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HomeMy WebLinkAboutA Theoretical Investigation Kachemak Bay by Bradlkey Lake 1981=:.-. ·:) 14 .... , -·- A THEORETICAL INVESTIGATION OF THE POTENTIAL MODIFICATION OF ICE FORt4ATION IN KACHEMAK BAY BY THE BRADLEY LAKE HYDROELECTRIC POWER PROJECT J. P. Gosi nk and T. E. Osterkamp Fi na 1 report of r ·esearch performed under contract number OACA 89-81-K-0001 for the U.S. Army CRREL Hanover. N.H. ABSTRACT Construction of the Bradley Lake hydroelectric facility is expected to result in substantially increased winter discharge of fresh water into Kachemak Bay. This study assesses the expected changes in ice production associated with that increase. A thermodynamic model of ice production in the bay has been devised . which simulates the rate of fee production as a function of net surface heat loss, the added heat from relatively warm water discharge at the tailrace, and the effects of mixing on freezing point depression. The model was combined with prevailing environmental and oceanographic condi- tions and with present information regarding the Bradley Lake Power Project. Ice production rates were found to be sensitive to the fresh water- sea water mixing conditions at the head of the bay. Three cases were examined: complete mixing, no mixing and mixing or dilution as measured in the studies of Colonell (1980) and Knull (1975). The latter case represents a best estimate in the sense that it includes the appropriate fresh water dilution rates, mixing conditions, predicted heat losses, and fresh water lens area for the projected winter operation of the facility. It is concluded, based on this information and the thermodynamic model, that the projected winter ice production due to discharge from the hydroelectric power system is expected to decrease relative to present winter fee production due to discharge from the Bradley River. i INTRODUCTION The Alaskan Power Administration designated the Bradley Lake-Bradley River system as being suitable for hydroelectric power development in its 1974 survey of potential hydroelectric sites in Alaska (Alaska Regional Profiles, 1977). Bradley Lake is located on the east side of the head of Kachemak Bay at an elevation of 1090 feet. It discharges into the Bradley River which flows into the head of Kachemak Bay. The proposed power plant will be located near tidewater suggesting at least a 1000 foot head. Discharge will be channeled into a small unnamed creek = l/2 mile I~E of Sheep Point. Kachemak Bay is one of the major fisheries in the Cook Inlet Managment Area. Fishermen live primarily at Homer and on the east side of the bay. In the past, ice formation in the bay has occasionally caused problems for the fishermen apparently by impeding boat traffic and by damaging port and dock facilities (Bredthauer, personal communication). Substantially increased water discharge may be expected during winter operation of the Bradley Lake hydroelectric facility. Construction of a dam at Bradley Lake is required to insure the maintenance of hydro-potential during the winter months when peak electrical demand is expected. r~ximum electrical power can then be maintained by increasing the discharge from the lake through the turbines. According to the Alaska Corps of Engineers (Bredthauer, personal communication}, a maxi~um regulated discharge of 1400 cfs may be expected during severely cold weather. This contrasts with present unregulated winter discharge, which ranges from 40-100 cfs. The net ice production in the upper bay should be related to the fresh water input from several rivers which discharge into the bay, including Bradley River, Fox River, and Sheep Creek. These rivers contribute to the total ice production, first by the discharge of frazfl fee in the river l water which may amount to 10\ or more of the total discharge,. second, by the mixing with ambient salt water which raises the freezing point of the mixed water, and finally, by interfacial ice production which occurs when a lens of fresh water overlies colder salt water. Since these ice production mechanisms are all related to the winter discharge of fresh water and that discharge may be expected to increase substantially, as noted above, then there is a potential for an increase in ice production, associated with the hydroelectric development, in Kachemak Bay. However, the possibility of increased ice production is not so straightforward to assess as the above considerations would suggest because of the fact that the increased winter discharge from Bradley Lake will be warm water (probably about +3 to +4°C). The purpose of this study is to assess the expected changes in ice production in Kachemak Bay caused by the increased winter discharge of relatively warm fresh water from the Bradley Lake Power Project. The general approach will be to combine models of ice production with prevailing environmental and oceanographic conditions and with present information available on the Bradley Lake Power Project. The study will be restricted to the upper portion of Kachemak Bay, that portion above Homer Spit. Background information concerning ice production and movement in Kachemak Bay has been summarized in a recent report by Gatto (1981). In the present study, a conceptional model of ice production in the bay is formulated. Three possible cases have been included: complete mixing of the fresh water discharge and sea water, no mixing (spreading of a fresh water lens}, and an estimate based on measured salinity and mixing condi- tions. The analysis has required that estimates be made of several para- meters, including the temperature of the outflow at the tailrace, the 2 expected extent of a fresh water lens~ and the heat transfer associated with interfacial ice production. These values have been obtained from appropriate laboratory and field studies. For the worst case, that of no mixing, we have used limiting values of the parameters in order to establish a maximum ice production rate. 3 PREVIOUS RESEARCH AND PHYSICAL SETTING Since the Alaskan Power Admfn1station designated Bradley Lake as a potential hydroelectric power site, there have been a number of studies of the Kachemak Bay area which are particularly relevant to the problems associated with hydroelectric development. One of the most recent of these is the study of dispersion in upper Kachemak Bay by Colonell (1980). The dispersion and dilution of a slug of dye over several tidal cycles was followed using Rhodamine WT dye as a tracer. Dispersion was found to be dominated by the tides, which have a range of no less than 11 feet near the mouth of Kachemak Bay. Colonell (1980) also described damming of Bradley River water during the incoming flood tide, followed by rapid mixing during the next tidal cycle. Within two tidal cycles, it was found that the mixing was virtually complete. Gatto (1981) summarized historical evidence of ice formation and circulation as recorded in Landsat imagery. Most of the ice in the Inner Bay formed at the mouths of the Fox and Bradley Rivers and Sheep Creek. Wind and currents coupled with the Coriolis force tend to move the ice southwestward along the north shore toward Coal Bay and Homer Spit. Table 1 lists the dates of ice observations (Gatto, 1981), the type of ice obser- vation and the average wind speeds and air temperatures for the day of the ice event and the previous day. The meteorological data describing Kachemak Bay were gathered from the Local Climatological Data of NOAA for Homer 1968-1980. The average dai1y minimum winter temperature during this period are plotted in Figure 1. The average minimum temperature for these years was consistently greater than 5°r. The coldest winter of the 1968-1980 period occurred in 1972 (Figure 4 Table 1: Ice observations in Kachemak Bay (Gatto, 1981) and average wind speed and air temperatures measured at Homer for the same day and the previous day. ICE AVERAGE WINO AVERAGE AIR DATE OBSERVATION SPEED (mph) TEMPERATURE (°F) Same Previous Same Previous Day Day Day Day 8 Mar 73 Possible ice 4.8 1.3 31 30 stringer from Coal Bay Possible ice accumulation in Coal Bay 4 Mar 74 Ice on tidal 14.2 21.9 9 22 flat Ice stringer north of Homer Spit 22 Mar 74 Ice on tidal 5.2 9.4 37 40 flat 26 Jan 76 Ice build-up 6.5 12.1 36 35 in Coal Bay 24 Feb 76 Fast ice in 3.6 6.6 12 16 Coal Bay 5 Feb 79 Ice in Coal