Internal hydraulics and mixing in a highly stratified estuary

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 105, NO. C6, PAGES 14,215 14,222, JUNE 15, 2000 Internal hydraulics and mixing in a highly stratified estuary Robert J. Chant Institute for Marine and Coastal Sciences, Rutgers University, New Brunswick, New Jersey Robert E. Wilson Marine Science Research Center, State University of New York at Stony Brook Abstract. Shipboard acoustic Doppler current profiler and conductivity-temperaturedepth data obtained during highly stratified conditions in the Hudson River estuary along a section of variable width and breadth are presented. The observations emphasize tidal period asymmetries in the vertical structure of current and salinity. However, these asymmetries exhibit significant along-channel structure which is determined by channel morphology. During the ebb the flow is linearly sheared, and steep halocline slopes in the vicinity of channel contractions are maintained by momentum advection. A minimum in vertical shear across the pycnocline occurs in channel contractions. During food the pycnocline sharpens and flattens with a middepth velocity maximum embedded in the pycnocline which separates a stratified surface layer from a bottom mixed layer. The along-channel structure in vertical shear is consistent with a lateral vorticity equation. Estimates of Richardson numbers suggest that vertical mixing across the pycnocline is enhanced downstream of channel contractions. 1. Introduction Early work describing density-driven circulation in estuaries focused on tidal residual fields. The seminal observational work of Pritchard [1954, 1956] suggested that a mean baroclinic pressure gradient is balanced by a vertical stress divergence, with advection of momentum playing a smaller yet potentially significant role. A balance between baroclinicity and vertical stresses was assumed in the theoretical work of Hansen and Rattray [1966], whose work produced widely cited estuarine circulation diagrams and an estuarine classification scheme. Hansen and Rattray [1996] suggest that estuaries be classified by two nondimensional parameters, a stratification parameter and a circulation parameter, both of which, in theory, could be obtained observationally. Despite the simplicity of this scheme, actual application of the theory was complicated by difficulties in obtaining representative tidal mean quantities [Dyer, 1997]. Although both Pritchard [1956] and Hansen and Rattray [1966] recognized that tidal residual stresses were largely associated with tidal motion, the tidal motion itself was treated implicitly. More recent work has explicitly discussed the interaction between tides and stratification [Jay and Smith, 1990a, b; Geyer and Smith, 1987; Geyer and Farmer, 1987]. Based on field observations Geyer and Farmer [1987] discuss the tidal period advance and breakdown of a bottom salt front. They characterize the flow in terms of a two-layer Froude number and conclude that the salt wedge is destroyed during the ebb tide when the flow becomes supercritical and vertical shears, augmented by bottom stress, increase until stratification is broken down by vertical mixing. Jay and Smith [1990a, b] describe the interaction between stratification and tidal forcing based on a two-layer analytical model with an interface of finite thickness. They suggest that Copyright 2000 by the American Geophysical Union. Paper number 2000JC900049. 0148-0227/00/2000JC900049$09.00 tidally mean motion arises from a flood to ebb asymmetry in vertical mixing which occurs because of tidal period variations in stratification. They emphasize the importance of tidal period internal motion in characterizing the estuary and use an internal Froude number to define the stability of the salt wedge. However, in contrast to Geyer and Farmer [1989], Jay and Smith [1990a] conclude that the salt wedge can be destroyed on the flood if the Froude number exceeds unity. Jay and Smith [1990a, b] neglect bathymetric effects, which they justify by scaling arguments. However, long ago, Stommel and Farmer [1952] recognized that topographic effects can control halocline structure and exchange in estuaries. Stommel and Farmer s [1952] work was extended by Armi and Farmer [1986] and Farmer and Armi [1986] to develop a two-layer internal hydraulic theory which includes effects of channel morphology and barotropic forcing. Gan and Ingram [1992] used a similar two-layer