Entrainment and flushing time in the Fraser River estuary and plume from a steady salt balance analysis

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1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 116,, doi: /2010jc006793, 2011 Entrainment and flushing time in the Fraser River estuary and plume from a steady salt balance analysis M. Halverson 1 and R. Pawlowicz 1 Received 7 November 2010; revised 10 May 2011; accepted 31 May 2011; published 23 August [1] A passenger ferry equipped with oceanographic instrumentation made eight transects across the Fraser River plume in the Strait of Georgia, British Columbia, each day for 4 years. Estimates of average plume salinity and volume were made from the observations and used in a quasi steady budget to estimate the net upward flux of ambient salt water into the plume. This entrainment flux increases from a minimum of 17,000 m 3 s 1 at low river flow to a maximum of about 27,000 m 3 s 1 at intermediate flow. The entrainment flux then becomes essentially independent of river flow, although we argue on physical grounds it must eventually decrease. Entrainment velocities of a few mm s 1 were estimated from the entrainment flux when assuming that mixing primarily occurs along the salt wedge estuary or in the near field plume. These values are consistent with previous estimates. This, in turn, suggests that entrainment is not important over most of the plume s full extent and that bulk plume properties are set in the near field. Finally, the observations are used to estimate a plume freshwater flushing time of 2.2 days, with essentially no dependence on river discharge, even though discharge varies seasonally by almost an order of magnitude. This value lies in between the expected time scales for near field and far field plumes and is long enough for rotation to play an important role. There is no evidence, however, of a coherent bulge as observed in other systems. Citation: Halverson, M., and R. Pawlowicz (2011), Entrainment and flushing time in the Fraser River estuary and plume from a steady salt balance analysis, J. Geophys. Res., 116,, doi: /2010jc Introduction [2] One driver of primary production in deep estuaries and estuarine like systems is the upwelling flux of nutrient rich water [e.g., Pawlowicz et al., 2007]. During entrainment in these systems, deeper water is brought to the surface. This process often brings nutrient rich waters to the photic zone, promoting algal growth. In a river plume, this upwelling flux of deep water may significantly augment the nutrients brought by the river itself [DeMaster and Pope, 1996]. In low nitrate river systems, entrainment into the plume is the primary source of nitrate [Harrison et al., 1991; Lohan and Bruland, 2006]. Quantifying the upwelling entrainment velocity is important in, for example, biophysical models [e.g., Collins et al., 2009], but it is generally too small to measure directly and therefore must be inferred from other measurements. [3] Entrainment occurs throughout the estuary/river plume system but it varies in nature and intensity [Hetland, 2005]. In a salt wedge estuary or near field river plume, entrainment is caused by vertical shearing of the horizontal velocity. In an estuary, the shear is established by estuarine 1 Department of Earth and Ocean Sciences, University of British Columbia, Vancouver, British Columbia, Canada. Copyright 2011 by the American Geophysical Union /11/2010JC circulation [Hansen and Rattray, 1965], but modified by tidal variations, and is generally maximized at some point during the ebb [Partch and Smith, 1978; Geyer and Farmer, 1989]. In some cases, entrainment is localized to channel constrictions or bathymetric changes which accelerate the flow [Geyer and Farmer, 1989]. Dynamics in the near field differ from the estuary because the buoyant outflow is not constrained by lateral boundaries. This allows the near field plume to spread, which in turn causes it to thin and accelerate [Wright and Coleman, 1971]. Acceleration enhances entrainment, which alters the thickness and density anomaly of the plume [Hetland, 2010]. Turbulent dissipation in the near field is potentially high, perhaps the highest among oceanic shear flows [MacDonald and Geyer, 2004]. The mixing vertically redistributes salt and momentum and thus the outflow characteristics of the water which becomes the river plume [Hetland, 2005]. [4] Mixing continues farther from the estuary and nearfield plume, but at lower intensities. Shear and stratification are weaker, and wind becomes increasingly important for mixing. Mixing intensity (e.g., vertical salt flux) is lower, but the amount of work done by mixing can be higher because it occurs over a larger area [Hetland, 2005]. It is not clear whether the vigorous entrainment processes occurring over a small area near the river mouth, or the weaker processes occurring over the whole area, ultimately govern bulk plume characteristics. 1of15

2 [5] Tracking freshwater provides a means of quantifying the relative importance of mixing and advection processes. MacCready et al. [2002] present a framework to quantify advection and mixing by analyzing the volume of water between isohalines. The advantage of this formulation is that the processes which create water of a particular salinity are transparent. The pitfall of the isohaline budget is that it requires detailed knowledge of the salinity field, and to date has been applied only to numerical studies. Monsen et al. [2002] review three basic quantities used to characterize mixing and advection in a semienclosed system: residence time, age, and flushing time. Residence time is the time it takes for a water parcel to exit a system, while age is the amount of time a water parcel has spent since entering the system. They are both quantities assigned to individual water parcels, and quite often they will vary throughout the domain because of spatially variable mixing and advection. By their nature, residence time and age are difficult to evaluate with field measurements. Flushing time, however, is a simple bulk estimate of the exchange characteristics of the domain [Monsen et al., 2002]. While it obfuscates physical processes, it is generally straightforward to compute. [6] Choosing boundaries to define a volume is somewhat arbitrary for a river plume because it lacks solid boundaries. One meaningful approach is to define the volume with surfaces important for mixing and entrainment, or