## 1. Introduction

The deep oceans are partitioned by submarine ridges into basins that are often connected by deep passages. The water mass exchange between these basins is an important process in the global overturning ocean circulation. For example, overflows of dense water across the Greenland–Scotland Ridge feed the lower limb of the Atlantic meridional overturning circulation (AMOC). The overflow transport is controlled by the height of topography and by the width of the passages. It has been shown that the flow is hydraulically controlled at the topographic sills that lie within the passages, meaning that the sills exercise some degree of influence over the volume flux and upstream stratification (Pratt and Whitehead 2008). These choke points are strategic places to monitor the volume flow rates and ultimately estimate the state and variability of the thermohaline circulation (Helfrich and Pratt 2003).

The pathway and the volume flux of the water approaching a sill are affected strongly by the geometry of the sill and Earth’s rotation. Rotating hydraulic theory provides insight into the pathway of the flow as it approaches the sill as well as establishing formulae for calculation of volume flux. Stern (1972) pioneered the use of rotating hydraulics in oceanography. Whitehead et al. (1974) developed the first analytical model of hydraulic behavior in a steady rotating channel flow with rectangular cross section and performed a set of laboratory experiments to test the transport relations. The assumption in their work was that the flow is fed from a deep quiescent reservoir (with infinite layer thickness) and therefore, zero potential vorticity (pv). Gill (1977) introduced a unifying framework for rotating hydraulics for a finite basin with constant (nonzero) pv. The Gill model provided insight about how the water approaches the sill from the upstream basin. For a sufficiently wide reservoir (relative to the Rossby deformation radius), the upstream flow is divided into two independent boundary layers. For a given sill geometry and rate of supply of fluid to the reservoir, the Gill model provides ranges of steady controlled solutions (Gill 1977; Pratt and Whitehead 2008). Whitehead (1989, 1998, 2003, 2005), Whitehead and Salzig (2001), and Borenäs and Nikolopoulos (2000) modified hydraulic models in order to make the transport relation easier to apply. Killworth and McDonald (1993, 1994) developed bounds on transport in terms of upstream flow measurements. None of these studies considered various types of inflow and their effect on the controlled flow in the channel, however, nor did they consider effects of dissipation or friction. Helfrich and Pratt (2003) developed an idealized model of a basin–strait system with three different inflows introduced into a bowl-shaped basin in order to investigate how the pathways of overflow are affected by the basin circulation. They considered boundary inflows as well as localized, and uniform downwelling. Their simulations also accounted for bottom friction. The different inflows resulted in different circulation patterns and interface heights within the basin, but their study revealed that the flow in the channel is remarkably independent of the type of inflow in the basin and thus the potential vorticity in the channel (a key quantity in rotating hydraulics) is also independent of the location of the upstream mass source. This simplification allowed them to relate the transport to an upstream measurement of interface height. As opposed to the Gill model, the upstream measurement is made at the entrance of the channel rather than in the basin interior which is advantageous considering the dependence of interface height to the type of the circulation within the basin. Their study was restricted to a channel with rectangular channel cross section, bounded by vertical walls, as in most of the previous studies. Despite mathematical convenience in considering a rectangular cross section, it becomes difficult to apply the results in real oceanic channels, where the topography varies smoothly and where the layer thickness vanishes at the edges. Moreover, the physics of the hydraulic processes potentially changes when the layer depth goes to zero at the edges: for one thing, the character of the edge waves that transmit hydraulic control to the upstream basin changes from that of a Kelvin wave to that of a slower frontal wave (Stern 1980; Pratt and Whitehead 2008). The parabolic channel cross section greatly increases the chance of flow instability (Pratt et al. 2008) and flow reversal (Borenäs and Lundberg 1986).

In this study, we follow up on Helfrich and Pratt (2003) by using a more realistic strait geometry and considering comparisons with nonzero uniform pv theory. We address the following questions:

What are the transport pathways from the mass source through the basin, into the narrow strait and over the shallow sill? How do these pathways change as the location of the mass source changes? Does it remain true that the flow in the strait depends primarily on the volume flow rate, and is independent of the circulation in the upstream basin as in the case of rectangular channel geometry? How far into the strait does the fingerprint of the upstream circulation penetrate? What features characterize velocity distribution and interface shape in the region where the flow encounters the obstacle, spills over the crest of the obstacle, and descends downstream? What are some possible monitoring strategies that take advantage of the fact that hydraulic control and hydraulic criticality exist at or near the sill? What is the best location to conduct monitoring with fixed instruments? Are there any trade-offs in terms of convenience and accuracy between possible strategies?

We start by elaborating on the theory formulated by Borenäs and Lundberg (1986) for the parabolic channel with uniform pv and then compare the theory with our numerical simulations.

## 2. Theory

The hydraulics theory for steady flow of a 1.5-layer (reduced gravity) model with uniform nonzero pv in a rotating channel with rectangular cross section was described by Gill (1977). In the Gill model, the flow was assumed to come from a wide upstream basin and funnel through a narrow channel into a downstream basin. Borenäs and Lundberg (1986) extended the Gill theory into a more geophysically realistic case by considering a parabolic cross section resembling the rounded nature of real-world sea straits. In this case, it is no longer necessary to distinguish between flows that are attached to or separated from the vertical sidewalls of the rectangular channel. In the following subsection, we elaborate on the theory by solving the critical condition and providing graphical tools to interpret the solution. Then in the results section, we compare the predictions from the theory with the results of the numerical simulations for a range of parameter values. The theory described in this section is for Northern Hemisphere applications and can be easily adapted to the Southern Hemisphere.