Bay; 10.9 11 9 possible ice floes in head of bay 6 Feb 79 Ice in Coal Bay 8.3 10.9 9 11 and along north shore of bay 7 Feb 79 Ice in Coal Bay 16.5 8.3 7 9 14 Feb 79 Ice along north 14.2 8.2 6 9 shore; shorefast ice in Coal Bay out to Coal Point 23 Feb 79 Ice along north 4.3 5.8 20 18 shore; degrading shorefast ice in Coal Bay 5 2). A minimum temperature of -l8°F was recorded on January 10, 1972. It appears that cold spells of less than -4°F are uncommon, and of short duration, in general, less than 4-5 days. The daily average wind velocity over the 1968-1980 period is plotted in Figure 3. Winds rarely exceeded 10 mph at Homer during this period; however, it is likely that much stronger winds exist near the head of the bay. During a field trip to Homer on February 24, 1981, severe winds and heavy gusting were encountered over the mouth of the Bradley River during two reconnaissance flights. It is likely that severe winds are character- istic of the head of Kachemak Bay due to the local topography which is favorable for both katabatic winds and winds driven by the venturi effect. Katabatic winds would tend to channel down the steepest portions of the Bradley River basin, increasing water surface heat losses and frazii ice production. Construction of the tunnel for the hydroelectric facility would divert the Bradley River eliminating it or reducing its importance as a source of frazil ice. Downstream of the tailrace, strong winter winds should help to mix the fresh water with the ambient water of the bay. Earlier studies of circulation and mixing processes in Kachemak Bay include those by Knull (1975), Wright (1975) and Burbank (1977). These reports tend to confirm the circulation studies of Gatto (1981), at least in a general sense, of counterclockwise circulation in upper Kachemak Bay. Knull (1975) has provided useful information on tidal exchange rates in the bay, which suggests that the mixing half life for river water is 13.5 days. Knull (1975) also reported that a July, 1969, study of fresh water content in the bay indicated a 5.7~ fresh water content. This corres- ponds to a reduction in average salinity of sea water entering and leaving 6 the bay from 31.6 parts per thousand (ppt) to 30 ppt. These numbers are in reasonable agreement with the measured salinities given by Wright (1975), where summer water ·sal i n1 ties ranged from 25-28 ppt at the surface to about 31.5 ppt at the bottom. Bradley River discharge statistics were compiled by Carlson et al. (1977) and by Knull {1975) and are also reported in the studY by Colonell (1980). Of significance to the present investigation is the relative contribution of Bradley River to the fresh water input into the bay during winter. Colonell (1980) suggests that the Bradley River contributes one- fourth of the fresh water inflow to Kachemak Bay. All of the earlier mentioned studies conclude that the Bradley River discharge is mainly con- trolled by glacial melt (i.e. by temperature) and hence reaches its minimum value in mid to late winter. Typical discharge values during this time range from 40-100 cfs. Regulation of the river for hydroelectric power production, in contrast, may raise the winter discharge to as much as 1400 cfs (Bredthauer, personal communication). 7 THERMODYNAMIC MODELS AND CALCULATIONS General Considerations Estimates of the rates and accumulations of ice in Kachemak Bay are at best deductions reached from theoretical analyses. Empirical evidence to support these deductions is limited. The objective is to devise conceptional models of ice production in the bay based on the available information regarding the mixing processes and thermodynamics of the fresh water-sea water-ice system. Ice in Kachemak Bay can be a result of ice discharge from the rivers and streams flowing into the bay, anchor ice growth on the tidal flats, sheet ice growth and frazil ice growth. Frazil ice can be formed directly from sea water, in a lens of fresh water floating over colder sea water and from a mixture of sea water and fresh water. Ice discharge from the rivers flowing into the bay may consist of frazil ice, sheet ice and anchor ice. The amount of fee discharged into the bay by the rivers will not be affected by the Bradley Lake project except for the Bradley River. Since Bradley Lake appears to be the primary source of water for the river it is thought that the ;ce discharge from the river will be negligible once the Power Project is ;n operation. Thus, the Bradley Lake Power project will probably eliminate or substantially reduce the ice discharge from the Bradley River into the bay. Anchor ice growth on. the tidal flats could be increased by the operation of the pow'r project. However, there is no information available on this fee production mechanism. In addition, it is thought that the anchor ice would remain fixed in place until warm weather and/or warm water melted its bonds to the sea bed. If it were to become mobile at this time then it would be expected to melt rapidly in the warm water of 8 the bay. Therefore, no attempt will be made to analyze this mode of ice production in the bay in greater detail. Very little sheet ice is thought to form in the bay except possibly in protected areas along the shores. Photographs supplied to us by Mr. s. Bredthauer suggest that most of the ice photographed in the bay during the 1980-81 winter was floating frazil ice. It does not appear that sheet ice growth will be changed by the fresh water discharge from the power project. The frazil ice mode of ice production in the bay appears to be the only way in which present ice production rates could be significantly altered by a hydroelectric facility with a winter discharge much larger than the present discharge. Several factors may be expected to affect frazil ice production in the bay. These include dilution of salt water with fresh water, tempera- ture of the fresh water, degree of mixing or stratification, lateral spreading from the fresh water source as well as others. The temperature of the fresh water discharge from the power project is not known at this time. However, water temperature measurements made at the Eklutna power station and the Bennett Dam (British Columbia) suggest that a discharge temperature of +3 to +4•c may be expected. Temperature profiles (Fig. 7, 8 and 9) obtained in the present Bradley Lake (U.S. Army Corps of Engineers) under an ice cover also suggest that +3 to +4•c would be a good estimate of the water discharge temperature. Another factor of some importance is the stratification of the mixed water. Present data from the Corps of Engineers and Colonell {1980) yield information about high flow -low stratification {summer) conditions, and low flow -high stratification (winter) conditions. The projected 9 regulation of river flow could result in high flow -high stratification conditions during peak demand periods associated with severly cold weather. Since a density difference of about 3S may be expected9 the fresh water may tend to stratify as a surface layer over the more saline bay water. Existing three dimensional models of horizontal buoyant discharges are presently limited to physical conditions described by simple temporal and spatial boundary conditions. A partial listing of these models may be found in Table 2. The second column in Table 2 specifies model type, whether integral or numerical. Column three specifies whether the effects of stratification are included in the analysis of dispersion. It should be noted that none of the models consider the transient case, and none consider tidal effects. The reason for this omission is straightforward. The discharge of a buoyant flow into an even partially enclosed basin is basically elliptic (i.e. downstream velocity and temperature conditions affect upstream conditions). Thus, a numerical solution can only be found by iteration on assumed boundary conditions. If a transient, or tidal case were to be considered, this would require successive iterations for each time step, a prohibitally expensive procedure. The dispersion studies of Colonell (1980) and of Knull (1975) in Kachemak Bay clearly show that tidal mixing is the dominant dispersion mechanism. This conclusion might well be anticipated by noting the 5 m tidal range at Homer. Knull (1975) suggests that this tidal range amounts to 10% of the total volume of the