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

Larry J. Pratt Department of Physical Oceanography, Woods Hole Oceanographic Institution, Woods Hole, Massachusetts

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Heather E. Deese Department of Physical Oceanography, Woods Hole Oceanographic Institution, Woods Hole, Massachusetts

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Stephen P. Murray Coastal Studies Institute, Louisiana State University, Baton Rouge, Louisiana

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William Johns Rosenthal School of Marine Science, University of Miami, Miami, Florida

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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 N2(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−1 and c1 of the two waves belonging to the first internal mode obey c−1 < Umin < Umax < c1, where Umin and Umax 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 Umin < 0 < Umax, 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−1 = 0 and/or c1 = 0) or even supercritical (c−1 and c1 have the same sign) with respect to the first mode. Similar findings follow for the second baroclinic mode phase speeds (c−2 and c2). 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 Umin and Umax. 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 <1/4, a result consistent with the presence of mixing over portions of the water column.

Corresponding author address: Dr. Larry J. Pratt, Woods Hole Oceanographic Institution, Dept. of Physical Oceanography, Woods Hole, MA 02543.

Email: lpratt@whoi.edu

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 N2(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−1 and c1 of the two waves belonging to the first internal mode obey c−1 < Umin < Umax < c1, where Umin and Umax 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 Umin < 0 < Umax, 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−1 = 0 and/or c1 = 0) or even supercritical (c−1 and c1 have the same sign) with respect to the first mode. Similar findings follow for the second baroclinic mode phase speeds (c−2 and c2). 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 Umin and Umax. 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 <1/4, a result consistent with the presence of mixing over portions of the water column.

Corresponding author address: Dr. Larry J. Pratt, Woods Hole Oceanographic Institution, Dept. of Physical Oceanography, Woods Hole, MA 02543.

Email: lpratt@whoi.edu

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