1. Introduction

The general characteristics of the thermohaline circulation in the Red Sea and neighboring Gulf of Aden strongly suggest that the exchange flow in the connecting strait, the Bab al Mandab (BAM, Fig. 1), is hydraulically controlled. The Red Sea thermocline is very shallow, lying mainly above the (∼165 m) sill depth (Maillard and Soliman 1986; Garrett et al. 1995) and the outflowing Red Sea water spills though the BAM and down into the Gulf of Aden (Fedorov and Mechanov 1988; Bower et al. 2000) in a manner reminiscent of flow spilling over a dam. The importance of understanding this presumed control (its location, the wave mode or modes involved, the locations of hydraulic jumps and other mixing agents, etc.) has been mentioned in connection with a variety of recent work on the Red Sea (Maxworthy 1997; Tragou and Garrett 1997; Tragou et al. 1999; and especially Maxworthy 1997 and Smeed 1997, 2000).

Pratt et al. (1999, hereafter referred to as PJMK) recently assessed the internal hydraulics of the flow in the BAM by fitting a three-layer idealization of the exchange flow to density and velocity data. The reader is referred to the introduction of that paper and to Smeed 1997 for a more thorough discussion of the geometry of the strait and the seasonal character of the exchange flow. The data was collected (Murray and Johns 1997) as part of the first deployment of an April 1995–November 1996 field program employing CTD moorings and bottom mounted ADCPs, deployed at the Hanish Sill section and Perim Narrows section of the strait (Fig. 1). The three-layer model was used to calculate the long-wave propagation speeds of first and second internal modes at the sill and narrows. Surprisingly, the results indicated that the flow was substantially subcritical with respect to the first internal mode at the narrows, the location of strongest exchange velocities. As it turns out, the subcritical character of the flow is due to the large layer depths at the narrows, a feature that tends to enhance the speed of internal waves relative to the background flow. At the sill, where the velocities are weaker but the layer depths shallower, the nonsummer flow was found to be marginally subcritical with respect to the first mode, possibly indicating a section of critical flow (and hydraulic control) nearby. Another surprise finding was that the flow during nonsummer months was nearly critical with respect to the second internal mode at the sill (and possibly the narrows). The vertical structure of this mode suggests that the corresponding hydraulic control might regulate the composition of the Red Sea outflow, particularly the relative contributions of upper Red Sea deep water and Red Sea intermediate water.

The study of PJMK raised a number of issues and was subject to several limitations, many of which are addressed in the present study. First of all, the velocity and density structure in the BAM does not have a distinct layer character. The velocity varies continuously and the density, while relatively uniform near the top and bottom of the water column, varies continuously and smoothly in the interior. There are no obvious interfaces and PJMK were forced to vary the positions of their model interfaces over what was believed to be reasonable bounds in order to make uncertainty estimates in the computed phase speeds. Second, there were a number of gaps in the hydrographic data that were filled from extrapolation or historical sources. Finally, all calculations were based on monthly mean vertical profiles, so little information about instantaneous flows and the effects of tides and other sources of time-dependence was found.

Since PJMK was published, data from the second deployment (April through November 1996) has become available. Although this dataset is limited by the fact that no hydrographic measurements were taken at the narrows, the measurements at the sill are more complete than those from the first deployment. (This is fortuitous since the results of PJMK seem to indicate the sill as the most likely location of hydraulic control with respect to the first mode.) The completeness of the velocity and hydrographic measurements at the sill allows us to compute the first and second continuous dynamical modes and their phase speeds. In order to do so we first extend the theory for calculating long-wave modes to include the possibility of variable cross-channel bottom topography. We suspect that topography is important in the BAM because the outflowing Red Sea Water is confined to a narrow deep channel running along the center of the strait, as shown in Fig. 2a. The computed phase speeds allow us to make determinations regarding the hydraulic character (subcritical, supercritical, or critical with respect to different modes) of the flow. We do this using both monthly mean profiles and a selection of “instantaneous” profiles. The latter are really 2-h average profiles that coincide with maximum tidal inflow and outflow and with different states of net (barotropic) inflow/outflow due to longer period disturbances. The calculated phase speeds may imply vertical mixing, and we discuss this in connection with calculations of the Richardson number over the water column.

