Time-varying underflow into a continuous stratification with bottom slope

Time-varying underflow into a continuous stratification with bottom slope By Rocío Luz Fernandez1 and Jörg Imberger2, M. ASCE

Abstract: Results are presented from a laboratory investigation of a continuous discharge gravity current moving down an inclined plane into a linearly stratified fluid; the density of the inflow decreasing linearly with time, initially larger and finally smaller than the bottom ambient density. The inflowing water was observed to follow both underflowing and intrusive flow regimes. Hence, during the time in which the inflow was denser than the water in the stratified reservoir, an underflow was observed to descend down the sloping bottom with a speed that was consistent with that given by the theory for a buoyancy-conserving gravity current on gentle slopes. However, the continuous decrease of the density at the source leaded shortly to an unstable density distribution within the initial underflow, which then collapsed into an intrusion that travelled as a horizontal gravity intrusion. Scaling arguments were used to identify both the position and time to the break up of the underflow. To the end of the experiment, multiple intrusions were established successively at different depths in between the initial underflow and the surface buoyant plume.

CE Data base subject headings: unsteady, underflow, linear stratification, intrusion, slope angle, buoyancy, front velocity, shear waves, density.

1

Postgraduate student, Ctr. for Water Res., Univ. of Western Australia, Perth 6009, Australia. 61(0)864883666.

Fax: 61(0)864881015. E-mail: [email protected] 2

Prof. and Chair, Ctr. for Water Res., Univ. of Western Australia, Perth 6009, Australia. 61(0)864883085. Fax:

61(0)864881015. E-mail: [email protected]

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INTRODUCTION A frequent occurrence in nature is when the bulk density of a river inflow is larger than the fluid density on the surface of the reservoir leading to an underflow down the drowned river valley. A simplification often made in the modeling of such flows is to assume that the density of the inflowing water is constant in time, even if there is field evidence that river inflows more commonly have a hydrograph that has cooler and or saltier water in the rising part of such plot, as shown by Figure 1 for the Helena River into Mundaring Reservoir (Fernandez and Imberger 2006). Further, the receiving quiescent water is commonly assumed to be homogeneous even though recent studies on gravity currents into linearly stratified environments have shown that internal gravity waves are excited by underflows. Maxworthy et al. (2002), Ungarish and Huppert (2002) have observed two, quite different flow regimes for inflows into a stratified ambient receiving water. First, a subcritical regime where the internal wave speed of the interior first vertical wave mode moves faster than the underflow and second, a supercritical regime were the underflow propagates down the sloping bottom faster than the internal wave speed. Subcritical flows often set up intermediate depth intrusions into the metalimnion of the lake and the intruding water generates shear waves that in turn influence the speed of the intrusion. Shear waves produced by such intrusions have been observed by Browand and Winant (1972), Maxworthy (1972), Manins (1976), and McEwan and Baines (1974) who all illustrated that shear waves can move out ahead of the inflowing front or intrusion, reflect from the downstream boundary (dam wall in the case of a reservoir) and travel back to intercept the oncoming underflow or intrusion. Baines (2001) experimentally investigated the behaviour of an underflow from a constant buoyancy source into a very strong ambient stratification. He showed that in the limit when the stratification length scale approached the thickness of the underflow, the interface between the underflow and the ambient flow exhibited both entrainment into the underflow and detrainment from the underflow into the ambient fluid. Fischer and Smith (1983) documented an unsteady inflow into Lake Mead and they illustrated the tendency for the inflowing water to form, what appeared to be, multiple middepth intrusions whenever the inflow reached a level of neutral buoyancy during its course down the drowned river valley. However, apart from this brief description of an unsteady inflow there does not appear to be any description, in the literature, of an unsteady inflow into a stratified ambient water body.

