1. Introduction

In the classical hydraulics for the steady inviscid discharge of a single layer of dense fluid from a reservoir into a long channel of transverse width L(y) and bottom elevation M, a steady quasi-laminar downstream (y) velocity υ(x) and cross-stream thickness h(x) are assumed, so that the Bernoulli function on each streamline is equal to its reservoir value; furthermore, the potential vorticity (PV) is assumed to be uniform. Besides the volume discharge Q, there is (at least) one more unknown [say h(0)] than equation, so that some form of the Hugoniot control condition is required to close the problem and to determine Q as a function of the reservoir head (relative to the sill). In one version of the Hugoniot condition, a topographic change (say δL) is made and δh(0)/ δL is computed. If it is finite, the flow is not controlled at this section, because δL merely produces a local perturbation without affecting Q. Otherwise, if a bifurcation point (δh/δL = ∞) exists, the flow Q is controlled¹ (although not necessarily at the section in question). Because δh/δL = ∞ implies δh ≠ 0 for δL ≡ 0, the branch-point condition is equivalent to the stationary-wave condition in a straight channel. These alternate conditions are also equivalent to a Q extremal; that is, ∂Q/∂h(0) = 0 (Killworth 1995). It is important to note, however, that some free discharges (see the appendix) are “controlled” in the ordinary sense of the word, even though the stationary-wave condition is irrelevant. Such simple examples point out the need for a wider conception of what constitutes control.

The stationary-wave condition has also been carried over to highly unsteady exchange flows such as exist in the Bosphorus [see Gregg and Özsoy (2002) for comprehensive field observations and numerical calculations pertaining thereto]. This exchange problem is beyond our scope, however, and only one-way flow (nonreversing) over a simple sill or through a narrows is considered.

The hydraulic approach has been extended to the free discharge over a rotating sill (Whitehead et al. 1974; Stern 1974, 1975; Gill 1977; Shen 1981; Borenas and Lundberg 1986; Helfrich and Pratt 2003) with application to the important problem of the overflow of the cold bottom water, such as occurs in an oceanic strait (Fig. 1). In these theoretical studies (typically in a rectangular channel), there is a dynamically active dense layer in the upstream reservoir, and the overlying slightly lighter layer is resting; thus a reduced-gravity (g′) model applies. Upon initiating the discharge, the dense fluid flows over the sill and establishes a steady flow with a uniform PV equal to that assumed in the reservoir. As previously mentioned, the problem is closed by applying the Hugoniot condition. With certain other conditions (Gill 1977) this gives Q as a function of Coriolis parameter f, reduced gravity g′, reservoir head H∗, and the channel dimensions. Mention should be made of the Killworth (1994, 1995) papers, which show that for general PV and M(x) the value of Q has an upper bound; this bound is realized in the PV = 0 calculation of Whitehead et al. (1974). Killworth showed that for a very general class of PV the Hugoniot condition is equivalent to extremizing Q. Killworth (1992) also considered the nonuniform PV problem for a very special Bernoulli function in the upstream reservoir.

For consistency in these long-wave theories, υ(x) ≥ 0 is assumed, and then the negative definite relative vorticity requires the minimum channel width to be less than a certain multiple of the radius of deformation (g′H∗)^1/2f⁻¹. In the case of wide ocean straits (Fig. 1) there is another limitation because each (synoptic) realization contains large-scale eddies (Spall and Price 1998) and dynamically significant cumulative bottom friction, which modify the fluid in its long passage from the reservoir to the sill. Thus, the time-average Denmark Strait flow (Fig. 1) consists of a jet whose Bernoulli function and PV are not known a priori (Spall and Price 1998). Because of the turbulence, the mean flow poses a statistical mechanical problem in which the mean channel velocity υ(x) is underdetermined. As a consequence, it is necessary to articulate a control principle (section 2a) so that it may be applied to the mean of the turbulent flow. The spirit of this investigation is similar to that in turbulent thermal convection (cf. Howard 1972) and turbulent shear flow (Stern 1980); the goal is to approximate certain integral features of a real and time-dependent channel flow.

In section 2a the generalized control principle is stated, and the implication is that the channel discharge (Q) is to be extremized subject to certain constraints; most notable is the requirement that the mean υ(x) contains a branch point (section 2b). Note that the foregoing two conditions are equivalent in classical hydraulics. When the first (Q) of these two conditions is taken alone, the result is illustrated in section 2c. When the second condition is taken alone, the result is illustrated in sections 3 and 4. When the two conditions are taken together, a problem in the calculus of variation (section 3) is obtained (but the latter is not solved here).

