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Physical Model of Niagara River Discharge
J. Great Lakes Res. 20(3):583-589 Internal. Assoc. Great Lakes Res., 1994
Physical Model of Niagara River Discharge Joseph F. Atkinson, Guoqing Lin, and Maneesha Joshi Department of Civil Engineering State University ofNew York at Buffalo Buffalo, New York 14260
ABSTRACT. Development of a physical model study for the Niagara River discharge into Lake Ontario is described. The model is designed on the basis of (distorted) Froude and Rossby similarity and the results demonstrate clearly the effects of rotation on the flow. Important features of the plume are shown to be reproduced by the model, including its anticyclonic (right-turning) flow and the subsequent formation of a near-shore current along the southern coastline. A scaling analysis of the integrated longitudinal momentum equation is suggested as a framework for analyzing this type offlow. This analysis considers three regions of the flow: (/) a near-field, where inertial forces are balanced by pressure and bottom drag; (2) an intermediate-field where the balance is between inertia and buoyancy; and (3) a far-field characterized by nearly geostrophic flow (balance between buoyancy and rotation). This scheme is then tested against conditions characteristic of the Niagara River discharge during spring flows and also with observations from the physical model tests, showing reasonable agreement. One particular result of interest is that the distance at which rotation starts to turn the flow scales approximately with the internal Rossby radius. Results of the study help to distinguish between the relative effects of buoyancy and rotation. INDEX WORDS: Niagara River.
Buoyant jet, Co riolis force, physical model, Lake Ontario, buoyancy, hydrodynamics,
some distance Coriolis accelerations become significant and turn the plume clockwise. The main emphasis here is to demonstrate the effect of rotation, in conjunction with buoyancy, in determining the path followed by the discharge. A scaling analysis is proposed to identify regions of the flow controlled by different force balances and this helps to provide a general framework for describing the flow behavior. Several results from a physical model experiment are also presented. In general, discharges by rivers and estuaries are capable of modifying the circulation of receiving waters, both through baroclinic pressure gradients induced by buoyancy input and through the initial momentum flux of the discharge. The momentumdominated region near the discharge point is characterized by vertical homogeneity, relatively high velocities and lateral entrainment. Depending on the discharge Froude number (see below), the jet may be attached to the bottom for some distance and is therefore affected by bottom friction. After detachment the flow is increasingly buoyancy-dri-
INTRODUCTION The Niagara River (43° latitude, 79° longitude) is an essential link in the Great Lakes system, carrying flow from all the upper lakes and Lake Erie. The average discharge is nearly 7,000 m 3/s and represents about 85% of the total inflow to Lake Ontario (Masse and Murthy 1990). The 58 km-long river also passes through a highly industrialized region containing a number of "areas of concern" identified by the U.S. Environmental Protection Agency (EPA) and the International Joint Commission (UC) for potential loading of hazardous materials. Because of the relatively shallow depths for Lake Erie, its temperature may be 3 to 4°C warmer than in Lake Ontario over much of the spring and early summer (Masse and Murthy 1990). The present study is aimed at understanding the physical transport processes which govern the fate of pollutants carried by the river into Lake Ontario, particularly under these buoyant discharge conditions which result in a well-defined surface plume. The discharge spreads by inertia and buoyancy and after 583
Atkinson et 81.
584
ven and there is a transition zone where some vertical entrainment may occur. After this the flow approaches the state of a pure plume, spreading radially by buoyancy. Figure I illustrates these different regions. Masse and Murthy (1990) have documented this type of jet/plume structure for the Niagara River and have also shown qualitatively how Coriolis effects modify this flow. Previous work related to the present study includes physical modeling experiments conducted by Li et al. (1975) to gain understanding of general circulation patterns in Lake Ontario and also a number of studies of the Niagara River plume (see, for example, Journal of Great Lakes Research, vol. 9 (2), 1983; Masse and Murthy 1990). However, little has been done to examine the physical dynamics of the flow in great detail. Numerical models of buoyant plumes have been developed (e.g., O'Donnell and Garvine 1983, Chao and Boicourt 1986, Garvine 1987), but a major problem is faced in trying to validate circulation models for very large systems since comprehensive synoptic data sets are lacking. An advantage of well-posed physical models is that it is possible to generate data that may be used to calibrate numerical models. Although this is not a goal of the present study, it provides motivation for further development of the physical model. SCALING ANALYSIS The following scaling analysis is proposed to establish a general framework for describing a largescale buoyant discharge. The results may be used to distinguish between three regions of the flow: (1) a
far field
I transition I
near field
_ buoyant spreading
H
o
u o
near-field, where inertial forces are balanced by pressure and bottom drag; (2) an intermediate-field where the balance is between inertia and buoyancy; and (3) a far-field characterized by nearly geostrophic flow (balance between buoyancy and rotation). Referring to Figure 1, the discharge flow has initial velocity Uo' depth Ho and relative buoyancy, or reduced gravity g' = g~p/p, where g = gravitational acceleration, p =ambient density, Po =density of the discharge, and ~p = p - po. Beyond the discharge point the bottom has slope Sb' assumed constant. Using characteristic scaling quantities for velocity (U), vertical length (H), and buoyancy (g'), the densimetric Froude number is defined in the usual way, Fr = U/(g'H)1I2, and Fro is the initial value of Fr. For now it is assumed that Fro > 1. Steady conditions are assumed, as well as high flow Reynolds number, Re = Reynolds number = URh/v, where Rh = hydraulic radius (approximately equal to depth for wide flows) and v = kinematic viscosity. Assuming a normal right-handed Cartesian coordinate system with z the vertical (positive upward) direction, the longitudinal momentum equation is
au
au
au az - Iv
1 ap 2 a2u =- p ax + Ah V h U + Az az2 '
(1)
where u, v and w are the velocity components in the x, y and z directions, respectively, t =time, p =pressure, f = Coriolis parameter = 2rosinq>, ro = angular velocity of the earth's rotation, q> = latitude, Ah = horizontal diffusivity, Az = vertical diffusivity and Vh2 = a2/ax 2 + a2/ay2. A similar equation could be written for the transverse direction, but the main concern here is for the flow direction. The vertical pressure distribution is approximated as hydrostatic. In the near field the jet is still attached to the bottom and it is assumed that Coriolis effects are not yet important. Upon introducing previously defined scaling quantities and incorporating the hydrostatic assumption for pressure, a vertically integrated form of (1) may be obtained as
u2
-""g
Ln
FIG. 1. Definition sketch for bouyant discharge, side view.