### a. Parabolic channel cross section

*α*(

*y*) is the dimensional parabolic curvature, a parameter that can vary along the channel. If we scale the

*h*

^{*}with layer depth scale

*H*(this can be considered to be the sill height from the bottom of the basin),

*x*

^{*}with the Rossby radius of deformation (

*r*(

*y*) =

*f*

^{2}/

*g*′

*α*(

*y*) is the nondimensional radius of curvature. Pratt et al. (2008) interpret

*r*(

*y*) in the following way: suppose that the fluid is at rest, so that the interface is horizontal, and that the channel is filled to a depth

*D*, measured at the center

*x*= 0. Then the half-width of the interface is

*L*is then

_{d}*r*which is small for wide channel and large for narrow channels.

*υ*is geostrophically balanced as the channel width and bottom elevation vary gradually, but the cross-channel velocity

*u*is not. This is the so-called semigeostrophic approximation. With this approximation, the nondimensional shallow water equations in the

*x*direction reduce to the velocity in the

*y*direction:

*x*direction can be derived:

*q*is constant (uniform pv with no forcing or dissipation), the solution has the form

*y*-dependent coefficients

*C*

_{1}and

*C*

_{2}can be determined by the boundary conditions at

*x*= −

*a*,

*b*where the interface intersects the bottom of the parabola (see Fig. 1c). Therefore,

*d*(−

*a*) =

*d*(

*b*) = 0, leading to

*d*(

*x*,

*y*) in Eq. (3), leads to the corresponding geostrophic velocity:

*h*

_{0}(

*y*) =

*h*(0,

*y*) is the topographic height at the center of channel and only depends on

*y*. The Bernoulli function is conserved along streamlines, including the streamlines corresponding to the edges of the flow. Evaluating the Bernoulli function at the intersections of interface height with the parabolic bottom (

*d*= 0) leads to

*y*value) of the channel is provided by the variables

*a*(

*y*) and

*b*(

*y*), along with the potential vorticity and the local geometric parameters

*h*

_{0}(

*y*) and

*r*(

*y*). The evolution of the flow in the

*y*direction is governed by the statements of conservation of the volume flux

*Q*and the Bernoulli function

*G*

_{1}and

*G*

_{2}to be the volume transport relation, and the average Bernoulli function. The two dependent variables describing the state of flow are considered to be

*γ*

_{1}=

*a*+

*b*which is the wetted width and

*γ*

_{2}=

*a*−

*b*which is twice the centerline position of the flow at any cross section. Therefore, the transport

*G*

_{1}and the Bernoulli function

*G*

_{2}can be rewritten in terms of variables

*γ*

_{1}and

*γ*

_{2}and parameters

*q*and

*r*as

*G*

_{1}and

*G*

_{2}with respect to

*γ*

_{1}and

*γ*

_{2}, and substituting them in Eq. (18) results in

*G*is set to zero) when the problem is formulated in terms of a single variable. Their algebra is lengthy and does not appear in their manuscript. They did not realize that the critical condition can be obtained more easily using the multivariate approach described above. Equation (19) implies a relationship between

*γ*

_{1}and

*γ*

_{2}at the critical section for given values of

*q*and

*r*. Therefore, a reduction in the number of independent measurements is established. From the multivariable approach, the critical section is shown to be located at the sill section (Pratt and Whitehead 2008). In the next section, we solve for the critical solution for any given sets of

*q*and

*r*matching our theoretical simulations.

## 3. Numerical solver

*f*plane just as in Helfrich and Pratt (2003) but the coordinate system is rotated (270° counterclockwise) to be consistent with the theory. The momentum and continuity equations are

*is the horizontal gradient operator,*

_{h}*h*(

*x*,

*y*) and interface height

*d*(

*x*,

*y*) are consistent with Fig. 1. The horizontal velocity vector

**u**(

*x*,

*y*) is scaled with

*w*(

*x*,

*y*) < 0 is scaled by

*Hf*, and the lengths in the horizontal

*x–y*plane and in

*z*direction are scaled with deformation radius (

*H*, respectively. The time

*t*is scaled with

*f*

^{−1}. The term

**D**is the friction operator, which is a linear bottom drag

**D**= −

*ϵ*

**u**with constant coefficient

*ϵ*(scaled with

*f*). The term

**M**is the momentum flux due to downwelling in the basin with

**M**=

*w*

**uΘ**(−

*w*)/

*d*. The Θ(

*x*) is a step function which is equal to 1 for

*x*< 0 and 0 for

*x*> 0. The scaling is consistent with the theory; however, the numerical model includes extra terms such as friction. The dissipation is necessary to reach a steady state solution and make comparisons with the theoretical derivations discussed earlier.