The connection between long waves and hydraulic behavior is well understood in layered models, and the reader is referred to Smeed (2000) and PJMK for discussion of this issue in relation to the BAM. In continuously stratified flows, which exhibit more complex behavior and are more difficult to model, this connection is not as well established. Nevertheless, it is clear that signals propagating from the Indian Ocean into the Red Sea potentially influence features like the structure of the Red Sea thermocline, the composition of the outflow, and perhaps even elements of the horizontal circulation. It is therefore important to assess the ability of these signals to reach the Red Sea. If certain types of signals are unable travel northwest through the BAM, then the Red Sea possesses a degree of isolation and self-determination that goes hand-in-hand with hydraulic control. If all signals can travel in this direction, than the BAM flow would not be hydraulically controlled, and one might expect features like the depth of the Red Sea thermocline to be imposed largely by the vertical density structure of the Gulf of Aden and Indian Ocean. In brief, there is sufficient motivation for calculating the phase speeds of these signals at the sill even though one may not have a particular model of hydraulic behavior for a continuously stratified exchange flow in hand.

A number of calculations and results are new and distinguish this work from PJMK. First, we introduce a method for calculating the continuous, dynamical, internal long-wave modes in the presence of arbitrary, cross-channel topography by deriving an extended version of the Taylor–Goldstein equation for long waves. Second, we employ a few idealized profiles of channel topography, background velocity and N² to illustrate some of the basic features of long-wave modes expected in exchange flows over typical topography. Third, we document certain observed features, including monthly mean and selected 2-h average velocity, stratification, and Richardson number for the second deployment (April–November 1996). (PJMK used data from the first deployment, which spanned June 1995–March 1996). Fourth, we compute the lowest two dynamical modes, their phase speeds, and the corresponding uncertainty bars for the second deployment monthly mean sill flows and use the results to asses the criticality of the exchange flow. (In order to overlap the first deployment and develop a full year of results, we also perform the calculations for a block of first-deployment months.) Finally, to gain some insight into the effects of time dependence, we perform the same calculations for a selection of instantaneous flows.

2. An extended Taylor–Goldstein equation for long waves

The BAM outflow is largely confined to a deep central trough whose width is much less than the width of the strait (PJMK). This topographic feature can be seen in the Hanish Sill cross section shown in Fig. 2a. We formally analyze the general effects of cross-channel topographic variations by considering the propagation of long, internal gravity waves in a channel with laterally varying but longitudinally uniform bottom elevation. The waves are linear and propagate in a stratified background flow with velocity U(z) and density ρ₀(z). The channel is aligned in the x direction and the bottom elevation h(y) has a single minimum with respect to the cross-channel coordinate y. As shown in Fig. 2a, the width of the channel at any elevation z is denoted b(z). If several minima in h(y) are present, then b(z) is understood to represent the sum of the widths of the individual topographic troughs.

Let ρ, u, and p denote small perturbations from the density, x velocity, and hydrostatic pressure of the background flow, and let υ and w denote the associated lateral and vertical velocity components. Employing the Boussinesq approximation, the linearized, inviscid, hydrostatic equations of motion describing these fields are then given by

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We seek waves for which ρ, p, u, and w are uniform in y, implying that the isopycnal surfaces rise and fall uniformly across the channel. Such solutions are dynamically consistent only in the limit of long wave length compared to channel width.¹ Integrating (2.5) across the channel at any z and applying the conditions w = υ(dh/dy) at the two side walls leads to

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where T(z) = b⁻¹db/dz = d(lnb)/dz.

Now let (u, w, ρ, p) = Re[(ũ(z), w̃(z), ρ̃(z), p̃(z))e^ik(x−ct)], so that (2.1), (2.3), (2.4), and (2.6) become

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Eliminating all variables in favor of w̃ and neglecting derivatives of ρ₀(z) unless multiplied by g leads to

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where N² = −ρ⁻¹₀g dρ₀/dz.

The geometry of the cross section is contained in the parameter T(z). If the cross section is rectangular (T = 0), the Taylor–Goldstein equation for long waves is recovered. For the calculations presented later, we will assume a rigid-lid boundary condition at the upper surface (z = z_T) and thus w̃(0) = w̃(D) = 0, where z = 0 denotes the elevation of the deepest point in the channel and z = D denotes the upper surface elevation.