au
ai+ u ax +v dy +w
, H
- + 'l'-
Ln
pH'
(2)
where 't = bottom shear stress, and Ln is a horizontal length scale providing a measure of the distance over which the jet remains attached. In other words, inertial momentum is balanced by pressure and bottom friction. If't =CDPU2 is assumed, where CD is an av-
Physical Model ofNiagara River Discharge erage bottom drag coefficient, then (2) may be solved for L n to obtain
Ln '"
JL(I- _1_) . CD
Fr 2
(3)
This shows that Fr > 1 is a necessary condition for there to be any attachment at all. This is consistent with our assumption (if Fr = Fro) and also with results of classical analyses, such as the arrested wedge model of Schijf and Schonfeld (1953). More precisely, from experimental observations, Fro must be greater than about 2.5 in order for attachment to occur (Safaie 1979). After separation bottom friction disappears and the momentum balance is between inertia and buoyancy (or pressure). This region is referred to as the intermediate-field, where the jet undergoes a transition to become more plume-like. Although some entrainment may occur (from underneath as well as from the sides), turbulence is generally dying out and a buoyant spreading layer forms. An internal hydraulic jump may form (Chu and Jirka 1986). The parameterized momentum balance for this region is the same as in (2), but without the bottom friction term. Thus, Fr = 1, which relates the speed of propagation of the buoyant layer to the relative buoyancy and depth of the layer. For the far-field plume, rotation becomes important as inertial forces decay. The flow is still buoyant and the momentum balance becomes approximately geostrophic, (4)
Where L f represents a measure of the distance between the detachment point and the location at which Coriolis effects become significant. Using the definition of the Rossby number, Ro = U/fLf , (4) may be rearranged to obtain Fr2 = Ro, or n Lf",_lFr '
(5)
where ri := (g'H)1I2/f is the internal Rossby radius of deformation (e.g., Garvine 1987). Since Fr is expected to be about 1 at or near the detachment point, (5) shows that L f scales approximately with r·. This is consistent with previous interpretations of ~i. For example, results from Garvine's (1987) model indicate that rotation effects become apparent at a distance of about 0.6 ri , while Masse and Murthy (1990) suggest that rotation becomes important for the Niagara River discharge at distances of about (2-3)r. (this latter estimate refers to distance from th~
585
shore). Also, for constant U and H, increasing buoyFr, so L f is larger. ThIS IS conSIstent wIth expenmental observations presented in the following section. an~y ~g') inc~eases r i ~nd decre~ses
EXPERIMENTS Results from experiments performed in the Rotating Laboratory Facility (RLF) at SUNY/Buffalo were used to test some of the above results, specifically the effect of rotation on the Niagara River discharge. This facility consists of an enclosed room with a 3.5 m by 5 m floor space, which is rotated smoothly on a single thrust bearing at speeds up to 6 revolutions per minute (rpm). A schematic model of Lake Ontario, originally designed and constructed by Li et al. (1975), was refurbished for these tests. Because of the large length-to-depth ratio of the lake, the model length scaling is distorted, with the horizontal scale ratio being 1:100,000 and the vertical ratio 1:800. It is constructed of a rigid polyurethane foam with an epoxy coating. Overall dimensions of the model are 3.3 m long, 1.3 m wide and 0.67 m deep. An inlet section corresponding to the Niagara River inflow is provided, along with outflow to the St. Lawrence River. Other flows are neglected. A calibrated pump was used to provide the necessary flowrate and an adjustable V-notch weir controlled the outflow. Salt was mixed into the "lake" water to provide buoyancy for the fresh water river flow. A small amount of dye was added to the river water and the resulting flow patterns were continuously recorded by an overhead video camera revolving with the model. If the Reynolds number is high enough (to ensure turbulence in the model), the scaling analysis of the previous section suggests that Fr and Ro are the important dimensionless parameters for determining dynamic similarity between the model and prototype. This leads to (6)
and (7)
where subscript r indicates a ratio of the model value to the respective prototype value. Li et al. (1975) assumed that g'r = 1 and showed that this scaling resulted in a model rotation rate of 1.71 rpm. If a time scale for horizontal motions is defined as
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T = LIU, then it can be shown that 1.02 seconds of model time corresponds with 1.0 hour in real time. Thus, to model a flow over a season requires a 35-40 minute test. Experimental results are shown in the series of photographs, Figures 2 and 3. The thicker grid lines in these pictures have a spacing of 40 cm, which corresponds with 40 km in the lake. The finer grid lines have a spacing of 5 cm. The time reading in the lower left corner indicates real time, but with an arbitrary starting point. These tests were performed by first filling the lake water with a slightly salty solution (approximately 0.1 % NaCl, by weight). The model was then slowly accelerated until the desired angular velocity was reached, after which it was rotated at constant speed for about an hour. Preliminary tests using dye streak observations showed that a period of 40-45 minutes was sufficient for the model to reach an equilibrium condition. After this "spin-up" period the Niagara River flow (dyed) was started and the resulting flow patterns were recorded by the camera. Figure 2 shows the flow during a test in which the model was not rotated, to serve as a base case for comparison with rotating tests. The discharge is 10-