The basin is bowl shaped with topographic height *h*(*x*, *y*) as in Fig. 1. The bottom of the basin is smoothly connected to the channel through a Gaussian slope region between −5 < *y* < 0. The basin’s back wall is located at *y* = −15 (see Fig. 2). The basin width is 8 along the *x* direction and it narrows with a Gaussian shape centered at *y* = 0 to a channel with parabolic cross section. The sill is located at *y* = 6. The numerical domain is necessarily finite. The parabolic cross section is tall enough that the active layer never reaches the edges of the domain. The boundaries of the numerical grid are symmetric about *y* = 0 marked by black boundaries along the channel in Fig. 1a). As the layer thickness changes along and across the channel, the dynamic width of the flow changes as well. The edges of the dynamic width along the channel are shown as blue dotted dashed lines in Fig. 1a. The interface height at an arbitrary upstream section is shown in Fig. 1c. To avoid abrupt geometric changes at the channel entrance, the parabolic channel has a slowly varying radius of curvature *r*(*y*) ramping up from zero to a constant value *r*(*y*) = *r*_{0} with an inverse Gaussian function at the mouth of the channel entrance (0 < *y* < 1) and then the curvature remains constant throughout the channel afterward [*r*(*y* ≥ 1) = *r*_{0}].

The flow can enter the basin in three different ways similar to Helfrich and Pratt (2003) to mimic conditions in nature: uniform downwelling or localized downwelling to represent convection, or inflow through a segment of the basin boundary, representing drainage from another basin. The uniform downwelling *w* < 0 is imposed throughout the basin (*y* < 0), the localized downwelling is placed at the upstream wall of the basin with uniform *w* in the *x* direction and a Gaussian function of width *y* = −15. The boundary inflow enters through the upstream sidewall at *y* = −15 and −0.1 < *x* < 0.1. The total flux is distributed uniformly from *x* = −0.1 to *x* = 0.1.

The numerical model solves Eq. (20) in flux form and Eq. (21) using a second-order finite volume method that handles complexities such as grounding (fluid depth going to zero), shocks and hydraulic jumps. The details about the numerical algorithm and its implementation are explained in Helfrich and Pratt (2003) and Helfrich et al. (1999). The model solves for the velocities on an orthogonal generalized quadrilateral grid. The solution is mapped back to a Cartesian grid using conformal mapping. The transformation is described by Knupp and Steinberg (1993) and the details of its algorithmic implementation which solves the velocity field in the mapped domain are explained by Bell et al. (1989). The numerical width of the channel varies in different simulations depending on *r*, but because the variations are small, the number of grid cells does not need to be changed to sustain the numerical stability and the Courant–Friedrichs–Lewy (CFL) condition. The number of grid cells for all simulations is the same with 220 cells in the *y* direction and 80 cells in the *x* direction. The average grid cell sizes are Δ*y* ≈ 0.11 and Δ*x* ≈ 0.11. The flow exits the domain at *y* = 9 through an open boundary set by the Orlanski radiation condition (Orlanski 1976; Helfrich et al. 1999). No-flux conditions are set where the interface contacts the rigid walls.

The numerical solutions are initialized with the basin filled with motionless fluid up to the sill crest as shown in Fig. 1b. The specified imposed volume flux enters the basin at *t* = 0 and the model is integrated until a steady solution is obtained. For certain parabolic curvatures, *Q* and *ϵ*, a steady state may not be achieved.

## 4. Results

### a. Simulations

We ran several experiments with ranges of parameter values. The range of imposed nondimensional mass flux is 0.01 ≤ *Q* ≤ 0.1. The range of drag coefficient is 0.001 ≤ *ϵ* ≤ 0.05, but most cases shown here are run with *ϵ* = 0.01. All experiments have the same basin geometry and sill height but different parabolic channel curvatures. For a parabolic channel the width of the flow is not specified by the width of the channel (as opposed to rectangular channel), and it is a dynamically determined variable depending on the value of the imposed *Q* and the curvature of the parabola. We discuss two specific geometries: a relatively wide and a relatively narrow channel which represent the parameter space we explored. The first (relatively wide) case study has *r*_{0} = 2/3 which resembles the bottom topography at Faroe Bank Channel (Borenäs and Lundberg 1988). The relatively narrow case has *r*_{0} = 1/6 resembling bottom topography of narrow gaps such as fracture zone canyons along mid-Atlantic ridges (Clement et al. 2017), and the western pathway of the Samoan passage (Alford et al. 2013) in the southern Pacific Ocean.

Figure 2 shows the numerical solutions of interface height and transport (depth-integrated) velocity (or equivalently gradient of streamfunction) for *r*_{0} = 2/3 with *Q* = 0.05 introduced in the form of boundary inflow, uniform and localized downwelling at the steady state (*t* = 6000). The Gill theory predicts that the inflow splits into two boundary currents that flow around the basin boundary to rejoin at the strait. Unlike the Gill model, the potential vorticity in the numerical simulations is nonuniform, in part due to the bowl shaped topography of the basin. Although boundary currents are present in the basin, we see additional features.

