1. Introduction

Hydraulic criticality and control are concepts that date back to work by Reynolds (1886) and Hugoniot (1886) in the area of gas dynamics. The same ideas hold for steady, shallow flow over a dam and have more recently been applied to deep ocean overflows. In these applications, “critical flow” refers to a steady state that supports stationary nondispersive waves of small amplitude. In most cases the nondispersive waves are “long” waves, but there are exceptions (e.g., Baines and Leonard 1989). Changes in topography at a section of hydraulically critical flow lead to changes in the overflow transport and other properties. “Critical sections” often occur at points of constriction such as the crest of a dam or the narrowest section of a wind tunnel. It is also normal for the flow upstream of the constriction to be “subcritical,” meaning that long-wave propagation in both up- and downstream directions is permitted, and for the downstream flow to be supercritical, meaning that propagation only in the downstream direction is permitted. The transition between subcritical and supercritical states occurs at or near the most constricted section and the flow there is said to be critical. Generalizations have been made to situations where the flow is bidirectional and “upstream” and “downstream” are less clearly defined (e.g., Armi 1986).

Hydraulic transitions and the occurrence of critical flow have profound implications for the physics of the flow. A hydraulic transition implies a downstream region of supercritical flow subject to enhanced mixing and entrainment and the formation of hydraulic jumps. Upstream effects are also important and include partial blocking of the transport and regulation of the stratification in the upstream basin. The critical condition can be used in principle to write down a “weir” formula relating the transport to hydrographic properties of the upstream flow (e.g., Whitehead 1989; Borenäs and Nikolopoulos 2000). Such formulas potentially serve as a basis for long-term monitoring of the transport (e.g., Hansen et al. 2001). Weir formulas traditionally assume that critical flow occurs at the sill, but nonconservative processes such as bottom drag and entrainment can cause the critical section to lie elsewhere (Pratt 1986; Gerdes et al. 2002) with consequences for the weir formula. All of these factors make the diagnosis of flow criticality and the location of the critical section important in the analysis of deep ocean overflows and of other hydraulically driven flows of geophysical relevance.

For nonrotating, homogeneous, shallow, free-surface flows, the standard indicator of the hydraulic state at a given cross section is the Froude number υ/(gd)^1/2, with υ denoting the depth-independent horizontal velocity, d the fluid depth, and g the gravitational acceleration. This result can be generalized to a deep layer flowing underneath an inactive upper layer, and the appropriate Froude number is obtained by reducing g in proportion to the relative density difference between the two layers. The flow is subcritical, critical, or supercritical depending on whether υ/(gd)^1/2 is <1, =1, or >1, an interpretation valid where the variation of υ and d across the flow is weak. Although this assumption may be valid for certain streams and open channels there are other applications where such variations are large and where the appropriate Froude number is not known. Examples include the deep ocean overflows such as those of the Denmark Strait, the Faroe–Bank Channel, the Jungfern Passage, and the Vema Channel. The combination of rotation and complicated cross-sectional geometry lead to significant variations in υ and d across the passage in question.

The belief that subcritical-to-supercritical transitions take place is founded on the observed “overflow” character itself: the spilling of dense water from one ocean basin into another. The upstream/downstream asymmetry and resemblance to flow over a dam has led to the presumption that hydraulic transitions are taking place, but this conjecture has almost never been verified by direct measurement in settings where rotation is strong. Theories for homogeneous, rotating-channel flow (i.e., Whitehead et al. 1974; Stern 1974; Gill 1977; Borenäs and Lundberg 1986) have resulted in the formulation of generalized Froude numbers, but application is often limited by inherent idealizations such as restriction to a rectangular cross section or to uniform potential vorticity. Here we will present two new measures of criticality that are less restricted. Both are based on a multiple streamtube representation of an overflow, constructed by dividing observed flow at a particular section into transport and energy conserving subsections. The first measure is obtained by asking whether a stationary wave can exist at the section in question. Development of the formal criterion requires that one generalize the Gill (1977) approach (reviewed in section 2) to a system with an arbitrary number of degrees of freedom (section 3). The general form of the critical condition and associated compatibility condition for such a system are derived and applied to the streamtube model (section 4). As an aside, it is shown by example that Gill’s original critical condition can fail to capture all possible critical states, but that this problem is fixed when the multivariable approach is used. As it turns out, the criterion is well suited in application to an analytically or numerically modeled flow, but requires more data than are typically available in field studies. A second condition based on direct calculation of long-wave speeds is then developed (section 5) and shown to be better suited to oceanic data. The wave speed calculation is based on a time-dependent version of the multiple streamtube model and the result is directly tied in to the extended Gill formulation for steady flow.

