Non-rotating sill exchange flows
Previous experimental studies  had attempted to define the parametric conditions under which saline intrusion blockage occurred for non-rotating conditions only (i.e. f = 0). This preliminary analysis demonstrated that, for the range of conditions tested, saline blockage occurred at a specific value of the densimetric Froude number F12 = q12/(g′0hb3) > ~ 0.125 for the freshwater flow across the sill (i.e. where layer thickness h1 = submergence depth hb), irrespective of the relative magnitude of the source fresh and saline volume fluxes (i.e. Q* = Q1/Q2 = q1/q2). This suggested that saline intrusion blockage requires a specific combination of a large freshwater volume flux q1 (= Q1/B) and/or lower submergence depth hb and reduced gravity g′0. Specifically, the two experimental runs from Cuthbertson et al.  in which full saline blockage occurred (i.e. Runs 6 and 7, see Table 1 in ) were initiated at Q* = 8.90 and 3.75, respectively, for otherwise identical conditions (i.e. g′0 = 0.046 m.s−2,hb = 0.35 m). It is interesting to note here that a similar freshwater densimetric Froude number F0 was defined by Sargent and Jirka  in their analysis of saline wedge formation (i.e. an arrested saline intrusion) generated by counterflowing fresh and saline water masses along a horizontal, rectangular channel. Overall, they found that the flow dynamics in the saline wedge were controlled primarily by the freshwater overflow, with a narrow range of Froude numbers (F0 = 0.389−0.491) at which the stationary salt wedges formed. Within the current study, full blockage of the saline intrusion layer across the sill was measured at F1 = 0.383 and 0.392 (for Runs 6 and 7, respectively, see Table 1 in ), where there was also clear evidence from the density profiles [i.e. Figure 7(a) for Run7(f)] that an arrested saline wedge developed on the sill between x/L = 0.0 and −0.5.
For the equivalent non-rotating BOM simulations, the formation of an arrested saline wedge along the horizontal sill crest was demonstrated clearly both from velocity fields [e.g. Figure 6(b)] and density profiles [Fig. 7(b)]. However, a primary motivation of conducting these numerical simulations was to investigate in greater detail the sensitivity of the saline blockage condition to the relative sill submergence depth hb/H over a wider range (i.e. hb/H = 0.075−0.634) than considered in the experimental study (i.e. hb/H = 0.411−0.474, Table 1). In this context, the numerical results shown in Fig. 8(a) indicated that full saline blockage (i.e. an arrested saline intrusion with Q2,sill/Q2 = 0) occurred when Q* = 1.15 → 9.0 for hb/H = 0.075 → 0.462 (i.e. hb = 0.07−0.43 m for H = 0.93 m). The corresponding upper freshwater Froude numbers F1 for these blockage conditions ranged from F1 = 1.29 → 0.66 (i.e. Q1 = 8.0 → 62.6 l.s−1; g′0 = 0.05 m.s−2), which are considerably higher than observed in the experiments. It is considered likely that these differences arise largely from BOM treatment of interfacial mixing and entrainment fluxes generated between the counterflowing water masses across the sill, as well as its representation of the inflowing and outflowing boundary conditions specified in basins M and I. These points are discussed further below.