cated near the number "2" in the lower left of the picture and the plume is seen as a slightly darker water mass extending offshore. The relative density difference was ~p/p = 3.8 x 10-4, which corresponds to a temperature difference of about 2°C. The Reynolds number was Re = 220. The initial momentum of the flow carries it out into the lake, aided by buoyancy which also causes some lateral spreading. As the flow approaches the far coastline, adverse pressure gradients turn it to either side, with most of the flow moving eastward. This is the situation shown in Figure 2, about 7 minutes after the flow was started. Although Re is fairly small for this discharge, there appears to be some lateral entrainment, as evidenced by the inward (clockwise) turning of the plume. Vertical entrainment was not observed. Figures 3a and 3b show results for the same experimental conditions as used in the test of Figure 2, except that the model was rotated. In Figure 3a the initial development of the plume can be seen, about one minute after initiating the discharge, while Figure 3b shows the flow at a later stage, about nine minutes after starting. There is clearly an effect of rotation on the resulting flow pattern. These figures
FIG. 2. Test result without rotation, about 7 minutes after the Niagara discharge was started; the flow is dyed and the discharge point is near number "2" in the lower left of the photo.
Physical Model ofNiagara River Discharge (a)
587
(b)
FIG. 3. Test results with rotation, (a) about 1 minute after initiating dyed discharge, the plume immediately exhibits a characteristic right-turning motion; (b) about 8 minutes later in the experiment, a concentrated, rotating region is seen near the river mouth and a shoreline current has begun to flow eastward along the southern shore.
show a flow pattern which is qualitatively similar to satellite images of the plume (see Masse and Murthy 1990, or Atkinson and Masse 1990), including the strong turning of the flow and the development of a shoreline current along the southern coast, seen in Figure 3b. Although not shown here, this test was conducted until the plume reached the St. Lawrence outlet, which occurred about 42 minutes after the flow was started. This corresponds with a period of about 3 months in real time, considerably shorter than the average retention time for the lake (around 7 years). One potential problem with the present experimental setup involves the discharge conditions. The sand bar at the mouth of the river is not represented in the model (flow would be only about I mm deep over the bar) and the discharge aspect ratio, A = BdHo' where B o is the discharge width, is not optimal. This is because the model was originally designed to study general circulation patterns within the lake and not necessarily the details of the river discharge. At the mouth the river width is about 1,250 m and average depth is 22 m (Masse and Murthy 1990). The discharge width in the physical model is consistent with the horizontal scaling, but
its depth is about 1 cm, only one-third of the value it would have if the actual vertical scaling were used. McClimans and Saegrov (1982) have suggested a model flow with a minimum depth of 1 cm for a salt-stratified experiment in order to avoid problems with surface tension. Using a non-dimensional form of the integrated momentum equation, they also showed that model distortion is a necessary condition for simulating frictional effects and that A should be at least 10-100. Thus, the present model satisfies the minimum depth constraint, but is probably not reproducing frictional effects well for the discharge. This is compounded by the low Re of the model, Re == 200. This is in the laminar flow regime and frictional effects are probably not well reproduced in general. Although bottom and side-wall friction are not necessarily of direct importance for modeling the plume dynamics at some distance from the mouth, the lack of turbulence in the model also implies that frontal mixing and entrainment are not well represented. McClimans and Saegrov (1982) suggested that, in addition to the above constraints, the model flow should have Re > 500 in order to accommodate this feature. Of course, even higher Re values are desirable. Future tests will have to be
S88
Atkinson et ale