In further analysis it will be convenient to nondimensionlize (2.11) by scaling U and c by an unspecified background velocity scale U₀, w̃ by an unspecified vertical velocity scale W, and z, h, and y by D. (Thus, dimensional N² is replaced by N²D²/U²₀ and dimensional T by DT.) The dimensionless problem for the vertical structure of the modes then becomes

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with

w̃w̃(2.13)

In case studies of geophysical interest, where b smoothly goes to zero as z → 0, T(z) typically has a z⁻¹ singularity. For example, let h(y) ∝ α̂y^μ near y = 0, so that b = αz^m, where m = μ⁻¹ and α = 2α̂^−1/μ. It follows that T(z) = (d/dz) ln(αz^m) = m/z. Figure 3 shows some idealized shapes with corresponding T(z). The topographic shapes shown in Figs. 3c,d have finite b at z = 0 and thus avoid the singularity. For numerical purposes, the singularity can be avoided by approximating the actual bottom topography at z = 0 by a short horizontal section, as suggested in the inset of Fig. 2a. This subject discussed further in appendix A.

The broken dashed line in Fig. 2b shows an approximation to the Hanish Sill topography (solid curve) used in our calculations. The deepest part is represented by a 100-m long horizontal section at 160-m depth. The slopes of the side walls are varied in a piecewise-constant manner in order to fit the actual topography. Above a level z₀ the side walls become vertical (T = 0). The dimensionless shape function describing the approximate topography is

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where z₀ = 15/16.

3. Examples of modes in idealized systems

Since our assessment of the hydraulics of the BAM sill flow is centered on signal propagation, it is important to understand in advance what types of signals are possible and what restrictions exist on their phase speeds. We therefore begin this section by reviewing the properties of solutions to the original Taylor–Goldstein equation. Unless noted otherwise, these results apply directly to our extended, long-wave version of Eq. (2.12) provided that the singularity in T(z) at z = 0 is no worse than 1/z. That is, the cross-sectional geometery of the channel has not been found to alter basic theorems about phase speed bounds and the Richardson number criterion for instability. Proofs showing the extension of previous results [such as Howard’s (1961) semicircle theorem] to the new equation have been developed by J. Deng, L. Howard, L. Pratt, H. Deese and C. K. R. T. Jones (unpublished manuscript) but these are beyond the scope of the present paper. The influence of topography on the structure and propagation of wave modes becomes evident through specific examples, and we present several later in this section.

As summarized by Drazin and Reid (1981) and Baines (1995) the solutions of the (original) Taylor–Goldstein equation with N² > 0 depend crucially on the minimum value of the Richardson number

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over the water column. If (Ri)_min > 1/4, then there is a countable infinity of discrete neutral modes. These can be thought of as internal gravity waves that have been modified by the background shear. Denoting the corresponding eigenfunctions and eigenvalues by [w̃_j(z), c_j], it can be shown that

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and c_j → U_min as j → −∞ and c_j → U_max as j → ∞. Thus the wave speeds lie outside the range of the basic velocity but within a finite factor (π⁻¹N_max for long waves) of the velocity extremes. Finally, it can also be shown that the jth eigenfunction w̃_j(z) has exactly |j| − 1 zero crossings in 0 < z < 1.

The discrete neutral modes are important as they provide a connection with layered systems. An n-layer hydraulic model has n normal modes and these are analogous to the first n discrete modes of a continuous system. This analogy can break down when (Ri)_min < 1/4 in which case the flow may be unstable (Miles 1961). Neutrally stable discrete modes can still exist but their number may be finite, even zero. Their phase speeds continue to lie outside the range of the basic velocity. In addition, there may be a finite number of nonsingular unstable modes (with conjugate damped partners). According to the semicircle theorem, the real part of the phase speed for such a mode will lie within the range of the background velocity. Finally there can be a finite number of marginally stable² neutral modes that have real phase speeds c in the range of the background velocity and therefore have critical levels z_c such that U(z_c) = c. If the value of the Richardson number at the critical level is >1/4, w̃(z) oscillates rapidly in the neighborhood of z_c (Booker and Bretherton 1967).

In addition to the discrete modes, a continuous spectrum of neutral modes will exist with critical levels. At these critical levels, w̃(z) will have a singularity no worse than a discontinuity in the first derivative. To the best of our knowledge, the role that these waves might play in the hydraulic behavior of the flow is unknown. However, one can easily examine the behavior of Couette flow (which has a continuous spectrum but no descrete modes) in the presence of an obstacle and show that there can be no hydraulic control. This calculation is presented in appendix B.