model to describe intratidal variability in halocline structure in the Manitounuk Sound of the southern Hudson Bay, and related the adjustment of this structure to vertical mixing associated with internal bores and solitary waves. Partch and Smith [1978] discuss the importance of internal hydraulic jumps on mixing in the Duarmish estuary in Washington State, and conclude that these jumps are associated with channel morphology. In this paper we describe the effect of channel morphology on density and velocity structure in a highly stratified estuary and discuss the implications this structure has on vertical mixing. This description is based on data collected in a 1993 field program conducted in the Hudson River estuary (Figure 1). Observations were made along a highly stratified 10 km reach with pronounced variations in channel cross-sectional area (Figure 2). The amplitude of the M2, N2, S2, K1, and O1 tidal constituents at the Battery (Figure 1) are 66, 14, 13, 10, and 5 cm. The survey extended from neap to spring tidal conditions. The Hudson has an annual mean discharge of 550 m 3 /s, with the highest monthly mean discharge of 1000 m 3 /s occurring during the month of April. 14,215

14,216 CHANT AND WILSON: HYDRAULICS AND MIXING IN A STRATIFIED ESTUARY Figure 1. Map of survey area. Dots show locations of conductivity-temperature-depth (CTD) stations. Along-channel sections were run along these transects. Cross-channel sections were across the channel at stations 5, 6, 7, and 8. Battery is denoted with B. In section 2 of this paper we describe the field program. In section 3 the data are used to describe the spatiotemporal structure of instantaneous velocity and density fields. In section 4 we present estimates of terms in the along-channel momentum balance. Richardson number estimates are made in section 5. In section 6 a discussion places these observations in context with a two-layer model and a lateral vorticity equation. Concluding comments are made in section 7. 2. Field Program Between April 29 and May 3, 1993, an acoustic Doppler current profiler (ADCP) and conductivity-temperature-depth (CTD) survey was conducted in a 10 km reach of the Hudson River estuary (Figure 1). The survey included along channel sections on April 29 and May 3 with cross-channel sections run the intermediate days. This reach is characterized by a large fractional change in channel cross section (Figure 2). River discharge of approximately 2000 m 3 /s produced strong stratification throughout the survey, though increasing tidal currents during the survey tended to weaken stratification. Winds were calm during the survey. Instrumentation included a 1200 khz narrowband RD Instruments ADCP and an Applied Microsystems CTD with a sampling rate of 5 Hz. The ADCP had a right-angle head and was towed abeam on a catamaran, placing the instrument near the surface. Current measurements were made at 1 m intervals, with the uppermost bin centered at 1.8 m and the deepest reliable bin at 85% water depth. Additionally, we used a 200 khz Furino depth sounder to resolve details in halocline structure. Position was recorded from a Global Positioning System (GPS). The survey was designed to provide data suitable for time series analysis at a fixed point while providing quasi-synoptic sections. Maximum tow speed of the catamaran was approximately 6 knots (11.1 km/h) through the water. Along-channel surveys consisted of running repeated transects approximately every 1.5 hours over a 12 hour period. Each transect consisted of 10 CTD casts at selected stations (Figure 1). Stations are separated by approximately 1 km and were selected to resolve halocline variability associated with variations in channel morphology. ADCP data were processed and averaged into 1 min ensembles, which typically resulted in 200 m ADCP footprints. Details of the lateral surveys are given by Chant and Wilson [1997]. Channel constrictions are located at the George Washington Bridge at station 7 (km 18 on Figure 2) and between stations 3 and 4 (km 12). A channel expansion is located between stations 5 and 6 (km 16). The channel tends to deepen in contractions and shoal in expansions. South of the survey region the channel tends to increase in width and cross section. However, the increase is not monotonic. Local minimums in channel cross section occur 1 and 6 km north of the Battery (Figure 2). Figure 2. Channel cross-sectional area. Region surveyed is indicated by thick line.