specific isohalines [e.g., MacCready et al., 2002]. River plumes are generally strongly stratified, and the base of the stratified layer is expected to be a dynamically meaningful surface because mixing and entrainment may be important here. The seaward extent of a plume is often bound by a zone of strong flow convergence resulting in a salinity front. Here, mixing is important as plume water reaches the front and is forced downward [Garvine and Monk, 1974]. This simple configuration, with addition of water by entrainment from below, and a loss of water near the front, can be described by a simple set of volume and salt conservation equations. In fact, the quasi steady limit of these equations yields a relationship analogous to the Knudsen relations used for estuaries with solid boundaries. [7] In this paper we address mixing and dynamics by modeling the river plume as a 1 D slab[e.g.,stronach, 1977; McCabe et al., 2008; Hetland, 2010]. Equations describing the salinity of the plume are developed, and the key unknown parameters maintaining the salinity balance in the quasi steady limit are estimated using spatially resolved observations of the Fraser River plume. The data was collected by instruments on a passenger ferry in the Strait of Georgia, B.C., Canada. The track of this ferry passed directly through the plume of the Fraser River eight times per day. This paper is a companion to Halverson and Pawlowicz [2008] (hereafter HP08), where time series techniques were applied to the same data to understand the physical driving factors important in the Fraser plume and estuary. As such, this paper begins with a brief summary of HP08 in section 4. Next, the flushing time and salinity budget equations are developed and applied to the ferry data. Section 5 contains a discussion of the quantities derived from the budget: entrainment flux, entrainment velocity, and vertical salt flux. Next, the implications of the freshwater flushing time are discussed and compared to other systems. The work is summarized in section Strait of Georgia and the Fraser River [8] The Fraser River discharges into the Strait of Georgia (hereafter SoG), a midlatitude semienclosed coastal basin situated between mainland British Columbia and Vancouver Island. The SoG is a relatively deep, fjord like system with two significant entrances to the Pacific Ocean. Circulation in the SoG has been described by a number of authors [Waldichuk, 1957; LeBlond, 1983; Crean et al., 1988; Thomson, 1994; Masson and Cummins, 2004; Pawlowicz et al., 2007]. Tides in the SoG are of the mixed type and characteristic of the temperate eastern Pacific. Both spring/ neap (i.e., lunar phase) and tropic/equatorial (i.e., lunar declination) cycling produce a sizable fortnightly modulation. The minimum daily tidal range over the study period was 1.9 m, while the maximum was 5.0 m. [9] Figure 1 shows a map of the lower SoG with 551 nm reflectance measured using the Moderate Resolution Imaging Spectroradiometer (MODIS). The Fraser River plume is formed by discharge from each of its three arms. The southernmost or Main arm, carrying 87% of the total river flow [Crean et al., 1988], is 15 km due south of Vancouver. The SoG is approximately 30 km wide here, and observations show the plume can span the SoG under some conditions [Tabata, 1972]. Before reaching the SoG, the river follows a 9 km channel through tidal mud flats ending at the Sand Heads meteorological station. The channel is periodically dredged to a navigable depth of about 10 m but has a few pools of roughly 18 m depth. From the seaward extent of the mud flats the plume is detached from the bottom as the depth of the SoG increases quickly to greater than 100 m. Salt water penetrates a considerable distance upstream of Sand Heads within the river, creating a salt wedge estuary during much of the year [Geyer and Farmer, 1989; Tedford et al., 2009]. Although mixing does occur within the river, salinities in the surface of the SoG can be less than 5 for some distance away from the river mouth in summer. [10] The Fraser River is the single largest point source of freshwater into the SoG, and it contributes 50 85% of the total freshwater flux [Waldichuk, 1957; Crean et al., 1988; Pawlowicz et al., 2007]. The river is not controlled by dams, so that it exhibits natural fluctuations in response to weather and seasonal forcing. The annual discharge cycle is driven by early summer snowmelt, providing freshet flows which peak in early June. The lowest flows are typically observed in fall and winter. However, winter rain storms can cause the discharge to more than double over the span of a few days. 3. Methods and Data 3.1. Ferry Sampling [11] The primary data set used in this paper was acquired from a collection of sensors aboard the British Columbia Ferry Services Inc. vessel, the M.V. Queen of New Westminster. For a month in spring 2003, another instrumented vessel, the M.V. Queen of Alberni, serviced this route. The 2of15

3 Figure 1. Map of the lower Strait of Georgia plotted on a MODIS 551 nm 1 km resolution image. The image was taken 19 July 2005, while the river discharge was about 6,000 m 3 s 1. The plume appears as dark shades signifying a high reflectance. The small white points are three months of subsampled GPS fixes from the ferry. Westbound tracks sail north of eastbound tracks. STRATOGEM hydrographic stations S2 3, S3, and S4 1 are marked by triangles. ferry makes four round trips between the Tsawwassen and Duke Point terminals each day (Figure 1). The track essentially runs along the strait, oriented to the northwest/ southeast. The first sailing departs Tsawwassen at 0515 local time, and the last sailing departs Duke Point at 2345 local time. A complete transect covers 70 km and takes two hours. [12] Ferry data began on 13 January 2003, and continued until 29 October The data set contains 8,502 transects. Every winter the ferry was removed for an annual refit, creating monthlong data gaps. Occasional instrumentation problems caused additional gaps in the data record, the longest being a month in May Other gaps typically last a few days and occur infrequently and sporadically through the time series. [13] The instrument suite consists of chemistry free sensors to measure several oceanographic variables. In this paper, we will only make use of salinity, measured by a Seabird SBE45, and GPS position. The thermosalinograph samples at 5 s intervals and GPS fixes were taken at 10 s intervals. The