This work also addresses several issues closely connected to the concept of hydraulic criticality. One concerns the significance of the local Froude number F in cases where this quantity varies across the flow (section 6). We discuss the differences between local propagation of free disturbances and hydraulic control with respect to a normal mode. A result of this discussion will be the conjecture that F must equal unity at some point across the section in order for the flow to be hydraulically critical with respect to a long normal mode. We also discuss the conditions for hydraulically criticality in certain cases in which traditional hydraulic approximations are invalid (appendix A).

2. Hydraulics in a single variable: The Gill (1977) approach

If “hydraulic criticality” is defined as a steady flow state that supports stationary long waves of infinitesimal amplitude, then the methodology of Gill (1977) can be used to write down conditions for critical flow. The approach applies to a steady, conservative flow in a conduit or waveguide that varies gradually in the longitudinal (y) direction. Conserved properties might include volume and/or mass flux, energy, density (or potential density), and vorticity (or potential vorticity). It was assumed by Gill that the corresponding conservation laws can be combined, reducing the number of dependent variables to a single unknown γ(y). For flow in a rotating channel of rectangular cross section γ(y) is often chosen as the average of the layer thicknesses on the two side walls. Knowledge of γ(y) at a particular “section” y of the conduit determines all the flow characteristics (depth, pressure, etc.) at that section. The conservation laws relate γ(y) to the local geometric properties of the conduit h(y), w(y), and so on by an equation of the form

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where C is a constant. For example, h(y) and w(y) could represent the bottom elevation and width of a channel of rectangular cross section. The relation (2.1) also depends on parameters B, Q, and so on, corresponding to the values of the conserved quantities. By definition

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which is often just the differential form of a momentum or continuity equation.

Now consider the conditions under which free, stationary long waves of small amplitude exist. By “long waves” we mean disturbances that vary gradually in the y direction, just as the steady flow does. By “free” we mean disturbances that occur spontaneously and are independent of any forcing mechanism, including bottom slope. When a steady flow becomes hydraulically critical at a particular section y = y_c, it can support a stationary disturbance at that section. In other words, the steady state can be locally altered without changing either the conduit geometry or the upstream conditions. The altered flow must therefore have the same volume flux, energy, and so on, as the undisturbed flow and G must have the same value. In formal terms, there must exist a small disturbance that preserves the value of G and is free of variations in h, w, and so on:

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Here and elsewhere the notation ()_x,y,... means that x, y, . . . are held constant while the operation in parentheses is carried out.

As an example, consider a nonrotating, two-dimensional, free-surface flow passing over a bottom of variable elevation h (Fig. 1). If the flow at the sill is hydraulically critical, a localized, infinitesimal disturbance of the fluid depth d and velocity υ may exist there. The amplitude and shape of the disturbance may be regarded as arbitrary as long as it is recognized that the disturbance is small. The disturbed flow (dashed line) has the same upstream state and therefore the same volume flow rate per unit width Q = υd and energy per unit mass (Bernoulli function) B = (υ²/2) + gd + gh as the original. Substituting for υ in the expression of the Bernoulli function leads to a conservation law in the single variable d:

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with B and Q constant. Applying (2.3) with d now playing the role of γ yields the well-known critical condition Q = g^1/2d^3/2, or υ = (gd)^1/2.

The flow at a critical section is particularly vulnerable to external forcing. In the case of the 1D flow governed by (2.4) the stationary wave is a long (and therefore nondispersive) gravity wave. At a critical section, the long wave will be resonantly excited by stationary forcing. Since the group velocity of the stationary wave is zero, disturbance energy is unable to escape and will increase with time, presumably leading to a dramatic change in the background flow. This behavior distinguishes stationary long waves from lee waves, which are stationary waves of finite length. The latter are dispersive and allow energy to propagate away even though the phase speed is zero.

The threat of long-wave resonance suggests that critical flow can only occur where the topography does not force the flow. This idea can be put on formal ground by evaluating (2.2) at a critical section (∂G/∂γ)_y=yc = 0 and assuming that the flow remains smooth as it passes through the critical section (dγ/dy exists and is bounded at y_c). It follows that

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This “regularity” condition restricts the locations y = y_c at which critical flow can occur and allow smooth computation of solutions through those locations. These locations are sometimes called control sections. Applying (2.5) to (2.4) leads to the conclusion that critical flow can occur only where dh/dy = 0, as at a sill.

