Continuous Dynamical Modes in Straits Having Arbitrary Cross Sections, with Applications to the Bab al Mandab*

Abstract

The continuous dynamical modes of the exchange flow in the Bab al Mandab are computed in an attempt to assess the hydraulic character of the flow at the sill. First, an extended version of the Taylor-Goldstein equation for long waves that accounts for cross-channel topographic variations, is developed. A series of calculations using idealized background velocity U(z) and buoyancy frequency N(z) are presented to illustrate the effects of simple topographic cross sections on the internal modes and their speeds. Next, hydrographic and direct velocity measurements from April to November 1996 using moored CTDs and a bottom-mounted ADCP are utilized to construct monthly mean vertical profiles of N 2 (z) and at the U(z) sill. An analytical approximation of the true topography across the strait is also constructed. The observed monthly mean profiles are then used to solve for the phase speeds of the first and second internal modes. Additional calculations are carried out using a selection of ''instantaneous'' (2-h average) profiles measured during extremes of the semidiurnal tide. The results are compared with a three-layer analysis of data from the previous year. Many of the authors' conclusions follow from an intriguing observation concerning the long-wave phase speeds. Specifically, it was nearly always observed that the calculated speeds c Ϫ1 and c 1 of the two waves belonging to the first internal mode obey c Ϫ1 Ͻ U min Ͻ U max Ͻ c 1 , where U min and U max are the minimum and maximum of the velocity profile. An immediate consequence is that neither wave has a critical level. For monthly mean profiles, each of which have U min Ͻ 0 Ͻ U max , the flow is therefore subcritical (the phase speeds of the two waves have opposite signs). For instantaneous profiles this relationship continues to hold, although the velocity profile can be unidirectional. Thus the flow can be critical (c Ϫ1 ϭ 0 and/or c 1 ϭ 0) or even supercritical (c Ϫ1 and c 1 have the same sign) with respect to the first mode. Similar findings follow for the second baroclinic mode phase speeds (c Ϫ2 and c 2). The authors conclude that hydraulically critical flow is an intermittent feature, influenced to a great extent by the tides. It is noted that the phase speed pairs for each mode lie very close to U min and U max. As suggested by the analysis of idealized profiles, this behavior is characteristic of flows that are marginally stable, perhaps as a result of prior mixing. This suggestion is supported by Richardson number (Ri) profiles calculated from the monthly mean and instantaneous data. Middepth values of Ri were frequently found to be O(1) and sometimes Ͻ1/4, a result consistent with the presence of mixing over portions of the water column.

Two-Layer Rotating Exchange Flow between Two Deep Basins: Theory and Application to the Strait of Gibraltar

“…As shown by Pratt and Helfrich (2005), the condition of hydraulic criticality for such systems is given in terms of the generalized Jacobian of the functions…”

“…In reality, the thicknesses of the overflows across the sills of the Denmark Strait and Faroe Bank Channel are comparable with those of the overlying fluid and Froude numbers based on the upper-layer properties are not always small (D. Sutherland and J. Girton 2004, personal communication). The importance of the upper layer is acknowledged in other applications such as the exchange flows of the Strait of Gibraltar (e.g., Armi and Farmer 1988) and the Bab al Mandab (e.g., Smeed 2000; Pratt et al 2000). These straits have widths on the order of or less than the Rossby radius of deformation and the effects of rotation are generally neglected.…”

Flow and Hydraulics near the Sill of Hood Canal, a Strongly Sheared, Continuously Stratified Fjord

“…This calculation considers local channel shape and uses as cross-channel average profiles of velocity U(z) and buoyancy frequency N(z). The resulting dynamical modes are solutions to the extended Taylor-Goldstein equation (Pratt et al 2000),…”

Determining the Stratification of Exchange Flows in Sea Straits

“…The formulation of bounds on the phase speeds in terms of properties of the basic flow often allow the investigator to determine the hydraulic state without having to explicifly solve for the modes. This work is contained in Pratt, et al (2000) and Deng et al (2003). A demonstration of the use of the theorem appears in Figure 2.…”