thermosalinograph precision is ±0.005 PSU, but regular lab and factory calibrations revealed that fouling occasionally freshened the salinity by up to 0.6 PSU. Because the plume was identified by relative changes in salinity along a transect, and because the salinity can vary by 10 PSU in a single transect, we do not correct for fouling. Salinity is given according to the PSS 78 scale. Additional details regarding instrumentation are given by Halverson [2009]. [14] Our system, located in the engine room, draws a continuous stream of water from the ship s engine seawater cooling system. Engine cooling water is pumped into a sea chest, located midship, from a depth of 3.5 m. This may vary by up to 40 cm depending on payload. A ship s hull will disturb the surface waters as it moves, potentially changing the depth of the source water. When sampling a highly stratified system, even a difference of a meter can be significant. The details of how the effective sampling depth was determined are documented in Appendix A. In the case of the Queen of New Westminster, the effective sampling depth was estimated to be 2 m. [15] The ferry travels at roughly 20 knots making the highest achievable spatial resolution 40 m for a 5 s sampling interval. However, there is a delay of about three minutes between the time water enters the sea chest and the time it passes through our instruments. While in the sea chest, water can mix and effectively decrease the spatial resolution. If we take three minutes as the minimum temporal resolution, the spatial resolution becomes 1.9 km. [16] Data quality control and processing began with discarding data obtained while in port, when the ship s seawater pumps are run intermittently, and also discarding data obtained during equipment servicing, when the source water was disconnected. The data was binned to 30 second intervals, and then gridded according to along strait distance by projecting the tracks onto a line running along the center of the SoG (Figure 1). The transect was then divided into plume and nonplume sections (section 3.4). Spatial averages of these sections are taken to form plume and nonplume time series. The time series are then binned by day to remove tidal variations. 3of15

4 3.2. Fraser River Discharge [17] The nearest gauged station which does not experience tides is near the town of Hope, B.C., located 120 km upstream of the river mouth. Numerous tributaries downstream of Hope, however, can contribute a substantial amount of water [Pawlowicz et al., 2007], and the difference is important to quantify for this paper. The method used to account for the additions downstream of Hope is presented in HP08. Briefly, two relatively large gauged rivers downstream of Hope are added to the Hope discharge, and a single small river is scaled up to account for the numerous smaller rivers CTD Profiles [18] Monthly hydrographic surveys were conducted from April 2002 until June 2005 as part of the STRATOGEM project to study coupled physics and biology in the SoG. During spring blooms, and for a short period in summer 2003, cruises were held weekly or biweekly. Nine stations were sampled for a range of physical and biogeochemical parameters. In this paper, we will use profiles from S2 3 and S3, representing the plume, and S4 1, representing SoG water (Figure 1). Full depth continuous profiles of a range of parameters were taken, but in this paper only profiles of salinity are used. Salinity is accurate to within ±0.01, and is given in terms of the PSS 78 standard Defining the Plume [19] It is important to explicitly detail how the plume is identified from within a ferry transect. In nearly all of the transects, there was a low salinity region roughly centered on the part of the track nearest to the river mouth, which we identify as the river plume. While the spatial structure of this low salinity region can vary substantially among transects, it was always bound by saline water at the northwestern end of the transect. In this region, salinity is less variable in both space and time than in the plume water closer to the river mouth. Typical transects of salinity, taken during times of moderate and low river discharge (3,800 m 3 s 1 and 1,600 m 3 s 1, respectively), are shown in HP08. [20] There is generally a sharp salinity gradient between the plume and the ambient water at the northwestern edge of the transect. It is not clear, a priori, if this is a front because the ferry cannot resolve features with length scales of tens of meters [Garvine and Monk, 1974; O Donnell, 1998]. Based on the results of Horner Devine et al. [2009], who observe a front bounding the bulge of the Columbia plume from its far field, and HP08, who suggest that the Fraser plume sampled by the ferry is on a scale where rotation is important, we suggest that the sharp gradient is a front. This does not imply that the Fraser plume shows a distinct bulge region, only that the plume within the front farthest from the river mouth is a dynamically distinct region of the Fraser River estuary and plume system. [21] Given the amount of data, it was desirable to develop an automated method to choose which part of the transect was occupied by the plume. We initially expected that the front separating the plume from ambient water could be identified by simply choosing the steepest salinity gradient in a transect. However, many of the transects contained multiple instances of steep salinity gradients, potentially because there were multiple fronts [e.g., Luketina and Imberger, 1987], so that choosing the maximum gradient did not necessarily locate the farthest plume front. [22] Thresholding the transect according to salinity proved to be a more robust method because it generally separated the low salinity plume region from ambient waters while ignoring spurious gradients. The threshold salinity was allowed to vary in time to account for seasonal changes in the ambient water salinity. A reference salinity was estimated for each track by taking the spatial mean of salinity between the 45 and 50 km points on the transect, which lie outside the plume (see Figure 3 of HP08). This water will be referred to as Strait of Georgia (SoG) surface water. The plume was then defined as the part of the transect with a salinity below the threshold, S thresh = S ref S offset, where S offset was chosen to be a linear function of the reference salinity: S offset ¼ 4:8 0:14 * S ref This quantity varies between 0.4 to 1.8 for the range of observed S ref, and is largest when the reference salinity is low. If, for example, the salinity offset was too high during low flow periods, then much of the plume would be missed because the difference between plume and reference salinity is small. The salinity is low pass filtered in along strait distance by a running mean filter having 10% of the width of the transect, and the plume boundary is then chosen as the point where S thresh intersects the filtered salinity curve. 