bay. Accordingly, rather than attempt to predict dilution rates, we have devised a model which includes the mixing term as a parameter in the problem. Thus, measured dilution rates, as determined by Colonell (1980) or Knull (1975), may be used directly in the model. The transient problem 10 Table 2 HOrizontal buoyant jet models Investigator Stolzenbach & Harleman (1971) Prych ( 1972) Spraggs & Street (1977) Waldrop & Farmer ( 197 3) Gos ink {1979) McGuirk & Rodi ( 1 978) Type integral integral numerical numerical numerical numerical 11 Stratification included yes yes no no yes yes of tidal effects on dilution is bypassed by considering only long-term dilution over several tidal cycles. Dilution of salt water with fresh water can significantly affect the formation of ice in the bay. Mixing of the fresh water in the ambient saline water facilitates the formation of ice by raising the freezing point of the mixed water. The importance of dilution rates of salt water with fresh water may be illustrated by considering two limiting cases. First, consider complete and immediate mixing of salt water with fresh water. In this instance the fresh water to sea water ratio is extremely small, or dilution is infinite and there is no alteration in the freezing point of the bay water. Consequently, the ice production rate is identical to that ocurring in "pure" sea water (i.e. it is un- affected by the presence of the fresh water). The other limiting case occurs when there is no mixing of fresh water with salt water. This may be visualized as the spreading of a lens or thin layer of fresh water over the surface of the bay. In this instance, there is heat loss from the fresh water surface layer both to the air above and to the underlying relatively cold salt water (at -l.8°C). Martin (1981) and McClimans (1978) have discussed the latter form of heat transfer. McClimans (1978) suggests that when there is no relative motion between the fresh water and the underlying salt water there is a predictable rate of heat loss from the warmer to the cooler layer. The heat transfer between moving layers of water may be expected to be greater than the conductive heat transfer associated with two layers of water with no shear. However, no theoretical model has been devised to predict heat loss when there is relative motion between the upper and lower layers. Heat transfer from the upper surface of the fresh water due to 12 ambient air conditions will produce ice at a faster rate than the rate at which ice is formed in pure sea water. This is due to the 1 .8°C freezing point depression of salt water. Hence, the spreading of a thin layer of fresh water over the surface of the bay is a limiting case probably yielding the maximum rate of ice production. We consider this limiting case in some detail subsequently. In reality, dilution of the fresh water with bay water is neither infinite nor negligible. It is in fact relatively fast, and dominated by tidal processes as indicated by the dispersion studies of Colonell (1980). Thus, it is necessary to consider a wide range of mixing conditions ranging from negligible to complete. We have attempted to do this through a mixing model which allows for a wide range of dilution rates. In particular, we shall consider three cases: negligible mixing, complete mixing and an estimate based upon the dilution measurements of Colonell (1980) and Knull (1975). Since the model is one based on a control volume and long term steady state approach, it describes the net ice production rates under equilibrium conditions. While it is recognized that this type of quasi-steady state analysis will not answer the question of transient ice formation in the immediate vicinity of the tailrace, it does provide insight into the far-field ice conditions, particularly the ice production and flow rate past Homer or into Coal Bay. In a subsequent section we address the problem of near- field ice production over time scales less than one tidal cycle. The first analysis provides the long term and/or far-field ice production rate. An important point to note is that the far-field ice production rate is the critical control parameter for the bay, since, even if local 13 ice production is significant, by the second law of thermodynamics, the net ice production is controlled by far-field conditions. Thermodynamic model of ice formation in Kachemak Bay Consider the conservation of mass and energy for water flow into and out of Kachemak Bay. Assume that some mixing of fresh and salt water does occur, that there is surface heat loss due to ambient air conditions, that no ice is discharged into the bay from the rivers, and that a long term steady state condition is found to exist (over several tidal cycles). It is also assumed that conditions are 11 Suitablen for ice formation in the control volume (i.e., the nucleation temperature of the water has been attained and sufficient nuclei are available to nucleate ice in the water) or that some ice already exists in the control voluMe. In the first case, the model describes the ice growth and in the second case, ice growth or melting. Precipitation fn the form of snow and rain and enthalpy changes caused by heat flow from the sea bed to the control volume (or possibly heat flow into the sea bed in the tidal flats) are not included in the model. The vertical temperature profile in the control volume is assumed to be constant. Then an integrated control volume analysis over the inner bay (above Homer) may be described as follows: Figure 4. Mass balance in the bay 14 mfw • mass flux of fresh water entering the bay [kg/s] msw = mass flux of "pure" sea water entering the bay [kg/s] This mass flux represents the net sea water entering the bay to mix with river water over several tidal cycles • . msw' = mass flux of mixed fresh and sea water leaving the bay [kg/s] mi = mass flux of fee leaving the bay [kg/s] For a steady state condition, conservation of mass implies: {1) Figure 5 depicts the flow of thermal energy into and out of the bay. For long term conservation of energy, assume that kinetic and potential energy changes are negligible. QA Figure 5. Thermal energy conservation in the bay. 15 hfw hsw hsw hi Q q A AL = 2 = = = = = = enthalpy of fresh water entering the bay [kJ/kg] enthalpy of sea water entering the bay [kJ/kg] enthalpy of mixed fresh and sea water leaving the bay [kJ/kg] enthalpy of ice leaving the bay [kJ/kg] surface heat flux (kJ/s-m2] interfacial heat flux from fresh to salt layer [kJ/s-m2] surface area of the bay [m2] surface area of fresh water lens [m 2] For a steady state condition, conservation of energy requires: mfw hfw + msw hsw = Q A+ q AL+ mi h; + msw hsw (2) Note that the interfacial heat transfer does not transport heat across the boundaries of the volume, and therefore is not normally included in a control volume approach. However, it is included here for two reasons. First, the lower layer or the water basin is effectively an infinite sink for heat flux from the upper layer, i.e •• the relative heat loss from the surface is much more important than the relative corresponding heat gain to the lower layers. Second, the inclusion of interfacial heat loss in thfs form increases the net ice production rate assuring that the ice production rate will be a conservative estimate. By solving equation . (1) for msw and substituting into equation (2), we find: 16 m i • Rate of ice pro- duction . QA + qAL I Ice production due to heat loss across top and bottom surfaces + msw x II Ice decay due to entering warm fresh water hsw -hsw ( 3) III Ice production due to raising the freezing point of sea water The rate of ice growth or melting is controlled by three terms: . . I) (QA + qAL}/{hsw-hi) This term represents ice production due to surface heat flux. Note that hsw -h; • 80 kcal/kg is the latent heat of fusion in sea water. Ice production proportional to Q in the bay occurs even when there is no fresh water influx (i.e. it is the rate of ice production in a basin of sea water for cold ambient air temperatures). (Methods of assessing Q as a function of air temperature and wind velocity are discussed subsequently). Ice production proportional to q occurs only when a fresh water lens overlays a saline water basin at a temperature less than o•c . . II) -mfw (hfw -hsw>l(hsw -hi) This term represents ice melting due to the intrusion