Their nondispersive nature is just one of the properties that makes long waves (as opposed to other stationary disturbances) centrally important in hydraulics. Another is that long waves project exactly on the steady flows that they modify. In the case of the shallow flow governed by (2.4) both the steady flow and the long waves have depth-independent horizontal velocity, whereas surface gravity waves of finite length have depth-decaying horizontal velocity. A second reason is that the long-wave resonance described above makes steady flow particularly vulnerable to changes the conduit constriction. Imagine a steady flow over a dam and consider the consequences of raising the crest of the dam slightly. If the depth and velocity of the crest flow is initially d₀ and υ₀, then υ₀ − (gd₀)^1/2 is initially zero at the crest. The change in crest elevation excites a stationary long waves and its energy builds until the disturbance acquires finite amplitude. Its speed can now exceed the linear value υ₀ − (gd₀)^1/2 and it can break free and propagate upstream, altering the upstream flow. Numerous demonstrations of this and similar processes can be found in the literature, beginning with Long’s (1954) towing experiments and including many numerical extensions Baines (1995) and Pratt et al. (2000).

In most applications, critical flow occurs at a section (or sections) y = y_c marking the transition between states supporting wave propagation in different directions. Strictly speaking, the flow is able to support stationary disturbances only at y_c and not at points immediately upstream and downstream as suggested in Fig. 1. The stationary disturbances are therefore possible in theory but are difficult to visualize in most applications. They should not be confused with stationary lee waves, which involve waves of finite length.

The form (2.1) of the functional G used by Gill is based on the presumption that the flow state at a given y depends only on the characteristics of the conduit at that y (and on the values of the parameters Q, B, etc.). Nonlocal dependence on y can occur in systems that exhibit hydraulic behavior, either as a result of dissipation or of variations in geometry that are nongradual. In appendix A, we give two examples of such systems and show how the Gill approach can be extended and used to find a critical condition. One involves a system in which the short, rather than long, waves are nondispersive.

3. Extending the Gill approach to systems with multiple variables

Reduction of the problem to the single-variable format envisioned by Gill (1977) is not always easy. It is often more convenient, and sometimes necessary, to work with two independent relations in two variables γ₁ and γ₂:

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and

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The existence of a stationary wave now requires that small variations dγ₁ and dγ₂ can be found such that G₁ and G₂ both remain fixed. Thus

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and

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The critical condition is just the solvability condition for (3.3) and (3.4):

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first obtained by Pratt and Armi (1987). Stationary waves then involve the tangent displacement (dγ₁, dγ₂) as given by (3.3) or (3.4):

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where dγ₁ is small but arbitrary. The displacement vector contains information about the cross-sectional structure of the stationary wave.

The generalization of the regularity condition (2.5) can be found by writing out the identities dG₁/dy = 0 and dG₂/dy = 0:

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and

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Solving for dγ₁/dy leads to

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where [∂()/∂y]_γ1,γ2 = [∂()/∂w](∂w/∂y) + [∂()/∂h](∂h/∂y) + · · · is a derivative taken with γ₁ and γ₂ held constant. Critical flow requires that the denominator vanish and the numerator must then vanish if the flow is to remain well behaved. The regularity condition is thus

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(The i = 2 version, which follows from developing an expression for dγ₂/dy, is not independent of the i = 1 version.)

Dalziel (1991) showed that two-layer hydraulics with no rotation can be formulated using two functionals of the forms (3.1) and (3.2). He derives the well-known critical condition (e.g., Armi 1986) equivalent to setting a composite Froude number to unity (e.g., Armi 1986); however, (3.5), (3.6), or (3.9) are never explicitly written down. The stationary disturbance in this case is an internal wave and the relationship (3.6) linking dγ₁ and dγ₂ determines the ratio of the amplitudes of the wave in the two layers.¹ In Smeed’s (2000) treatment of three-layer exchange flow, the hydraulics problem is again reduced to two functional relations of the required form and (3.5) and (3.9) are derived.

The machinery can be extended to problems governed by N relations for N independent variables:

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The condition for stationary waves is now

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and the corresponding solvability condition is the vanishing of a generalized Jacobian:

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where

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The tangent displacement vector (dγ₁, dγ₂, . . .)_c, which is computed from any member of (3.11), again determines the transverse structure of the stationary wave.

A generalized regularity condition for critical flow can also be obtained. Since dG_i/dy = 0 it follows from (3.12) that

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where (∂/∂y)_γ denotes a derivative with γ₁, γ₂, and so on, held fixed. From Cramer’s rule, it follows that

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where (∂G_i/∂γ_j)|(∂G_i/∂y)_γ is the matrix obtained by replacing the ith column of (∂G_i/∂γ_j) by

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Critical flow occurs when the denominator of the right-hand side of (3.15) vanishes. To insure that the derivatives on the left-hand side remain bounded, the numerator must also vanish:

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This is the regularity condition.