Now consider some examples of long-wave modes in idealized systems, starting with the case U(z) = 0 and N = const. Here the eigenvalues arise in pairs, designated j = ±1, ±2, etc., with c_j = −c_−j. If the channel cross-section is rectangular (T = 0) then c_j = ±N/jπ and w̃_j(z) = a_j sin(jπz). The vertical structure function w̃_j(z) is the same for each of the waves having the same |j|, but the phase relation between w̃_j(z) and the pressure p̃_j(z) differ for each pair. When a background velocity is present and (Ri)_min > 1/4, it is still possible to group the waves in pairs. Although w̃_j(z) generally differs from w̃_−j(z), the number of zero crossings is the same for each. Thus, we will consider each |j| as designating a vertical mode, recognizing that each such mode is associated with two waves.

Next, consider the topographic function T(z) = 1/(z + α), α = y₀/(y₁ − y₀), corresponding to a trapezoid with bottom and top dimensions y₀ and y₁. A triangle results from the limit α → 0, a rectangle from α → ∞. Solutions to (2.12) with U = 0 and N² = const take the form

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where J₁ denotes the Bessel function of first kind and Y₁ denotes Weber’s Bessel function of the second kind, both of order 1. Application of the boundary conditions (2.13) yields a transcendental relation for the eigenvalues c_j, and Table 1 summarizes these speeds for rectangular, trapezoidal (with α = 1), and triangular topography. As one moves from the case of a rectangular channel (for which c_j = ±N/jπ) to a trapezoid, and eventually to a triangle, the phase speed for each mode decreases. This trend is not surprising: the average depth of the fluid decreases in each case. There is also a tendency for the wave modes, particularly the higher ones, in trapezoidal or triangular geometry, to be intensified near the bottom, as evidenced by the decay of the Bessel functions for large arguments: J₁(z) → 2/(πz) cos(z − 3π/4) and Y₁(z) → 2/(πz) sin(z − 3π/4) as z → ∞.

Some insight into the influence of a background flow can be gained by considering a linear shear flow U(z) = U_max (2z − 1) with N² = const, contained in a rectangular channel (T = 0). The value of Ri is uniform and for Ri < 1/4 the flow is stable and has no discrete modes (Case 1960). For Ri > 1/4 the discrete modes are given (Baines 1995) by

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where

c_jjπμj

and where μ = (Ri − 1/4)^1/2. The phase speeds for |j| = 1, 2, 3, and 4 are plotted as a function of Ri in Fig. 4a. As Ri → 1/4 from above, the positive phase speeds converge to the value U_max (and the negative phase speeds to −U_max) so that the propagation speeds accumulate around the upper and lower limits of the velocity range. This behavior is quite important in interpretating later results for the BAM data.

Figure 4b shows w̃_j(z) for j = 1, 2, and 3 at values Ri = 4.25 and 16.25. Decreasing the value of Ri causes the zero crossings to migrate upward in the water column so that the wiggles become confined near the top boundary. (For the negative j modes, the wiggles would become densely packed at the bottom of the water column.) The rapid oscillations are associated with the emergence of a critical level at z = 1 (or z = 0), which occurs for each mode as Ri → 1/4. This behavior, which was elluded to above, clearly raises an important issue concerning resolution of the higher modes. As Ri is reduced toward 1/4, the eigenfunctions for different j look alike over most of the depth range (as shown by the upper trio of solutions in Fig. 4b). The differences can be distinguished only in a narrow band at the top (or bottom) of the water column.

A final example combining topographic elements with shear effects arises from consideration of

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with N² = const. This background state consists of an exchange flow in a channel with a triangular cross section. The phase speeds of the first few modes have been calculated numerically and the results are presented in Figs. 5a,b. As the minimum value of Ri (which here occurs at middepth) is reduced from large to O(1) values, the phase speeds of the modes converge to ±U_max in a manner similar to those of the case just described. Figure 5a shows this behavior for the phase speeds of the wave pairs for the (j = ±1, ±2, and ±3). Unlike the constant shear, rectangular geometry case, however, these modes do not disappear when (Ri)_min = 1/4 but, instead, continue to (Ri)_min = 0. The vertical structure of the modes (not shown) is similar to what was observed in the previous example with the zero crossings migrating toward the level of U_min and U_max, here the bottom and top of the channel, as (Ri)_min → 0.