of relatively warm water. The term is negative whenever hfw-hsw > 0, i.e., whenever the enthalpy of the entering fresh water is higher than that of the mixed fresh and sea water. For example, if the sea water is at -1.8•c and the fresh water at 4•c, then this term becomes -mfw (5.8/80) defining the net melting rate of ice due to the added warm fresh water. The term may be positive, indicating added fee formation if hfw < hsw but this does not appear to be a physically interesting case. 17 III msw (hsw-hsw)/(hsw-hi) This term represents ice production associated with the entrainment of fresh into sea water, raising the freezing point of the sea water. The amount of sea water whic~ is mixed with fresh water defines the amount by which the freezing point of the sea water is elevated. In other words, hsw -hsw• the net change in sea water enthalpy, increases when the mixing process is incomplete, and decreases as more sea water is entrained into the fresh. Obviously, if ice is present in the bay and term II exceeds the sum of terms I and III then the ice will melt. Determination of the net surface heat loss Heat losses from the water surface to the atmosphere occur as a result of net radiative heat transfer, latent and sensible heat transfer, and heat removal or addition associated with falling snow or rain. These forms of heat transfer are assumed to occur uniformly across the water surface. Models of heat transfer from the water surface include those by Gotlib and Gorina (1974), Goryonov and Perzkinskey (1967), Asvall (1972), Pailey, Magagno and Kennedy (1974), Dingman, Weeks and Yen (1967) and Dingman and Assur (1969). All of the above models were devised for sub- zero atmospheric conditions. Although surface heat transfer is a very complex term, each of the models contains simplifications especially of the radiative and evaporative heat transfer, such that the surface heat transfer is 0 (cal/m2 -sec) where Q is at most a function of atmospheric temperature, wind speed, cloud cover and the difference between water and air temperature. Thus all models allow for a straightforward compari- son with published meteorological data. 18 In a recent unpublished manuscript, we compared all of the above models with meteorological data and the lengths of open water reach in two Alaskan and Canadian rivers. The model by Dingman and Assur (1969), employing linearized forms of the surface heat loss coefficients originally devised by Rimsha and Oonchenko (1957), proved to be both the most con- servative and the most accurate of all models tested. Apparently the critical distinction between the latter model and the others is the prediction of heat transfer under low wind conditions. Heat transfer under these conditions is substantially underestimated with all but the Dingman and Assur {1969) model. Hence in our further calculations, we have assumed that the formulae given by Dingman and Assur (1969) are the most accurate available. It should be noted that Larsen (1978) completed an analysis of heat loss terms in the Dingman and Assur (1969) formulae, and suggested that heat loss from turbulent river reaches may be under- estimated. HOwever, in the present study we are concerned with heat loss from the bay surface where water velocities are substantially less than in turbulent river flow. The Dingman and Assur (1969} formulae are given below: 1) Clear sky conditions 0 = Qo + e0 (Tw -Tair) (cal/cm2 day) where Tw = water temperature (°C) Tafr = air temperature (•c) a 0 = 105.18 + 23.14 W (cal/cm2 day) a0 = 35.08 + 4.24 w {cal/cm2 day•c) W = wind speed (m/sec) 19 (4) 2) Cloudy sky conditions Q =-a1 +a, (Tw -Tair) (cal/cm 2 day) where a1 = -72.85 + 9.08 W (cal/cm2 day) a1 = 37 • 1 0 + 4 • 58 w ( ca 1 1 cm2 day • c) (5) In a subsequent section, calculations employing these expressions for Q will be used to determine local temperature distributions from the tailrace. These calculations are meant to suggest the local or transient ice production within a tidal cycle. For the present control volume approach, it is sufficient to consider the magnitude of the surface heat loss term in comparison with other terms in the governing energy balance, Equation (3). Under severely cold conditions, with air temperatures of -2s•c and wind velocities of 6 m/sec (10 mph), the Dingman expressions suggest a surface heat 1 oss of about 250 cal /cm2 day. For fresh water at the freezing point this magnitude of heat loss implies an ice formation rate of 3.2 em/day. Interfacial heat transfer Another form of surface heat loss which is not considered in the Dingman analysis is heat transfer at the fresh-salt water interface. This form of heat loss from the overlying fresh water to the bottom saline layer may be expected to exist whenever the salt water is colder than the fresh water. Nansen (1897) made estimates of the ice production rate at the interface indicating an ice production of 0.35 em/day. The same rate was also measured by Martin and Kauffman (1974) in a laboratory experiment simulating the formation of ice at the interface in an initially quiescent basin. A growth rate of 0.35 em/day is equivalent to a heat loss rate of 20 28 cal/cm2 day. tence, during a severely cold period the dominant mechanism for ice formation is heat loss to the air. Under certain conditions, including for example, cloudy skies, low winds, and fresh water temperatures of o•c, the interfacial heat transfer may be dominant, but maximum ice production rates are produced by severely cold air temp- eratures and high winds. Calculations for the case of complete mixing of fresh water with sea water The case of complete mixing is suggested as a limiting case yielding the minimum ice production rate. By complete mixing we imply that the relative volume of sea water entrained with the fresh water is very large in the immediate vicinity of the tailrace. Thus, the properties of the mixed water are identical to those in the pure sea water. This implies that Tsw = Tsw• or, hsw = hsw· In the limiting case of complete mixing, the final term in Equation 3 is zero and there is no interfacial ice production; therefore, Equation 3 becomes m = QA hfw -hsw (6) i hsw -hi I I I Equation 6 shows that the net ice production rate depends only on the heat flux across the water surface and the heat associated with the influx of fres~ water. Ice will grow when term I exceeds term II and will melt when term I is less than term II. The difference between ice production rates with and without the hydroelectric facility depends, in this approximation, only on term II, the heat provided by the influx of fresh water. Term II depends on mfw and on the temperature of the fresh water influx. As noted previously, regulation of the Bradley Lake dis- 21 charge during winter operation of the hydroelectric facility suggests a possible ten-fold or more increase in mfw· This increase in winter flow would be routed from the lake via a tunnel to an unnamed creek about 1/2 mile east of Sheep Point. The discharge temperature would be sub- stantially higher than present discharge from the Bradley River {o•c during winter}. Specifications of the hydroelectric power plant include withdrawal of lake water at depth during winter operation. Temperature profiles below the ice surface were measured by the Corps of Engineers and are presented in Figures 6, 7, 8 and 9. It is clear that Bradley Lake bottom temperatures remained between 2• and 4•c during the period of obser- vation. This agrees with measured temperatures at the tailrace of the Eklutna Power Station near Palmer, Alaska (Wilde, personal communication). Wilde stated that water temperatures at the tailrace remained constant at 4•c throughout the winter, even when drawdown in the reservoir was sufficient to bring the ice surface to within twenty feet of the reservoir outlet. Since the Eklutna facility, like the Bradley Lake project, includes a thousand foot tunnel from the reservoir to the tailrace, it appears that the assumption of constant fresh water temperature equal to 3• or 4•c at the tailrace is warranted. The ten-fold or more increase in mtw and an increase of = 4•c in the temperature of the fresh water discharge suggests that term II in Equation 6 will be much larger during winter operation of the hydroelectric facility .. as compared to present conditions. Therefore a net decrease in the total ice production rate in the bay may be expected once the project is in operation. It must be remembered that this theoretical conclusion is based on the assumption of complete mixing which will probably not be achieved in reality. 