For the case of boundary inflow shown in Figs. 2a and 2d, the flow in the basin approaches the strait as boundary currents, but additionally we see the closed contours of interface height. The branch approaching the strait along the right wall (*x* > 0) crosses to the left wall as it approaches the channel entrance. The flow approaching the strait entrance is still rather weak and therefore geostrophically balanced and strongly steered by topography. Accordingly, the flow in the right-wall boundary layer is topographically steered over to the left wall, which now becomes a dynamical “western boundary,” and forms a boundary current there. Although the planetary vorticity is constant, the vortex squashing set by the decrease in layer thickness causes an increase in the relative vorticity of the flow close to the left wall. This veering to the opposite wall is also observed in laboratory experiments and is believed to account for the observation that the overflow carried by the North Icelandic Jet (NIJ) upstream of the Denmark Strait sill hugs the Iceland coastline, its dynamical western boundary (Pratt and Whitehead 2008). The separated East Greenland Current steering toward the Iceland coastline can potentially be explained by this veering as well.

For the case of uniform downwelling (Figs. 2b,e), the flow in the basin is mostly anticyclonic. The flow approaches the channel mostly from the left wall. At the channel entrance, some of the flow returns back to the basin forming an asymmetric anticyclonic circulation. When the downwelling is localized at the back wall (Figs. 2c,f), the flow is again fed to the channel from the left wall with very little reverse flow into the basin. The basin circulation is still predominantly anticyclonic, but is localized near the back wall. In general, the basin circulation has some qualitative similarities to what Helfrich and Pratt (2003) showed in their simulations for the rectangular channels; however, the flow in the channel especially at the entrance is different.

In all three cases, the flow enters the parabolic channel from the left wall, but is diverted toward the channel center forming a recirculation upstream of the sill and localized near the entrance (1 < *y* < 2.5) as is evident from Figs. 2d–f. This feature is persistent regardless of the inflow type in the basin. In the case of uniform downwelling (Figs. 2b,d) the flow approaches the channel entrance along the left wall. Some enters the channel and the remainder bypasses the channel and forms a reverse flow, back into the basin, along the right wall. Eventually, the transport velocity profile in the channel becomes nearly the same for all three cases downstream of the localized recirculation (*y* > 2.5). Several experiments with wider channels showed similar behavior but with a stronger recirculation and flow reversal near the entrance. Helfrich and Pratt (2003) also see a flow reversal at the entrance but it is much weaker. Previous theoretical studies showed that the flow reversal happens for channels with *r*_{0} ≥ 2/3 (Borenäs and Lundberg 1986; Pratt and Whitehead 2008) consistent with our simulations, though the location of the reversal is not always specified.

The interface height profile along the centerline of the channel and at two upstream cross sections are shown in Fig. 3. The depression in the interface height due to recirculation at 1 < *y* < 2.5 is visible for all three inflow types, but it is larger for uniform downwelling. The height profiles for all three cases coincide with each other after about *y* > 3.6. Notice that at the *y* = 2 cross section (Fig. 3b), although the height profiles differ in the middle of the channel, the heights are nearly identical at the right and left edges for all three types of inflow. This is also true all along the channel after *y* > 3.6. Although the basin circulation intrudes into the entrance, it does not affect the channel flow further downstream.

Plots of potential vorticity (pv) for these cases (Fig. 4) show that at the flow edges, where the layer depth approaches zero (e.g., *d* < 0.08), the pv becomes unrealistically large, therefore we removed the contours near the boundary from the plot, and we do so in all subsequent plots wherever the layer thickness *d* < 0.08. As expected, the pv distribution is nonuniform along and across the channel and it varies from 1.4 near the center of the channel to 4.25 closer to the edges of the flow. The pv contours get closer together as the flow narrows along the channel. The influence of mass source type can be seen again near the channel entrance. In the case of uniform downwelling (Fig. 4b), because of the return flow discussed earlier (Fig. 2e), some of the contours are diverted from the left wall back into the basin. Note that the pv contours downstream of the entrance (e.g., at *y* > 3.6) are nearly identical for all three mass source types.

Parabolic cross sections with higher curvatures (narrower channels) show a different regime of behaviors. We ran several experiments with radius of curvatures such as *r*_{0} = 1/3, 1/6, and 1/15. But we only show figures from our second case study with *r*_{0} = 1/6 as it is representative of all narrow channels considered. Figure 5a shows the interface height for the case of boundary inflow. We omit showing the plan view of the height contours for the case of uniform downwelling and localized downwelling as they are similar to that of boundary inflow with the difference that the flow approaches the channel mainly from the left boundaries.

Figure 5b shows the interface height profile at *y* = 2 for three mass source types. The three interfaces are indistinguishable and independent of mass source type. They also have a higher cross-channel slope in comparison to the first case study (*r*_{0} = 2/3, Fig. 3b) which has the same *Q* but a wider stream. Also the interface height slopes more linearly than the first case study which had a depression in the middle of the cross section.

Comparing the transport velocities in Figs. 5c–e for three mass source types with those of Figs. 3d–f, we see that the flow in the narrow channel is diverted toward the middle of the channel soon after it passes the entrance. Therefore, the basin circulation does not intrude into the channel further than *y* = 0.5, and the mass source type does not affect the flow further in the channel. In addition, the flow reversals in the mouth of the channel on the right wall (−1 ≤ *y* ≤ 0.5) are localized and quite smaller in comparison to the first case study. For the experiments we ran with even narrower parabolas (e.g., *r* = 1/15), the channel flow is independent of basin circulation even at the entrance (e.g., *y* ≤ 0.5).