4. Results 4.1. Plume and Strait of Georgia Water Salinity [23] The full time series of plume salinity at 2 m and SoG water salinity are shown in Figure 2, along with the Fraser River discharge at the mouth. Mean salinity is defined here as the spatial average of the section of a ferry transect determined to be the plume (section 3.4). The largest fluctuations in plume salinity generally occur on seasonal time scales, and are forced by variations in the Fraser River discharge such that plume salinity decreases with increasing river discharge. When fortnightly and shorter cycles are suppressed with a 25 day Hamming filter, the mean plume salinity decreases quasi linearly with increasing river flow at 1.4 per 1000 m 3 s 1 (HP08). On shorter time scales, there is significant energy at biweekly periods which are thought to be caused by fortnightly cycling in the strength of tidal mixing. The daily binned time series shows that fluctuations around the long time scale behavior are larger during times of high river discharge compared to times of low river discharge. Although not evident in Figure 2, individual tidal variations can be quite strong, particularly in summer. [24] The SoG surface water salinity is less variable than plume salinity. The largest variations are correlated with seasonal cycles in freshwater input from the Fraser and other rivers. When high frequencies are suppressed with a 25 day Hamming filter, SoG salinity decreases quasi linearly with river flow at 0.7 per 1000 m 3 s 1, or half the rate observed in the plume (HP08). Like plume salinity, it exhibits relatively high frequency fluctuations. These fluctuations are smaller ð1þ 4of15

5 Figure 2. Full time series of mean daily plume salinity at 2 m (black line) and SoG water salinity (gray line). The heavy lines are daily salinity filtered with a 25 day Hamming window. The shaded curve is the Fraser flow at the mouth. Four years of data are shown: (a) 2003, (b) 2004, (c) 2005, and (d) in magnitude than those of plume salinity, and are not correlated with river discharge Plume Freshwater Fraction [25] Estimating the fraction of the plume volume consisting of freshwater is necessary to estimate the freshwater flushing time, and it also becomes useful in the salinity budget analysis. The freshwater fraction is simply: f ¼ 1 S p S o where S p is the volume averaged plume salinity and S o is a reference salinity. ð2þ [26] A few assumptions and corrections must be made to arrive at a volume averaged salinity from the ferry salinity record. First, we assume that the track average 2 m salinity is equal to the area averaged 2 m salinity. Second, because the plume is strongly stratified, we must specify a vertical structure to estimate the depth averaged salinity from the 2 m salinity. These corrections represent the simplest possible approach, and the potential bias introduced by incorrectly estimating plume salinity is discussed in section 5.1. [27] Hydrographic profiles can be used to estimate the plume stratification and thickness, and to see if these quantities vary with respect to changes in river flow or distance from the river mouth. In Figure 3, 144 salinity profiles from three hydrographic stations are shown (Figure 1). Station S2 3 Figure 3. Salinity profiles from stations S2 3, S3, and S4 1. Stations S2 3 and S3 are representative of the plume, while S4 1 is representative of ambient Strait of Georgia water. The thin gray lines are individual profiles, while the thick lines represent the mean profiles during times of high river discharge (solid, >3,500 m 3 s 1 ) and low river discharge (broken, <3,500 m 3 s 1 ). 5of15

6 Figure 4. Time series of 7 m CTD salinity at S2 3 (solid line, open circles) and 2 m salinity at S4 1 (broken line, filled circles). is 4 km seaward of the river mouth, so that it represents the near field plume. Station S3 is 18 km seaward of the mouth, so that it generally represents the plume outside of the near field, while S4 1 is 36 km seaward so that it represents ambient SoG water. Stations S2 3 and S3 show a high degree of variability over the top 10 m, while S4 1 shows somewhat less variability. In general, the salinity variation in the plume stations becomes relatively small at a depth of about 10 m. The profiles taken during low river flow periods show very little depth variation below about 4 m. [28] The plume thickness relevant to this study is the distance from the surface to the base of the strongly stratified surface layer. A number of automated methods, such as searching for changes in the vertical gradients, were used to find this distance. However, they did not yield results which were significantly correlated with river discharge. The river discharge clearly influences the near surface salinity, and therefore stratification and freshwater content. This is readily apparent in the mean profiles from high discharge periods (>3,500 m 3 s 1, 15 profiles) and low discharge periods (<3,500 m 3 s 1, 33 profiles) (Figure 3). The variability in plume thickness may have been due to tides, which would have a particularly large effect at S2 3. [29] In light of the apparent poor correlation between river discharge and plume thickness, and because this paper is concerned with subtidal time scales and plume wide averages, we will simply assume that the plume thickness does not change substantially in time or space. Clearly, on tidal time scales, accounting for the variability would be necessary. The mean salinity profiles from station S3 (Figure 3), which represents the majority of the plume, show that the stratified layer begins to weaken at about 7 m. Thus the plume will be approximated as a 7 m thick linearly stratified layer. [30] The 2 m salinity can now be used to obtained a depth averaged salinity: S p ¼ 7 10 Sz¼ ð 2Þþ 3 10 Sz¼ ð 7Þ ð3þ We will use SoG water salinity as it is calculated in section 3.4 as the salinity at 7 m. This assumes that the water at 2 m outside of the plume has the same