made with these adjustments in mind, but the present results provide some useful information regarding the effect of rotation on the flow from the river and also a test of the scaling analysis presented in the previous section. Since the sand bar at the mouth is absent, there should be no bottom attachment in the model. The Froude number for the river, assuming a temperature difference of 1-4°C and flow velocity of 1.0 mis, is Fro ~ 2.5-5. According to the scaling analysis, eq. (3) gives an estimate for the distance of attachment as Ln = 4.2 kIn, assuming CD =0.005. This distance compares closely with the length of the bar, about 5 kIn (Masse and Murthy 1990). Alternatively, if the bar is considered as an extension of the river (noting that average flow depths over the bar are about half the depth at the river mouth), then Fr = 0.9-1.5 is estimated at the edge of the bar, using U =0.2 mis, H = 10m and the same temperature range as before. These values are too low for attachment to occur past the bar, consistent with the observations of Masse and Murthy (1990). The results can be used to check, at least in a qualitative sense, the scaling analysis for far-field conditions where rotation becomes important. The internal Rossby radius for the discharge, using the same parameter estimates as before and noting that f === ~0-4 sec- 1 for this location, is r j = 2-4 kIn. With estimates for Fr as above, this implies that rotation effects should be seen at about 1.3-4 kIn (past the bar). It is difficult to check this estimate directly with data, though it appears to be of the right orderof-magnitude, compared with the satellite images of Masse and Murthy (1990). For the model, f = 0.36 sec-1 and rj = 1.7 cm. Then, with Fr = 2.9, (5) shows that rotation should start to affect the flow at a distance around 0.6 cm. From Figure 3a it is difficult to determine a precise point at which the flow starts to tum, although it is evidently within the first centimeter or two. Considering the approximate nature of scaling analyses, this is a reasonabie comparison. One other rotating test was performed using the same discharge as in the previous tests, but with a much higher relative buoyancy, ~p/p = 2.6 x 10-2 . This corresponds with a freshwater discharge in a coastal environment, with a salinity difference of 3.4%. Although this is not representative of conditions in Lake Ontario, it provides a second test for eq. (5) which gives L f "" 40 cm for Fro =0.35 and r· =14 cm. This was approximately the same value ob~ served in the experiment (around 35 cm), though at this distance there was also probably some effect of the far coastline.
CONCLUSIONS In order to model the fate and transport of pollutants carried by the Niagara River into Lake Ontario it is necessary to understand the dynamics of the flow as determined by the various forces that act on it. The present study represents a first step toward this goal by examining the importance of rotation in affecting the discharge trajectory. The scaling analysis appears to provide correct order-of-magnitude estimates for distinguishing between the different regimes of the flow, based on comparisons with observations of the Niagara River plume as well as results from the physical model. The model also is shown to reproduce certain basic features of the discharge and demonstrates the importance of rotation in the dynamics of the flow. Future tests will have to be made with an improved discharge geometry in order to further refine this sort of analysis. For example, experiments should be performed to determine whether rotation may start to have an effect on the flow before it detaches from the bottom. Other features, such as cross-flow or surface wind stress, were not considered here and these should be considered in future experiments. ACKNOWLEDGMENTS This paper is a result of research funded by the NOAA Office of Sea Grant, U.S. Department of Commerce, under Grant #NA90AA-D-SG078 to the New York Sea Grant Institute. The U.S. Government is authorized to produce and distribute reprints for governmental purposes notwithstanding any copyright notation that may appear hereon. The views expressed herein are those of the authors and do not necessarily reflect the views of NOAA or any of its subagencies. The first author was also partially supported as a Lady Davis Fellow at the Technion-Israel Institute of Technology during part of this study. The authors thank Hasan Pourtaheri for setting up and filming the experiments, and Dr. Ann Masse for her help in conducting the tests. REFERENCES Atkinson, J.F., and Masse, A.M. 1990. Physical modeling study of the Niagara River plume. ASCE Hydraulics Division Spec. Conf., San Diego. Chao, S-Y, and Boicourt, W.C. 1986. The onset of estuarine plumes. Journal of Physical Oceanography 16:2137-2149. Chu, V.H., and Jirka, G.H. 1986. Surface buoyant jets and plumes. Chapter 25. In Encyclopedia of Fluid
Physical Model ofNiagara River Discharge Mech., vol. 6, pp. 1053-1084, ed. N. Chermisinoff. Houston, TX: Gulf Publ. Co. Garvine, R. W. 1987. Estuary plumes and fronts in shelf waters: A layer model. Journal of Physical Oceanography 17:1877-1896. Li, C.-Y, Rumer, R., and Kiser, K. 1975. Physical model study of circulation patterns in Lake Ontario. Limnology and Oceanography 20:323-337. Masse, A.K., and Murthy, C.R. 1990. Observations of the Niagara River plume (Lake Ontario, North America). J. Geophys. Res. 25:16,097-16,109. McClimans, T.A., and Saegrov, S. 1982. River plume studies in distorted Froude models. J. Hydraul. Res. 20:15-27.