The convergence of c_j and c_−j just above and below U_min and U_max as the stability of the flow is reduced can be anticipated by writing (3.1) is a slightly different form. Consider a velocity profile of fixed shape in that U_max can be increased or decreased but remains proportional to U_min. Dividing (3.1) by U_max or U_min leads to the dimensionless bounds

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where R has been set to zero and

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As the overall Richardson number R_bulk is reduced, all of the c_j approach U_max from above and all of the c_−j approach U_min from below. (Of course, some or all of the discrete neutral modes may cease to exist as R_bulk is reduced.)

The background state (3.3) also has an unstable normal mode, and we briefly note the effect of topographic variations on this instability. For a rectangular cross section [T(z) = 0] a mode with complex c first appears at (Ri)_min ≅ 0.17. The upper dashed curve of Fig. 5b shows Im(c) as a function of (Ri)_min and reveals a maximum Im(c) ≅ 0.32U_max at (Ri)_min ≅ 0. With the triangular topography, the unstable mode appears at a lower (Ri)_min (≅0.09) and has a smaller maximum (≅0.09U_max). The implication is that the topography T(z) = 1/z has a stabilizing influence on long waves, although this has not been tested in connection with other velocity profiles.

This section contains quite a bit of information, and we now reemphasize some of the behavior that could be of importance in the interpretation of results from the BAM or other straits. The first concerns the general effect of “realistic” topography, meaning any cross section containing significant decreases in width with depth. With U = 0 and N² = const, realistic topography causes a moderate reduction in c below the case of a rectangle (Table 1). The topographic influence on c in the presence of nontrivial U(z) is more difficult to describe simply. In any case, one should proceed under the assumption that topography will be at least moderately important in most straits of geophysical interest. Second, propagating modes in the presence of topography are subject to the same phase speed bounds that govern modes in a rectangular channel. For instance, the speeds of discrete neutral modes must lie outside the extremes U_min and U_max of the velocity profile. As the bulk Richardson number becomes small, these speeds accumulate just outside of U_min and U_max. Under these conditions each mode nearly has a critical level. An important consequence of the phase speed bounds for an exchange flow (U_max > 0 and U_min < 0) is that all c_j are nonzero. Therefore, such a flow cannot be hydraulically critical (c_j = 0) with respect to any discrete neutral mode. If the velocity profile becomes temporarily unidirectional through the action of a barotropic tide, say, then hydraulic criticality is possible. The occurrence of phase speeds just outside the range of the basic velocity and the occurrence of critical flow only when the velocity profile is unidirectional is exactly the situation we observe in the BAM.

4. Monthly mean stratification, velocity, and Richardson number

We calculated monthly mean ADCP velocity profiles with 10-m vertical resolution extending from 10-m to 130-m depth as shown in Fig. 7. We linearly extrapolated from neighboring points to obtain velocities at 0 m, 140 m, and 150 m (represented by crosses in Fig. 7). During the nonsummer months the exchange flow is characterized by moderate surface inflow (positive velocity) and significantly stronger deep outflow. As noted by PJMK, the inflow velocity is weaker because it is spread over a much larger cross-sectional area than the outflow, the latter being confined to the central topographic trough. In fact, the inflow and outflow volume transports nearly cancel, as required by the overall mass balance for the Red Sea.³ During the “summer” (months 6, 7, 8, and 9) the deep outflow is attenuated while the inflow lies lower in the water column. Only during months 8 and 9 is there a clear surface outflow.

The Richardson numbers for each month are displayed in Fig. 8 along with uncertainty bars.⁴ (At depths where the uncertainty is quite small, the central estimated value is represented by a dot.) In addition to analyzing the second deployment months (April–November 1996), we reappraise months November 1995–March 1996 of the first deployment. Since we do not have stratification data from the first deployment, we utilize N²(z) from both April 1996 and November 1996 in our calculations of Ri, resulting in larger uncertainty bars for these months.

Although instability is formally possible only for

values of about one or less can be suggestive of prior mixing. Such values occur during all months except June, July, August and September 1996, all four of which are characterized by relatively weak flows. Comparison of Figs. 7 and 8 shows that the smallest Ri for a given month generally occur within 10–20 m of the mean interface (the zero crossing of mean velocity profile). The position of this mean interface between inflowing Gulf of Aden water and deeper outflowing Red Sea water ranges from 35 m in winter to 110 m in summer. (The latter roughly corresponds to the depth of the lower maxima in the August N², as shown in the middle left panel of Fig. 6.) Although some very low estimates of Ri appear in the top 20 m and bottom 30 m of the water column in several months, the uncertainty is magnified at these depths because both the shear and stratification are weak and the velocity data is extrapolated (see appendix C).