salinity as the water underneath the plume nearer to the river mouth. We can show that this is a reasonable approximation using profiles taken at S2 3 and S4 1. Figure 4 shows a time series of 7 m salinity at S2 3 and a time series of 2 m salinity at S4 1. The salinities covary and are of nearly the same magnitude. The largest differences occur in summer during high river flow periods. The CTD data suggests that, on average, using SoG salinity underestimates the reference (or entrained) salinity by 1.0 ± 0.6 (95% CI for salinity difference). [31] After making the approximations, Figure 5 shows the fraction of the plume volume occupied by freshwater relative to SoG water as function of river discharge for the 25 day low pass filtered time series. The time series has been decimated into points 25 days apart so the markers represent approximately independent estimates. At the lowest river flows, the plume is only about 5% freshwater, while at 8000 m 3 s 1, the plume contains 25% freshwater. The increase is very close to linear, increasing by 2.9% every 1,000 m 3 s 1. Figure 5. Fraction of the plume composed of freshwater versus river discharge. The solid line is a best fit straight line, where a = and b = (m 3 s 1 ) 1, and the dashed line is a nonparametric LOESS fit for comparison. 6of15

7 Figure 6. (a) The daily time series of plume freshwater volume (dots) along with Fraser River discharge (gray). The thick line is the freshwater volume after filtering with a 25 day Hamming window. (b) The time series of freshwater flushing time (dots), formed by dividing the river discharge into the freshwater volume. The thick line is the flushing time after filtering the daily time series with a 25 day Hamming window Plume Surface Area [32] The next step in estimating the freshwater flushing time is to estimate the surface area. The surface area was estimated and discussed in HP08. To summarize, the variance of plume surface area is dominated by high frequency energy in the 8 to 15 day band. It is not clear what forces these frequencies, but it was hypothesized to be a combination of advection of the plume relative to the ferry track by wind and tides, and errors in the plume finding algorithm. Smoothing with a 25 day Hamming window removes the high frequency variance. The remaining variance is forced mostly by river discharge. Surface area increases with river discharge from 200 to 500 km 2 at low flows to between 1,000 and 1,500 km 2 at high flows (HP08). [33] When calculating the vertical entrainment velocity from the salinity budget (section 4.5), we require the surface area as a function of river discharge. In this case, we will use a parameterization of the relationship. Nonparametric smoothing of the relationship between river flow and plume surface area by the LOESS algorithm [Cleveland, 1979] reveals a slight curvature, such that the area varies proportionately less with increasing flow (HP08). Two theoretical relationships between river discharge and plume surface area were discussed in HP08, A / Q 1/2 and A / Q 1. The linear relationship more accurately predicted the magnitude of the surface area, while the square root more accurately reproduced the shape of the relationship. In this paper, we chose to parameterize the surface area with Q 1/2 because it captures the observed curvature better than Q 1. Fitting the data thus yields the following expression: A ¼ Q 1=2 where g is 11.1 ± 0.5 km 2 (m 3 s 1 ) 1/ Plume Freshwater Volume and Flushing Time [34] Surface area, percent freshwater, and plume thickness can be combined to estimate the total volume of freshwater ð4þ in the plume. Formally, freshwater volume is the integral of freshwater fraction over the volume of the plume. A simplified form, written in terms of the plume freshwater fraction, surface area, and thickness is: V fw ðþ DA t ðþf t ðþ t where D is the plume thickness. Inserting the ferry measurements, and taking the thickness to be 7 m, yields a time series of freshwater volume (Figure 6a). The freshwater volume, as with the plume area and salinity, shows a high degree of variability, primarily caused by the 8 15 day variance in surface area, so a 25 day Hamming window was again used to smooth the time series. During times of high river flow, the fresh water volume can reach 3.0 km 3, and during times of low river flow the freshwater volume is typically 0.5 km 3. [35] Dividing the unsmoothed plume freshwater volume by the river discharge yields the freshwater flushing time, t fw (Figure 6b). As with the freshwater volume, the flushing time is noisy, so it too is smoothed with a 25 day Hamming window. The smoothed flushing time varies from less than 1 day to just over 4 days, with a mean of 2.2 days. Plotting freshwater flushing time versus river discharge (not shown) suggests that flushing time might increase weakly with river flow. However, the 95% confidence interval about the average flushing time is ±1.2 days, meaning that any dependence on river flow would be weak on statistical grounds. We also note that systematic variations in plume depth, which could range by ±50% from 7 m (section 4.2), would change the fresh water volume and flushing time by a similar percentage (equation (5)) Plume Salinity Budget Model and Framework [36] The salinity budget is based on a single layer slab plume with surface area, A, and thickness, D, illustrated by the schematic diagram in Figure 7. Mass introduced into the ð5þ 7of15

8 Figure 7. Schematic diagram of a plume used to derive the volume (equation (6)) and total salt (equation (7)) budgets for a plume of surface area A and thickness D. Q is the river flow, w e A is the upward flux of ambient water, and F f is the volume flux lost at the plume front. With respect to the salt budget, w e A brings in ambient water S o, and F f exports plume salinity S p. The river has zero salinity. plume is mixed instantaneously and uniformly [e.g., Monsen et al., 2002]. Both freshwater input by the river, Q, and entrainment of seawater at the base of the plume add volume, while brackish plume water is entrained out of the plume front [Garvine, 1974; Stronach, 1977] at a rate, F f. The volume flux of seawater entrained into the plume is written as the product of the plume surface area, A, and an entrainment velocity, w e. The volume encompassing the plume is filled with a uniform salinity, S p. The equations for the time evolution of plume volume, V p, and of total salt mass, r p V p S p, for this model