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Here we examine an inflow, down an inclined bottom, from a source with a decreasing density into a stratified ambient water body. Results show that the variability in the buoyancy of the released fluid leads to an unstable density distribution within the initial underflow, that then collapses and detrains at an intermediate level. This typically resulted in the development of a more complicated horizontal structure of shear waves supported by the ambient stratification and subsequently in the progressive establishment of multiple intrusions that in turn interact with the shear waves. THE EXPERIMENT Experimental facility and methods A schematic of the experimental facilities, the dimensions and the coordinate system used are given in Figure 2. The 8 m long (effective length L = 7.2 m), 0.4 m deep and 0.6 m wide glass-walled tank was tilted with a slope angle θ = 2.5° for all the experiments. Two trolleys were placed on rails on top of the tank; one served as a mounting platform for the stratifier, while the other held one of the conductivity-temperature probes. The x coordinate was directed downslope tangential to the channel bed and the z coordinate was assumed to be directed upward normal to the bed. The tank was filled with water stratified with salt, as described by Imberger (1972). In brief, at the start of an experiment, the tank (in horizontal position) was filled with a bottom layer of salty water overlaying by a layer of fresh water. A series of three plates, mounted on one of the trolleys, were then moved through the tank to generate a linear profile by the mixing of this two-layer system. Once the linear stratification was established, the tank was tilted to the desired angle. The conditions at the inlet were carefully controlled to minimize any momentum input by the released water and to ensure that the inflow was uniform across the width of the tank. This was achieved by placing foam in front of a set of five diffusers, all contained in a discharge chamber. The varying salinity of the inflow was achieved by using the ‘two-tank’ method as proposed by Oster (1965). Dye and an amount of ice were also added to the inflowing fluid in order to produce differences in color and temperature of the released water, providing a visual and measurable tracer. However, the temperature differences were kept small so that the buoyancy effects were negligible and the temperature difference acted essentially as a passive tracer. An experimental run was typically completed in about 20 minutes. Two accurate Precision Measurement Equipment low response probes were used to record temperature-conductivity measurements of the inflow: one of 3

the probes was mounted on the trolley, while the other was installed inside the discharging chamber and remained fixed during the experiment. The trolley mounted probe allowed the profiling along the length of the tank; a stepper motor arrangement was used to move the probe vertically. Vertical velocity profiles were obtained by dropping dyed sugar crystals into the tank to produce columnar dye-lines. The distortion of these lines was videotaped, and the image frames were analyzed to estimate the fluid velocity. In some experiments, neutrally buoyant particles (instead of the crystals) were placed in different sections of the tank and an image processing software package was used to obtain data on the particle motion and thus the water velocity. However, it is to be stressed that the objective of these experiments was to gain a first order insight into the bulk behaviour of the multiple intrusions and not to study any detailed part of the flow; for this reason dye visualization sufficed. In addition, the motion of the dyed intrusions was recorded by a combination of photography and video for later analysis. A complete summary of all the experimental conditions is provided in Table 1. Given the typical averaged velocities, u, and high of the current, h0, the Reynolds number defined as Re = (uh0 )/ ! was moderately large (of about 1900 with a dynamic viscosity ~ 1.12-6 m2 s-2) and hence, the effect of viscous forces on the flow properties remained small (e.g., Maxworthy et al. 2002).

Initial ambient stratification and entry flow conditions Typical profiles of the linearly stratified water column in the tank are shown in Figure 3 for a section located at x/L = 0.4. The strength of the ambient stratification characterized by its intrinsic frequency N is listed in column 2 of Table 1; with N = (" g ! a ) [d! a (z ) dz ] , where ! a (z ) is the ambient density at the level z, ! a 0.5

denotes the ambient density averaged over the depth, and g is the gravitational acceleration. The imposed flow was varied to produce a linearly time decreasing density of the released fluid, namely

# r ( t ) = # rmax " ! b t

(1)

where ! r ( t ) is the time-dependent density of released fluid, ! r max is the maximum inflow density that occurred at time t = 0, and ! b is the constant density decrease per unit time of the inflow, typically O(10-2) kg m-3 s-1. The

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starting density of the released water was such that initially an underflow (heavier inflowing fluid) formed, while the final inflow density was comparable to the density on the surface of the reservoir to allow the occurrence of an overflow (lighter inflowing fluid). Figure 4a shows the characteristics of the released fluid for a typical run. In all the experiments, the inflow was released at a steady rate of q = 7.5 10-4 m2 s-1. A constant inflow temperature was maintained during the experiment, typically ~ 4 °C less than that of the tank in order to be able to distinguish between ambient and released fluid with temperature profiling. The associated buoyancy flux B(t) = qg’(t) is shown in Figure 4b, where g' (t) = g [! r ( t ) " ! w ] ! w is the reduced acceleration due to gravity in which ! w is the reference surface density (~ 1000 kg m-3). The Froude number for such a reversing buoyancy of the intruding fluid, says Fr = (q / h0 ) [g' ( t )h0 ]!1 2 , is displayed in Figure 4c, where h0 is the height of the current at x/L = 0.