CHANT AND WILSON: HYDRAULICS AND MIXING IN A STRATIFIED ESTUARY 14,217 Figure 3. Vertical profiles of velocity (solid line) and density (dashed line) from station 5 and station 7 near maximum ebb and maximum flood. 3. Observations 3.1. Vertical Structure Vertical profiles of along-channel current and density during maximum ebb and maximum flood in a channel constriction (station 7) and a channel expansion (station 5) are presented in Figure 3. During ebb in the expansion, currents and stratification are linear with depth. Currents 2.8 m below the surface are ebbing at approximately 140 cm/s while at 20 m, currents are ebbing at 15 cm/s. Surface and bottom densities are 2 t and 16 t, respectively. Vertical structure differs markedly during the flood (Figure 3, top right plot). Currents have a middepth maximum at approximately 10 m below the surface where currents are flooding at approximately 70 cm/s. Vertical shears are strongest above the middepth maximum. Vertical shears above the middepth velocity maximum column are commensurable to those during the ebb. Vertical stratification during the flood is characterized by a two-layer system. Density varies by 12 t units across 5 m thick halocline which separates a weakly stratified surface layer from a bottom mixed layer. In the channel constriction, vertical shear during the ebb is considerably weaker than in the channel expansion. This reduction is associated with increased ebbing velocities at depth while surface currents, despite variations in channel width, remain fairly constant. Stratification during the ebb in the contraction is bottom intensified. During the flood, velocity in the constriction has a middepth maximum, approximately at the same depth as at station 5. In the bottom mixed layer, vertical shears in the contraction and expansion are of similar magnitude. However, in the stratified fluid above the middepth velocity maximum, vertical shear in the contraction is substantially weaker than in the channel expansion. 3.2. Along-Channel Density Structure Along-channel sections of t are presented in Figure 4 during maximum ebb and maximum flood on April 29. During maximum ebb the halocline exhibits large-amplitude excursions over the reach. The two regions of depressed halocline are associated both with contractions and with deepening of the main channel. Associated with these excursions are

14,218 CHANT AND WILSON: HYDRAULICS AND MIXING IN A STRATIFIED ESTUARY Figure 4. Density from CTD: (top) the first, (middle) the fourth and (bottom) the fifth transects. Maximum ebb occurred during the first transect. Maximum flood occurred between the fourth and fifth transects. changes in halocline thickness; the halocline thickens as it descends in the water column and sharpens as it rises. The halocline remains intact throughout this reach with very little variation in surface to bottom salinity over the 10 km stretch. The lower two plots in Figure 4 present concurrent transects taken around the time of maximum flood. Flood is characterized by a rapid growth of the bottom mixed layer. Neph and Geyer [1996] suggest that this thickening is primarily due to overstraining whereby the horizontal density gradient is overturned by the vertical shear during flood. Halocline slopes tend to weaken during the flood. 3.3. Along-Channel Velocity Structure A contour of instantaneous along-channel velocity during maximum ebb and around maximum flood on April 29 is shown in Figure 5. Although a tidally driven eddy develops between stations 5 and 7 [Chant and Wilson, 1997], it is not present in these two fields during this time. Isotachs during the ebb exhibit similar behavior to the density field. Isotachs spread in the channel constrictions and compress in channel expansions. Despite the twofold change in channel width, surface currents exhibit only weak along-channel variability. In contrast, currents at depth accelerate in channel contractions. At slack water the strong pycnocline slopes internally adjust. During flood, strong vertical shear is found above the halocline south of the constriction at station 7 with very weak shear north of this point. Bottom currents are strongly convergent in the southern third of the domain during the flood. The reduction of shear in the vicinity of the contraction at section 7 suggests that the tidal period internal motion is blocked by the channel contraction. 3.4. Froude Number The state of the flow can be defined by the two-layer Froude number U 2 1 ( g H 1 ) U 2 2 ( g H 2 ), where U 1, U 2, H 1, and H 2 are the upper and lower layer velocities and upper and lower layer depths, respectively, and g is reduced gravity ( g / ). Estimates of the Froude number were made by