When hydraulic models are formulated directly from the differential form of the equations of motion, as is sometimes done with two-dimensional, multilayer flow, a constraint of the form (3.15) is obtained directly (e.g., Engqvist 1996). The critical condition is then identified as the vanishing of the denominator of the right-hand side, implying (3.16). This provides a link with the Gill formulation. One must be prepared to explain why the vanishing of the denominator implies critical flow; here an insightful person might cite the resonance condition described earlier.

In some cases, there may be an advantage to working with a number of dependent variables, even though that number can be reduced to one by algebra. As an example, consider the hydraulics of a rotating layer of zero potential vorticity flowing along a wall and over a horizontal bottom of elevation h (Fig. 2). As shown by Stern (1980) the flow state at any section can be described by its width w_e and the value of the alongshore velocity υ_e at its free, offshore edge. The alongshore evolution of these quantities can be calculated from conservation of volume flux:

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and energy

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(both in nondimensional form). It can also be shown by direct calculation of the long-wave speeds for this system that the flow is hydraulically critical when w_e = υ_e or when υ_e = 0. (However, the latter corresponds to a limiting case where the water depth and velocity go to zero all across the flow.)

An unusual feature of (3.18) is that it appears to satisfy all the requirements for a Gill functional (2.1) in a single the variable (υ_e). It would appear that one could solve the hydraulic problem using (3.18) alone. However, if one defines G(υ_e; h) = (½)υ²_e + h and applies the Gill critical condition ∂G/∂υ_e = 0, the solution υ_e = 0 misses the most pertinent condition, w_e = υ_e. If instead (3.17) and (3.18) are treated as a two-by-two system and (3.5) is applied, both critical conditions follow. The physical explanation behind the apparent failure of Gill’s original approach is that the stationary wave in question is manifested by an offshore excursion dw_e in the position of the free edge (Fig. 2) with no corresponding variation in the free edge velocity, (i.e., dυ_e = 0). A search for this stationary wave using the requirement ∂G/∂υ_e = 0 comes up empty.

As discussed in appendix B, the above formalism can be adapted to a continuous system. The index j is replaced by a continuous variable such as the streamfunction ψ. A functional expressing conservation of a property such as energy exists for each ψ. We show that the critical condition for such a system is that a nontrivial solution to a particular homogeneous equation exists. The equations contain coefficients that depend on the flow state, that these coefficients must be specialized in order for a nontrivial solution to exist. The result is analogous to the solvability condition for (3.7) and examples pertaining to continuously stratified, nonrotating, and rotating flow have been documented by Killworth (1992, 1995). Unless the homogeneous equation is quite simple, however, there is no established analytical procedure for specializing the coefficients. Progress then requires one to consider a discrete approximation to the continuous system and to find a solvability condition for the resulting finite set of equations, an exercise tantamount to solving (3.12).

4. Assessing hydraulic criticality using a steady, multiple streamtube model

Consider a steady, reduced-gravity flow at a particular cross section of a deep, rotating strait (Fig. 3a). The elevation of the interface is denoted η(x), where x is the cross-stream coordinate, and the along-strait velocity υ is geostrophically balanced:

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Density stratification is the most commonly measured physical property of such flows and we will assume that several such measurements have been taken at discrete positions x₁, x₂, . . . across the section in question (Fig. 3b). By identifying the density interface at each position the values η(x_n) are found. We will also assume that the topographic height h(x) and the positions x = x₁ and x = x_N+1 of the left and right edges of the lower layer are known. Then it is convenient to divide the observed flow into N segments (Fig. 3b), with the nth element extending from x_n to x_n+1.

We wish to examine the possibility that free, stationary, long disturbances can exist in the observed flow. In so doing, we will think of the sides of each element as streamlines and the elements themselves as streamtubes. The introduction of a stationary disturbance alters the positions of the streamlines but not the volume flux nor energy within each tube. In other words, the sides of the elements may move laterally and the corresponding values of η_n may change but the volume flux Q_n and energy B_n across each element remain fixed. The statements of volume flux conservation may be approximated as

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where the average velocity υ for a given element has been related to the interface elevation using (4.1). Conservation of energy takes the approximate form