EXPERIMENTAL RESULTS Qualitative description of a time-varying buoyancy inflow Flow visualization was used to gain a qualitative description of the evolution of the inflow. The sequence of side-view photographs in Figure 5 shows a typical experimental development in which the underflow degenerated into multiple layers of varying depth. For early times, Figure 5a shows that the incoming water underflowed the tank with what appears to be a uniform thickness. The underflow front did not have a raised head, as normally is the case for an underflow into a homogeneous receiving fluid (e.g. Simpson and Britter 1979), since the stratification within the tank flattened the frontal roller. The underflow was observed to have a well defined wedge shape front with a uniform wake. In addition, the vertical blue-dye lines placed along the tank, ahead of the inflow, clearly revealed that shear waves were moving fluid ahead of the underflowing front. As seen in Figure 5b, when the underflow was two thirds across the tank (~ t = 250 seconds), its front part thickened and started to form multiple intrusions. The newly formed upper intrusion developed rather quickly, as seen from the transition from Figure 5c to Figure 5e and 5d, and by the time t = 320 seconds the intrusion was seen to have overtaken the primary underflow front, which had almost stopped propagating down the slope only to start moving again as both intrusions approached the end wall. At the time t = 450 seconds, the midlevel intrusion thickened in an analogous fashion to that of the underflow at t = 320s, and formed yet another new intrusion clearly seen in the photo taken at t = 600 seconds

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(labeled as ‘new intrusion’ in Figure 5f). The previous intrusion (named ‘midlevel intrusion’ in Figure 5f) slowed as the fluid input to that layer decreased (owing to the density-variability of the source) and subsequently it was advected upstream by the returning shear waves. Two features emerge from this discussion. First, the variability of the source gives birth to a series of intrusions and second, the intrusion speeds are strongly modulated by the shear waves, generated by the inflow and intercepting the intrusions upon reflecting from the upstream end wall.

Quantitative description of the flow dynamics observed The analysis of the flow pattern described above may be broadly divided into the three following distinct stages: the underflow regime, during which the effects of the ambient stratification are absent, the collapse of the underflow, and the subsequent formation of the multiple intrusions, stages in which the ambient disturbances play a key role.

The underflow stage For the flow configuration just described, the front position of the underflow can be written in terms of the experimental parameters

x f = f ( t , q , g '0 , N ,! ) ,

(2)

where t is the time elapsed, q is the discharge, g '0 is the initial reduced gravity, N is the ambient buoyancy frequency and θ is the slope angle. During the initial phase of the experiment, and for an underflow much thinner than the scale for the stratification, the dependence on N in (2) may be neglected, and so from dimensional analysis follows that the propagation rate of the current is proportional to the initial buoyancy flux per unit width in the form

( )

x f ~ qg '0

13

t.

(3)

Equation (3) has certainly been amply justified to describe the motion of a turbulent buoyancy flux conserving gravity current from a constant source into both a non-stratified (Britter and Linden 1980, Turner 1986) and a

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stratified environment (Monaghan et al. 1999, Baines 2001). The general validity of equation (3), as applied to nonconservative buoyancy flux currents of clay suspensions, have been later demonstrated by Altinakar et al. (1990), as well as Graf and Altinakar (1995) and Hürzeler et al. (1996). The numerical factor in (3), named herein α, has been determined to depend, although weakly, on the slope angle (e.g. Simpson 1987, Hopfinger 1983). Particularly on gentle slopes ( ! < 5° ), Graf and Altinakar (1995) compiled data from previous experiments involving both saline and turbidity gravity currents to found that the range of variability of such a coefficient is 0.7 ! u f ( qg '0 )1 3 ! 1.5 . A plot x f