The most interesting, and oceanically relevant, problem should take into account a variable cross-stream bottom topography that is intersected by the interface of the density layer. The simplest version of this free-streamline problem (see Fig. 6 below) is considered in section 4 in which the stationary-wave condition is obtained, but the incorporation of the Q extremization is left for subsequent work. The results obtained are used in section 5 to argue that the observed mean flow in the Denmark Strait is indeed controlled by the topography.

Sections 3 and 4 may be read independently as specific examples of the effect of nonuniform PV [due to variations in h(x), υ(x), or M(x)] on the stationary-wave condition.

2. The control principle

a. Generalized Hugoniot condition

A definition can be given, as follows: if all small local topographic changes (±δL, ±δM) at a given section in a strait merely result in a local υ(x) perturbation without changing the reservoir state (e.g., the discharge Q) then the upstream state is not controlled at this section; otherwise the flow is controlled (either at this section, or some other one).

This definition has operational content; in principle one could make topographic changes and see whether merely local flow changes are produced. The theoretical content of the definition emerges with the assumption that the straits in question do control Q and with the assertion that the underdetermined mean velocity υ(x, λ₁, … , λ_n), which contains more free parameters λ₁, … , λ_n (e.g., Fourier coefficients) than known dynamical constraints, should satisfy the control condition. By a cross-stream integration of υh one relates the reservoir discharge,

QQλ₁λ_nM,L(2.1)

to the straits. In classical hydraulics for a single layer flow, there is one more unknown (λ) than dynamical equations, and its value is obtained by the Hugoniot branch-point condition; in the turbulence problem, however, there are many more λ. We now assume that all the underdetermined λ solutions have finite a priori probability and that the most probable λ solution satisfies the control condition. The meaning of this is as follows.

At a given topography (M, L) consider a possible λ solution with the specific values (λ⁰₁, λ⁰₂, … , λ⁰_n)and then examine the effect of changing the local topography by (δM, δL). All concomitant changes δλ₁, δλ₂, … , δλ_n are also a priori possible. If the various derivatives ∂Q/∂λ exist (on both sides of λ⁰₁, … , λ⁰_n) then the change in transport is

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Now if ∂Q/∂λ_i ≠ 0 for any λ⁰_i, say λ⁰₁, then there certainly exists a solution with δλ₂, … , δλ_n = 0 and with

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for which

δQδM,δL

Because this violates our control hypothesis, the solution with λ⁰₁, λ⁰₂, … , λ⁰_n is not realized. On the other hand, our solution may satisfy the control condition if

Qλ_ii(2.4)

or if λ_i is an endpoint at which the derivative does not exist. Both of these contingencies are illustrated in Fig. 2 (see also the Hele-Shaw weir in the appendix for an illustration of an endpoint “extremum”). Thus, the control condition serves to select a discrete number of realizations from the continuum presented by the turbulence problem. Perhaps our theory is best stated by saying that because the solution (2.4) satisfies more requirements it is more probable than the other λ_i.

c. Illustrative example of δQ = 0

Suppose we have a (globally) turbulent flow passing through a channel with triangular bottom topography M(x); the two sides in Fig. 3 slope at angles α₀ and α₁, respectively. The current has “left-hand” support (h = 0) on α₀ > 0. The layer interface h(x) intersects the sides at elevations H₀ and H₁, respectively, and L₀ and L₁ are the respective distances of these intersections from the vertex of the triangle. Let us maximize Q by (temporarily) ignoring the stationary-wave constraint and by constraining υ(x) to be (uniformly) equal to the x average of the real geostrophic mean flow. Then

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If s = tanα₁/tanα₀ and λ = H₀/H₁, then a straightforward calculation gives

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where λ > 0 is a free parameter. For fixed H₁, Q is maximized at

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In particular, for s = 1 or α₀ = α₁, we obtain

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Equation (2.9) predicts an approximate value of the mean channel velocity in a controlled flow. In a subsequent variational approximation a uniform shear might be added to υ, and the stationary wave constraint might supplement the δQ = 0 condition.