CHANT AND WILSON: HYDRAULICS AND MIXING IN A STRATIFIED ESTUARY 14,219 defining reduced gravity and layer velocities with the mean quantities above and below the 7 t isopyncal. During maximum ebb, Froude numbers exceed 3 throughout the reach, indicating that the flow is appreciably supercritical at this phase of the tide (Froude number estimates during ebb remain appreciably supercritical when chosing other reasonable isopycnals to define the interface). During the flood the two-layer Froude number is just below unity, suggesting that the flow is subcritical. In summary, these observations emphasize tidal period asymmetry in the vertical structure of current and density. During the ebb the vertical structure is characterized by a nearly linear velocity and density profile. In contrast, the velocity profile during the flood has a middepth maximum embedded in a sharp halocline. The halocline separates a bottom mixed layer from a stratified surface layer. The vertical structure, however, exhibits significant along-channel variability. The along-channel variability is most pronounced during the ebb. In particular, vertical shears are weaker within channel contractions. Reduced vertical shears in channel contractions are associated with increased lower layer velocities while upper layer flows remain fairly constant along the reach. Estimates of the two-layer Froude number indicate that the flow is appreciably supercritical during the ebb and subcritical on the flood. 4. Momentum Balance Strongly convergent horizontal flow indicates that advection contributes to the momentum balance. To assess this, we begin with the momentum equation u t g x g z x z uu x z 0, (1) where t is time, x and z are the along-channel and vertical directions, u is the along-channel current, is the surface elevation, g is the acceleration due to gravity, and are the Reynolds stresses. By neglecting stresses near the surface a momentum balance at the surface is u0 t g x u0u0 x 0. (2) The barotropic pressure gradient is removed by subtracting 2 from 1 to form a shear equation u u0 t g z x z uu x u0u0 x z 0. (3) Figure 5. Along-channel velocity obtained from ADCP during the same transects presented in Figure 4. The second and third (uu x u0u0 x ) terms represent a density-driven shear term and an advectively driven shear term, respectively. These two terms are estimated between each station during maximum ebb and around maximum flood on April 29, and the result is plotted in Figure 6. Since this estimate is made near the time of maximum current, the time-dependent term should be small. The sign of the pressure gradient has been switched to facilitate comparison. During the ebb both the baroclinic and advective shear term are enhanced relative to the flood. More importantly, the sign and magnitude of these two terms tend to be the same, particularly in the vicinity of the channel contractions. Although obvious discrepancies are evident, Figure 6 indicates that during the ebb there is a tendency for baroclinically driven shears to be balanced by advective shears. Discrepancies in the terms are probably related to contributions by vertical stresses, particularly in the bottom mixed layer, and by secondary circulation which is active in this channel [Chant and Wilson, 1997]. 5. Richardson Number Estimates Geyer and Smith [1987] suggest that halocline thickness in a highly stratified estuary during a supercritical ebb tends to remain close to an equilibrium value whereby the flow is marginally stable with active but intermittent vertical mixing. Slight increases in shear trigger instabilities which vertically mix both momentum and salt, thus reestablishing stable conditions. Our observations suggest that vertical shear exhibits an alongchannel structure which is related to channel morphology. To assess if the observed longitudinal structure in vertical shear influences mixing, we compare estimates of Richardson numbers in the contraction at section 7 to those in the channel expansion at sections 5 and 6. Since we are interested in the development in shear instabilities within the halocline, Richardson number estimates are restricted to locations where stratification exceeds a specified criterion. Since stratification weakened over the survey, the

14,220 CHANT AND WILSON: HYDRAULICS AND MIXING IN A STRATIFIED ESTUARY Figure 6. Estimates for the second and third terms in equation (3). The solid line is the baroclinic term, while the dashed line is the advective term. Horizontal ticks are at increments of 0.01 cm 2 /s. criteria decreased from 0.5 t /m on April 29 to 0.25 t /m on May 3. These values were chosen to maximize the number of estimates without being influenced by the near-zero Richardson numbers in the bottom mixed layer. Histograms of Richardson numbers for sections 5 7 are plotted in Figure 7. Richardson numbers tend to be smaller in the channel expansion at sections 5 and 6 than in the contraction at section 7. On one hand, it may appear counterintuitive to have reduced mixing within channel constrictions. However, data presented here indicate that vertical shears are strongest downstream of channel contractions, and subsequently, we expect to find reduced Richardson numbers downstream of channel contractions. 