(B t ) versus Nt is then shown in Figure 6 for the experiments listed in Table 1. We find that 13 0

equation (3) holds along the initial distance of propagation of the underflow, with a coefficient of proportionality of about ! = 0.71 . As expected, the ambient stratification does not affect the motion of the current at early times. This is consistent with the behaviour of a purely buoyancy-driven flow far away of its neutrally buoyant condition for which the surrounding stratification, N, plays a negligible role (Baines 2001, Maxworthy et al. 2002). However, the failure of the collapse of the data by Nt ~ 100 illustrated by Figure 6, clearly suggests that ambient motions become to dominate the underflow propagation at later stages of the experiment.

The collapse of the unstable underflow Figure 7 shows the time-variability of densities associated with the propagation of the underflow. Here, ! r is the density of the fluid released at the inlet, ! f is the front density of the underflow, and ! opp is the ambient f density opposite to the front of the underflow as it flows down the sloping bottom. Included in the figure are the bottom densities within the underflow ( ! u ), as measured with the probe fixed at three different locations: x = 0.2L, 0.35L and 0.5L. Significantly, these density records suggest that the underflow is almost uniform in density during its propagation and, at 160 seconds from the start, the frontal density of the underflow (for the experiment shown in Figure 4) is approximately equal to the density of the fluid released at the inlet by that time, i.e.

" f ~ " r ~ 1014 kg m !3 . Further, the front density recorded with the probe fixed at x = 0.5L (named as point 3 on Figure 7) indicates that the underflow eventually reaches a section at which the density within the current is less than

~ 1016 kg m !3 . Because of this the ambient density in front of the head of the underflow, i.e. " f ~ 1014 < " opp f density deficit in the frontal region of the current, the underflow water may be no able to support the negative

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buoyancy of the overlaying ambient fluid, and subsequently collapses by detraining fluid about its level of neutral buoyancy. The scaling for the distance to the lift off of the intrusion may thus be obtained by noting that

! opp = ! bx =0 + f

"!x f L

(4)

where "# = # bx = L ! # bx =0 , with ! bx =0 and ! bx = L being the bottom ambient densities at x = 0 and x = L, respectively. By substituting from (3) and rearranging yields

Lf L

= 1+

1 1.4" b L

(5)

"!B01 3

where L f L is the non-dimensional distance from the source to the detachment point of the intrusion, and ! b is the constant density decrease per unit time of the inflow as defined earlier. By rearrangement and using (4) and (6), the associated non-dimensional time to the detaching event is thus given by

Nt f =

1.4 LN " L + b "!

(6)

B01 3

In the present experiments, for values of B0 and N that varied over the ranges 6 < B0 < 14 x10-5 m3s-3 and 0.59 < N < 0.79 s-1, equations (5) and (6) give the ranges 3.5 < L f < 4.3 meters and 110 < tf < 140 seconds, respectively, which are in good accordance with the experimental observations (see Table 1). The distance-time for the formation of multiple intrusions is also indicated in Figure 6, where it is seen that the scaling correctly describes the order of the phenomena.

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In addition to the bottom density records, vertical profiles were also taken during the experimental runs. Figure 8 shows the time evolution of the vertical structure of the flow for run (A) in a section located at x = 0.4L, i.e. upstream of the point in which the first intrusion forms. The density-stratified ambient fluid shown in Figure 8a at the time t = 0 seconds, is modified in Figure 8b when shortly after starting the fluid release the density profile begins to show a bottom layer (~ the lower half of the underflow) with a density that is approximately uniform in depth. This basal layer becomes much thicker with time, and by t = 200 seconds the profiles of Figure 8d suggest a nearly homogeneous well-mixed basal layer within the underflow, of about 0.1m deep, and with a density value typically of ! u ~1014 kg m-3. These profiles suggest that by the time the first intrusion forms, the underflow water column is nearly homogeneous. Indeed, the vertical density profiles of Figures 7c and 7d show a smaller density gradient within the turbulent underflow than in the column of surrounding ambient fluid (initially linearly-stratified). Given the unstable nature of such a homogeneous column of water, the time that it can exist next to the stratified ambient fluid is limited, and this unstable condition is easily disturbed by the impinging shear waves.