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Equations (4.2) and (4.3) form a system of 2N equations in terms of 2N variables. The latter are composed of the N + 1 streamline positions x_n and the N − 1 values of η₂, η₃, . . . , η_N. [Note that the grounding of the flow at either edge means that the corresponding elevations η₁ = h(x₁) and η_N+1 = h(x_N+1) are not independent of x₁ and x_N+1.] Applying (3.12) therefore leads to the critical condition:

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where

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We consider two applications, both to flow in rotating channels with parabolic cross sections:

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If the potential vorticity of the fluid is uniform then solutions along the whole length of channel can be found analytically from expressions developed by Borenäs and Lundberg (1986). It is also possible to write down a generalized Froude number giving the criticality of the flow at a given section. More details are given in appendix C. We have calculated several solutions for hydraulically controlled flow corresponding to different upstream conditions. The critical section occurs at a sill corresponding to a maximum in h₀. Cross sections of the flow for a case in which α is held fixed at 0.25 are shown in Fig. 4. Upstream of the sill (top panel) the flow is subcritical and relatively deep. The interface slopes up with positive x over most of the section, indicating positive velocity, although there is a narrow band of reverse flow on the right wall. At the sill the flow is critical, the overall layer thickness is smaller, and there is a more prominent band of reverse flow along the right wall. At the downstream section, the layer thickness is quite small, the velocity is positive all across the section, and the current is strongly banked against the right wall.

To evaluate the critical condition (4.4), the flow at each section is partitioned into three segments of equal width (N = 3), leading to a 6 × 6 matrix 𝗯. The value of det[𝗯] is evaluated at a number of sections extending from upstream to downstream of the sill. In principle det[𝗯] should change sign where the Froude number F_p [from (C.1)] crosses through unity, and Fig. 5a shows that this is very nearly the case. [The actual crossing is at F ≅ 0.93.] Similar results hold for solutions calculated with α = 1 and α = 4, with the zero crossings at F ≅ 0.98 and F ≅ 0.99, respectively; N = 3 therefore appears to provide reasonably good resolution.

The magnitude of det[𝗯] is quite small in the subcritical (F < 1) range relative to the supercritical range (Fig. 5a), also characteristic of the other cases analyzed. If the analysis had been based on field or laboratory data with accompanying noise, it is quite possible that the value of det[𝗯] would be judged indistinguishable from zero at all upstream sections. This situation would cloud interpretation of the results. An alternative is to calculate the eigenvalues χ₁, χ₂, χ₃ , . . . , χ_2N of 𝗯. Since

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at least one of χ_j must be zero where the flow is critical:

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In the case of the Fig. 5a, the magnitudes of some of the χ_j are finite but quite small upstream of the sill, rendering det[𝗯] small. However, the magnitude of the eigenvalue (χ₆) that becomes zero at the sill remains relatively large upstream and downstream. As shown in Fig. 5b, its zero crossing is clearly defined.

A slightly more ambitious example of application of (4.4) or (4.8) involves a numerical simulation of an overflow in a parabolic channel (Fig. 6). Details of the numerical model and its use in studies of similar hydraulic flows in rectangular channels can be found in Helfrich et al. (1999) and Pratt et al. (2000). In short, the model solves the single-layer, shallow-water equations using a finite volume flux-limiting scheme that is designed to handle the complexities of rotating hydraulic flows (e.g., shocks, jumps, and layer outcropping). For the run in Fig. 6 the model is initialized in a uniform parabolic channel with α = 4 and h₀ = 0 [see (4.6)] with a flow with uniform potential vorticity (q = 1) and semigeostrophic Froude number F_p = 1.5 [see (C.1)]. Between t = 0 and 2 a bump with amplitude h₀ = 0.5, centered at y = 0, is introduced into the flow. Figure 6a shows the free surface elevation at t = 60 after the introduction of the bump. Disturbances have propagated both upstream and downstream leaving a new hydraulically controlled flow in the vicinity of the bump. Figure 6b shows F_p calculated from (C.1) for the numerical solution at t = 60 assuming uniform q = 1. The jagged quality of F_p (and of the curves in Fig. 7) is a numerical artifact caused by the discrete representation of the edges of the flow on the numerical grid. While the numerically computed flow does not have uniform q in the wake of the upstream and downstream propagating disturbances, the Froude number based on uniform q indicates a hydraulic transition from sub to supercritical flow just downstream of the sill crest. The upstream/downstream asymmetry of the flow also indicates the presence of a hydraulic transition over the bump.

A direct determination of the critical section can be made by introducing a small disturbance at some section y in the Fig. 6a flow and integrating the model forward in time. Upstream and downstream propagation of the disturbance indicates locally subcritical flow while downstream-only propagation indicates supercritical flow. For this example, the critical section was found to lie between 0 ≤ y ≤1, which is in good agreement with the uniform q prediction.