22 Calculations for little or no mixing A second limiting case, in principle, occurs when there is little or no mixing of the sea water with the fresh water discharge from the power plant. In reality it does not appear that this case will be important. The large tidal range c~ 14 ft) and associated currents coupled with wind driven turbulence should be sufficient to mix the fresh water with the sea water in a relatively short time. Colonell's (1980) study suggests that only a few tidal cycles would be required. Crude calculations also suggest that if the fresh water discharge were to spread in a lens over the head of the bay that its thickness would be negligible (a few hundredth inch). In this instance the very thin fresh water layer would be subject to heat loss from above and below to effectively infinite heat sinks. Heat input of the fresh water discharge would be negligible in comparison to the atmospheric losses if the salt water were at -1 .a•c and the weather cold and windy. Hence, the total discharge would produce ice at a rate proportional to Q + q which implies a fifteen-fold increase in ice produc- tion rate from Bradley River water or a five-fold increase for ice produc- tion from all the rivers at the head of the bay (other rivers 0.75, plus 15 x 0.25 for the Bradley River). However, as noted above, the mixing conditions appear to be sufficient to prevent a thin layer from forming and from turning all the discharge into ice (as assumed by this calculation). While some stratification of the fresh water discharge, may occur, it would probably be restricted to a bounded fresh water lens area. Beyond this region, the fresh water would become relatively well mixed taking on the characteristics of the ambient sea water. Thus, direct application of Equation 3 is somewhat academic. Our approach will be to focus on the fresh water lens, the area it occupies, its thickness and the distance from the tailrace that would be 23 required to cool the "lens" water to o•c. These results then allow approximate conclusions to be made regarding the ice formation in the lens. Fresh water lens area A simple calculation can be made to illustrate the importance of the fresh water lens area with respect to interfacial ice formation. This mode of ice formation is expected to be significant in the case of no mixing. The interfacial heat loss from the fresh water layer is given by q AL (cal/day). renee a stationary fresh water layer above salt water at the freezing point would form ice at a rate propportional to q AL· However, with a warm water discharge, there is an added heat . flux of mtw (hfw -hsw> which serves to slow the interfacial ice production rate. A comparison of the net heat flux for the present and projected . . cases [q AL -mfw (hfw -hsw)], suggests that the total area increase required to achieve equality of net heat flux is ·29 square miles. That is, a fresh water lens area twenty-nine square miles larger than that found during present winter conditions would be necessary for interfacial ice production to equal or exceed present rates. Hence the extent of the fresh water lens area is of considerable importance to the total ice produciton. The present analysis of the extent of AL will consider both theoretical predictions of the extent of a buoyant discharge, and Colonel1 's {1980) salinity measurements for varying discharge conditions. Calculations to determine the maximum fresh water lens area with projected winter discharge are presented in the appendix. These calculations indicate that at most AL would increase by a factor of five from present conditions. 24 Best estimate -thermodynamic model The final case represents what appears to be a best estimate including the effects of mixing of sea water entrained past Homer into upper Kachemak Bay with Bradley River water. The control volume is the whole inner bay, fresh and salt water, and we assume a long term stea~ state (over . several tidal cycles) Since the "pure" sea water, msw• is mixed with • 1 • the fresh water, mfw• to form mixed water, msw , plus ice, mi, the conservation of mass and energy is given by Equation 1 and 2. As in the earlier section, the conservation equations are written for the Bradley River -Kachemak Bay system only. Winter discharge and ice production from Sheep Creek and Fox River is assumed to remain constant before and after construction of the hydroelectric facility. Equation (3) may be rewritten: net ice production due to fresh water discharge = interfacial ice production 1 1 mfw (hfw -hsw )/L + msw (hsw -hsw)/L ice decrease due to relatively warm water inflow ice production due to raising of freezing point of mixed water where Lis the latent heat of freezing (80 cal/g). The entrained pure sea water entering the inner bay has a ••typical" salinity S. (ppt). This refers to a depth averaged salinity value of the sea water before mixing with the fresh water. Similarly a typical mixed water salinity S (ppt) may be defined for water leaving the bay. For example, Knull (1975) has estimated that the entering sea water during July, 1969, had salinity typical of Cook Inlet water (S. • 31.6 ppt) and the mixed water leaving the bay had a salinity of S 2 29.9 ppt. The conservation of salt equation written for flow passing Homer is 25 ( 10) . . msw s. = (mfw + mswl S and the entrained sea water . . msw = mfw S/ ( S. -S) (11) Knull (1975) uses the conservation of salt equation to suggest that during July, 1969, the fresh water in the bay amounted to 5.7% of the total, ; . e. : mtw = S..-S = 31.6-29.9 = 0.057 s 29.9 This is a summer estimate of the fresh.water discharge into the bay. Judging by the data of Carlson (1977), the fresh water influx decreases by about 15 during the winter. 1-ence, the percent of fresh water presently found in the bay should be about 5.7/15 = 0.38% during winter. This corresponds to a salinity of 31.48 ppt. Since construction of the hydro- electric facility implies a maximum fifteen-fold increase in Bradley River discharge, and assuming Bradley River represents 25% of the total fresh water, the projected winter fresh water content of the bay is ( .75)( .38) + 15( .25)( .38) = .0171 implying a typical projected salinity of 31.04 ppt. The enthalpy of fresh water is hfw = c Tfw where c = specific heat of water (1 cal/g •c) and Tfw = temperature of entering fresh water c•c) The enthalpy of the sea water and the mixed water is· also a function of temperature only. We assume that the sea 26 water is at its freezing point, i.e., at the temperture of a oody of water with salinity S. (Doherty and Kester, 1974) T5w • (-0.0137-.05199 s.-.00007225 s!J t•c) If ice is being formed from the mixed water, then it must also be at the freezing temperature of a body of water with salinity of the mixed water, S. Hence, I 2 2 Tsw -Tsw = [.05199(5.-S) + .00007225 (S. -s )] (•c) . Then, multiplying by msw from equation (11), • I • msw (hsw -hswl = c mfw S [.05199 + .00007225 (S. + S) ]. Substituting into Equation (10) m; QA/L = net ice production minus in situ production from salt water c mfw (Tfw + .0137 -.00007225 S S.)/L (12) net ice production due to entering fresh water This equation specifies the ice production due to the discharge of entering fresh water. The right hand side accounts for ice production due to interfacial effects, temperature of incoming water, and the effect of dilution upon the freezing point. To establish the difference between present and projected winter ice production, it is necessary to evaluate the right hand side for the two conditions. We have summarized these calculations in table 3. (Note that T' is defined as Tfw + .0137 - .oooo7225 s s.) 27 . TABLE 3. Calculation of qAL/l-cmfw T'/L Present AL = 2 sq. mi. AL = 5.18 • 10lO cm2 . q = 28 cal/cm2 day . mfw 2 100 cfs = 2.446 • 1011 g/day . qAL/L = 1 .8125 • 1010 g/day cmfw/L = 3.059 · 109 g/day •c s = 31.48 ppt Tfw = o •c T' = -0.05187 qAL/l Cmfw T'/L = 1 .83 • 1olO g/day Projected AL = 10 sq . mi. AL = 25.9 • 10 10 cm2 . q = 28 cal/cm2 day mfw 2 1500 cfs = 3.67 · 1012 g/day qAL/L = 9.065 • 1010 g/day cmfwiL = 4.588 · 1olO g/day 0 ( s = 31.04 ppt Tfw = 4 •c T' = 3.943 qAL/L -Cmfw T' /L = -9.0255 • 1010 g/day Hence, the projected winter ice production due to fresh water input is expected to decrease relative to present ice production. Furthermore, assuming that the temperature of the discharge at the tailrace is 2 •c, we find T' = 1.943 •c for the projected flow. Using this assumption it may be shown that the fresh water ice production amounts to 1.505. 