The formation of the multiple intrusions Additionally, the temperature profiles on Figure 8 indicate that the peeling of the underflow water (by t ~ 200 seconds) is preceded by a significant excitation of columnar disturbances, which are zero-frequency internal wave modes with finite propagation velocity c = NH/π (e.g. McEwan and Baines 1974, Manins 1976, and others), typically in this study ~ 0.055 m s-1. Although these perturbing modes appear as dominant features in the temperature profiles in Figures 7c-7f, they do not greatly affect the ambient stratification in that the density profiles are seen to be only slightly modified by such modes. Note here that the mode wave functions are also included in Figure 8 to contrast with the shape of the temperature measurements. A better visualization of the velocity field was obtained by dropping buoyant particles during the experiments. The oscillation of the resulting velocity profiles, shown in Figure 9, clearly reveals the domain of shear waves in a section located at x = 0.35L, upstream of the intrusion. At early times, Figure 9a shows that the underflow moves with a mean velocity of about 0.02m s-1, while an upper relatively low velocity layer travels in opposite direction with a speed of about ~ 0.005 m s-1. The evolution of this profile to the fluctuating vertical structure illustrated for later times by the rest of the curves shown in Figure 9, suggest that the shear fronts may strongly interact with the propagation of the intrusions.

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In particular, it is seen that as soon as the first intrusion forms, typically in this study at t ~ 200 seconds from the start of the experiments, the excited upstream propagating shear fronts reflect from the endwall of the channel in a period of about t ~ 2.5Lπ/NH = 300 seconds. By this time, i.e. t ~ 500 seconds from the start of the experiments, note that the inflowing water becomes less dense than the water at the bottom of the reservoir,

! r < ! bx =0 ~1009 kg m-3 (see Figure 3a). The arriving of the shear fronts are found then to disturb the unstable inflowing water column with the subsequent formation of a new intrusion, as suggested by Figure 5e for a typical experiment.

DISCUSSION The coefficient of proportionality, α ~0.7, is consistent with the motion of a continuous fed gravity current on gentle slopes ( ! < 5° ) for which 0.7 ! u f ( qg '0 )1 3 ! 1.5 , as indicated by Graf and Altinakar (1995). Earlier, Altinakar et al. (1990) had found that the value of α may diminish as the slope decreases below ! < 5° , and gradually approach a value defined by the relative depth ratio, h/H. In this study the relative submergence varied as h/H ~ 0.4-0.5. Based on these findings, the front propagation velocity of the underflow at early times for a range of experiments, appears to be in good agreement with that given either for a conservative (saline) gravity current (e.g. Britter and Linden 1980) or a non-conservative (weakly depositing) turbidity current (e.g. Altinakar et al. 1990), provided neither the density agent is nearly conservative and the stratification is not too strong; which is the case for most practical situations found in natural lakes and reservoirs. Following this initial stage of the flow propagation, it is found that an unstable condition was established within the underflow that leaded to their physical collapse in the form of a multiple intrusion. Two processes were found to contribute to this phenomenon. First, the analysis of densities associated with the propagation of the underflow illustrated that the current reached a section along the slope at which the front of the current was less dense than the surrounding ambient fluid. This density deficit within the underflow could have provided a mechanism by which the current became temporarily unstable, rose at the front and intruded at the neutrally buoyant level. Second, the near constant density of the underflowing column of water rendered it unstable to impinging shear waves that had reflected off the front wall of the tank. This is similar to the circulation forced by oscillating grids (e.g. Browand et al. 1987), in which the resulting intrusions that move outward present a density distribution nearly