When we consider the measured local flux ratios Q*sill = Q1,sill/Q2,sill generated across the sill in Run 2 (blue data in Fig. 3), it is interesting to note that, at both Q* = 0.0 and 0.43, the experimental data typically lies above the straight line suggesting Q*sill > Q*. Thus, under net-barotropic conditions generated in the lower saline intrusion layer (Q* < 1), a counterflowing motion must be induced in the upper layer that enhances the freshwater outflow across the sill [as shown in Fig. 4(a)(i) and (ii)]. By contrast, at Q* = 1.15, 1.73 and 3.03, the local flux ratio Q*sill < Q* suggesting that the fresh and saline fluxes across the sill are reduced and increased, respectively, in comparison to the source volume fluxes entering the channel. Such conditions may result from a number of scenarios or combination of internal flow mechanisms, including: (i) internal energy losses due to bottom friction and interfacial shear between the counterflowing layers (e.g. [38, 39]), resulting in net entrainment of freshwater from the upper layer into the bottom intruding saline layer , (ii) the formation of an internal hydraulic control across the sill limiting the freshwater outflow and thus promoting recirculation within basin I , and (iii) the saline intrusion spill into impoundment basin I generating sufficient mixing (e.g. via the internal hydraulic jump forming at the bottom of the sill incline) to initiate freshwater entrainment and recirculation in basin I, thus limiting the freshwater outflow flux across the sill. Furthermore, when we consider the fractional reduction in the intruding saline volume flux across the sill, Q2,sill/Q2, as presented in Fig. 8(a), we observe that Q2,sill/Q2 typically decreases monotonically with increasing Q* values. This clearly indicates that the saline intrusion across the sill is also lower than the salt water volume flux into basin M, again most probably associated with internal friction losses across the sill obstruction, the potential formation of a hydraulic control at the sill, thus limiting salt water intrusion, and consequently the development of salt water recirculations in basin M. Referring back to Fig. 3, at higher Q* values of 3.75 and 4.32, the sill flux ratio is now Q*sill > Q*, indicating that the freshwater outflow becomes dominant (i.e. strong net-barotropic flow in the upper layer that controls the bidirectional exchange flow across the sill). This local sill flux condition again may again develop from a number of possible scenarios, including: (i) a net entrainment of saline water from the lower intruding layer into the upper freshwater layer due to strong interfacial mixing, and (ii) partial blockage and recirculation of the inflowing saline water flux in basin M due to the strength of the freshwater outflow across the sill. This saline water blockage and circulation in basin M may also be promoted by the pumped abstraction (i.e. outflow) of the upper freshwater layer at the end of the channel, directly above the salt water inflow into basin M (Fig. 1). The effect of interfacial mixing and the entrainment of saline water by the dominant upper freshwater outflow can be shown by comparing Q*sill values at different x/L locations along the sill (different symbols in Fig. 3). While this effect is not entirely clear in Run 2, it was demonstrated in other runs that Q*sill values typically increase along the sill length (from basin M → basin I) under higher Q* conditions [see Fig. 8 in Cuthbertson et al. ].
Comparing again the experimental data with equivalent data obtained from BOM simulations for Run 2 (red data in Fig. 3), we see that, fundamentally, the same trends in local sill flux ratio Q*sill versus source flux ratio Q* are observed to those described above (i.e. Q*sill > Q* at Q* = 0.0, 0.43 and > 3.75; and Q*sill < Q* at Q* = 1.15 and 1.73), albeit these trends are more subtle in the BOM results. The disparity between laboratory measurements and numerical simulations is also demonstrated clearly when comparing equivalent runs from Fig. 8(a) [e.g. grey circles (hb/H = 0.462–Run 2) and red dashed line (hb/H = 0.462–BOM)]. Here, the measured local saline flux across the sill with respect to the saline water inflow is shown to reduce by approximately 30% (to Q2,sill/Q2 ≈ 0.7) at a relatively low Q* = 1.73, before stabilising as Q* increases up to its maximum value of 4.32. By contrast, the equivalent BOM experiment reveals that Q2,sill/Q2 remains above unity until Q* ≈ 4, before decreasing steadily to Q2,sill/Q2 = 0 (i.e. full saline blockage) at Q* = 9. These variations between the laboratory and numerical model results are most likely to arise from differences in the BOM treatment of internal friction (i.e. interfacial mixing and bottom boundary roughness) across the sill and the imposed inflow/outflow boundary conditions at the edges of the model domain within basins M and I (see Sect. 2.1) that are clearly different from the laboratory set-up (see Sect. 2.2). In a sense, the semi-enclosed laboratory configuration of basin M and the open boundary of basin I is the opposite of what might typically be expected in a real fjordic setting (where a submerged sill separates a semi-enclosed fjordic basin from the open sea boundary.
In summary, within both the non-rotating laboratory and numerical experiments, the saline intrusion across the sill is first restricted and then fully blocked with increasing Q* values (i.e. increasing freshwater flow Q1), while the key topographic controlling parameter is the sill submergence depth hb. Within Fig. 8, where the effect of hb on saline intrusion blockage is clearly demonstrated, this topographic sill parameter has been normalised by the total basin depth H. However, it could be argued that the sill length L would be expected to have more influence on the exchange flow dynamics [e.g. through boundary friction effects [1, 40]] than the overall basin depth H. In the current laboratory and numerical experiments, the sill length L was kept constant at 2 m and thus the variability in L/hb (= 4.44−5.71 and 3.39−28.6, respectively) was associated solely with changes to submergence depth hb. This compares with L/H = 4.68 for the dimensions of the rectangular channel used in the arrested salt wedge experiments by Sargent and Jirka .