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O'Donnell, J., and Garvine, R.W. 1983. A time-dependent, two-layer frontal model of buoyant plume dynamics. Tellus 35A:73-80. Safaie, B. 1979. Mixing of buoyant surface jet over sloping bottom. Journal of the Waterway, Port and Coastal Division, ASCE 105:357-373. Schijf, J.B., and Shonfeld, J.C. 1953. Theoretical consideration of the motion of salt and fresh water. Proc. Minnesota International Hydraulics Convention, IAHRA:32 1-333. Submitted: 6 July 1993 Accepted: 24 May 1994
Physical Model of Niagara River Discharge Joseph F. Atkinson, Guoqing Lin, and Maneesha Joshi Department of Civil Engineering State University ofNew York at Buffalo Buffalo, New York 14260
ABSTRACT. Development of a physical model study for the Niagara River discharge into Lake Ontario is described. The model is designed on the basis of (distorted) Froude and Rossby similarity and the results demonstrate clearly the effects of rotation on the flow. Important features of the plume are shown to be reproduced by the model, including its anticyclonic (right-turning) flow and the subsequent formation of a near-shore current along the southern coastline. A scaling analysis of the integrated longitudinal momentum equation is suggested as a framework for analyzing this type offlow. This analysis considers three regions of the flow: (/) a near-field, where inertial forces are balanced by pressure and bottom drag; (2) an intermediate-field where the balance is between inertia and buoyancy; and (3) a far-field characterized by nearly geostrophic flow (balance between buoyancy and rotation). This scheme is then tested against conditions characteristic of the Niagara River discharge during spring flows and also with observations from the physical model tests, showing reasonable agreement. One particular result of interest is that the distance at which rotation starts to turn the flow scales approximately with the internal Rossby radius. Results of the study help to distinguish between the relative effects of buoyancy and rotation. INDEX WORDS: Niagara River.
Buoyant jet, Co riolis force, physical model, Lake Ontario, buoyancy, hydrodynamics,
some distance Coriolis accelerations become significant and turn the plume clockwise. The main emphasis here is to demonstrate the effect of rotation, in conjunction with buoyancy, in determining the path followed by the discharge. A scaling analysis is proposed to identify regions of the flow controlled by different force balances and this helps to provide a general framework for describing the flow behavior. Several results from a physical model experiment are also presented. In general, discharges by rivers and estuaries are capable of modifying the circulation of receiving waters, both through baroclinic pressure gradients induced by buoyancy input and through the initial momentum flux of the discharge. The momentumdominated region near the discharge point is characterized by vertical homogeneity, relatively high velocities and lateral entrainment. Depending on the discharge Froude number (see below), the jet may be attached to the bottom for some distance and is therefore affected by bottom friction. After detachment the flow is increasingly buoyancy-dri-
INTRODUCTION The Niagara River (43° latitude, 79° longitude) is an essential link in the Great Lakes system, carrying flow from all the upper lakes and Lake Erie. The average discharge is nearly 7,000 m 3/s and represents about 85% of the total inflow to Lake Ontario (Masse and Murthy 1990). The 58 km-long river also passes through a highly industrialized region containing a number of "areas of concern" identified by the U.S. Environmental Protection Agency (EPA) and the International Joint Commission (UC) for potential loading of hazardous materials. Because of the relatively shallow depths for Lake Erie, its temperature may be 3 to 4°C warmer than in Lake Ontario over much of the spring and early summer (Masse and Murthy 1990). The present study is aimed at understanding the physical transport processes which govern the fate of pollutants carried by the river into Lake Ontario, particularly under these buoyant discharge conditions which result in a well-defined surface plume. The discharge spreads by inertia and buoyancy and after 583
Atkinson et 81.
584
ven and there is a transition zone where some vertical entrainment may occur. After this the flow approaches the state of a pure plume, spreading radially by buoyancy. Figure I illustrates these different regions. Masse and Murthy (1990) have documented this type of jet/plume structure for the Niagara River and have also shown qualitatively how Coriolis effects modify this flow. Previous work related to the present study includes physical modeling experiments conducted by Li et al. (1975) to gain understanding of general circulation patterns in Lake Ontario and also a number of studies of the Niagara River plume (see, for example, Journal of Great Lakes Research, vol. 9 (2), 1983; Masse and Murthy 1990). However, little has been done to examine the physical dynamics of the flow in great detail. Numerical models of buoyant plumes have been developed (e.g., O'Donnell and Garvine 1983, Chao and Boicourt 1986, Garvine 1987), but a major problem is faced in trying to validate circulation models for very large systems since comprehensive synoptic data sets are lacking. An advantage of well-posed physical models is that it is possible to generate data that may be used to calibrate numerical models. Although this is not a goal of the present study, it provides motivation for further development of the physical model. SCALING ANALYSIS The following scaling analysis is proposed to establish a general framework for describing a largescale buoyant discharge. The results may be used to distinguish between three regions of the flow: (1) a
far field
I transition I
near field
_ buoyant spreading
H
o
u o
near-field, where inertial forces are balanced by pressure and bottom drag; (2) an intermediate-field where the balance is between inertia and buoyancy; and (3) a far-field characterized by nearly geostrophic flow (balance between buoyancy and rotation). Referring to Figure 1, the discharge flow has initial velocity Uo' depth Ho and relative buoyancy, or reduced gravity g' = g~p/p, where g = gravitational acceleration, p =ambient density, Po =density of the discharge, and ~p = p - po. Beyond the discharge point the bottom has slope Sb' assumed constant. Using characteristic scaling quantities for velocity (U), vertical length (H), and buoyancy (g'), the densimetric Froude number is defined in the usual way, Fr = U/(g'H)1I2, and Fro is the initial value of Fr. For now it is assumed that Fro > 1. Steady conditions are assumed, as well as high flow Reynolds number, Re = Reynolds number = URh/v, where Rh = hydraulic radius (approximately equal to depth for wide flows) and v = kinematic viscosity. Assuming a normal right-handed Cartesian coordinate system with z the vertical (positive upward) direction, the longitudinal momentum equation is
au
au
au az - Iv
1 ap 2 a2u =- p ax + Ah V h U + Az az2 '
(1)
where u, v and w are the velocity components in the x, y and z directions, respectively, t =time, p =pressure, f = Coriolis parameter = 2rosinq>, ro = angular velocity of the earth's rotation, q> = latitude, Ah = horizontal diffusivity, Az = vertical diffusivity and Vh2 = a2/ax 2 + a2/ay2. A similar equation could be written for the transverse direction, but the main concern here is for the flow direction. The vertical pressure distribution is approximated as hydrostatic. In the near field the jet is still attached to the bottom and it is assumed that Coriolis effects are not yet important. Upon introducing previously defined scaling quantities and incorporating the hydrostatic assumption for pressure, a vertically integrated form of (1) may be obtained as
u2
-""g
Ln
FIG. 1. Definition sketch for bouyant discharge, side view.