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uniform, and opposite to them, a return flux in between the intrusions. In a prototype lake any high wave number internal waves would be sufficient to impose their length scale on the downflow. Based on the experimental results reported here it appears that the observed lift-off mechanism may be of the same type as that observed on particle-driven gravity currents which exhibit reversing-buoyancy phenomena (see Sparks et al. 1993, Hürzeler et al. 1996 , Maxworthy 1999, Hogg et al. 1999, and others). In those experiments, the lift-off distance is taken as he downstream distance at which the nose of the current first becomes buoyant, relative to the ambient fluid, and leaves the underlaying boundary towards the surface; no further propagation of the front of the current occurs along the bottom boundary. The major difference between the two types of behaviour is that in our case, the ambient fluid is stratified and the collapse of the underflow is also due to the interaction with the reflected generated shear wave, which causes the negatively buoyant fluid to propagate as an horizontal intrusion (and not to the surface), as fluid is being fed from the source into that layer. Subsequent past investigations have identified the establishment of shear waves within stratified environments as the features illustrated by the vertical temperature profiles in this study. They have been observed to be established by a sink flow (Pao and Kao 1974, Imberger et al. 1976), by a single intrusion (Browand and Winant 1972, Maxworthy 1972, McEwan and Baines 1974, Manins 1976), and even by an artificially produced underflow (Baines 2001, Maxworthy et al. 2002). The present experiments clearly identify that such shear waves are relevant in generating multiple intrusions within closed environments. In this context, the experiments give some indications that led to the conclusion stated by previous work (Faust and Plate 1984, Ungarish 2005) that intrusions into a linearly stratified environment behave differently from the theory. The major difference is that the ambient can not be considered quiescent and internal waves can not be neglected. Lastly, it appears that the dynamics suggested by the transport paths in the field-scale experiment in Lake Mead, as discussed by Fischer and Smith (1983), resulted from exactly such as wave-intruding fluid interaction.

CONCLUSIONS New features of the development of an unsteady underflow into a continuous density-stratified reservoir are described from a laboratory investigation. In particular, the experimental results show that: (i) the flow initially propagates with a velocity proportional (~0.7 times) to the cube root of its initial buoyancy flux.

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(ii) The continuous reduction of the density at the source leads to an unstable density distribution within the initial underflow, which then collapses and detrains its water at an intermediate depth. (iii) The collapse of the underflow appears due to the existence of a density deficit within its front, and also, because of a smaller density gradient within its body than that in the ambient surrounding fluid. (iv) The unsteady collapse of the underflow typically results in multiple intrusions which are postulated to be formed by internal waves, in this case shear waves excited by the flow itself, interacting with the downflow.

ACKOWLEDGMENTS The present work was performed while the first author was the recipient of an International Postgraduate Research Scholarship and a University of Western Australia Postgraduate Award. Financial support which made these experiments possible was provided by the Centre for Water Research. We thank C. Dallimore for the comments on the earlier versions of the manuscript, and G. Atwater who assisted in some early calibration tasks. We also thank the anonymous referees for their very positive and constructive reviews of the manuscript.

APPENDIX I. REFERENCES Altinakar, S., Graf, W. H. and Hopfinger, E. J. (1990). “Weakly depositing turbidity current on a small slope”. J. Hydr. Res. 28, 55-80. Baines, P. G. (2001). “Mixing in flows down gentle slopes into stratified environments”. J. Fluid Mech. 443, 237270. Britter, R. E. and Linden. (1980). “Experiments on the dynamics of a gravity head”. J. Fluid Mech., 88, 223-240. Browand, F. K. and Winant (1972). “Blocking ahead of a cylinder moving in a stratified fluid: An experiment”. J. Geophys. Fluid Dyn., 4, 29, 53. Faust, K. and Plate, E. (1984). “Experimental investigation of intrusive gravity currents entering stably stratified fluids”. J. Hydraulic Res., 22, 315-325. Browand, F. K., Guyomar, D. and Yoon, S.-C. (1987). “The behaviour of a oscillating grid”. J. Geoph. Res., 92(C5), 5329-5341.