It is also interesting to compare these experimental L/hb values with equivalent values for topographic sills within Norwegian and Scottish fjords. In Scotland, Loch Etive has a total of six sills along the length of the loch, averaging 15 m in depth. The largest of these sills, separating the deep inner basin (maximum depth 145 m) from the shallower outer basins (maximum depth ~ 65 m), has a maximum sill depth hb = 13 m and crest length L = 210 m (i.e. L/hb = 16.2) [1, 2]. The exchange across this relatively shallow sill is controlled by tidal barotropic forcing and results in a dominant mode-I baroclinic response within the stratified upper basin . High freshwater runoff into the upper basin also contributes to the strong basin stratification, contributing to periods of restricted basin circulation and reduced tidal intrusion across the sill, which limits deep water renewal and increases the likelihood of hypoxic/anoxic conditions in bottom basin waters. Similar hypoxic/anoxic basin waters are also commonly experienced in Norwegian fjords, including, for example, Iddefjorden at the border of Norway and Sweden . Here, the mouth of the Iddefjord basin has two narrow sills (L ~ 70 m) with submergence depths hb ~ 7 m (i.e. L/hb ≈ 10), separated by a wider and deeper small basin . Again, tidal exchange flow dynamics across the sills are controlled by a combination of bottom friction effects, barotropic form drag and the baroclinic response in the basin. Recent simulations in Masfjorden (Norway) by Aksnes et al.  have also indicated a decrease in the rate of deep water renewal events (i.e. ventilation of basin waters) associated with reduced tidal intrusion across the submerged sill (hb = 70 m; L ~ 500 m; L/hb ≈ 7) at the entrance to the deep Masfjorden basin (maximum depth ~ 494 m). The simulations suggest this reduction is associated with the ongoing warming of North Atlantic Waters (NAW) leading to a reduction in the density of the seawater intrusion, which is expected to result in prolonged anoxic conditions in Masfjorden basin within 7–12 years . In all these cases, the exchange flow dynamics (and seawater intrusion, in particular), along with the resulting impacts on basin stratification, circulation and water quality, are thus strongly controlled by the sill topography, with the aspect ratios L/hb of the fjordic sills found to coincide with the range of L/hb values tested in the current laboratory and numerical simulations.
Rotating sill exchange flows
Within most fjordic scenarios, the overall exchange flow dynamics are unlikely to be affected to any significant extent by Earth rotation effects (i.e. Coriolis forces), as this requires the width of fjordic basin (and sill region) to be larger than the Rossby radius of deformation . In the context of larger scale oceanographic flows, however, Coriolis forces are known to deflect counterflowing water masses passing through sea channels and ocean straits [e.g. Gibraltar Strait , Baltic Sea Channel ] or deep water density currents flowing down the continental margins of oceans [e.g. Faroe Bank Channel , Darelius and Fer ]. Many experimental and analytical studies have been conducted into the fundamental effects of rotation on salinity-driven gravity currents, making important contributions in defining the geostrophic adjustment of topographically-constrained currents (e.g. [11, 12, 18, 47, 48],the flow velocity structure and turbulence intensities (e.g. [17, 49, 50],and the role of Ekman boundary layers in the development of secondary flow circulations (e.g. [14, 15, 51, 52]. Nonetheless, there remains open questions regarding the effects of rotation on the dynamics of bi-directional exchange flows within topographically-constrained settings (e.g. across submerged sills), particularly in relation to (i the transverse distribution of counter-flowing water masses across the sill; (ii geostrophic adjustment in bi-directional exchange flow layers; (iii secondary circulations and boundary (Ekman layer development; and (iv the saline blockage conditions under strong net-barotropic flows in the upper freshwater layer. For this purpose, the numerical experiments within the current study were extended to consider the effect of Coriolis forces on all these aspects of exchange flow dynamics in the same idealised sill-basin configuration as tested in the non-rotating experiments.