5. Criticality of the monthly mean sill flow with respect to the first and second internal modes

Winters and Riley (1992) developed a numerical code to solve the Taylor–Goldstein equation in finite difference form. Using the same second-order difference operators we have modified the code to include the topographic term present in our extended, long-wave version of this equation (2.12). Given the fitted monthly mean N² profiles (Fig. 6) and the monthly mean velocities (Fig. 7) specified at 15 levels, we have calculated the vertical modes using 15 interior grid points. This number is sufficient to resolve the first and second vertical modes as verified by our tests using analytical profiles with known solutions. Since the horizontal velocity profiles have at most two zero crossings, the first two vertical modes appear to be the ones of greatest importance for the hydraulics of the exchange flow. In the following description we continue to use the convention that j = +n and j = −n denote waves belonging to mode n whose speeds in the absence of a background velocity obey c_n = −c_−n.

Figure 9 shows the propagation speeds for j = ±1 and j = ±2 modes for each month. The speeds are represented by arrows which point to the right (left) if the wave propagates towards the Red Sea (Gulf of Aden). For a given mode, a pair of arrows pointing in opposite (the same) directions indicates subcritical (supercritical) flow with respect to that mode. If one or both of the arrows has zero length, the flow is critical with respect to that mode. The two circles on each set of arrows represent the extremes (U_min and U_max) of the mean velocity profile for that month. Speed uncertainties are within ±8 cm s⁻¹ for each month. Our estimates of uncertainty are due to extrapolation of the velocity profiles, fitting of the density profiles, the presence of a residual barotropic flow in the total transport, and expected numerical errors (see appendix C for full discussion).

We first draw the reader’s attention to the results based on second deployment (months 4–11 in 1996) data, which are shown in the unshaded part of Fig. 9. In all second deployment months the flow is subcritical with respect to both modes, although there are some instances when flow is moderately close to a critical state with respect to one or both modes. This can be seen during months 4, 5, 10, and 11; c₁ and c₂ (the speeds of the two waves propagating into the Red Sea) are considerably reduced in relation to their partners c₋₁ and c₋₂. For example, in month 11 c₋₁ is −1.37 m s⁻¹ and c₊₁ is 0.34 m s⁻¹ so that signal propagation in the direction of the Red Sea is relatively slow. For the second mode, the speeds are −1.04 and 0.24 m s⁻¹, respectively, leading to a similar interpretation. During the summer months (6–9) there is an imbalance favoring the wave propagating toward the Red Sea. For example, month 8 has speeds (c₋₂ = −0.34, c₂ = 0.58) indicating relatively slow propagation towards the Gulf of Aden. As noted by PJMK and explained in the introduction, it is somewhat surprising to find subcritical flow at the sill as it appears to be the most likely point of hydraulic control in the BAM.

For completeness we have recomputed c_±1 and c_±2 from the first-deployment months (Nov 1995–Mar 1996) using our continuous model. This calculation incorporates direct, monthly mean ADCP data along with the stratification from April 1996 and November 1996. (Complete stratification during the first deployment was unavailable.) The results are shown in the shaded region of Fig. 9. Again, we see that the flow is subcritical with respect to both modes, but that the speeds of the waves attempting to enter the Red Sea are small, possibly indicating a nearby control section. These results imply a slightly subcritical flow, whereas PJMK indicate that the flow is right at the critical point with respect to mode 2.

The modes for each month display a common behavior in their relationship to the measured velocity profile. The phase speeds always lie outside, in some cases barely outside, the window of the velocity extremes. This can be seen in Fig. 9, in both the shaded and unshaded regions, as each arrow stretches beyond the circles representing the background velocity extremes. Since the phase speeds are never inside that window, there are no critical levels with respect to these modes.

In summary, the flow is subcritical with respect to each mode, though barely so in the winter months. The phase speeds based on monthly mean data lie outside the extremes of monthly mean velocity for each month, therefore critical levels do not arise. The mismatch in the magnitude of the negative and positive extremes of background velocity lead to similar mismatches in the propagation speeds for the two waves belonging to a particular mode. For the second mode, the phase speeds for most months lie very close to these extremes, suggesting incipient critical level behavior. This behavior is reflected in the positions of the zero crossings of w̃(z)(not shown), which lie close to the levels of the velocity extremes.