are: dv p dt ¼ Q þ w e A F f d p S p V p ¼ o w e AS o p F f S p dt where S o is the salinity of the water entrained into the plume from below and r is density. We are assuming that the entrained water has a salinity equal to the SoG water salinity. The first term on the r.h.s. of equation (6) is the addition of river water, the second is the addition of salt water from below by entrainment, which we refer to as entrainment flux, and the third term is the loss of plume water due to mixing processes at the front. In equation (7), salt is gained from below by term 1 on the r.h.s. and lost at the plume front by the second term. [37] An expression for the time evolution of plume salinity was obtained by combining and simplifying equations (6) and (7). First, plume and SoG water density were disregarded because their difference is assumed to be very small. Next, we eliminated the loss term from the equations, and wrote the volume as the product of surface area and thickness. Finally, the l.h.s. of equation (7) was expanded with the chain rule to arrive at the balance for plume salinity: ds p dt ¼ Q AD S p þ w e D S o S p The first term on the r.h.s. of equation (8) is always negative and is therefore a sink, while the second term is always ð6þ ð7þ ð8þ positive because S o > S p, and is therefore a source of salinity. The derivative term on the l.h.s. is referred to as the storage term, which is important when source and sink do not balance Quasi steady Balance [38] Rapid changes in river discharge and fortnightly cycling in tides may cause plume salinity to decrease by up to 4 per day or increase by as much as 6 per day (Figure 2). Wind may cause mixing, but it can also change the estimate of plume salinity if it advects the plume relative to the ferry track. The plume salinity balance (equation (8)) does not explicitly account for tides and wind, though their effects are implicit in w e. In this way, w e is a catch all mixing term which incorporates the average net effects of wind and tidal mixing in addition to intrinsic mixing processes in the estuary and plume such as the vertical shear driven by estuarine circulation. [39] The terms in equation (8) must be smoothed to achieve a quasi steady balance which, in turn, provides a very straightforward computation of entrainment flux. Here, we demonstrate that a steady salinity balance is achieved by convolving the plume salinity and budget terms with a 25 day Hamming window. One might anticipate this to be sufficient because the same filter was implemented in HP08 to remove both wind and tide forced variability in plume salinity. On appropriately long time scales, the sink and source terms of equation (8) will balance and keep the salinity storage term small. The storage and sink terms are readily evaluated with existing data, however the source term contains w e, which is unknown. To evaluate the sink term, Q and S p are filtered with a 25 day Hamming window, A is parameterized with equation (4), and D is fixed at 7 m. The result is that the sink term varies from 1.8 to 0.9 d 1, and has a mean value of 1.3 d 1. In contrast, the storage term is comparatively small after smoothing. Only 25% of the storage time series has an absolute value exceeding 0.13 d 1 (10% of the mean sink), and the extremes ( 0.4 to +0.3 d 1 ) are less than half of the minimum sink. Thus, we conclude that the plume salinity is quasi steady at time scales of 25 days or greater, and the form of equation (8) is valid in the sense that all variations in salinity can be modeled with dilution by river input and enrichment by entrainment. [40] In the quasi steady limit of equation (8), the sink and source terms balance each other, yielding a Knudsen relation for the upward flux of salt water driven by river flow: Q ¼ w e A S o 1 ð9þ S p The steady balance in this form allows the entrainment flux, w e A, to be readily determined from salinity and river flow measurements Entrainment Flux [41] Equation (9) may be rearranged to solve for the entrainment flux, w e A, given the measurable quantities, Q, S p, and S o : w e A ¼ Q S o S p 1 ð10þ 8of15

9 Figure 8. Entrainment flux as a function of river discharge. The circles are direct estimates made with equation (10), while the thick solid line uses the freshwater fraction parameterization of equation (12). The LOESS smoothed curve (thick dashed line) and the 2s error bounds (thin dashed lines) elucidate the relationship. The thin dash dot line represents the case where a = 0. Figure 8 shows the entrainment flux as a function of river discharge. While the entrainment flux shows a fair amount of scatter, it is clear that it begins at about 17,000 m 3 s 1, and increases with river flow until reaching a maximum in the range of 25,000 to 35,000 m 3 s 1 at higher flows. Fitting a LOESS nonparametric curve clarifies the relationship, and suggests that the entrainment flux reaches a broad maximum of 28,000 m 3 s 1 at a river discharge of about 4,500 m 3 s 1. The 2s bootstrapped error bounds on the LOESS curve are wide enough such that the existence of a maximum in w e A is questionable, but it is clear that it initially rises before reaching a plateau at about 4,500 m 3 s 1. [42] Plume and SoG water salinity were needed to estimate the entrainment flux, meaning equation (10) is not prognostic because, intuitively, plume salinity is expected to depend on the entrainment flux. Plume and SoG salinity are each correlated in a relatively simple way with river flow, and if they can be parameterized with river discharge then an empirical expression can be derived in which entrainment flux depends only on river discharge. [43] Plume and SoG water salinity will be parameterized in terms of the freshwater fraction (equation (2)). The denominator of equation (10) can be written as f(1 f ) 1, so that the parameterized entrainment flux becomes: w e A ¼ Q 1 f 1 ð11þ This form is particularly useful because f is very well approximated by a linear function of river discharge (Figure 5). A fit of a + bq to the freshwater fraction, with bootstrapped 95% confidence intervals, yields a = (2.97 ± 0.63) 10 2 and b = (2.49 ± 0.21) 10 5 (m 3 s 1 ) 1. Strictly speaking, the linear model is unrealistic because it allows the freshwater fraction to exceed one at large river flows. An example of a more realistic model is 1 exp( (a + bq)). However, in practice, both models fit the data equally well over the range of observed river flow. Inserting the linear