The main effects of rotation on the transverse density structure of the exchange flows [Figs. 9(a) and 10(a)] are demonstrated by the inclination of the density interface [represented by the Δρ′ = 2.5 kg m−3 (ρ′ = 0.5) isopycnal] and convergence of isopycnals in the positive y direction across the sill. Both of these effects are indicative of the geostrophic adjustment within the sill exchange flows. This is confirmed by comparing the transverse density interface inclination angle αρ′=0.5 (i.e. inclination of the ρ′ = Δρ′/(ρ2 − ρ1) = 0.5 isopycnal) with the expected transverse slope αg at which the exchange flow is geostrophically balanced, αg ≈ f.(ū2–ū1)/(g′)0 (where ū1 and ū2 are representative average velocities in the upper and lower layers) (Fig. 12). It should be noted that although αρ′=0.5 ≈ αg in the majority of the runs, at higher f values (corresponding to lower Ro values) the agreement between αg and αρ′=0.5 values reduces, especially at higher Q* values. This is at least partly due to the average velocity ū1 calculated for the upper fresh layer including a flow region [where u1 ≈ 0, e.g. see Fig. 10(b)(iii)] that appears to have little or no influence on the geostrophic adjustment in the lower saline intrusion layer. For such runs, better agreement is gained with αg ≈ f.(u2–u1)max/(g′)0 (as plotted for these runs in Fig. 12). The observed isopycnal convergence or “pinching” in the positive y direction has been described in previous experimental studies of dense water outflows through topographically-constrained channels (e.g. [11, 12, 18] and in field survey measurements (e.g. [45, 53–55], of dense water overflows at the sill constriction in the Faroe Bank Channel. Johnson and Sanford  suggested that this wedged-shaped density field (and isopycnal pinching) was induced by secondary cross-channel circulations and mixing driven by the development of bottom and interfacial Ekman layers.
Evidence of these bottom Ekman layers is detected in all modelled rotating sill exchange flows in the current study [e.g. Figs. 9(b) and 10(b)], with a thin cross-channel sill boundary flow directed in the positive y direction (i.e. to the left, looking downstream). By contrast, the presence of the interfacial Ekman layer in the same cross-channel direction is only detected in certain runs [e.g. Fig. 9(b)(iv) and (v)]. Furthermore, the interior of the lower saline intrusion layer is shown to flow in the negative cross-channel y direction (i.e. y → 0), representative of the main sill intrusion flow being directed to the right (looking downstream) (as expected for rotating sill overflows in the Northern Hemisphere), which acts as a volume balance to the boundary Ekman layers. Qualitatively similar secondary circulations, as described in detail in Cossu et al. , have been observed in several laboratory experiments considering density current flows along rotating channels and canyons (e.g. [14, 15, 49, 52] and reported in field studies of ocean gravity currents (e.g. [56–58]. In the bi-directional exchange flows considered in the current study, the outflowing upper freshwater layer is also directed to the right (looking downstream) at lower Q* values [e.g. Fig. 9(b)(ii) and (iii)], while as Q1 increases, large scale secondary circulation cells develop [e.g. Fig. 9(b)(iv) and (v)] with the freshwater sill outflow following spiral motions. This finding is again consistent with the previous experimental study of Johnson and Ohlsen , who found that the secondary helical circulations generated by friction and rotation effects (i.e. interfacial and solid-boundary Ekman layers) led to a reduction in the two-layer exchange through a semi-circular channel, compared to the equivalent non-rotating, two-layer exchange flow case. Similar measurements of the transverse velocity structure have been made for density currents flowing along a rotating straight rectangular channel  and within rotating V-shaped laboratory-scale ridges and canyons [14, 49]. The studies also showed that the secondary circulations generated in the density currents, due to rotation and friction effects, resulted in the flow being substantially slower than in equivalent non-rotating currents due to the helical motions of dense water parcels (i.e. longer flow paths taken) as they travel along the channel/canyons.