The value of det[𝗯] based on N = 3 is calculated over a range of sections upstream and downstream of the sill (Fig. 7a). As before, this value crosses from positive to negative values slightly downstream of the sill with relatively small magnitudes over the upstream range. On the other hand, the zero crossing of the associated eigenvalue (Fig. 7b) is less ambiguous. However, both calculations agree quite well with the estimates of the position of the critical section using F_p and by observation of wave propagation.

5. Direct calculation of the wave speed

The critical condition (4.4) will be most useful in analytical models, whereas (4.8) will be better suited to the evaluation of output from a numerical simulation. Application of (4.8) to the ocean, where data are typically collected at a moderate number of sections, is likely to be more problematic. The value of a particular eigenvalue χ_j at a section determines only whether the flow is critical or possibly not critical. The critical section (where one of the χ_j values crosses through zero) will almost certainly fall between two of the observed sections. One would therefore have to calculate the value of χ_j from one observed section to the next, making sure that the same eigenfunction (the same j) is followed. The bookkeeping is straightforward when the sections can be spaced closely enough so that χ_j varies gradually from one to the next, as in a numerical model. It is more difficult when the sections are widely spaced and the eigenfunctions undergo large changes. A procedure that is less elegant but better suited to observational data is direct calculation of the linear long-wave speeds c_j of the system. The result allows one to judge the flow at a particular section as subcritical or supercritical depending on whether the wave in question has positive or negative phase speed. Moreover, the eigenfunctions have a physical basis as wave modes that make them more identifiable from one section to the next. The following discussion assumes that the wave speed is real for the hydraulically relevant wave mode.

A method of calculation of c_j that is convenient and provides a direct connection with the Gill approach is based on a time-dependent version of the multiple streamtube model. Let the observed edge positions (Fig. 3b) and the corresponding interface elevations be given by γ_b = (x₁, x₂, . . . , η₁, η₂, . . . )^T. The positions x_i of the cell edges and the corresponding elevations η_i are now allowed to vary about these background positions and the resulting fluctuations γ′ = (x′₁, x′₂, . . . , η′₁, η′₂, . . . )^T are assumed to have y − ct dependence. As shown in appendix D, γ′_b obeys the linear eigenvalue relation

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and thus the wave speeds c_j are the eigenvalues of the matrix 𝗮⁻¹(γ_b)𝗯(γ_b). Both 𝗮 and 𝗯 are defined in appendix D. The matrix 𝗯 is identical to that defined in section 4, providing a link with the extended Gill formulation. Since

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the critical condition (4.4) implies that at least one of the speeds c₁, c₂, and so on, is zero.

The calculation of c_j based on the streamtube model should be equivalent to a calculation based on the discrete representation of the continuous equations for a linear normal mode. Our approach has a direct tie-in with Gill’s approach. It also has built in variable cross-stream resolution, should one portion of the flow require higher resolution.

6. The significance of the local Froude number

The “local” Froude number is defined as

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In traditional, one-dimensional (u = 0, ∂/∂x = 0) models, hydraulic criticality corresponds to F = 1. The significance of F for hydraulic control in a flow with transverse variations (∂/∂x ≠ 0) is less clear, but this has not prevented its appearance in discussions of models and data. For example, Rydberg (1980) based his theory of deep-water, rotating channel flow on the assumption that the F = 1 all across the critical section. In their report on a numerical simulation of the Strait of Gibraltar exchange flow, Izquierdo et al. (2001) present two-dimensional maps of the composite Froude number (the two-layer version of F) based on time-averaged fields. They show that this quantity falls above and below unity across certain sections; the Tarifa Narrows being one. They refer to such sections as “fragmentary” controls and write

the term “control” is not appropriate to that situation because such a fragmentary control cannot provide efficient blocking (of) interfacial disturbances within a subcritical flow region and, hence, cannot completely determine the exchange rate in this region.

By “subcritical” they mean regions where the local composite Froude number falls below unity.