The friction coefficient *ϵ* in the cases discussed above was 0.01. We performed a sensitivity study to the friction coefficient by running experiments with 0.0001 ≤ *ϵ* ≤ 0.05. The flow becomes unsteady for small values of friction coefficient. The critical *ϵ* which causes the flow to transition to an unsteady state depends on mass source type similar to what Helfrich and Pratt (2003) found. For example, for the case of *r*_{0} = 2/3, *Q* = 0.05, and (uniform or localized) downwelling mass source with *ϵ* < 0.005, the flow does not reach steady state, whereas for boundary inflow the cutoff for transition to unsteady regime is *ϵ* < 0.0025. The steady-state flow at the entrance has some sensitivity to the friction coefficient. As we saw earlier in Fig. 3, the steady-state flow at the entrance is sensitive to the mass source type. As *ϵ* increases (e.g., from 0.01 to 0.04), the sensitivity to the inflow at the channel entrance decreases, and the interface height slopes more linearly. Similar to the interface height, the pv contours become more uniform along the channel for larger *ϵ*. Also, sensitivity to *ϵ* is less in narrow parabolas. Further study of the properties of the unsteady flow is beyond the scope of this paper.

### b. Comparison of simulations with theory

Although the theory for constant pv flow in a parabolic channel is known, graphical tools that help with interpreting the solutions have been lacking. To better understand hydraulic behavior of the theoretical flow, we turn to a diagram that shares some features to the Froude number plane developed by Armi (1986) for two-layer flows. Instead of the Froude number plane, we use the (*γ*_{1}, *γ*_{2}) plane as in Fig. 6a.

Recall that *γ*_{1} can be interpreted as the width of the flow and *γ*_{2} as twice the centerline position of the flow, respectively, at any given cross section. There are three sets of theoretical curves in Fig. 6a. The thick black dashed line is the solution of the critical condition [Eq. (19)] for a specified parabolic geometry with radius of curvature, *r*, and a fixed *q* value, *q*. The blue and red curves correspond to the contours of constant *Q* and *r* and *q*, as evaluated using Eqs. (16) and (17). Note that the critical solution and the *γ*_{1} axis, and the *Q* contours are antisymmetric (the mirror image for *γ*_{2} < 0 is not shown).

The graph can be used to develop intuition about the behavior of the flow as the fluid passes through the channel and encounters a varying bottom elevation by tracing along the constant *Q* curves. The position on any constant *Q* curve is determined by the local topographic elevation at the channel center, as given by the *h*_{0}(*y*) as *γ*_{1} and *γ*_{2} vary along the channel. Notice that the contours of constant *Q* and *h*_{0} is not changing in *y*, i.e., at the sill. Additionally, the two sets of curves suggest that the critical flow occurs where *h*_{0}(*y*) reaches a maximum (i.e., at the sill). Below the dashed line the flow is subcritical and above it the flow is supercritical. Here, “supercritical” flow implies that the two long waves of the system, of which is trapped to the left edge and the other to the right edge of the current, both propagate in the downstream direction. For subcritical flow, the wave trapped to the left edge propagates upstream while the other wave propagates downstream.

Tracing a constant *Q* curve, the dynamic width of the flow tends to be wide at the channel entrance (large *γ*_{1}), and it narrows as the flow moves along the channel, passes over the sill, and then becomes supercritical. Also, the centerline position of the flow (*γ*_{2}/2) tends to increase as the fluid moves downstream. Thus, the flow veers to the right and rides higher up on the right-hand bank of the channel.

*r*=

*r*

_{0}= 2/3 and constant

*q*varies within the basin and along the channel. We replace the constant

*q*in the theory by a representative average value

*q*near the entrance is different from the average

*q*near the sill section. We choose to use the average

*q*both near the channel entrance region 2.5 <

*y*< 3.5 away from the recirculation region, and around the sill section at 5.5 <

*y*< 6.5 to compare the differences. Since the flow width dynamically changes along the channel, the integration width in the cross-channel direction changes as well. As discussed before, the

*q*value becomes unrealistically large near the wetted edges. To avoid this, the integration is calculated only in the portion of the channel width where the layer thickness

*d*is larger than 0.08:

*Q*= 0.05 the average values of

*q*become

*q*near the sill to construct Fig. 6a. The simulation with imposed

*Q*= 0.05 results in the interface height profiles shown in Figs. 6b–d near the entrance at

*y*= 2, the sill section at

*y*= 6, and a downstream section at

*y*= 7. Pairs of

*γ*

_{1}and

*γ*

_{2}are calculated by finding the intersection of the interface height with the bottom topography at these three sections and are mapped on to the theoretical graph as green, yellow, and cyan circles in Fig. 6a. Note that as the flow moves along the channel from the upstream section to the downstream section (Figs. 6b–d), the flow regime changes from subcritical to supercritical. Also, the three circles in Fig. 6a show that the stream narrows (

*γ*

_{1}decreases) and the centerline of the flow shifts to the right (

*γ*

_{2}increases) consistent with theoretical predictions. The circles fall along (or close to) the theoretical curve of constant

*Q*= 0.05 demonstrating the ability of the theory to predict the transports, given pairs of gammas measured at any section along the channel. From the simulation, we know the topographic height difference between the three sections. For example, the topographic height of the entrance section (Fig. 6b) is 0.8 and that of the sill section (Fig. 6c) is 1. By looking at the location of the green and yellow circles on the curves of constant