In the current rotating sill exchange flow experiments, a significant reduction in the saline intrusion flux ratio across the sill Q2,sill/Q2 was also observed (Fig. 11) as both the source flux ratio Q* and the Coriolis parameter f increased (i.e. the latter corresponding to a reduction in Ro). It is interesting to note that when Ro ≈ 1 (corresponding to the f = 0.0267 s−1 run in Fig. 11), the effect of rotation results in only a relatively small reduction in the saline intrusion flux ratio Q2,sill/Q2 compared to the equivalent non-rotating run (i.e. f = 0.0 s−1, Fig. 11), with full intrusion blockage (i.e. Q2,sill/Q2 → 0) occurring at the same source flux ratio Q*. This clearly suggests that for Ro > 1, the geostrophic adjustment in the sill exchange flow will have only marginal effects on the resulting flow properties (e.g. saline intrusion fluxes), which is broadly in agreement with the flow regime transition, defined by Cossu et al. , where Ekman boundary layer dynamics become less important. It is also important to note that this Ro = O(1) condition is representative of the geostrophically-controlled Mediterranean dense water outflows through the Strait of Gibraltar (e.g.  and through the Bosporus Channel into the Black Sea , as well as in the Faroe Bank channel overflow at the threshold sill (e.g. [15, 45].
In the current experiments, the relative influence of increasing the Coriolis forces (i.e. through increasing f) and, hence, reducing the Rossby number to Ro < < 1, is shown to have a clear impact on the magnitude of the saline intrusion flux ratio Q2,sill/Q2 across the sill. In particular, this can be demonstrated in Fig. 11 where the same overall reduction in Q2,sill/Q2 occurs at significantly lower Q* values when the Coriolis parameter f is increased (i.e. Ro is reduced). This Rossby-number-induced blocking effect is shown clearer in Fig. 13(a–d), where the cross-channel flow structure of the bi-directional sill exchange flows at Q* = 3.75 are plotted with Ro values decreasing from ~ 1 → ~ 0.5 → ~ 0.25 → ~ 0.125. [Note: the corresponding Q2,sill/Q2 values reduce from 0.92 → 0.77 → 0.45 → 0.13, respectively (see Fig. 11), with Q2,sill/Q2 ≈ 1 for the equivalent non-rotating exchange flow condition]. This finding again appears to be general accord with Cossu et al.  and others who suggest that, when Ro < < 1, rotating gravity current flows are “substantially slower” than their non-rotating counterparts, and that Ekman boundary layer dynamics should be included in models to describe rotating flow properties when Ro < 2.
It is also apparent from Fig. 13, however, that, although the saline intrusion flux across the sill clearly diminishes as Ro decreases (as described above), both the maximum and cross-section average velocities u2,max and ū2 within the saline intrusion layers remain relatively consistent over the majority of Ro conditions tested. Indeed, the mean and maximum saline intrusion layer velocities ū2 and u2,max are shown to only reduce by ~ 1.1 and ~ 1.5%, respectively, as Ro reduces from ~ 1 to ~ 0.25 [Fig.13(a–c)], while ū2 only shows a more significant reduction (~ 66%) at Ro = ~ 0.125 (i.e. close to the full saline intrusion blockage condition). It is also interesting to observe that the equivalent ū2 and u2,max values calculated at the mid-sill location within the equivalent non-rotating sill exchange flow experiment [i.e. Q* = 3.75, Fig. 4(b)(iii)] are ~ 2% and ~ 15% lower than within the rotating sill exchange run with Ro = ~ 1. This suggests that, although the overall saline intrusion flux is larger in the non-rotating sill exchange flow (i.e. Q2,sill/Q2 ≈ 1), ū2 and u2,max are reduced due to the larger cross-channel flow area of the saline intrusion across the sill. This finding appears to be consistent at all Q* values tested and suggests a deviation from the hindered flow behaviour of rotating gravity currents described by Cossu et al.  and others. This potentially indicates that additional dynamical effects associated with the bi-directional, sill exchange flows considered here result in an increase in the intruding saline layer velocities between the non-rotating and rotating runs, whilst constraining the overall exchange in the latter case (e.g. Johnson and Olsen ). It can be hypothesised that the upper fresh layer flow structure and secondary circulations generated, in particular, must impose a strong influence on both the transverse distribution and extent of the saline intrusion flow region across the sill. This effect appears most noticeable at lower Ro values [i.e. Ro = ~ 0.25 and ~ 0.125, Fig. 13(c) and (d)], where the maximum outflowing upper layer velocities u1 occur in the confined flow region directly above the counter-flowing saline intrusion layer. By contrast, upper layer velocities diminish u1 → 0 in the positive y direction and can even reverse in direction at the opposite side of the sill [i.e. y > 1.2 m, Fig. 13(d)].