What is the significance of the local Froude number and how does it relate to stationary long-wave normal modes? Is it really true that a section in which F falls above and below unity provides only fragmentary and inefficient blocking? To resolve these issues, it is helpful to review a simple result from shallow water theory, as laid out by Courant and Friedrichs (1948) using the gas dynamics analogy. Consider a steady, shallow flow for which the value of F exceeds unity in a certain region (and in which the steady shallow water equations are therefore hyperbolic). If the velocity and depth at a particular point p within the region are given by u = 0, υ = υ₀, and d = d₀, then a localized disturbance generated at p will initially spread out in a widening circle as it is advected downstream (Fig. 8a). The radius of the circle will initially grow at rate (gd₀)^1/2 while the center of the circle will initially move forward at speed υ₀. If F = υ₀/(g′d₀)^1/2 > 1 the disturbance will spread within a wedge of influence that spans the angle 2A, where

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The “Froude” angle A and the edges of the wedge are analogous to the Mach angle and Mach lines of supersonic flow. The edges also define two of the three characteristic curves of the steady flow, the third being the steady streamlines. If the flow is nonuniform, as will generally be the case under rotation, the cone of influence still takes on the angle given by (6.2) near p but becomes distorted farther from p. The essential point is that localized disturbances propagate downstream. If F < 1, the disturbance circle spreads upstream and downstream (and the governing equations are no longer hyperbolic).

While the above description applies to a localized disturbance, hydraulics is generally concerned (for reasons already discussed) with long waves. These waves satisfy the sidewall boundary conditions and therefore have a cross-strait modal structure. Normal modes are important because the control results from a choking effect in which the entire flow is squeezed from below and from the sides. The wave that is instrumental in the control and blocking of the upstream flow must therefore sense the boundaries and satisfy the boundary conditions. It is easy to find examples where such waves can propagate upstream or remain stationary even where a portion of the flow across the section in question has F > 1. A simple example is the nonrotating flow with uniform depth and shear:

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confined to a channel that spans −w/2 < x < w/2. A straightforward calculation show that this flow supports long, free surface gravity waves with speeds

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and thus the flow is hydraulically critical for υ = (gd₀ + s²w²/4)^1/2. It is also easy to find cases where the overall flow is hydraulically subcritical (c₋ < 0 even though F exceeds unity across part of the channel cross section. An example of such a case (Figs. 8b, c) shows a band of flow with F > 1 on the left side of the channel (facing downstream) and F < 1 on the right.

The example of Fig. 8b may seem paradoxical. The long wave, which has a normal mode structure extending all across the channel, clearly propagates upstream even though localized disturbances must propagate downstream in the F > 1 band along the left wall. In addition, the wave is nondispersive and its energy must propagate upstream at the phase speed. The situation may be rendered less mysterious by noting that only the net (width integrated) disturbance energy is required to propagate upstream for the long wave. Positive energy flux in the F > 1 region can be offset by a larger negative energy flux in the F < 1 region. Figure 8c shows the local, along-channel, disturbance energy flux S^(y) = gD〈υ′η〉 + ½V(D〈υ′²〉 + g〈η²〉) (appendix E) plotted across the channel; S^(y) is positive where F > 1, as expected from our prior discussion. However, S^(y) is negative over the whole right-hand portion of the channel, where F < 1.

Further insight into the role of localized disturbances, their ability to affect the upstream flow and their relationship to normal modes can be gained from two numerical simulations (Figs. 9 and 10). In the first example, we pose an initially steady flow of the form (6.3), with υ, s, and d₀ chosen to make c₋ < 0 (subcritical flow) and such that F > 1 (< 1) to the left (right) of the channel centerline x = 0 (Fig. 9). At t = 0, a localized disturbance of small amplitude is introduced into the flow on the left side (facing downstream) of the channel, where F > 1. As predicted, the disturbance initially spreads in the downstream direction (t = 1 frame of Fig. 9). Eventually, some of the disturbance energy begins to leak into the right side of the channel, where F < 1 (t = 2) and some of this energy moves upstream (t = 5). By this time, the disturbance has become aware of both sidewalls and a normal mode structure is becoming evident in the upstream portion. The normal mode is characterized by phase lines roughly perpendicular to the channel axis and extending all across the channel. As the disturbance evolves, this structure becomes more evident (t = 20, 100). The background flow is subcritical with respect to the long-wave normal mode and this is reflected by propagation upstream of the original site of the disturbance. Upstream propagation is allowed all across the channel because disturbance energy can propagate upstream along the right wall and leak into the left side of the flow as it does so.

Figure 10 shows a similar example in which the initial flow is supercritical c₋ > 0. Regions with F > 1 and F < 1 still exist on the left and right sides of the channel, but the F > 1 is expanded slightly from the previous case. The localized disturbance initially develops as before and a hint of upstream penetration of disturbance energy is observed along the right wall at t = 20. This is not surprising; after all it is this region where F < 1. However, as the normal mode structure emerges this trend is reversed and the entire disturbance moves downstream. Localized upstream propagation of information is possible within the F < 1 region, but only over the time scale required for a normal mode to form. This scale is the time required for a free disturbance to traverse the channel width, reflecting off the channel walls several times, and therefore should equal 3 or 4 times w/(gd₀)^1/2. This scale equals 12–16 dimensionless time units of the simulation, about the time it takes for the normal mode structure to emerge. Thereafter upstream propagation is controlled by the speed c₋ of the long normal mode, which in this case is >0.