109 g/day, a ten-fold decrease from present ice production. This estimate is conservative: it is based on a maximum predicted fresh water lens area, zero input of frazil ice from Bradley River, and measured dilution rates in the bay. Even if the temperature of regulated flow at the tailrace is as high as 2 •c, the total ice production should still decrease by a factor of ten. Local Effects In this section some consideration is given to local effects in the vicinity of the tailrace. The temperature distribution is calculated to 28 show the extent of the "warm 11 water area. The effective fresh water lens area for interfacial ice formation should be reduced by this warm water area. Following Dingman and Assur {1969), an expression may be written to calculate the extent of an open water area downstream from a warm water discharge. Figure 10 depicts the flow geometry for a cross section of jet flow. Q coUhbT h co UhbT + a( ecUhbT) ~x ax Figure 10 -Heat balance for section of fresh water discharge Heat transfer into section = Heat transfer out of section. CpUhbT a CpUhb T + a ( pCUhbT) !J.X + Q b !J.X ax 29 where c = specific heat of water [1 cal/g •c] p • density of water [1 g/cm3] u = velocity [em/day] h = 1 ayer thickness [em] b = 1 ayer width [em] X • longitudinal distance [em) Q = surface heat loss [caltcm2 day] Assuming no mixing, the discharge (Uhb) remains constant and pcUhbdT = -Qb dx (13) The surface heat loss, Q, according to Dingman and Assur (1969) is pcUh dT = -(a+ B[T-Ta;r]} ax ( 14) where a and a are defined by Equation 4. If b is constant an analytic solution for this equation may be written for boundary conditions of T(x = o} = T0 , T-Tair +a/a= (T 0 -Ta1r +a/B) exp [-ax/pc Uh] (15} Equation 15 gives the temperature as a function of distance from the outfall for a prescribed surface heat flux formulation, and is shown in Figure 11 as the dotted line. The heat flux coefficients a and s were chosen to represent severe atmospheric conditions at Homer, specifically, Ta;r = -2o•c and wind velocity = 3m/sec. The discharge is 1400 cfs and temperature at the tailrace is 4•c. The width, b, of the outflow is constant and equal to 500 feet. Equation 13 may also be solved for a variable b, in particular when b is a linear function of distance from the outfall. As discussed in the appendix, the spread for the buoyant jet. is controlled by the outfall Froude number. Accordingly, we have plotted the decay of fresh water 30 temperature with distance from the tailrace for two width formulations: b • 0.214 x and b • 0.107 x. The latter spread is typical non-buoyant jet spread (Schlichting 1968) and the former chosen simply to double the standard jet spreading rate. The wider jet reaches a temperature of o•c two miles downstream from the tailrace~ where the width is about 0.5 mile. Figure 12 depicts the spread of the jet and the location of the 0°C iSOtherm with the prescribed heat lOSS expression assuming the 0.214 X spreading rate. Equation 15 may be inversed to determine the water area with temperature greater than 0°C for a variety of surface heat loss expressions . . Q is primarily a function of Tair· Figure 13 shows the warm water areas for constant wind (W = 6 m/sec), constant outflow temperature (T 0 = 4°C), and variable discharge and air temperature. The minimum calcu- lated warm water area is about 0.28 sq. miles under the most severe of atmospheric conditions with Tair = -2s•c. This area is outlined on Figure 12 and corresponds to a region just upstream of Sheep Point. Since the site of the proposed barge docking facility is downstream of Sheep Point. there are instances when the facility will be subject to local ice growth. However, according to our earlier estimates, the net ice production for the upper Kachemak Bay should decrease as a result of the regulated flow. Hence~ we expect that local accumulations of ice near the barge docking facility will occur during severely cold atmos- pheric conditions, but they will be transitory in nature. Conclusions The conclusions reached by this study must be prefaced by the reminder that this is largely a theoretical research approach. There are no studies to our knowledge that chronicle the ice production rates before and after 31 construction of a hydroelectric facility. However, in each instance where assumptions were to be made, the appropriate parameters were chosen from recent laboratory and field studies. A summary of the conclusions for this investigation follows: 1} In the case of complete local mixing of fresh and sea water, increased discharge of fresh water having a temperature higher than present Bradley River water would result in a net decrease in ice production. 2) In the case of no mixing, a thin layer of fresh water would spread over the surface of the bay. If the weather were cold and the bay water temperature below zero, then the layer would be subject to effectively infinite heat sinks. Hence the pro- duction rate would increase proportional to the increase in discharge. However the conditions required for no mixing do not exist in Kachemak Bay. Judging by Colonell 's (1980} disper- sion studies, the tidal range information, and observations of winds at the head of the bay, dilution of the fresh water from the tailrace should be rapid. 3) The final case represents our best estimate and includes both mixing and interfacial ice production. In this case, if a tempera- ture of 4•c is assumed at the tailrace, the increased discharge will melt ice. If a temperature of 2•c is assumed at the tail- race, there will be a ten-fold reduction in ice produced from Bradley River over present conditions. 4) ~ring severely cold periods, there will be local ice growth in the vicinity of the barge dock. However, it is expected to be less than that presently found under similar atmospheric conditions and rapidly dispersed by tidal mixing. 32 ACKNOWLEDGMENTS We wish to thank personnel from the u.s. Army Corps of Engineers, Alaska District, for helpful discussions and suggestions. Funds were pro- vided by the U.S. Army CRREL under contract number OACA 89-81-K-0001. We also wish to thank Janis Zender for her careful preparation of the figures. 33 REFERENCES Abraham, G., 1960. Jet Diffusion in liquid of greater density, J. ~draul. Div., ASCE, Vol. 86, No. HY6, Proc. Paper 2506, pp. 1-13. Anwar, H. 0., 1970. Spread of buoyant jet at the free surface, J. ~draul. Div., ASCE, Vol. 96, No. HYl, pp. 288-291. Asvall, 1972. Proc. of Banff Symposia on the role of snow and ice in hydrology. Burbank, D., 1977. Circulation studies in Kachemak Bay and lower Cook Inlet, Environmental Studies of Kachemak Bay and Lower Cook Inlet, Vol. III, Ed. L. Tratsky et al., Alaska Dept. Fish and Game, Marine/ Coastal Habitat Management, Anchorage, Alaska. Carlson, R. F., Seifert, R. D. and Kane, D. L., 1977. Effects of season- ability and variability of streamflow on nearshore coastal areas, Inst. Water Resources, Rept. IWR-78, University of Alaska, Fairbanks. Colonell, J. M., 1980. Circulation and dispersion of Bradley River water in upper Kachemak Bay, Final Rept. to Dept. of the Army Alaska District, Corps of Eng., Anchorage, Alaska, 35 pp. Dingman, and Assur, 1969. CRREL Res. Rept. 206, Pt. II. Dingman, S. L., Weeks, W. F., Yen, Y. C., 1967. The effects of thermal pollution on river ice conditions--!: A general method of calculation, u.s. Army Cold Regions Res. and Engin. Lab., Res. Rept. 206, Pt. I, (AD 666205). Doherty, B. T. and Kester, 0. R., 1974. Freezing point of sea water, Jour. of Marine Res., Vol. 32, No.2, p. 285-300. 