6. Discussion 6.1. Two-Layer Model Certain aspects of the spatial and temporal density and velocity structure can be understood in light of a relation for an invicid two-layer flow in rectangular channel of variable width and depth. u 2 u 1 t g 1 G 2 H 1 x g F h 2 2 x 1 B B x u 1 2 u 2 2, (4) where u 1, u 2, H 1, and H 2 are upper and lower layer velocities and thicknesses, h is the height of the bottom above a reference level, B is channel width, G 2 is the two-layer Froude number, and t and x are time and distance, which are positive up estuary. Neglecting time dependency, (4) can be rewritten to express halocline slope as a function of channel morphology and the flow state F h 2 H 2 1 x x 1 B g B x u 1 2 u 2 2 1 G 2. (5) R. E. Wilson et al. (Influence of channel morphology on halocline behavior and trapping of fine-particles in a stratified estuary, submitted to Journal of Physical Oceanography, 1999) demonstrated that halocline slopes along this reach during the ebb are consistent with this two-layer model and that flow along this reach is appreciable supercritical during the ebb. The bathymetric terms, B x and H x, tend to be of the same sign because the channel deepens in contractions and shoals in expansions. Thus the first two terms are of the same sign when surface layer velocities exceed lower layer velocities, as is the case during the ebb. Therefore effects of a deepening and contracting channel act in concert to thicken the surface layer during the supercritical ebb. Likewise, effects of a shoaling and expanding channel act together to thin the surface layer. In contrast, during the flood, lower layer velocities exceed those in the upper layer, and the first two terms in (6) are opposed. In summary, width and depth variations in this channel act

CHANT AND WILSON: HYDRAULICS AND MIXING IN A STRATIFIED ESTUARY 14,221 Figure 7. Richardson number estimates made in the pycnocline at sections 5, 6, and 7. together to produce steep halocline slopes during the ebb. In contrast, during flood these bathymetric effects are opposed. Reduced shears during the supercritical ebb in the contraction are also elucidated by the two-layer model. During the ebb the upper layer largely determines the two-layer Froude number and can be viewed as a single active layer. This causes surface ebb velocities to remain fairly constant along this reach, despite the factor of 2 change in channel width, because supercritical flow draws from its kinetic energy to squeeze through a contraction, in contrast to subcritical flow, which draws from its potential energy [Turner, 1973]. If the lower layer were motionless, surface flows would actually decrease within contractions, which itself would decrease shears in the contraction. However, the lower layer is in motion, and lower layer flows must increase because both the contracting channel and thinning bottom layer reduce cross-sectional area in the lower layer. Consequently, in the constriction the shear between the layers decreases during the supercritical ebb. 6.2. Shear Tendency During the Ebb The tendency for vertical shears to weaken in a contraction during a supercritical ebb can be demonstrated more directly by a lateral vorticity, or vertical shear, equation. This equation is obtained by first writing a steady, nonrotating, invicid momentum equation u u u w x z 1 P (6) x and a laterally averaged continuity equation ub wb 0, (7) x z where P is pressure, composed of both a barotropic and a baroclinic component, B( x,z) is the channel width, and u and w are the horizontal and vertical currents. Taking the vertical gradient of 6, assuming linear vertical shear, and substituting in (7) yields a shear tendency equation u x u z u z u B B w x z 1 x or, equivalently, a lateral vorticity equation for an invicid nonrotating flow though a channel of variable width. The first term on the right-hand side represents vortex stretching/squashing while the second term represents baroclinically driven shears. The term on the left-hand side represents the horizontal advection of shear. As a supercritical ebb enters a contraction the halocline descends in the water column, causing the baroclinic pressure gradient to change sign from the estuarine-scale along-channel baroclinic pressure gradient. This tends to reduce shear as the flow enters the contraction. Similarly, the vortex-stretching term tends to reduce vertical shears as the flow enters a contraction because both the horizontal flow and downward vertical motion squash the fluid into regions of reduced channel breadth. Consequently, both terms on the right-hand side act in concert to reduce shears in the contraction. In this invicid example the shear tendency is balanced by the horizontal advection of vertical shear which requires shears to weaken moving into the contraction. In contrast, downstream of the contraction the halocline rises, and both the baroclinic term and stretching terms change sign. Here the sign of the baroclinic term is the same as the estuarine-wide baroclinicity. Consequently, both of the terms on the right-hand side act to increase the estuarine shear (8)

