and vorticity, which appears to be equivalent to or higher than that produced at the interface between the two layers because of shear. Boundary layers then contribute rather to energy dissipation than to mixing Atlantic and Mediterranean waters in particular during spring tide. These findings may explain why similarly high turbulent dissipation rates levels are observed under both neap tide and spring tide conditions in observational data . They also underscore the critical role that topography and bottom boundary layers (friction) play in governing the overall flow dynamics and turbulence and the strong interaction with shear-induced instabilities in promoting water mixing and dilution, an effect that, in some cases, can surpass that of interfacial mixing alone, as already reported in and .

Figure  displays the Hovmöller diagrams of the pycnocline depth hp (ρ*=1/2) along the three considered transects (south to north from the top to the bottom panels, see the bathymetry on the left) as a function of the tidal cycle, evolving from maximal outflow to maximal outflow through maximal inflow. The top panels give the variation of the vertically averaged along-strait tidal velocity in both the Atlantic and the Mediterranean for neap tide (left) and spring tide (right) conditions (see the locations indicated in Fig. ). First of all, an increasing pycnocline depth in the Mediterranean at a given x position moving from the northern to the southern transect can be observed for both neap tide and spring tide conditions, due to the geostrophic slope induced by rotation within the Strait. A remarkable strong variation in time of the pycnocline depth is reported when passing from outflow to inflow (t>8 s) to the east of CS, suggesting that an internal bore is propagating to the East when the flow reverses. This is particularly evident during spring tide conditions in all three transects. The trapped bore is released when the flow criticality is attained. Across-strait variations of the tidal current, stratification conditions, and topography can modulate the shape of the released bore. evidenced that the semi-diurnal tidal wave M2 propagates southward in the Strait of Gibraltar. The author exhibited a 15° lag for M2, which corresponds to 29 mn delay between North and South for the external tide. confirmed that a counter current heading east precedes the tidal signal in the central part of the Strait of Gibraltar. Thus, the tide tilts the internal bore south-eastward, a feature generally preserved in the far field as illustrated by a selection of SAR images presented in . Note that differences in the values of phase shifts is attributed to the different considered locations in the different studies considering that some of them focus on the external tide instead of the internal tide as in our experimental study.

We attempted a quantitative estimate of the phase shift using our experimental data; however, it proved difficult to identify and was found to vary from one tidal cycle to another, preventing any robust conclusion. This issue is more appropriately addressed in a forthcoming paper focusing on internal solitary waves .

During outflow (t<10 and t>30 s), we see strong variations in the pycnocline depth around CS, suggesting the presence of hydraulic control for both neap tide and spring tide conditions in all three transects. During inflow, these gradients of hp are quite reduced, especially for the neap tide conditions, suggesting that the hydraulic control may have been lost at CS in all three transects. Note also the presence of quite high variations in the pycnocline depth in correspondence of the two other topographical features at x=-30 and x=-45 cm, as mentioned in the previous section, for both the neap tide and spring tide conditions.

The composite Froude number of the tidally averaged flow (see Appendix  and Eq. ) is displayed in the Hovmöller diagram of Fig.  for both neap tide and spring tide conditions (left and right panels, respectively). The top panels display the vertically averaged along-strait tidal velocity ub, whereas the bathymetry is shown on the left panels. The values G2=1 are given by the white dashed lines.

During neap tide conditions, hydraulic criticality is clearly present at CS for ≈88 % of the tidal cycle, and is lost during the maximum amplitude of the inflow 0.38<t/Ttide<0.5. This is consistent with the strong gradients of the pycnocline depth hp and the strong variations in the averaged vertical velocity w¯ reported in Figs.  and (and Figs.  given in the Appendix ), even though the composite Froude number G2 of the mean flow as given in Eq. () remains below unity at CS when considering the overall time average. In the southern transect (top panel) an area of G2>1 to the east of CS during inflow appears which may indicate a possible propagation of a bore to the east. Also, as reported previously, we observe locally G2≥1 at x=-20cm (x=-5km in the real ocean). During spring tide conditions, hydraulic criticality is clearly present at CS for ≈50% of the tidal cycle, and is lost during inflow. A clear propagation of the bore to the east when control is lost at CS at flow reversal is evident in all three transects under spring tides conditions, which is not evident for neap tide conditions. This difference reflects the different internal solitary waves propagating toward the east that are observed under neap and spring tides, which show to behave differently, an issue examined in the companion paper . Further supercritial zones appear during outflow at x=-30 and x=-45 cm (-7.5 and -11.25 km respectively for the real ocean) in correspondence of the other two topographic features during outflow to the west of CS.

As noted above, the frequent misalignment between the pycnocline and the region of maximum shear across tidal phases (cf. Figs. , and ) calls into question the conventional definition of the composite Froude number. Layer depths, h1 and h2, can be determined either from the density or the surface of maximum shear of the along-strait velocity, creating inherent ambiguity. Introducing a third layer could mitigate this issue, and a possible definition could delimit the third intermediate layer as the region bounded by the maximum velocity shear and the maximum density gradient, a departure from the more common definition based on intermediate density . Different computation methods using different definitions of the interface for both two- and three-layer systems are presented and discussed in detail in the Appendix , where we conclude that adopting the two-layer method with a local g′ is the most appropriate definition. For easier comparison with the literature, it is possible to retain the definition of the pycnocline as the intermediate density value (ρ*=1/2, approximately equivalent to Nmax⁡), although the definition based on Smax⁡ as the interface also remains equivalently valid.