au
ai+ u ax +v dy +w
, H
- + 'l'-
Ln
pH'
(2)
where 't = bottom shear stress, and Ln is a horizontal length scale providing a measure of the distance over which the jet remains attached. In other words, inertial momentum is balanced by pressure and bottom friction. If't =CDPU2 is assumed, where CD is an av-
Physical Model ofNiagara River Discharge erage bottom drag coefficient, then (2) may be solved for L n to obtain
Ln '"
JL(I- _1_) . CD
Fr 2
(3)
This shows that Fr > 1 is a necessary condition for there to be any attachment at all. This is consistent with our assumption (if Fr = Fro) and also with results of classical analyses, such as the arrested wedge model of Schijf and Schonfeld (1953). More precisely, from experimental observations, Fro must be greater than about 2.5 in order for attachment to occur (Safaie 1979). After separation bottom friction disappears and the momentum balance is between inertia and buoyancy (or pressure). This region is referred to as the intermediate-field, where the jet undergoes a transition to become more plume-like. Although some entrainment may occur (from underneath as well as from the sides), turbulence is generally dying out and a buoyant spreading layer forms. An internal hydraulic jump may form (Chu and Jirka 1986). The parameterized momentum balance for this region is the same as in (2), but without the bottom friction term. Thus, Fr = 1, which relates the speed of propagation of the buoyant layer to the relative buoyancy and depth of the layer. For the far-field plume, rotation becomes important as inertial forces decay. The flow is still buoyant and the momentum balance becomes approximately geostrophic, (4)
Where L f represents a measure of the distance between the detachment point and the location at which Coriolis effects become significant. Using the definition of the Rossby number, Ro = U/fLf , (4) may be rearranged to obtain Fr2 = Ro, or n Lf",_lFr '
(5)
where ri := (g'H)1I2/f is the internal Rossby radius of deformation (e.g., Garvine 1987). Since Fr is expected to be about 1 at or near the detachment point, (5) shows that L f scales approximately with r·. This is consistent with previous interpretations of ~i. For example, results from Garvine's (1987) model indicate that rotation effects become apparent at a distance of about 0.6 ri , while Masse and Murthy (1990) suggest that rotation becomes important for the Niagara River discharge at distances of about (2-3)r. (this latter estimate refers to distance from th~
585
shore). Also, for constant U and H, increasing buoyFr, so L f is larger. ThIS IS conSIstent wIth expenmental observations presented in the following section. an~y ~g') inc~eases r i ~nd decre~ses
EXPERIMENTS Results from experiments performed in the Rotating Laboratory Facility (RLF) at SUNY/Buffalo were used to test some of the above results, specifically the effect of rotation on the Niagara River discharge. This facility consists of an enclosed room with a 3.5 m by 5 m floor space, which is rotated smoothly on a single thrust bearing at speeds up to 6 revolutions per minute (rpm). A schematic model of Lake Ontario, originally designed and constructed by Li et al. (1975), was refurbished for these tests. Because of the large length-to-depth ratio of the lake, the model length scaling is distorted, with the horizontal scale ratio being 1:100,000 and the vertical ratio 1:800. It is constructed of a rigid polyurethane foam with an epoxy coating. Overall dimensions of the model are 3.3 m long, 1.3 m wide and 0.67 m deep. An inlet section corresponding to the Niagara River inflow is provided, along with outflow to the St. Lawrence River. Other flows are neglected. A calibrated pump was used to provide the necessary flowrate and an adjustable V-notch weir controlled the outflow. Salt was mixed into the "lake" water to provide buoyancy for the fresh water river flow. A small amount of dye was added to the river water and the resulting flow patterns were continuously recorded by an overhead video camera revolving with the model. If the Reynolds number is high enough (to ensure turbulence in the model), the scaling analysis of the previous section suggests that Fr and Ro are the important dimensionless parameters for determining dynamic similarity between the model and prototype. This leads to (6)
and (7)
where subscript r indicates a ratio of the model value to the respective prototype value. Li et al. (1975) assumed that g'r = 1 and showed that this scaling resulted in a model rotation rate of 1.71 rpm. If a time scale for horizontal motions is defined as
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T = LIU, then it can be shown that 1.02 seconds of model time corresponds with 1.0 hour in real time. Thus, to model a flow over a season requires a 35-40 minute test. Experimental results are shown in the series of photographs, Figures 2 and 3. The thicker grid lines in these pictures have a spacing of 40 cm, which corresponds with 40 km in the lake. The finer grid lines have a spacing of 5 cm. The time reading in the lower left corner indicates real time, but with an arbitrary starting point. These tests were performed by first filling the lake water with a slightly salty solution (approximately 0.1 % NaCl, by weight). The model was then slowly accelerated until the desired angular velocity was reached, after which it was rotated at constant speed for about an hour. Preliminary tests using dye streak observations showed that a period of 40-45 minutes was sufficient for the model to reach an equilibrium condition. After this "spin-up" period the Niagara River flow (dyed) was started and the resulting flow patterns were recorded by the camera. Figure 2 shows the flow during a test in which the model was not rotated, to serve as a base case for comparison with rotating tests. The discharge is 10-