These ideas also apply to a flow that is hydraulically critical. The implied nondispersive stationary wave must have zero group velocity and therefore zero net energy flux. If the fluid depth goes to zero at the edges (Fig. 11), and the velocity remains nonzero there, then F formally approaches ∞ at the edges. We would therefore expect to find bands of flow on either side of the channel in which F > 1. Since the characteristic curves of flow in these bands require downstream propagation of disturbance energy, there must exist a region with F < 1 in which upstream propagation of disturbance energy is permissible. The upstream flux of disturbance energy in this region must exactly cancel the downstream flux in the F > 1 regions, else the nondispersive wave cannot be stationary. These considerations suggest that for the situation shown in Fig. 11, the local Froude number F must pass through unity at some interior point in order that the flow be hydraulically critical. It also follows that the flow must be hydraulically supercritical if F ≥ 1 all across the section, for then not even local upstream propagation is possible.

The remarks made in this section may not apply if the flow is unstable.

7. Discussion

We can now recommend procedures by which an investigator can establish a critical condition or assess the criticality of a given flow. If the flow is specified by an analytical model, (3.12) can be used to formulate a critical condition in terms of the dependent variables. Equation (3.16) then establishes a regularity (smoothness) condition that restricts the location of the critical section. If the model is question is numerical and the flow is subject to the usual hydraulic approximations (gradual variations along the channel and conservative) then (4.4) or its less noise-sensitive sibling (4.8) can be used to isolate the critical section(s). These formulas are based on a description in which the cross section of the flow is divided into N sections (streamtubes). The optimal value of N based on our numerical simulations and on the Faroe–Bank Channel flow (J. Girton 2005, personal communication) appears to lie in the range 3–5. Application requires that a 2N × 2N matrix 𝗯 [see (4.5)] be calculated at a series of closely spaced sections. All results to this point have been derived by generalizing Gill’s method of treating hydraulically driven flows.

If the flow is observed and the observations have been made at a moderate number of sections, then there are two practical choices. A crude estimate of the criticality may be made by fitting the actual bottom topography to a parabola and calculating the parabolic Froude number F_p (Borenäs and Lundberg 1986; written out in our appendix C). The most problematic aspect of this approach is the estimation of the potential vorticity q of the flow (assumed constant in the theory), particularly if the observations are limited to hydrographic data. A more general approach is to calculate the phase speeds of the long, normal modes of the flow directly. We have laid out a procedure based on the streamtube description of the flow. The phase speeds are the eigenvalues of the matrix 𝗮⁻¹(γ_b)𝗯(γ_b) as defined in section 5. The type of wave is indicated by its cross-channel structure as reflected in the appropriate eigenfunction.

Of course, no observed flow is going to conform perfectly to usual hydraulic approximations. Perhaps the most serious departure for deep-ocean overflows is the presence of turbulence and dissipation. As suggested by the examples of appendix A, the extended Gill approach [as highlighted by (4.4)] continues to be valid in the presence of certain types of dissipation and forcing, provided that these effects enter the mathematical problem algebraically (and not as derivatives of the dependent variables). Turbulence and the closure problem present more formidable difficulties, but Stern (2004) has recently suggested a variational criterion for a hydraulically controlled state in the underdetermined system.

We have also made some remarks concerning the significance of the local Froude number F in rotating hydraulics. In regions where F > 1, the characteristic curves of the steady flow indicate information propagation solely in the downstream direction. However, we have also noted that the presence of F > 1 across a portion of a section does not in itself prevent the upstream propagation (or the arrest) of long normal modes. We have also reconciled these two facts by noting that the local energy fluxes for the normal mode are directed downstream where F > 1, even though the net energy flux is upstream. We have further argued that at a critical section with realistic topography, where the layer thickness vanishes on the sides, and where the velocity remains finite on at least one side, that F must =1 somewhere across the section. This feature may well be more general, as suggested by Stern’s (1974) criterion for control in a rectangular channel with positive υ:

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Again, F must = 1 somewhere in −w/2 < x < w/2. The property can also be inferred from Stern’s (2004) critical condition for flow over a sloping bottom. [The relationship following his (4.21) can be used to show that the right-wall Froude number is <1, whereas the Froude number on his left wall is infinite.]