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Fernandez, R. L. and Imberger, J. (2006). “Bed roughness induced entrainment in a high Richardson number underflow”. Accepted to J. Hydr. Eng. Fischer, H. B., List, E. J., Koh, R. C., Imberger, J. and Brooks. N. H. (1979). Mixing in Inland and Coastal Waters. Academic Press, Inc., pp. 483. Fischer, H. B. and Smith. R. D. (1983). “Observations of transport to surface waters from a plunging inflow to Lake Mead”. J. Limnol. Oceanogr., 28(2), 258-272. Graf, W.H. and Altinakar, M.S. (1995). “Courants de turbidité”. J. La Houille Blanche, 7, 28-38. Hogg, A., Huppert, H. and Hallworth, M. (1999). “Reversing buoyancy of particle-driven gravity currents”. Phys. Fluid., 11(10), 2891-2899. Hopfinger, E. (1983). “Snow avalanche motion and related phenomena”. Ann. Rev. Fluid Mech., 15, 47-76. Hürzeler, B. E., Imberger, J and Ivey, G. (1996). “Dynamics of turbidity current with reversing buoyancy”. J. Hyd. Eng.., 122(5), 230-236. Imberger, J. (1972). “Two-dimensional sink flow of a stratified fluid contained in a duct”. J. Fluid Mech., 43(2), 329-349. Imberger, J., Thomson, R. and Fandry, C. (1976). “Selective withdrawal from a finite rectangular tank”. J. Fluid Mech., 78, 489-512. Manins, P.C. (1976). “Intrusion into a stratified fluid”. J. Fluid Mech., 74, 547-560. Maxworthy, T. (1999). “The dynamics of sedimenting surface gravity currents”. J. Fluid Mech., 392, 27-44. Maxworthy, T. (1972). “Gravity currents with variable inflow”. J. Fluid Mech., 128, 247-257. Maxworthy, T., Leilich, J., Simpson, J. and Meiburg, E. H. (2002). “The propagation of a gravity current in a linearly stratified fluid”. J. Fluid Mech., 453, 371-394. McEwan, A. D. and Baines, P. G. (1974). “Shear fronts and an experimental stratified shear flow”. J. Fluid Mech., 63, 257-272. Monaghan, J. J., Cas, R. A. F., Kos, A. M. and Hallworth, M. (1999). “Gravity currents descending a ramp in a stratified tank”. J. Fluid Mech. 379, 39-70. Oster, G. (1965). “Density gradients”. Sci. Am. 213(2), 70-76.

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Pao, H.-P. and Kao, T.W. (1974). “Dynamics of establishment of selective withdrawal of a stratified fluid from a line sink”. J. Fluid Mech., 65, 657-688. Simpson, J. E. (1987). Gravity currents: in the environment and the laboratory. Cambridge University Press, pp. 244. Simpson, J. E. and Britter, R. E. (1979). “The dynamics of the head of a gravity current advancing over a horizontal surface”. J. Fluid Mech., 94(3), 477-495. Sparks, R., Bonnecaze, R., Huppert, H. Lister, J., Hallworth, M., Mader, H. and Phillips, J. (1993). “Sedimentladen gravity currents with reversing buoyancy”. Earth and Plan. Sc. Letters, 114(3), 243-257. Turner, J. S. (1986). “Turbulent entrainment: the development of the entrainment assumption, and its application to geophysical flows”. J. Fluid Mech. 173, 431-471. Ungarish, M. (2005). “Intrusive gravity currents in a stratified ambient: shallow water theory and numerical results”. J. Fluid Mech., 535, 287-323. Ungarish, M. and Huppert, H. E. (2002). “On gravity currents propagating at the base of a stratified ambient”. J. Fluid Mech., 458, 283-301.

APPENDIX II. NOTATION The following symbols are used in this paper B

buoyancy flux

Fr

densimetric Froude number

g

gravity acceleration

g'

reduced gravity acceleration

h

flow depth

H

reservoir depth

L

length of the reservoir

Lf

nondimensional distance to the intrusion

N

buoyancy frequency

q

flow rate per unit width

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Re

Reynolds number

t

time

T

water temperature

tf

nondimensional time to the intrusion

u

flow velocity

x

downstream coordinate

z

vertical coordinate

α

coefficient of proportionality

θ

bottom slope angle

!b

constant density decrease per unit time of the inflow

"!

difference in bottom densities between two sections of the tank

ρ

density

! opp f

ambient density opposite to the front current

The following subscripts are used in this paper a, r, b, u, w to denote: ambient, released, bottom, underflow and reference densities, respectively f

to refer to the front of the current

max

to refer to the maximum value

0

to refer to initial conditions

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