cated near the number "2" in the lower left of the picture and the plume is seen as a slightly darker water mass extending offshore. The relative density difference was ~p/p = 3.8 x 10-4, which corresponds to a temperature difference of about 2°C. The Reynolds number was Re = 220. The initial momentum of the flow carries it out into the lake, aided by buoyancy which also causes some lateral spreading. As the flow approaches the far coastline, adverse pressure gradients turn it to either side, with most of the flow moving eastward. This is the situation shown in Figure 2, about 7 minutes after the flow was started. Although Re is fairly small for this discharge, there appears to be some lateral entrainment, as evidenced by the inward (clockwise) turning of the plume. Vertical entrainment was not observed. Figures 3a and 3b show results for the same experimental conditions as used in the test of Figure 2, except that the model was rotated. In Figure 3a the initial development of the plume can be seen, about one minute after initiating the discharge, while Figure 3b shows the flow at a later stage, about nine minutes after starting. There is clearly an effect of rotation on the resulting flow pattern. These figures
FIG. 2. Test result without rotation, about 7 minutes after the Niagara discharge was started; the flow is dyed and the discharge point is near number "2" in the lower left of the photo.
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FIG. 3. Test results with rotation, (a) about 1 minute after initiating dyed discharge, the plume immediately exhibits a characteristic right-turning motion; (b) about 8 minutes later in the experiment, a concentrated, rotating region is seen near the river mouth and a shoreline current has begun to flow eastward along the southern shore.
show a flow pattern which is qualitatively similar to satellite images of the plume (see Masse and Murthy 1990, or Atkinson and Masse 1990), including the strong turning of the flow and the development of a shoreline current along the southern coast, seen in Figure 3b. Although not shown here, this test was conducted until the plume reached the St. Lawrence outlet, which occurred about 42 minutes after the flow was started. This corresponds with a period of about 3 months in real time, considerably shorter than the average retention time for the lake (around 7 years). One potential problem with the present experimental setup involves the discharge conditions. The sand bar at the mouth of the river is not represented in the model (flow would be only about I mm deep over the bar) and the discharge aspect ratio, A = BdHo' where B o is the discharge width, is not optimal. This is because the model was originally designed to study general circulation patterns within the lake and not necessarily the details of the river discharge. At the mouth the river width is about 1,250 m and average depth is 22 m (Masse and Murthy 1990). The discharge width in the physical model is consistent with the horizontal scaling, but
its depth is about 1 cm, only one-third of the value it would have if the actual vertical scaling were used. McClimans and Saegrov (1982) have suggested a model flow with a minimum depth of 1 cm for a salt-stratified experiment in order to avoid problems with surface tension. Using a non-dimensional form of the integrated momentum equation, they also showed that model distortion is a necessary condition for simulating frictional effects and that A should be at least 10-100. Thus, the present model satisfies the minimum depth constraint, but is probably not reproducing frictional effects well for the discharge. This is compounded by the low Re of the model, Re == 200. This is in the laminar flow regime and frictional effects are probably not well reproduced in general. Although bottom and side-wall friction are not necessarily of direct importance for modeling the plume dynamics at some distance from the mouth, the lack of turbulence in the model also implies that frontal mixing and entrainment are not well represented. McClimans and Saegrov (1982) suggested that, in addition to the above constraints, the model flow should have Re > 500 in order to accommodate this feature. Of course, even higher Re values are desirable. Future tests will have to be