34 Gatto, L. W., 1981. Ice distribution and winter surface circulation patterns, Kachemak Bay, Alaska, CRREL Rept. to the u.s. Army Eng. District, Alaska, Anchorage, Alaska, 57 pp. Goryunov and Perzzhinskfy, 1967. Soviet hydrology, Selected papers, Issue No. 4. Gotlib and Gorina, 1974. Gidrotekhnicheskoe, Stroitel 'stvo, No. 11. Gosink, J. P., 1979. A stuQy of turbulence in the horizontal buoyant surface jet, Ph.d. Thesis, Tech. Rept. HEL 27-6, University of California, Berkeley, ~draulic Engineering Lab. Knull. J. R., 1975. Oceanography of Kachemak Bay, Alaska--a summary of 1969 studies, Manus. Rept. File No. 113 NMFS-NW Fisheries Confer. Auke Bay, Alaska, 30 pp. Larsen, P., 1978. Thermal regime of ice covered waters, Int. Assoc. ~draul. Res., Pt. 3, pp. 95-117. Martin, S., 1981. Frazil ice in rivers and oceans, Ann. Rev. Fluid Mech., Vol. 13, pp. 379-397. t4artin, s .. Kauffman, P •• 1974. The evolution of under-ice melt ponds, or double diffusion at the freezing point, J. Fluid Mech., vol. 64, pt. 3, pp. 507-527. McClimans, T. A., Steen, J. E., Kjeldgaard, J. H., 1978. Ice formation in fresh water cooled by a more saline underflow, Int. Assoc. HYdraul. Res., pt. 2, pp. 20-29. McGuirk, J. J. and Rodi, w., 1978. Mathematical modelling of three-dimensional heated surface jets, SFB80/T/135, Sondertorschungsbereich 80, Ausbreitungs- und Transportvorglnge in StrBmungen, Universitat Karlsruhe. Nansen, F., 1897. Farthest North, Vol. 1, pp. 457-459, New York: Harper and Brothers. 35 Pailey, Magagno, Kennedy, 1974. J. of ~draul. Div., ASCE. Prych, E •• 1972. A warm water effluent analyzed as a buoyant surface jet, Swedish Meteorological and ~drological Institute, ~draulic Series Rept. No. 21. Rimsha, v. A. and Donchenko, R. V., 1957. The investigation of heat loss from free water surfaces in wintertime, Leningrad Gosudarstvennyi Gidrotogicheskii Institut, Trudy, Vol. 65, pp. 54-83 (text in Russian). Safaie, Bijan, 1978. Mixing of horizontal buoyant surface jet over sloping bottom, Ph.D. Dissertation, Univ. of Calif., Berkeley, Calif. Schlichting, H., 1968. Boundary layer theory, rkGraw-Hill, New York, 6th Ed. Selkregg, L. (coordinator}, 1976. Alaska regional profiles, Yukon Region, Publ. for State of Alaska, Juneau, by the University of Alaska, Arctic Environ. Data Center. Spraggs, L. D. and R. L. Street, 1977. Three dimensional simulation of thermally influenced hydrodynamic flows, Symp. on modeling of transport mechanisms in oceans and lakes, marine sciences directorate, Manus. Rept. Series, No. 43, Dept. of the Environ. Ottawa, Canada. Stolzenbach, K. D., and D. R. Harleman, 1971. An analytical and experimental investigation of surface discharges of heated water, Ralph M. Parsons laboratory for water Resources and HYdrodynamics, Dept. of Civil Enging., Mass. Inst. of Tech., Rept. No. 135, 212 pp. Tarnai, N., and R. L. Wiegel, and G. F. Tornberg, 1969. HOrizontal surface discharge of warm water jets, J. of the Power Division, proc. of the ASCE, Vol. 95, No. 2, pp. 253-276. 36 Tratsky, L. L., Flagg, L. B., Burbank, D. C., 1977. Environmental studies of Kachemak Bay andd Lower Cook Inlet, Vol. III, Circulation studies in Kachemak Bay and Lower Cook Inlet, 207 pp., Ak. Dept. of Fish and Game, Anchorage, Alaska. Waldrop, w., and Farmer, R., 1973. Three-dimensional flow and sediment transport at river mouths, Tech. Rept. No. 150, Coastal Studies Inst., Louisiana State Univ., Baton Rouge, Louisiana. Wennekens, M.P., Flagg, L. B., Trasky, L., Burbank, D. C., Rosenthal, R., Wright, F. F., 1975. Kachemak Bay, a status report, Ak. Dept. of Fish and Game, Anchorage, Alaska, 230 pp. Wiegel, R. L., 1964, Oceanographical engineering, Prentice-Hall, Inc., Englewood Cliffs, N.J., Fluid Mechanics SEries. Wright, F. F., 1975. Kachemak Bay--A status report, Sect. 1, Transport Mechanisms, Alaska Dept. of Fish and Game, Habitat Protection Section, Coastal Protection Programs, Anchorage, Alaska. 37 APPENDIX: EXTENT OF FRESH WATER LENS The spread of a horizontal buoyant jet over a quiescent basin has been studied by many investigators including Wiegel {1964), Stolzenbach and Harleman (1971), Anwar (1970), and Safaie (1978). For high Reynolds nu.mber flow, it has been found that the spreading rates and dilution are largely controlled by the outfall Froude number. For example, Tarnai et al. (1969) found that the temperature half-width (the distance from the jet centerline where the temperature is reduced by half) of a horizontal buoyant jet varies as b : x/F 1/4 0 where where b is the temperature half width x is the distance from the outfall F0 is the outfall Froude number F0 = U0 I Vg' h0 U0 is fresh water velocity h0 is fresh water depth g' = g 6p/p 0 is the reduced gravity, proportional to the fresh- salt water density difference, 6p Unfortunately this measure of the width of the jet is appropriate only in quiescent basins without tidal effects, and cannot be compared with the data from Colonell (1980). However, it is expected that the f~r- field dilution rates and areal extent of the fresh water lens will he largely controlled by the outfall Froude number. That is, the spreading rate in winter during operation of the hydro-facility should resemble the measured spreading rate at a similar outfall Froude number. Abraha~ (1960) investigated the dilution of a buoyant fresh water aischarge into a quiescent basin and found that the axial concentration could be expressed as: 38 C z F 2/3 x -5/3 0 Thus buoyant horizontal jet studies indicate that the areal extent of the fresh water discharge in a quiescent basin is controlled by the outfall Froude number. These studies also indicate that the width of the buoyant lens decreases with increasing Froude number, and the length (distance to a constant isohaline) increases with increasing Froude number. Since b is a measure of the width of the lens, and C is a measure of the length of the lens, these two studies suggest that the areal extent of the fresh water lens should vary as : Al • b • C or Al « Fo-1/4 • Fo2/3 Al « FoS/12 Finally, the depth of the fresh water lens should also depend upon the outfall Froude number. In the vicinity of the tailrace or the river mouth, the lighter fresh water (ap/p. • 3%) is found even to the bottom. The fresh water layer then thins as the flow moves away from the river mouth. For example, in August (see Table 4) low salinity water (S = 5 ppt) can be found at the 10 ft. depth at a station .8 mile from the river mouth, while 1.4 miles from the river mouth, the salinity at the 10ft. depth, is 8.5 ppt. A comparison of the three tables shows that the maximum penetration of the fresh water flow varies according to season. This corresponds to the theoretical model predicted by Gosink (1979) in which it was shown that the maximum penetration depth of a horizontal buoyant jet is hMAX = 0.9 ho Fol/2 where h0 is the depth of discharge at the outfall. 39 This suggests that a higher Froude number flow will produce a thicker lens. The three studies of horizontal buoyant jets imply that for a given volume flux, the fresh water lens will have an areal extent which varies as F0 511 2 and a depth which varies as F0 11 2• Hence the areal extent of a buoyant discharge is at worst linearly proportional to the outfall Froude number. The relation between outfall Froude number and fresh water mass flux is Oo = where Q0 = U0 b0 h0 • is the outfall volume flux and b0 is discharge width. A coMparison of the measured fresh water lens area with estimated river discharge or Froude number may be made using Colonell's (1980) study of dispersio in the bay. Salinity and temperature measurements during field experiment II in October, 1980, are compiled in table 5. The Bradley River discharge was particularly high at this time, 1275 cfs, almost three times typical October values. Hence a relatively large fresh water lens area is expected. The measured March 1980 salinity observations of the Corps of Engineers are typical winter conditions. These data are listed in Table 6, and indi- cate that the fresh water lens extends no greater than two miles from the Bradley River mouth. The corresponding outfall Froude number is Oo Fo = = 0.67 b 1/2 h 3/2 0 0 g•l/2 if Oo = 100 cfs bo = 100 ft estimated ho = 2 ft g' ,. gb..p/p,. = 10{.03) = 0.3 ft/sec 2 40 TABLE 4 Table 4 (from Colonell, 1980) Salinity-temperature -depth measurements by the Corps of Engineers, August, 1979. Distance is measured from inter- section of Sheep Creek with Bradley River. Tidal stage is given in minutes from high (H) or low (L) tide, assuming a tidal lag of 45 minutes from that at Seldovia. Distance Tidal stage Depth Salinity Temperature {mi) ( ft) (ppt) (oC} .4 H -48 0.6 0.3 10.3 6.0 b 0.3 9.8 .8 H -63 0.6 0.5 7.2 10.0 b 0.5 7.5 1.4 H -33 0.6 4.6 10.5 10.0 b 8.5 11.3 1.4 H -18 0.6 7.9 11.5 13.0 b 17.1 12.8 1.4 H - 3 0.6 3.1 10.0 13.0 b 20.0 13.3 1.4 H + 12 0.6 1.6 9.4 1.8 17.0 11.5 13.0 b 21.5 13.7 1.6 H -78 0.6 17.1 12.4 11.0 b 19.2 12.8 2.9 H -93 0.6 27.1 12.7 10.0 b 27.0 12.7 4.0 l + 150 0.6 2.5 10.2 3.6 b 20.4 13.1 4.3 L -60 4.5 6.1 9.8 4.5 L + 105 0.6 22.6 13.5 5.9 b 21.9 15.4 4.5 L + 120 0.6 22.0 13.2 5.0 b 20.5 13.4 4.7 L + 90 0\6 23.4 13.6 4.8 L -45 0.6 4.8 8.5 2.7 7.0 9.0 5.5 b 9.6 9.5 5.1 L -15 0.6 10.4 10.2 3.1 b 10.0 10.0 5.1 L + 80 0.6 21.9 13.8 5.4 L + 75 0.6 22.4 13.3 5.8 L 0.6 5.9 10.2 4.4 b 18.3 11.5 6.0 L + 60 0.6 8.0 10.1 5.0 23.3 13.1 6-14.5 b 28.1 11.3 6.1 L + 30 0.6 13.3 9.8 1.6 20.1 12 .1 2-8.3 b 21.8 12.1 6.5 L + 15 0.6 9.3 10.3 5.6 20.5 11.5 b: bottom depth 41 TABLE 5 Table 5 (from Co1one11, 1980) Salinity-temperature-depth measurements. October, 1980 • Distance is measured from the intersection of Sheep Creek with Bradley River. Tidal stage is given in minutes from high (H) or low (L) tide, assuming a tidal lag of 45 minutes from that at Seldovia. Distance Tidal Stage Depth Salinity Temperature (mt) ( ft) ( ppt) (oC) 1.5 l -76 0.5 0 6.8 1.8 l -6 0.5 0 6.5 1.9 l + 14 0.0 0 7.0 2.0 2. 