14,222 CHANT AND WILSON: HYDRAULICS AND MIXING IN A STRATIFIED ESTUARY downstream of the contraction. This is balanced by advection which requires shears to increase moving away from the contraction. More generally, (8) shows that for supercritical continuously stratified sheared flow, vertical shears are reduced in a channel contraction. Subsequently, we expect to find enhanced vertical shears and vertical mixing downstream of channel contractions. 7. Conclusions Observations from the Hudson River estuary emphasize a tidal period asymmetry in the vertical structure of density and current velocity. During the flood a sharp, relatively flat halocline separates a bottom mixed layer from a stratified surface layer. Embedded in this sharp halocline is a middepth velocity maximum. In contrast, during the ebb both velocity and density have a linear vertical structure. The vertical structure, however, exhibits significant along-channel variability. Steep halocline slopes occur during the ebb, with the halocline depressed where the channel deepens and elevated where the channel shoals and expands. This behavior is consistent with a two-layer model which emphasizes that the halocline is more responsive to channel morphology during the ebb because effects of channel deepening and contracting act in concert to produce steep halocline slopes. In contrast, both observations and the twolayer model indicate that the halocline is less responsive to variations in channel morphology during the flood. A shear tendency equation explains the observed reduction in vertical shears in channel contractions during the ebb. Surface currents exhibit weak along-channel variability, despite a factor of 2 change in channel width. In contrast, currents at depth accelerate in channel contractions and weaken in channel expansions. Observations indicate that the advection of shear contributes significantly to an instantaneous momentum balance. Although we do not estimate tidally mean terms, it is likely that the tidally mean terms reflect the ebb conditions, since halocline slopes and streamwise velocity gradients are more pronounced on the ebb. These 1 2 km scale fluctuating baroclinic pressure gradients are superimposed on a larger-scale but weaker estuarine scale baroclinic pressure gradient. Although the tidally mean momentum balance in the vicinity of these small-scale morphological features is probably imparted by advection, the estuarine-wide balance may still be between baroclinicity and mixing. However, these small-scale advective features indirectly influence the estuarine-wide momentum balance by driving localized regions of enhanced mixing. Consequently, we suggest that estimates of tidally averaged momentum balances in highly stratified estuaries are dependent on the length scale over which an estimate is made. Acknowledgments. R.E.W. acknowledges support from the National Science Foundation (OCE8917692) and from the Hudson River Foundation (01291A and GF0192). R.J.C. acknowledges partial support from New York State Sea Grant Institute. We also acknowledge John Brubaker, who collected and processed the ADCP. References Armi, L., and D. M. Farmer, Maximal two-layer exchange through a contraction with barotropic net flow, J. Fluid Mech., 164, 27 51, 1986. Chant, R. J., and R. E. Wilson, Secondary circulation in a highly stratified estuary, J. Geophys. Res., 102, 23,207 23,215, 1997. Dyer, K. R., Estuaries: A Physical Introduction, 2nd ed., 195 pp., John Wiley, New York, 1997. Farmer, D. M., and L. Armi, Maximal two-layer exchange over a sill and through the combination of a sill and contraction with barotropic net flow, J. Fluid Mech., 164, 53 76, 1986. Gan, J., and R. G. Ingram, Internal hydraulics, solitons and associated mixing in a stratified estuary, J. Geophys. Res., 97, 9669 9688, 1992. Geyer, W. R., and D. M. Farmer, Tide-induced variations of the dynamics of a salt wedge estuary, J. Phys. Oceanogr., 19, 1060 1072, 1989. Geyer, W. R., and J. D. Smith, Shear instability in a highly stratified estuary, J. Phys. Oceanogr., 17, 1668 1679, 1987. Hansen, D. V., and M. Rattray. New dimensions in estuary classification, Limnol. Occanogr., 11, 319 326, 1966. Jay, D. A., and J. D. Smith, Residual circulation in shallow estuaries, 1, Highly stratified, narrow estuaries, J. Geophys. Res., 95, 711 731, 1990a. Jay, D. A., and J. D. Smith, Residual circulation in shallow estuaries, 2, Weakly stratified and partially mixed narrow estuaries, J. Geophys. Res., 95, 733 748, 1990b. Neph. H. M., and W. R. Geyer, Intra-tidal variations in stratification and mixing in the Hudson Estuary, J. Geophys. Res., 101, 12,079 12,086, 1996. Partch, E. N., and J. D. Smith, Time dependent mixing in a salt wedge estuary, Estuarine Coastal Mar. Sci., 6, 3 19, 1978. Peters, Observations of stratified turbulent mixing in an estuary: Neapto-spring variations during high river discharge, Estuarine Coastal Shelf. Sci., 45, 69 88, 1997. Pritchard, D. W., A study of the salt balance in a coastal plain estuary, J. Mar. Res., 13, 133 144, 1954. Pritchard, D. W., The dynamic structure of a coastal plain estuary, J. Mar. Res., 15, 33 42, 1956. Rattray, M., and D. V. Hansen, A similarity solution for the circulation in an estuary, J. Mar. Res., 20, 121 122, 1962. Stommel, H., and H. G. Farmer, Abrupt change in width in two-layer open channel flow, J. Mar. Res., 11, 205 215, 1952. Turner, J. S., Buoyancy Effects in Fluids, 368 pp., Cambridge Univ. Press, New York, 1973. R. J. Chant, Institute for Marine and Coastal Sciences, Rutgers University, 71 Dudley Road, New Brunswick, NJ 08901. (chant@ imcs.rutgers.edu) R. E. Wilson, Marine Science Research Center, State University of New York at Stony Brook, Stony Brook, NY 11769. (Received February 17, 1999; revised October 20, 1999; accepted February 7, 2000.)