3. Influence of variable potential vorticity on stationary waves in a rectangular channel

The potentialities of our formalism are best illustrated in the case of a rectangular strait. Let υ(x) be the steady mean downstream (y) velocity in a wide flat-bottomed channel with vertical sidewalls (Fig. 4), the left-hand wall (looking downstream) being straight. If the right-hand vertical wall is slightly curved in the downstream direction, it will force a slowly varying perturbation of the undisturbed upstream current. (This wall curvature may eventually be reduced to zero to obtain the condition for the existence of a stationary wave in a completely uniform channel.) At the upstream end of the channel, the vertical walls are located at x = 0, L, and the assumed semigeostrophic mean velocity is

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where h is the time-averaged layer thickness. At the downstream end, the vertical walls are located at ξ = 0, L₁, and

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is the mean geostrophic velocity. Consider a streamline at a distance x from the left-hand wall at the upstream end, and at a distance ξ from the wall at the downstream end. The geostrophic volume flux between this streamline and this straight wall is given by

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and the equivalent flux at the upstream end is

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It follows that the downstream layer thickness is

h₁ξh²xa^1/2(3.3a)

where

ah²₁h²(3.3b)

As previously mentioned (section 2b) we assume a local Bernoulli equation υ²₁(ξ) + 2g′h₁(ξ) = υ²(x) + 2g′h(x) is satisfied, in which case

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When this and (3.3a,b) are substituted in (3.2) we obtain the first-order differential equation

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for the displacement ξ(x) as a function of υ(x) and h(x). The constant of integration a is obtained by integrating (3.4) across the entire channel. Because ξ goes from 0 to L₁ as h(x) goes from h(0) to h(L), (3.4) gives

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for a [= h₁(0) − h(0)] as a function of L₁ − L.

From (3.3b), we note that when a = 0 and h₁(0) = h(0) the integrand in (3.5) reduces to dh/υ = fdx/g′. Then the right-hand side of (3.5) reduces to fL/g′, and L₁ = L is recovered when a = 0. We now compute the value of a for small values of (L₁ − L) by differentiating (3.5) with respect to a and then setting this equal to zero:

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Because the derivative of (3.3b) is da = 2h₁(0)dh₁(0), we obtain

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This is positive and h₁(0) increases with L₁ − L if υ²(x) < g′h(x), but if the integral vanishes, that is, if

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then a branch point [∂h₁(0)/∂L₁ = ∞] occurs in the forced solution, and no undisturbed upstream solution will exist if the channel width decreases downstream. Thus, (3.7) is the critical Froude number condition for a controlled flow [as obtained previously by Stern (1975) and Pratt and Armi (1987)]. Equation (3.7) can also be interpreted as the condition for a stationary wave [amplitude dh₁(0) ≠ 0] in a channel in which both walls are straight (L₁ − L = 0).

If H₀ is the minimum layer thickness (at x = 0) and H is the maximum thickness (at x = L), then

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Equation (3.7) consequently reduces to

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According to the two parts of our control condition we extremize Q = (g′/f)(H² − H²₀) subject to (3.11), and the method of Lagrange multipliers implies that this is equivalent to extremizing

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with fixed Q, or fixed endpoints:

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Equation (3.11b) implies a second-order Euler–Lagrange equation for the structure of ĥ(η), and (3.11c) or ĥ²(L) − ĥ²(0) = 2Q(g′/f)⁻¹ supplies the boundary conditions. Equation (3.11b) serves to indicate the increased amount of information about a globally turbulent flow that is provided by our variational theory (section 2); without it, and with only the assumed Hugoniot condition (3.11a), we have only a single integral for ĥ(η). More important, our formulation of the general problem provides the basis for a variational approximation using suitable form functions for υ(x), as is illustrated below.

a. Constant υ(x)

Consider first the case in which υ is approximated by a uniform current (equal to the x average of the real υ), in which case ĥ varies linearly across the stream so that

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The transport

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depends on the maximum layer thickness H and on β. Then (3.11) gives two elementary integrals:

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whose evaluation yields

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The main result [from (3.13)] is that an arbitrarily wide (L) strait can provide a branch-point condition provided that β is large and positive; that is; the current is supported on the left-hand side wall, with H₀ = H(1 + β)⁻¹ > 0. We note in passing that the Froude number υ²/g′H₀ based on the minimum thickness (1 + β)β⁻¹ ln(β + 1) > 1 is “supercritical” whereas the right-hand value based on the maximum thickness Froude number β⁻¹ ln(β + 1) < 1 is “subcritical.” Note that the mean isopycnals in Fig. 1 do intersect the bottom on both sides of the strait, suggesting that the stationary-wave criterion may be relevant here. This conclusion is not foregone, however, because the flow might have separated from the left-hand side, as occurs in the numerical calculation of Spall and Price (1998) in which the overflow banks on the right-hand side of the channel. To further explore this qualitative point (of left-hand support) we will introduce a sloping bottom in section 4. First we want to investigate the effect of relative vorticity in the rectangular channel, however.