7.5 3.0 b 7. 8.0 2.0 L + 94 0.0 0 5.0 2.0 0 5.0 4.0 12. 6.5 6.0 16. 7.5 8.0 b 15. 7.5 2.2 l + 34 0.0 0 6.0 2.0 0 6.0 4.0 8.5 6.0 5.0 b 16.5 7.0 5.0 H + 16 0.0 23. 8.5 2.0 24. 8.5 4.0 25. 8.5 6.0 26. 8.5 8.0 27.5 9. 10.0 29. 9. 12.0 29.5 9. 6.0 H + 61 0.0 19. 7.5 2.0 22.5 7.5 4.0 25. 8.0 6.0 26.5 a. 8.0 27 .5 9. 10.0 27.5 9. 7.4 H -134 1.0 29.0 2.5 3.0 29.5 3.0 6.0 30.5 3.1 9.0 30.5 3.0 15.0 b 31.8 3.7 7.6 H -119 1.0 30.3 4.0 3.0 30.3 4.2 6.0 30.6 4.2 10.0 b 30.6 4.0 11.5 l + 127 o.o 33.5 3.8 5.0 b 33.5 3.8 b: bottom depth 42 TABLE 6 Table 6 (from Colonell, 1980) Salinity-temperature-depth measurements by the Corps of Eng1 neers, r~arch, 1980. Distance is measured from intersection by Sheep Creek with Bradley River. Tidal stage is given in minutes from high {H) or from low (l) tide assuming a tidal lag of 45 minutes from that at Seldovia. Distance Tidal Stage Depth Salinity Temperature (mi) { ft) ( ppt) (oC) o. H + 50 0.0 1.6 .3 2.0 1.9 .3 4.0 2.0 .3 .03 H + 110 0.5 10.0 1.3 .16 H + 115 0.5 11.5 1.3 .32 H + 140 0.5 11.5 1.0 .44 H + 170 0.0 15.0 1.0 2.0 b 15.0 1.0 .77 L -183 0.0 23.5 1.5 6.0 b 23.5 1.5 1.0 L -163 0.0 27.3 1.5 7.0 b 27.3 1.5 1.2 L -153 0.0 29.5 1.5 1.53 L -148 0.0 30.0 1.5 7.0 b 30.0 1.5 1.85 L -138 0.0 32.5 2.5 8.0 b 32.5 2.5 2.18 L -133 0.0 33.5 2.8 13 .a 34.0 3.0 2.5 L -123 o.o 33.0 2.8 12.0 b 32.5 3.8 b: bottom depth 43 On the other hand, the Froude number for the August 1979 study (Table 4) is since F0 = 2.7 Q0 = 761 cfs measured b0 = 100 ft. estimated h0 = 3 ft g' = 0.3 ft/sec2 Surface salinity for the three cases, October 1980, March, 1980 and August 1979, are presented in Figure 14. Measurements of salinity in March when the outfall Froude number is about 0.67 indicate that the sur- face salinity is always greater than 10 ppt less than 2 miles from the Bradley River mouth. In contrast, in August it appears that surface salinity may be as low as 10 ppt some 7 or 8 miles from the river mouth. The ratio of the outfall Froude numbers for these cases, is 4 {2.7/0.67 = 4}. Thus, the data indicate that a direct relation between outfall Froude number or fresh water discharge and fresh water lens area is warranted. This is also a conservative estimate of fresh water lens area, since it appears that the tidal effects in upper Kachemak Bay control the areal extent of fresh water. Hence it may be expected that an increase in winter outfall Froude number will at most proportionately increase the fresh water lens area, and consequently the interfacial ice production. If the Froude number for the projected hydroelectric outfall is similar to the August, 1980, value, then it is appropriate to assume that the spreading rates and areal extent of the fresh water lens will 44 resemble that occurring in August, 1980. The Froude number for the projected outfall condition is F0 = 3.5 when Q0 = 1400 cfs b0 = 14 ft. h0 = 14 ft. g' = 0.3 ft/sec2 Therefore the lens of fresh water formed during the August, 1980, study should approximate the fresh water lens associated with winter operation of the hydroelectric facility. The projected winter fresh water lens ·area should be five times (3.5/0.67) the present winter lens area. 45 List of Tables 1. Homer weather during ice occurrences (from Gatto 1981). 2. Horizontal buoyant jet models. . . 3. Calculation of q ALIL-c mfw T'/L. 4. Salinity -temperature-depth measurements by the Corps of Engineers -August 1979. 5. Salinity -temperature-depth measurements from Colonell (1980) -October 1980. 6. Salinity-temperature-depth measurements by the Corps of Engineers -March 1980. . 46 List of Figures 1. Average daily minimum winter temperature at Homer (1968-1980). 2. Minimum daily winter-temperature at Homer for 1971-1972. 3. Average wind speeds at Homer (1968-1980). 4. Mass balance in bay. 5. Thermal energy balance 6. Location of temperature profiles in Bradley Lake. 7. Temperature profile at station 12. 8. Temerature profile at station 13. 9. Temperature profile at station 14. 10. Geometry for heat balance. 11. Temperature decay with distance from the tailrace. 12. Fresh water spread in upper bay. 13. Warm water areas for varying atmospheric conditions. 14. Surface salinity measurements (from Colonell 1980) during August 1979, March 1980 and October 1980. 47 lL 0' 5 • ,, FIGURE 1. AVERAGE OAil Y MINIMUM WINTER TEMPERATURE. 11968-80). HOMER. • • • • • • • • • • • • • • NOVEMBER • ... • .. • • • .. • ' . • • .. • • • • DECEMBER ' • • • • • • •• .. ... t ... • • • • • • • . ... • • ' • . .. I I t .. ' • • JI\NIJ/\RY • • .. • • • . ' .. . .. • I ' . • • • • • • • FEORUI\RY • .. .. • • • • • • . . ... • •• • • MI\RCif .. • • • ' • • • • • • ...... • • ... • • . .. .. .1\PRil 0 m ("') m 3: a: m :::I '-> z c > :::I -< .., m ::I :::I c > :::I -< TEMPERATURE (°F) ("') 0 r- 0 m Cl) -( -< m > ::::l • 11 I • • • _10~ • ::r: a. I ::! • -9 0 I • w w a. V) •• 0 z I • 3: • • • w CJ 71-• <( • a: w > <( 6 • 5 NOVEMOER FIGURE 3. AVERAGE OAIL V WINO SPEED. WINTER (1968-801, HOMER. • • • • • • • • ' • • • • • OECEMOER • •• • • • ••• • • • • • • • • • • • • .. . • • • • • JANUARY • •• • • • • • • • • • • • • • • • • • • • • • • •• • • • • • •• • • FEBRUARY • • • • • • • • • • • • • • • • • • • • • • • • • •• • • • • • • • • • • • • ' • • • • • • • • • • • • • t MARCH APRIL FIGURE 6. I.OCATlON OF TEMPERATURE PROFILES IN BRADLEY LAKE. TEMPERATURE (°CJ 0 0.0 1.0 2.0 3.0 . 4.0 -I 1 I I ' Ho 10 0 0 20 0 FIGURE 7. BRADLEY LAKE TEMPERATURE PROFILE, STATION 12. 0 30 0 (IJ a: w 40 J.. 0 lu ~ I ' ~ ~501-0 11.. w 0 60 0 70 0 0 80 ( 0-. 0 0 0 10 0 20 0 30 U) ~ 40~ 0 0 w ~ - too UJ 0 FIGURE 8. BRADLEY LAKE TEMPERATURE PROFILE, STATION 13. 60 70 00 TEMPERATURE ( °C) 0.0 1.0 2.0 3.0 . 4.0 01 I 1 I I I %0 (I) 0:: 10 20 30 w 40 t; ::! ~00 0 60 70 00 0 0 0 0 0 FIGURE 9. BRADLEY LAKE TEMPERATURE PROFILE. STATION 14. 0 0 Ul a: :::> t- <( a: Ul 0.. :E 1 w t- J J <( I.L ~ 0 ·1 FIGURE 11. TEMPERATURE DECAY WITH DISTANCE FROM THE TAILRACE. .............. ........ , ', . . =---......... . "'.;. .. ', X X I ee X G G E) e G .8 1.6 ', ',, ' ', ' X e ', ', ' X ..... X e 2.4 3.2 4.0 DISTANCE DOWNSTREAM t MILES) T • Temperature at taifrece .. 4° C w • T air • ·20° C. Wind • 3m/sec, 0 0 • 1400 cfs b-14. + .2141c fhJ b • 14. + .107x fh) b-000. Cfd freezing point of mixed water C high discharge ) freezing point of mixed w1ter C low discharge J X ~ X 4.8 X X X 5.6 X X X X 6.4 FIGURE 12. FRESH WATER SPREAD IN UPPER DAY. I I I ' ' \ \ \ Kachemak Bay . I , N ~ ...... -...... ......... '· " . ' ' ' ' ' ....... , ~ Edqes of Jet If b ::!!! .2x ' ' ' 0° C Isotherm minimum open water area for T air --25° C 0 1000 1000 JOOO 4000 I I I I I Scalo ln fof!t .7 .6 .4 .3 .2 . , 0 FIGURE 13. WARM WATER AREA FOR VARYING ATMOSPtiERIC CONDITIONS. 5 10 Wind Speed • 6 m/sec Temperature at tailrac:a • 4° C 13E!O cfs !JJQ cfs 200 c:fs 15 20 -T AIR 25 ' -., 20 I I 10 8 -t: a.. -& t 2: ::i4 ~ I 2- FIGURE 14. SURFACE SALINITY MEASUREMENTS. DISTANCE IS MEASURED IN MILES FROM INTERSECTION Of BRADLEY RIVER WITH SHEEP CREEK fFROM COlONEll. 1980t. 0 0 0 0 0 0 X 0 xXCJcx X 0 0 0 X X X X X X August J 979 X o Mrrch 1900 X 0 October 1900 0 X 0 0 X X X X 11 I I I ' I I I I I I I I .1 .2 .J .4 .6 .8 1.0 ... --4.0 6.0 8.0 10.0 DISTANCE FROM BRADlEY RIVER MOUTH (MILES)