We ran several simulations with various imposed *Q* and mass source types. The results reveal that the theoretical curves in Fig. 6 are helpful in estimating the transport given pairs of *γ*_{1}, *γ*_{2} measured at any section along the channel. However, the experiments with different *Q* values tend to produce different *q* distributions, and the figure itself is constructed using one constant *q* value. Changing the constant *q* used to construct the graph to some other value (e.g., from

### c. Monitoring sill transport

*γ*

_{1}–

*γ*

_{2}space into a (

*q*,

*Q*) space for the critical values of gammas. To do so, Eqs. (16) and (19) are solved simultaneously to obtain pairs of critical gammas (

*γ*

_{1}

*,*

_{c}*γ*

_{2}

*) for a range of*

_{c}*q*and

*Q*values. The two equations are of the general form

*γ*

_{1}

*and*

_{c}*γ*

_{2}

*in (*

_{c}*q*,

*Q*) space is depicted in Fig. 7a for

*r*= 2/3.

The theory is derived for uniform *q*, therefore, from the theoretical point of view, the *q* at the sill is the same as anywhere else in the channel. However, this is not true in our simulations where variation in *q* is allowed. To compare the uniform *q* theory with our simulations, we compute the average *q* both near the sill [from 5.5 < *y* < 6.5 in Eq. (22)] and also upstream [from 2.5 < *y* < 3.5 in Eq. (22)] to study the sensitivity of the theoretical estimation to the choice of *q* in the (*q*, *Q*) space. The yellow symbols in Fig. 7a refer to the simulation data points when the averaged *q* is calculated at the sill. The green symbols refer to the simulation data points when the averaged *q* is considered near the entrance. Notice that there is a predictable drop in *q* from the entrance section to the sill section. The shapes of the symbols refer to the mass source type. The values of *γ*_{1}* _{c}* and

*γ*

_{2}

*for each*

_{c}*Q*are nearly identical regardless of the inflow type, implying the insensitivity of the flow at the sill to the basin circulation. The theoretical values of

*γ*

_{1}

*and*

_{c}*γ*

_{2}

*are then compared with the mapped simulation points in the (*

_{c}*q*,

*Q*) space. The results reveal that the simulated

*γ*

_{1}

*and*

_{c}*γ*

_{2}

*are well predicted by the theory as shown in Figs. 7b and 7c. The simulated*

_{c}*γ*

_{2}

*values are less than those of the theory for all the data points. This could be due to the effect of friction in the model resisting the flow tendency to pile up more on the right wall. The simulation–theory difference in estimating*

_{c}*γ*

_{1}

*and*

_{c}*γ*

_{2}

*is within 1%–11% and 10%–20%, respectively, regardless of whether the average*

_{c}*q*is considered at the upstream or at the sill section. Running simulations with different friction coefficients (

*ϵ*) revealed that the

*q*increases slightly as

*ϵ*increases. However, the values of the gammas at the sill remain nearly unaffected. This minimal sensitivity to the choice of averaged

*q*in (

*q*,

*Q*) space makes Fig. 7a more promising than Fig. 6a for the purpose of transport estimation at the critical section as it is less sensitive to the average

*q*.

Figure 7a suggests various possible monitoring strategies. For example, the first strategy is to measure *γ*_{1}* _{c}*,

*γ*

_{2}

*, as well as the averaged*

_{c}*q*to substitute in the transport relation [Eq. (23a)] without using the critical condition [Eq. (23b)]. To show the sensitivity of the transport estimate to the differences in

*γ*

_{1}

*and*

_{c}*γ*

_{2}

*from the model and theory, the values of transport from theory are compared with those of the simulations by substituting the modeled value of pairs of*

_{c}*γ*

_{1}

*and*

_{c}*γ*

_{2}

*in the theoretical formula for the transport along with a choice of averaged*

_{c}*q*. Figure 7d shows that the model–theory difference in estimating the transport is within 8%–18%.

The second monitoring strategy exploits the critical condition [Eq. (23b)] to eliminate *q*, and only measure the *γ*_{1}* _{c}* and

*γ*

_{2}

*(or equivalently*

_{c}*a*and

_{c}*b*). Therefore, by taking the gamma values from the model, we can estimate the corresponding

_{c}*Q*from the intersection point of the constant-gamma curves in Fig. 7a. Since the critical gammas are nearly identical regardless of mass source type, the closest intersections of gammas to the model data points are shown with green stars (the closest intersection to each of them results in the same point). Doing so results in Fig. 7e, which shows 7% difference between theoretical prediction of

*Q*and the modeled values. The second strategy only requires two quantities to be estimated and provides more accurate estimates than the first strategy by eliminating the need to measure

*q*.

The green symbols in Fig. 7a fall close to the extrema of the *γ*_{2}* _{c}* contours (shown by the thick black curve), suggesting that for a given

*Q*, the value of

*q*at the entrance is selected so as to maximize

*γ*

_{2}

*, the centerline position of the stream. Note that larger values occur when the stream is banking more highly on the sloping right-hand side of the parabola. This is true for other parabolic curvatures (both narrow and wide parabolas) we tried, and it suggests that (for reasons unknown) the flow tends to maximize its potential energy. Similarly, Helfrich and Pratt (2003) found that the flow chooses a*

_{c}*q*that maximizes an upstream parameter (upstream height on the right wall in their case). This is advantageous as it suggests a third monitoring strategy allowing for prediction of the overflow transport solely based on measuring one of the wetted edges,

*a*or

_{c}*b*.