b. Variable υ

It is of interest to see how the stationary-wave condition is modified for a jetlike υ(η) with maximum υ at η = 1/2. In (3.11a) let

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where λ is a free amplitude parameter. Equation (3.11a) is then written as

βIβ,λI₁I₂(3.15a)

where

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Figure 5 gives the numerical calculation of F ≡ f²L²/g′H for any β and for λ ≤ 0. Figure 5 shows that, for any given Q, H, and L, a stationary wave can exist for a jet with prescribed amplitude; for example, if F = 2 and β = 6 then λ ≃ −0.04. Although the simple jet parameterization [in (3.14b)] is insufficient to give a Q (or β) extremum, there is a (well known) upper bound Q = (g′/ 2f)H² at any F. This bound occurs at β → ∞ and λ → −1/(2π), [i.e., υ(0) = υ(1) = 0], because lines of constant F intersect all λ = constant curves in Fig. 5.

4. Sloping bottom

More interesting (and relevant) than the rectangular channel in Fig. 4 is one with a sloping bottom that is intersected by the interface of the density current, thereby providing a free-streamline condition for the stationary-wave problem; the problem is to predict the width of the current in a free discharge. In the simplest case (Fig. 6), a uniformly sloping (α) bottom M(x) intersects a vertical right-hand boundary, and the velocity is approximated by a uniform geostrophic velocity U equal to the η average of the realized velocity υ. At the (dimensional) transverse coordinate η = L the layer thickness is H(η), with H(0) = 0 and H(L) = H₁. This problem differs from the classical trapped coastal wave (LeBlond and Mysak 1978) for which there is no vertical wall and no sloping density interface. Our problem also differs from the Killworth (1992) “global” solution which is based on a small uniform current in a wide upstream reservoir.

The plan view (Fig. 6) of the steady long wave shows a streamline whose distance from the η = 0 origin is η at the upstream section and y(η) at the downstream section. Although such a displacement might be forced by a transverse displacement of the vertical wall to y = L − δL, we shall henceforth consider the unforced free wave problem (δL ≡ 0) on a uniform slope α. Therefore the bottom elevation at the downstream streamline is

MyMηαyη(4.1)

In this Lagrangian formulation, the corresponding geostrophic velocity and layer thickness are respectively given by u(y), and h(y). The mass conservation equation uhdy = UHdη may be written as

ϕudyUdη,(4.2a)

where

ϕhyHη(4.2b)

A second upstream/downstream equation is supplied by the Bernoulli relation

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The third relation is provided by the geostrophic equations, whose upstream value is

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and whose downstream value u = (g′/f)dh/dy − (g;ga)/f is expressed as

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where dH/dy = dη/dy(dH/dη) is used.

For infinitesimal perturbations (primed quantities) from the upstream state, the downstream state is given by

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The linearization of (4.6), (4.2a), and (4.3) gives a complete set of linear equations for ϕ′, u′, and y′:

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with nonconstant coefficients [in H(η)].

In the full nonlinear problem the free boundary condition is h[y(0)] = 0, and in the perturbation problem (4.2b) requires that ϕ′(0) be finite. Because y′(0) must also be finite, we have the free boundary condition

yϕη(4.9a)

and the other boundary condition for the stationary wave is

yL(4.9b)

By subtracting (4.8b) from (4.8a) and then from (4.8c), there results

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Now the foregoing equations are nondimensionalized using

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where M(0) ≡ εH₁ denotes the topographic elevation on the left-hand side of the upstream channel. The various coefficients in (4.10)–(4.11) then become

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where

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Thus the nondimensional differential equations become

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By substituting the power series

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and collecting the coefficients of z⁰, we obtain the indicial equations

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For n > 0, the coefficients of zⁿ satisfy

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Thus, the recursion relation is

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When the amplitude normalization condition b₀ = 1 is used, (4.15) gives

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and (4.16) then gives the remaining b_n. Because S_n < 0 for all n, b_n is an alternating series, and (4.14c) converges at all 0 ≤ z ≤ 1. The regular solutions (4.14c) satisfy the boundary condition (4.9a), and (4.9b) will be satisfied if

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This gives the value of εF as a function of ε. For the interesting case of ε ≪ 1, and εF ≡ r = O(1), we get

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Equation (4.17) then reduces to

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where G_m is the sum of the first m terms and m → ∞. For G₄ = 0, we get the approximate root r ≅ 1.45, or

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Because the discharge is Q = ULH₁/2 and U = (g′/ f)(H₁ − αL)L⁻¹, we obtain