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made with these adjustments in mind, but the present results provide some useful information regarding the effect of rotation on the flow from the river and also a test of the scaling analysis presented in the previous section. Since the sand bar at the mouth is absent, there should be no bottom attachment in the model. The Froude number for the river, assuming a temperature difference of 1-4°C and flow velocity of 1.0 mis, is Fro ~ 2.5-5. According to the scaling analysis, eq. (3) gives an estimate for the distance of attachment as Ln = 4.2 kIn, assuming CD =0.005. This distance compares closely with the length of the bar, about 5 kIn (Masse and Murthy 1990). Alternatively, if the bar is considered as an extension of the river (noting that average flow depths over the bar are about half the depth at the river mouth), then Fr = 0.9-1.5 is estimated at the edge of the bar, using U =0.2 mis, H = 10m and the same temperature range as before. These values are too low for attachment to occur past the bar, consistent with the observations of Masse and Murthy (1990). The results can be used to check, at least in a qualitative sense, the scaling analysis for far-field conditions where rotation becomes important. The internal Rossby radius for the discharge, using the same parameter estimates as before and noting that f === ~0-4 sec- 1 for this location, is r j = 2-4 kIn. With estimates for Fr as above, this implies that rotation effects should be seen at about 1.3-4 kIn (past the bar). It is difficult to check this estimate directly with data, though it appears to be of the right orderof-magnitude, compared with the satellite images of Masse and Murthy (1990). For the model, f = 0.36 sec-1 and rj = 1.7 cm. Then, with Fr = 2.9, (5) shows that rotation should start to affect the flow at a distance around 0.6 cm. From Figure 3a it is difficult to determine a precise point at which the flow starts to tum, although it is evidently within the first centimeter or two. Considering the approximate nature of scaling analyses, this is a reasonabie comparison. One other rotating test was performed using the same discharge as in the previous tests, but with a much higher relative buoyancy, ~p/p = 2.6 x 10-2 . This corresponds with a freshwater discharge in a coastal environment, with a salinity difference of 3.4%. Although this is not representative of conditions in Lake Ontario, it provides a second test for eq. (5) which gives L f "" 40 cm for Fro =0.35 and r· =14 cm. This was approximately the same value ob~ served in the experiment (around 35 cm), though at this distance there was also probably some effect of the far coastline.
CONCLUSIONS In order to model the fate and transport of pollutants carried by the Niagara River into Lake Ontario it is necessary to understand the dynamics of the flow as determined by the various forces that act on it. The present study represents a first step toward this goal by examining the importance of rotation in affecting the discharge trajectory. The scaling analysis appears to provide correct order-of-magnitude estimates for distinguishing between the different regimes of the flow, based on comparisons with observations of the Niagara River plume as well as results from the physical model. The model also is shown to reproduce certain basic features of the discharge and demonstrates the importance of rotation in the dynamics of the flow. Future tests will have to be made with an improved discharge geometry in order to further refine this sort of analysis. For example, experiments should be performed to determine whether rotation may start to have an effect on the flow before it detaches from the bottom. Other features, such as cross-flow or surface wind stress, were not considered here and these should be considered in future experiments. ACKNOWLEDGMENTS This paper is a result of research funded by the NOAA Office of Sea Grant, U.S. Department of Commerce, under Grant #NA90AA-D-SG078 to the New York Sea Grant Institute. The U.S. Government is authorized to produce and distribute reprints for governmental purposes notwithstanding any copyright notation that may appear hereon. The views expressed herein are those of the authors and do not necessarily reflect the views of NOAA or any of its subagencies. The first author was also partially supported as a Lady Davis Fellow at the Technion-Israel Institute of Technology during part of this study. The authors thank Hasan Pourtaheri for setting up and filming the experiments, and Dr. Ann Masse for her help in conducting the tests. REFERENCES Atkinson, J.F., and Masse, A.M. 1990. Physical modeling study of the Niagara River plume. ASCE Hydraulics Division Spec. Conf., San Diego. Chao, S-Y, and Boicourt, W.C. 1986. The onset of estuarine plumes. Journal of Physical Oceanography 16:2137-2149. Chu, V.H., and Jirka, G.H. 1986. Surface buoyant jets and plumes. Chapter 25. In Encyclopedia of Fluid
Physical Model ofNiagara River Discharge Mech., vol. 6, pp. 1053-1084, ed. N. Chermisinoff. Houston, TX: Gulf Publ. Co. Garvine, R. W. 1987. Estuary plumes and fronts in shelf waters: A layer model. Journal of Physical Oceanography 17:1877-1896. Li, C.-Y, Rumer, R., and Kiser, K. 1975. Physical model study of circulation patterns in Lake Ontario. Limnology and Oceanography 20:323-337. Masse, A.K., and Murthy, C.R. 1990. Observations of the Niagara River plume (Lake Ontario, North America). J. Geophys. Res. 25:16,097-16,109. McClimans, T.A., and Saegrov, S. 1982. River plume studies in distorted Froude models. J. Hydraul. Res. 20:15-27.
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O'Donnell, J., and Garvine, R.W. 1983. A time-dependent, two-layer frontal model of buoyant plume dynamics. Tellus 35A:73-80. Safaie, B. 1979. Mixing of buoyant surface jet over sloping bottom. Journal of the Waterway, Port and Coastal Division, ASCE 105:357-373. Schijf, J.B., and Shonfeld, J.C. 1953. Theoretical consideration of the motion of salt and fresh water. Proc. Minnesota International Hydraulics Convention, IAHRA:32 1-333. Submitted: 6 July 1993 Accepted: 24 May 1994