6. Criticality with respect to instantaneous flows

The sill flow in the BAM is strongly influenced by diurnal and semidiurnal tides and by subtidal disturbances. The subtidal disturbances typically have periods of a few days to a few weeks and are forced in part by wind stress variations over the strait (Seidler 1968). The detailed nature of the fluctuations are the subject of a separate investigation on the new Bab al Mandab data;here we simply note that they can be comparable in strength to the mean flows in the strait and at times can cause the top-to-bottom flow to be unidirectional. All of these features are apparent in the Fig. 10 time histories of ADCP velocity at three depths over months August–November 1996. We have made several phase speed calculations based on two-hour mean velocity profiles and instantaneous stratifications observed at various times during this period. These calculations are presented in order to show how dramatically the criticality of the instantaneous flow can depart from the monthly mean. A comprehensive study of time dependence is beyond the scope of this work.

We have computed phase speeds for ten realizations of the 2-h mean flow, corresponding to the daily extreme of the tidal ebb and flood on five selected days. These days are indicated on the Figs. 10a and 10b time series. The 2-h mean velocity profiles from two of these days are displayed in Figs. 11a,b. Figure 11a shows the results for 10 August 1996, which falls during a period of average tides and little subtidal activity. Even in the presence of such moderate tides, the maximum tidal flood (Red Sea inflow), which occurs at hour 17, is strong enough to reverse the normal deep and shallow outflows for August, creating inflow over the entire water column. Figure 11b shows the same information for 18 November, which falls during a period of strong outflow at 20-m depth due to a subtidal disturbance (see the Fig. 10b time record, in which 18 November is marked). In this case, top-to-bottom outflow occurs throughout the entire day.

The Richardson number profiles for the maximum flood and ebb on 18 November (hours 13 and 19) are shown in Fig. 12. Excluding values near the top and bottom (which are subject to large uncertainty due to extrapolation of the velocity), the lowest values range from 0.20 to 0.46 at maximum ebb, and from 0.08 to 0.29 at maximum flood. (The lowest value found for the monthly mean flow for this month was 0.25.) These values occur at depths from 30 m to 60 m. (Depths without marked values have Richardson numbers that are large enough to be off the horizontal scale of the figure.)

Figure 13 shows the phase speeds for instantaneous flows on the five selected days. For each day two results are shown corresponding to the maximum tidal flood (top set of arrows) and ebb (bottom set of arrows). Apart from 10 August critical and/or supercritical flows occur during the maximum ebb tide on all four days (the lower set of arrows). The circles in Figure 13 indicate the extremes of the 2-h mean velocity and it can be seen that in all cases but one (mode 2 on 13 November), the phase speeds continue to lie just outside U_min and U_max. Critical levels are thereby avoided, as was the case for the monthly means. These results indicate that the tidal and subtidal motions easily push the flow into a critical or supercritical regime with respect to both modes. The criticality of the flow is evidently a strongly intermittent feature.

Estimation of the average propagation speed based on the characteristic depth and velocity along the BAM axis yield values on the order of 1 m s⁻¹ for a first or second baroclinic mode. The corresponding average travel distance over the 12-h tidal period is ≅40 km, which is only a fraction of the 180 km length of the BAM. A signal propagating from the Gulf of Aden into the Red Sea would therefore spend many tidal cycles in the strait. The results of this section suggest that the direction of propagation can be temporarily, and perhaps permanently, reversed by intermittent supercritical conditions. The implications for hydraulic control are discussed further below.

7. Discussion

Our analysis of idealized profiles suggests that the observed frequent proximity of the phase speeds to U_min and U_max may be a characteristic of flows that are marginally stable, perhaps as a result of prior mixing. Richardson numbers calculated from the monthly mean and instantaneous profiles seem to be consistent with this hypothesis. Middepth values of R_i were frequently found to be <1 and sometimes <1/4.

When the flow is critical with respect to a particular wave, the vertical structure of that wave gives an indication of the manner in which the flow is being controlled. For example, we would expect that critical control with respect to a wave that is bottom trapped would involve regulation of the deepest part of the flow. PJMK found a nearly critical, second vertical mode that was bottom intensified in terms of the layer transports in their 3-layer model. They understood this structure to indicate the existence of a mode-2 control on the winter transport of Upper Red Sea Deep Water. Unfortunately, comparisons with the vertical structures of the continuous modes found here are difficult. For one thing, our mode-2 waves for the monthly mean winter flow are not critical. For another, the critical mode-2 waves that we have found for instantaneous conditions involve background velocity profiles quite different from the mean velocity profiles that PJMK based their results on. In brief, it is difficult to find a basis for comparison.