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The interesting result (4.20) predicts the width of the current at the control as a function of H₁; if α is sufficiently small, then L may be much larger than the radius of deformation. Also note from (4.13b) that g′αL/ U² = r, or

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For g′ = 0.1 cm s⁻², f = 10⁻⁴ s⁻¹, H₁ = 5 × 10⁴ cm, and α = (10⁻²)/4, we have L ≃ 2 × 10⁶ cm and Q = g′H²₁(2f)⁻¹(1 − 0.181). Of course, the cumulative effect of bottom friction will produce a jet with non uniform U(η); perhaps this effect may be taken into account in (4.8). With the increased parameterization of U(η) there may also be a Q extremum that predicts those parameters.

Also interesting in the context of the more general topographic problem is the case of α < 0 (Fig. 6b), which corresponds to a current that separates from the left-hand side of a channel (not shown) and banks on the right-hand side. In this case, ε < 0 and (4.18) cannot be satisfied; this condition means that the flow cannot be controlled at a section where “separation” occurs.

5. Summary and suggestions

The hope of all turbulence theories is to bypass the chaotic realization and predict the mean state; the Denmark Strait overflow provides a good example because of the strong geostrophic eddies observed therein. This paper addresses the qualitative question of whether such a wide current is actually “topographically controlled,” or whether the flow in the straits is merely locally forced without influencing the mean state of the Greenland Sea.

First of all it was necessary to develop a clearer conception of what constitutes “control,” such as is applicable to a wide class of fluid problems. Thus our generalization applies to the viscous Hele-Shaw weir (appendix) as well as to the quasigeostrophic problem that is the main focus of this paper. The distinguishing feature of the latter problem (as compared with classical hydraulics) is that there are many more unknowns (the “λ”) than dynamical conditions. Our generalization (section 2) of the classical Hugoniot idea to the case of a time-average flow requires that Q be extremized with respect to the degeneracy parameters and subject to a bifurcation condition on the mean flow. The scope of the new formulation is revealed by (3.11), which imply an Euler–Lagrange ODE for the structure of h(x). Additional constraints, as become known, may be necessary.

Although it will not be easy to solve for the stationary-wave condition, we may avail ourselves of the idea of a variational approximation, obtained by discarding conditions and/or constraining the form of the mean velocity profile [υ(x)]. Thus in section 2c we discarded the bifurcation condition, assumed that h(x) varies linearly, and maximized Q to predict the mean channel velocity (2.9). In section 3 we discarded the Q condition and evaluated the stationary-wave condition for arbitrary υ(x) (and PV) in a rectangular channel (Fig. 4). The main result (Fig. 5) is that a nonreversing and nonseparating jetlike υ(x) may exist at a hydraulic control section even if the width L is much larger than the radius of deformation. Because the mean hydrography of the Denmark Straits (Fig. 1) also has such left-hand support, it is suggested that the discharge could indeed be controlled.

Further investigation of this physically important point requires explicit consideration of bottom topography, extending the work begun by Killworth (1992). Perhaps the simplest relevant model (Fig. 6) is one with a uniformly sloping (α) bottom that intersects a vertical right-hand wall (at η = L), and which intersects the density interface at the (upstream) free streamline position η = 0. We also discarded the Q condition, constrained the form of υ to be uniform across the stream, and computed the condition for a stationary wave. Perhaps the most significant result of this paper is the prediction (4.20) of the width of the current, but when α < 0 (cf. Fig. 4b) there is no stationary wave. Because Fig. 4b corresponds to a flow that banks on the right-hand side of a strait, we conclude that this cannot be a control section; the latter condition requires the current to be supported on the left-hand side of the channel, that is, no separation.

These conditions may be extendable and testable in a laboratory experiment (see Whitehead et al. 1974) like that in Fig. 7, where the right-hand wall curves slightly into a wide upstream reservoir and the left-hand side of the current is supported on a uniformly sloping bottom. Geostrophic eddies in the reservoir may be produced by injecting a thin jet at middepth. Even if the resulting channel flow is quasi-laminar, it will not be possible to connect it (deterministically) to the source, in which case the experiment may provide a test of the ideas proposed herein.

Note added in proof. Why was U = constant chosen in section 4? For variable U(x) = g′/f∂(h + M)/∂η, the discharge is

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For fixed (α, L, H₁), the last term increases as ∫ ηU dη ≤ U_maxL²/2, and therefore maximum Q occurs for U(x) = U_max. This “endpoint” condition and the branch-point requirement (4.17) satisfy both control requirements (see section 2a and Fig. 2).