parameterization into equation (11) yields an expression for entrainment flux which is a function of river flow only: 1 w e A ¼ Q þ Q 1 ð12þ The curve predicted by this equation using the fitted empirical a and b parameters is included in Figure Discussion 5.1. Curvature in Entrainment Flux [44] The entrainment flux initially increases with increasing river flow until it plateaus at a river flow of about 4,500 m 3 s 1 (Figure 8). While this observed variation of entrainment flux is statistically significant, it is important to note that the magnitude and shape of the curve are each sensitive to the details of plume and SoG salinity. This can be tested by varying the empirical a and b coefficients in equation (12). For example, increasing (decreasing) a by 20% of its best fit value of (approximately the 95% confidence interval), while keeping b fixed at its original, empirically determined value of (m 3 s 1 ) 1, will increase (decrease) the entrainment flux by about 2,000 m 3 s 1 over all values of river flow. Increasing (decreasing) b by 10% (approximately the 95% confidence interval), increases (decreases) the entrainment flux by 1,000 m 3 s 1 at low river flow to 2,500 m 3 s 1 at high river flow. Thus, varying a and b within their estimated confidence intervals can change the magnitude of the entrainment flux. However, in all cases the entrainment flux increases with river discharge before reaching a plateau at a discharge of about 4,500 m 3 s 1. [45] The plume and SoG salinity changes implied by varying a and b can help determine the potential errors introduced by the assumptions needed to estimate a volume 9of15

10 averaged plume salinity from the ferry data. Increasing both parameters to their upper 95% confidence intervals corresponds to a decrease in plume salinity of 0.5 at low river discharge or 1.3 at high river discharge if the SoG salinity is unchanged. For example, if the plume salinity was overestimated by 1.3 at high river flow, the resulting entrainment flux would be underestimated by 3,500 m 3 s 1, or 14%. [46] The general shape of the entrainment flux curve shows an increase with discharge at low flows. The entrainment flux should vanish at zero discharge and extrapolation of the data does tend to agree with this assertion. The implication of this with respect to equation (11) is that the freshwater fraction must remain finite at low flow, so that a > 0 in equation (12). A zero a implies that w e A b m 3 s 1 at very small discharge. This is much larger than the measurements (Figure 8), and hence even a small plume must have a nonzero freshwater fraction. [47] The data do not make a definitive statement on the eventual high discharge behavior of entrainment flux because extrapolating it beyond a river discharge of 8,000 m 3 s 1 cannot be done with any confidence. The LOESS parameterization begins to decline at the highest river flows sampled, but the trend is only weakly significant because of the width of the 2s bounds. We also note that parameterizing the freshwater fraction with 1 exp( (a + bq)) has similar limiting behaviors when substituted into equation (11). [48] At sufficiently high discharge, the freshwater fraction should approach one. At f = 1, which represents a pure freshwater plume, the entrainment flux will be zero and there would necessarily be a maximum entrainment at some intermediate discharge. If f reaches some asymptotic value near one, then equation (11) shows that the entrainment flux would approach a value equal to some small fraction of the river flow. In this case there may or may not be an entrainment maximum at intermediate flow. [49] Although these theoretical arguments explain the existence of the plateau, they do not provide any guidance in determining why it occurs at a river flow of about 4,500 m 3 s 1. Perhaps coincidentally, this is approximately the river flow required to push the salt wedge to the mouth at low tide. According to Kostaschuk and Atwood [1990, equation (3)], the minimum discharge required to flush the salt wedge out of the river at low tide is 3,500 m 3 s 1 at Hope. This is 500 to 1,000 m 3 s 1 lower than the flow at the mouth (HP08), implying the salt wedge is flushed out of the estuary at a mouth discharge of 4,000 to 4,500 m 3 s 1. HP08 suggest that the location of the salt wedge influences plume salinity, which may imply a change in where most of the mixing occurs. Thus, a change in the entrainment flux might be expected, but the mechanisms determining its observed variation with river discharge are not understood. [50] While it is apparent there are some limitations in using the ferry data to estimate the terms in the quasi steady salinity budget, it is clear that the entrainment flux is m 3 s 1 to within 30% for a wide range in river flow. Furthermore, as shown in the section 5.2, the consistency with previous work supports the quantitative estimate of entrainment flux made here Entrainment Velocity [51] Entrainment velocity is a more fundamental quantity than entrainment flux because it does not depend on the dimensions of the system (i.e., surface area). It is determined by the local stratification and shear, and can be linked to a number of important turbulence parameters. [52] Entrainment velocity can be estimated from entrainment flux by dividing it by the surface area across which entrainment takes place: w e ðqþ ¼ w ea A e ð13þ With this formulation, entrainment velocity will generally depend on river discharge because the surface area for entrainment and the entrainment flux each depend on river discharge. Entrainment processes of different kinds are active over the entire estuary and river plume system, so that choosing an A e is not straightforward. Therefore, two idealized models will be considered. The first assumes that entrainment occurs uniformly over the whole plume, and the second assumes that entrainment occurs only in the salt wedge estuary and near field plume. The plausibility of each model is then assessed by comparing the resulting entrainment parameters to literature values Entrainment in the Plume [53] In this approximation, the river discharges freshwater directly into the SoG, and entrainment occurs uniformly over the river plume. The surface area for entrainment will be taken as the surface area of the plume as measured by the ferry, parameterized as a function of river discharge according to equation (4). The entrainment velocity is therefore a function of river discharge only, decreasing