This study raises a number of other issues that need to be addressed on a basic level. The most obvious concerns the significance of critical flow in a highly time-dependent setting. If it is indeed the case, as our results suggest, that the criticality of the BAM sill flow is highly intermittent, what does this mean for overall hydraulic control of the exchange flow?⁵ [One of the few modeling efforts to address this problem is Helfrich’s (1995) study, but the special nature of the flows involved make it difficult to relate the results to the BAM.] Clearly, the fundamental issues concerning signal propagation and regions of influence must still hold. If the long internal waves of a certain mode are unable to propagate into the Red Sea from the Gulf of Aden, then a certain degree of self-determination (a central feature of hydraulic control) is implied for the Red Sea. As we noted earlier, a first or second baroclinic wave propagating from the Gulf of Aden into the Red Sea would spend at least two or three tidal cycles in the strait. The direction of propagation might even be temporarily reversed by intermittent supercritical conditions, or the information carried by the wave might be lost due to encounters with hydraulic jumps. Clearly, a determination of whether long waves can propagate all the way through the strait in the presence of fluctuating currents is a matter that requires analysis of the flow along the entire strait, not just at the sill.

A second mystery, which is avoided in the present problem but which is of great concern for the critical control of steady exchange flows in general, concerns the presence of critical levels. If a steady, continuously stratified exchange flow is hydraulically controlled at some section, then the wave that is stationary there has a critical level z_c at the zero crossing of the velocity profile. Such a wave will generally be unstable. This situation is avoided in two-layer models of exchange flow, where the velocity is always nonzero, and in continuously stratified unidirectional flow (e.g., Armi and Williams 1993) in which stationary waves have no critical levels. Remarkably, long-wave critical layers are also avoided in our BAM observations; stationary waves only arise at the sill when the flow is unidirectional. Nevertheless, we might consider a hypothetical, steady exchange flow with continuously varying velocity and stratification. To what extent do the classical ideas about hydraulic control apply when stationary long waves occur in such a flow?

Some guidance might be gained from the literature on severe downslope winds in the atmosphere (e.g., Peltier and Clark 1979; Smith 1985) in which both hydraulic behavior and critical levels can arise. Instability and wave breaking about the critical level can result in the formation of a homogeneous wedge of fluid at and immediately downstream of the mountain crest. The base of the wedge acts as a perfect reflector of wave energy, dynamically insulating the underlying fluid from the flow aloft. This underlying fluid spills down the slope in the manner of a supercritical reduced-gravity layer. Hydraulic supercriticality can be diagnosed by solving the Taylor–Goldstein equation in this layer, subject to the condition for a pliant upper boundary. This upper boundary is simply the base of the homogeneous wedge. Laboratory and numerical demonstrations of the formation of the wedge appear in Wilkinson and Wood (1985) and Bachmeister and Pierrehumbert (1988), respectively. An analytical model for the evolution of the lower layer is discussed by Smith (1985).

In the BAM, the processes just described would presumably lead to a layer of homogeneous fluid about the zero crossing of the velocity profile. However, such homogenization is not observed; the zero crossing of the velocity tends to lie within the continuously stratified intermediate depths of the water column (e.g., Fig. 6 of PJMK), at least where velocity measurements have been made (the sill and narrows). Also, the development of a critical level into a perfectly reflecting boundary is a process that takes time and it is not clear how this development would occur in the presence of tides. Nevertheless, there are reasons that one might pursue the meteorological analogy further. First, Richardson numbers ⩽O(1) are observed at levels close to the zero crossing of the monthly mean velocity. Such values are consistent with prior mixing though, as noted above, they are not associated with outright homogenization. Second, there is a layer of relatively homogeneous fluid that forms on the downslope south of the narrows, namely the stationary wedge of Gulf of Aden Intermediate Water (GAIW) that exists for most of the year (Smeed 1997).⁶ This wedge bears remarkable similarity to the wedge of homogeneous fluid that forms around the critical level in meteorological simulations (e.g., Bachmeister and Pierrehumbert 1988), laboratory experiments (Wilkinson and Wood 1985), and other field studies (Farmer and Armi 1999). One difficulty with this analogy is that GAIW is colder than both the surface inflow and underlying outflow, so that it cannot have been formed as a result of mixing between the two. Also, estimates of wave speeds based on PJMK’s three-layer model suggest that the flow is substantially subcritical just upstream of this wedge.