_{c}In Fig. 8a, we replot Fig. 7a using contours of *a _{c}* or

*b*. The black curve which marks the extrema of

_{c}*γ*

_{2}

*intersects with contours of*

_{c}*a*and

_{c}*b*, which allows for estimating the transport based on only one of the wetted edges as in Figs. 7b and 7c.

_{c}The spread of simulation data points suggests that measuring *a _{c}* is more accurate but less precise, and measuring

*b*is more precise but less accurate in comparison to the theoretical result. The difference between model and theory in estimating transport is between 1% and 15% when the right wetted edge (Fig. 8b) is used and 5%–25% when the left wetted edge (Fig. 8b) is used to estimate the transport. Note that the third strategy stems from the assumption that the flow chooses a

_{c}*q*along the extrema of

*γ*

_{2}

*, as some of our simulations suggest. This emphasizes the dynamical importance of*

_{c}*γ*

_{2}

*as the centerline position of the flow for transport estimation.*

_{c}The reader may have noticed that in Fig. 8a, a (blue) curve of constant *a _{c}* may intersect a (red) curve of constant

*b*in two places. This implies that for a given choice of scales for any particular application, there may be two critical states, each with their own (nondimensional) potential vorticity

_{c}*q*and volume flux

*Q*, and its own velocity and depth profiles, that share the same edge positions. In such cases, factors such as stability or the presence of unrealistic features such as reversals in along-channel velocity, may preference one state over the other. This is an interesting dilemma, but one that we have not explored.

### d. Application of monitoring strategies to observations at FBC

The theoretical graphs in Figs. 7 and 8 can be used to estimate transport for a given sill section with *r*_{0} = 2/3 such as Faroe Bank Channel. Let us begin with the second monitoring strategy, which estimates the transport based on the *γ*_{1}* _{c}* and

*γ*

_{2}

*. To apply the theory, we first need to fit a parabola to the FBC sill section and find the wetted edges of the flow. Lake et al. (2005) made an estimate of interface height at the FBC sill section by fitting an idealized parabolic topography. The wetted edges from their estimate is*

_{c}*a*

^{*}= 5 km and

*b*

^{*}= 8 km. They also estimated

*q*from daily averaged ADCP records (see their Figs. 11 and 13). The corresponding scaling parameters for FBC are

*f*= 1.3 × 10

^{−4}s

^{−1}and

*g*′ = 0.004 m s

^{−2}. We use the depth scale

*H*= 1000 m. This results in a deformation radius of

*L*= 16.6 km that can be used to nondimensionalize the wetted edges of the parabolic fit to the Faroe Bank Channel sill section. Therefore,

_{d}*γ*

_{1}

*and*

_{c}*γ*

_{2}

*are 0.78 and 0.18, respectively. If the value of*

_{c}*γ*

_{2}

*is adjusted by 0.05, this pair of gammas intersect at*

_{c}*q*= 2.64 and

*Q*= 0.045. Therefore, the second monitoring strategy can be applied. We can recover the dimensional

*q*and dimensional transport as

^{6}m

^{3}s

^{−1}). These values are within the range of observed values at the FBC. Note that the value of adjustment is suggested by Fig. 7 and a study to explain the physical motivation (friction or otherwise) is beyond the scope of the paper.

The first monitoring strategy requires the measurement of *q* in addition to the wetted edges. The measured *q* within the course of 70 days varies which makes it inherently ambiguous to apply the uniform *q* theory. However, considering an average *q* of 3 × 10^{−7} m^{−1} s^{−1} yields the same results as the second monitoring strategy. Choosing a higher or lower averaged *q*, between 2 and 4 × 10^{−7} m^{−1} s^{−1} changes the estimated transport of 1.7 Sv only by about 5%.

The third monitoring strategy eliminates the need to measure both wetted edges. Let us assume that only the left wetted edge, *a*^{*} = 5 km, is provided from observations. Considering the same *L _{d}* = 16.6 km from the scaling parameters discussed earlier, the nondimensional wetted edge is

*a*= 0.3. Figure 8c gives

*Q*= 0.069, therefore,

*Q*

^{*}= 2.5 Sv. Similarly, if only the measurement of the right wetted edge is available, then

*b*= 0.48 yields

*Q*= 0.045 from Fig. 8b and the dimensional transport is

*Q*

^{*}= 1.7 Sv.

## 5. Summary and discussion

The dense flows in many deep passages in the ocean are the result of the funneling of much broader water masses through narrow gaps and are often hydraulically controlled. Both these factors render the sites advantageous for long-term monitoring of transport. One goal of hydraulic theory is to estimate transport using the minimal number of measurements. Previous theoretical studies have been restricted to zero or uniform *q* and to channels with rectangular cross sections. In this paper we explored the effect of continuously varying (parabolic) channel geometry on the rotating hydraulic behavior of a channel and basin system. To do so, we extended the existing theory for the parabolic channel (Borenäs and Lundberg 1986) by creating a graphical representation that shows the solution behavior more easily and by exploring different strategies for overflow monitoring. We also used a 1.5-layer, reduced gravity numerical model of coupled basin–strait flow, thus freeing ourselves from the restriction of uniform pv and semigeostrophic dynamics.