quasi linearly from about 4.5 m d 1 ( mm s 1 )at 1,000 m 3 s 1 to 2.2 m d 1 ( mm s 1 ) at 8,000 m 3 s 1 (Figure 9a). In terms of the salinity budget, the decrease of entrainment velocity reflects both the tendency for the plume surface area and the difference between plume and SoG salinity to increase with river discharge. [54] The entrainment velocity of O(10 2 )mms 1 is at least one order of magnitude smaller than previous estimates in the Fraser plume. Cordes et al. [1980] estimate a dimensionless entrainment coefficient, w e /U, of , which corresponds to an entrainment velocity of mm s 1 for a 1 m s 1 flow typical of their measurements. MacDonald and Geyer [2004] estimate a vertical entrainment velocity of 4 to 8 mm s 1. The factor of 10 or more difference between the estimate made here and the literature values arises from differences in the location of sampling with respect to the river mouth, where the entrainment velocity is expected to change rapidly [McCabe et al., 2008]. The Cordes et al. [1980] study was conducted within about 10 km of the river mouth, while the MacDonald and Geyer [2004] measurements were only a few kilometers downstream of the river mouth. Flows here are somewhat more energetic than those expected for the whole plume, and thus entrainment velocities would be higher. If we use the Cordes et al. [1980] entrainment coefficient together with a flow speed of 10 cm s 1,we obtain an entrainment velocity of mm s 1, which is on the order of those made for the whole plume in this paper. [55] Part of the utility of quantifying the entrainment velocity is that it can be used to compute turbulent flux parameters, such as the vertical salt flux, w e S o, and the 10 of 15

11 Figure 9. (a) The vertical entrainment velocity as a function of river discharge calculated with equation (13). The solid line is the entrainment velocity derived from the parameterized plume freshwater fraction. Entrainment was assumed to occur uniformly throughout the river plume. (b) The vertical salt flux, S o w e. The solid line uses the entrainment velocity parameterization from Figure 9a multiplied by a nonparametric fit of the SoG salinity (see Figure 5 in HP08). turbulent salt diffusivity, K s. Furthermore, the vertical salt flux can be computed as a function of river flow because reference salinity is correlated with river flow (section 4.1). As with the entrainment velocity, the salt flux decreases with river discharge (Figure 9b), from ms 1 at high flow to ms 1 at low flow. A turbulent diffusivity for salt can be estimated with the w e S o = K [e.g., McCabe et al., 2008]. As in section 4.2, stratification in the plume is approximated by a linear increase of salinity with depth. Turbulent salt diffusivity (not shown) decreases with increasing river discharge in a similar manner to entrainment velocity and salt flux, although the dependence is stronger. At 1,000 m 3 s 1, it is about m 2 s 1, and at 8,000 m 3 s 1, it is m 2 s 1. This is lower than the value of m 2 s 1 measured by MacDonald and Geyer [2004] in the near field of the Fraser River plume Entrainment in the Estuary and Near Field Plume [56] Assuming entrainment over the whole plume leads to vertical velocities lower than those observed near the river mouth. This, in turn, suggests that the total entrainment may be supported by larger velocities over a smaller area. We will now assume that mixing takes place primarily within the river, along the length of the salt wedge, or within the near field plume. [57] To estimate a surface area for entrainment in the estuary we consider the length of the salt intrusion. At low discharge the tip of the salt wedge may reach 30 km upstream [Kostaschuk and Atwood, 1990]. However, the actual length of the salt wedge, and thus the area for entrainment, is likely shorter because water at the mouth is brackish and less stratified. Hydrographic measurements in the estuary at low discharge suggest 15 km is a better length [Ages, 1979]. The entrainment velocity, with an entrainment flux of 17,000 m 3 s 1, a mean river width of 0.5 km, and a salt wedge length of 15 km, is 2.3 mm s 1 (200 m d 1 ). It would increase with river discharge, because the length of the salt intrusion decreases with increasing river discharge [Hansen and Rattray, 1965; Kostaschuk and Atwood, 1990], and because the entrainment flux increases over low to moderate river discharge (Figure 8). While there are no known estimates of entrainment velocity in the estuary at low discharge for comparison, this value is similar to the MacDonald and Geyer [2004] estimates at high flow in the Fraser near field plume. [58] During high river flow the salt intrusion is flushed from the mouth at low tide, and entrainment primarily takes place in the near field plume. A surface area for entrainment is more difficult to estimate in this case because the nearfield plume spreads as it leaves the mouth and its seaward extent is not known. Salinities as low as 10 have been recorded where the ferry track passes within 6 km of the river mouth (Figure 1), which we take as a reasonable estimate of the near field plume size. The width is assumed to be 3 km to account for spreading from the 1 km wide river mouth. The surface area for entrainment is 18 km 2, and the entrainment velocity at the observed peak entrainment flux of 27,000 m 3 s 1 is then 1.6 mm s 1 (130 m d 1 ). [59] The agreement with MacDonald and Geyer [2004], who estimate vertical velocities of 4 8 mms 1 in the Fraser near field plume during the ebb at high river discharge, is good given the crudeness of the area estimate. We expect the MacDonald and Geyer [2004] estimates to be higher because mixing tends to be very localized in the Fraser River estuary [MacDonald and Horner Devine, 2008], which would further reduce the surface area for entrainment. We also note that our values were tidally averaged, and it is known that mixing is tidally modulated in the Fraser River estuary [Geyer and Farmer, 1989; MacDonald and Geyer, 2004; Halverson and Pawlowicz, 2008] Where Does Entrainment Occur? [60] A number of observational studies, both in the Fraser [Cordes et al., 1980; MacDonald and Geyer, 2004] and in other systems [MacDonald et al., 2007; McCabe et al., 2008], have shown that entrainment is vigorous in the near field plume. Our results are consistent with these observations, but go one step further because they imply that entrainment is not important for bulk properties outside of 11 of 15