The question of the location and permanence of hydraulic controls in the Strait of Gibraltar is central to resolving the long-standing debate on whether the exchange flow is maximal or sub-maximal. If the Strait is bounded by two hydraulic controls, it is hydraulically isolated from both the Mediterranean Sea and the Atlantic Ocean. In that case, the exchange is maximal and does not depend on the pycnocline depth on either side of the Strait, but only on the local bathymetry and the density difference between the two water masses (). The Camarinal and Espartel sills, as well as the Tarifa Narrows, are often cited as likely locations for such controls (). In this paper, we only present measurements at the Camarinal sill section, and the presence of a dual permanent control is thus out of the scope of this paper. However, the speed above Camarinal see have been shown to be constant for almost 25 min (see Fig. ) despite of changing reservoir conditions, suggesting a maximal exchange regime.

These issues, however, are beyond the scope of the present paper and are therefore not discussed further here. Additional measurements of velocity and density, collected from west of CS to the eastern outer exit of the Strait, as well as over the Espartel and West Espartel sites and including both northern and southern channels, will allow a more comprehensive assessment of these questions and will be addressed in future work.

Under the Boussinesq approximation, the mean mass transport per unit cross section can be separated into four components. The first corresponds to the advection of the reference density ρ0 by the mean flow, hereafter referred to as the volume transport. The second arises from the advection of the mean density anomaly, Δρ‾=ρ-ρ0 by the mean flow, and physically represents the transport of salt. The third and fourth correspond to the transports of the tidal and turbulent flow. Accordingly, the mean transport per unit cross section can be expressed as: 16∫hb0ρu‾dz=ρ0∫hb0u‾dz︸volume transport+∫hb0Δρ‾u‾dz︸mean buoyancy transport+∫hb0ρ̃ũ‾dz︸tidal transport+∫hb0ρ′u′‾dz︸turbulent transport. The volume, buoyancy, tidal and turbulent transports are plotted from top to bottom, respectively, of the Fig. , for the three transects considered, north to south from left to right. The three tidal forcings are represented by different color's lines, as in the previous Fig. . The four terms are arranged from top to bottom according to the amplitude of their contributions to the mean transport, with the volume transport being the dominant component. Each subsequent term is at least one order of magnitude smaller.

In an idealized exchange flow without lateral variations, the volume transport (first row panels) should remain zero and constant along the strait axis. In contrast, the results show pronounced local variations at the main topographic constrictions, where strong cross-sectional velocity components develop to bypass the obstacles, as CS and those to the west. Overall, the transport is predominantly negative, indicating that, across the examined transects, a larger volume of Mediterranean water flows westward through CS than Atlantic water eastward. This imbalance is expected to be compensated by an enhanced transport of Atlantic water south of the transects, where the topography features an extended plateau. In addition, the geostrophic slope associated with rotation favors a greater westward transport of Mediterranean water in the northern part of the Strait (see Fig. b).

The vertically averaged mean buoyancy transport per unit cross section is displayed in the second row panel of Fig. . A significant decrease in transport is observed just upstream of CS for all tidal scenarios, suggesting again the presence of three-dimensional flow structures around the sill, with more water circumventing the sill than overflowing it above the summit. Mass transports are overall everywhere negative and decrease when moving from the southern to the northern transects because of the decreasing depth containing more diluted waters west of CS.

The tidal transport, shown in the next row of panels, exhibits distinct patterns under different tidal forcing conditions. During neap tide, the transport is negative east of the CS and approaches zero immediately west of the sill across all three transects. Under spring tide conditions, by contrast, the tidal transport remains close to zero both east and farther west of the sill, but becomes increasingly positive from north to south in the vicinity of and just west of the sill.

The turbulent transport, presented in the bottom row of Fig. , remains near zero for both tidal regimes in the northern transect and for spring tide conditions in the middle and southern transects as well. However, it becomes negative approaching and west of the sill for the neap tide conditions. This negative sign indicates that more diluted waters are advected downstream during neap tide conditions compared with spring tide, for which the transport is dominated instead by the tidal rather than the turbulent component. Figure  shows the tidal-mean, vertically averaged buoyancy transport over a full tidal cycle for neap tide (left column) and spring tide (right column) conditions. The volume transport component is not included, as we have shown in the previous Fig.  it is two orders of magnitude larger than the buoyancy transport and primarily driven by transverse variability linked to the Strait’s complex topography. buoyancy transport is again enhanced over the southern flanks, where water depth increases. The M2 tidal constituent modulates the mass transport, leading to a pronounced reduction of buoyancy transport during neap tide inflow, and even reversing the direction to positive buoyancy transport during spring tide inflow. Overall, tidal forcing promotes buoyancy transport from the Mediterranean toward the Atlantic.