The results revealed some similarities and some differences to rectangular channels in the flow structure and properties. Helfrich and Pratt (2003) found that the flow in the rectangular channel with a specified width and imposed *Q* is independent of the mass source distributions in the basin. In the parabolic channel the sensitivity of the flow to the mass source depends on the curvature of the parabola. The basin circulation intrudes more into the channel for parabolas with *r*_{0} ≥ 2/3 (we ran simulations with *r*_{0} = 1 and the results was similar to those of *r*_{0} = 2/3). The intrusion extends from the entrance to up to one-third the distance to the sill, but the flow further downstream and at the sill remains unaffected by the basin circulation, similar to the case of the rectangular channel. The flow characteristics in narrow parabolas are very much similar to the rectangular channel with little sensitivity to the basin circulation even near the channel entrance. The flow generally enters the channel from the left wall for both the rectangular and parabolic bottom topography. However, the parabolic geometry directs the flow to the center of the channel faster than the rectangular channel does [e.g., compare Helfrich and Pratt’s (2003) Fig. 3b with our Figs. 2d and 5c] so that a cell of recirculating fluid can arise in the left portion of the channel, just downstream of the entrance. Due to the grounding of the interface on the two edges, the flow in the parabolic channel experiences a range in *q* that is much larger than for the rectangular cross section.

We recast the volume flux formula and Bernoulli potential in terms of variables such as the flow width and the position of the flow centroid for a given geometry. This illuminates the flow evolution and hydraulic control theory in the channel. We establish a dynamical connection between the sill section and the overflow transport by proposing three monitoring strategies. The strategies require measuring the intersection of the overflow interface height with the bottom at the sill section. The first method does not involve solving the critical condition, but it requires computing *γ*_{1}* _{c}* and

*γ*

_{2}

*from the measurement of the two wetted edges and also an estimate of averaged*

_{c}*q*. The second method eliminates the need to provide

*q*measurements by solving the critical condition and only requires finding both wetted edges of the flow at the sill section. The third monitoring strategy requires only measuring one of the wetted edges. The comparison between simulation data points and the model suggests that the second monitoring strategy is more accurate. It is also more advantageous than the first monitoring strategy as it eliminates the error associated with inserting an average

*q*value in the uniform

*q*theory. The third monitoring strategy requires only one measurement, either the left or the right wetted edge. The third method is more accurate than the first method, and has the same range of accuracy as the second method. Measuring the right wetted edge yields a better agreement between simulations and theory than the left wetted edge. Although the third monitoring strategy is advantageous as it requires making only one measurement, it relies on the assumption that the

*q*of the flow is chosen (such that

*γ*

_{2}

*is maximized). Therefore, the second monitoring strategy makes fewer assumptions.*

_{c}The pioneers of rotating hydraulics theory focused on relating the volume flux to hydrographic measurements made upstream of the sill. The motivation was mostly to avoid the difficulty of making direct observations at the sill section itself. For example, early current meter moorings placed in the Denmark Strait, were not withstand the high velocities there (Worthington 1969). Given that for major sills and passages in the global ocean, the modern monitoring and measurements are made at the sill section, it can be more advantageous to focus on relating the transport to the measurements made at the sill itself. All transport relations suggested in the literature assume steady state condition. However, the conditions downstream of a sill are often turbulent and time varying. Therefore, one advantage of monitoring the overflow at the sill is that flow there is being squeezed through a very small cross section and therefore less coverage is required to resolve the stream. The second advantage of estimating transport based on sill measurements is that the flow is often hydraulically controlled at a large sill. As a result, theoretical predictions can provide a more accurate estimate of the transport at the sill than the transport formulas based on upstream measurements.

Our results revealed that for a continuously varying topography, the flow near the channel entrance is influenced by various mass source types and basin circulation. Therefore, relating the transport to upstream measurements may be less accurate compared to measurements made at the sill. Certain important passages such as the Denmark Strait contain hydraulically controlled flows that are influenced by rotation and that contain at least two active layers. Some theoretical development has taken place with such flows in mind [e.g., works by Hogg (1983), Timmermans and Pratt (2005), and Pratt and Spall (2008)], but no systematic study of monitoring the value of such theories has yet been undertaken. We applied the three monitoring strategies to the case of Faroe Bank Channel and the results showed that the estimated transports are within the range of observed values. One issue with all three monitoring strategies is the practicality of finding the interface height from observations, and deploying instruments close to the bottom especially for passages with complex geometries. Nevertheless, the theoretical progress still provides a transport estimate consistent with simulations and observed values with only 1–3 measurements and worth exploring in more complex models as a potential approach to enhance observing systems.

## Acknowledgments.

This work was financially supported by the U.S. National Science Foundation under Grants OCE-1657870, and OAC-1835640, and by NASA 80NSSC20K0823. Pratt was supported in part by KAUST through Award CRG-2017-2308. The work highly benefited from consulting with Shuwen Tan about the Samoan Passage.

## Data availability statement.

No observational datasets were generated or analyzed during the current study. The numerical model can be made available upon request.

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