Observations of River Plume Mixing in the Surf Zone

a. Mixing in river plumes

River plumes are frequently approximated as two-layer flows (Fong and Geyer 2001; Hetland 2010; McCabe et al. 2008; Kastner et al. 2018) in which stratification and shear are highest near the base of the plume layer, resulting in the collocation of turbulence and the salinity gradient. Vertical mixing in river plumes is typically driven by stratified shear instabilities along this interface, which occur when vertical shear is strong enough to overcome stratification (MacDonald and Geyer 2004; Geyer et al. 2010; MacDonald et al. 2007; Kilcher et al. 2012; Jurisa et al. 2016). The associated turbulent kinetic energy (TKE) dissipation values in river plumes are typically ε ∼ O(from 10⁻⁸ to 10⁻³) W kg⁻¹ (Nash and Moum 2005; MacDonald et al. 2007; Kilcher et al. 2012; Jurisa et al. 2016). River plume mixing has been shown to be predominantly a vertical process; lateral mixing in river plumes is small because the aspect ratio of a river plume is typically small (MacDonald and Geyer 2004; Chen and MacDonald 2006; McCabe et al. 2008; Horner-Devine et al. 2015).

b. River plumes in the surf zone

River water is often trapped in the surf zone at the Quinault River mouth and can be predicted based on the relationship between the surf zone width and the near-field plume length scale (Kastner et al. 2019). Once plume water is trapped in the surf zone, it is no longer forced by its initial momentum and is strongly influenced by surf-zone forcing (Wong et al. 2013; Olabarrieta et al. 2014; Rodriguez et al. 2018; Kastner et al. 2019; Jennings et al. 2020). In a saturated surf zone, waves break at a depth d proportional to their height H_s, such that γ = H_s/d is constant throughout the surf zone. Wave height therefore decreases with depth approaching the shoreline, resulting in gradients in wave energy flux and radiation stress (momentum flux) that cause the surf zone to be turbulent and energetic.

Wave energy flux gradients in the surf zone are related to local energy loss rates (Battjes and Janssen 1978; Thornton and Guza 1983). The TKE dissipation rate ε is high in the surf zone, with typical values of ε = O(from 10⁻⁴ to 10⁻²) W kg⁻¹ near the surface (Thornton and Guza 1983; George et al. 1994; Feddersen and Trowbridge 2005; Feddersen 2012a; Thomson 2012). For dissipative beaches, waves break consistently onshore of the break point, resulting in a flux of TKE through the water surface. The vertical structure of TKE dissipation rate has been found to scale with the wave height and water depth and inversely with distance from the water surface, such that dissipation is higher closer to the surface and closer to the break point (Longuet-Higgins and Stewart 1962; Thornton and Guza 1983; Feddersen 2012b).

Wave radiation stress gradients in the surf zone force alongshore currents with velocities on the order of υ from 0.5 to 1.5 m s⁻¹ (Longuet-Higgins and Stewart 1962; Thornton and Guza 1986; Spydell et al. 2007). In addition to mean currents, vertical vorticity introduced by short-crested breaking waves, wave groups, and shear instability cause the formation of energetic horizontal eddies (Peregrine 1998; Haller et al. 1999; Bowen and Holman 1989; Clark et al. 2012; Feddersen 2014). These eddies mix gradients in the cross- and alongshore directions, leading to large horizontal dispersion within the surf zone, with horizontal eddy diffusivity K_x ≈ O(from 0.1 to 10) m² s⁻¹ (Spydell and Feddersen 2009; Clark et al. 2010; Spydell and Feddersen 2012; Hally-Rosendahl et al. 2014). A variety of measurement techniques have been used to quantify surf-zone dispersion, including dye (Hally-Rosendahl et al. 2014) and drifting buoys (Spydell and Feddersen 2012). This high dispersion within the surf zone does not result in significant exchange with the neighboring inner shelf on short time scales (MacMahan et al. 2004; Hally-Rosendahl et al. 2014); however, rip currents can lead to significant exchange with the inner shelf, particularly on time scales of ∼1 day (Reniers et al. 2009; Clark et al. 2012; Moulton et al. 2017; Kumar and Feddersen 2017a,b).

Several recent studies have shown that surface wave breaking outside the surf zone (whitecapping) can affect plume mixing in deeper water. Whitecapping is generally a less energetic forcing mechanism than surf-zone wave breaking and tends to produce lower TKE dissipation rates (Terray et al. 1996; Feddersen et al. 2007; Gerbi et al. 2009). Breaking waves can significantly contribute to river plume mixing when the plume layer is shallow and slow moving far from the river mouth (Gerbi et al. 2015), or when strong wave breaking is collocated with a plume front (Thomson et al. 2014). In particular, Gerbi et al. (2015) show that breaking waves are most effective at mixing a surface layer when the layer is thin, suggesting that the spatial dislocation of the generation of turbulence from a density gradient can impact mixing rates. Breaking waves can impact the structure of a river plume far from the river mouth where the plume propagates as a buoyant coastal current (Gerbi et al. 2013), but these effects are less important closer to the mouth where river momentum is large (Akan et al. 2017; Kastner et al. 2018).

Prior studies of small plumes in the surf zone have suggested the importance of breaking wave-driven mixing increasing plume entrainment velocity (Kastner et al. 2019) and injecting TKE at the surface that mixes the plume (Rodriguez et al. 2018). Because of the higher TKE dissipation rate in the surf zone (from 10⁻⁴ to 10⁻² W kg⁻¹) relative to river plumes (from 10⁻⁸ to 10⁻³ W kg⁻¹), we expect wave-breaking processes to be the primary source of turbulence at the Quinault River mouth. However, a full analysis of wave-driven turbulence and stratification in shallow water has not been done, although some of the mechanisms discussed above for deeper water may be relevant.

In this study, we use novel in situ measurements to investigate the stratification and mixing of surf-zone-trapped river plume water that is exposed to wave-generated turbulence. The surf zone at the Quinault River mouth is shallow, wide, and saturated with energetic breaking waves, making it dangerous to conduct shipboard observations. For this reason our mixing estimates are based on measurements from drifters that follow the trapped plume water and moorings at the edge of the surf zone. These measurements allow us to make calculations of stratification, the material derivative of salinity, vertical eddy diffusivity, and TKE dissipation rate (section 2). These quantities are related to each other and to tidal variability in river volume flux such that tidal variability leads to changes in mixing (section 3). Last, we discuss the implications of these findings on stratified turbulence in the surf zone, examine the potential impacts of lateral dispersion in the surf zone, and compare surf-zone-trapped river plumes with more conventional plume systems (section 4).

a. Observations

The data presented here were collected as part of an observational campaign described in detail by Kastner et al. (2019) at the Quinault River mouth, on Quinault Indian Nation land by Taholah, Washington (Fig. 1). The observations were made with the permission and guidance of the Quinault Nation Division of Natural Resources. We focus in this work primarily on measurements from the SWIFT (Surface Wave Instrument Float with Tracking) drifting buoys deployed in the river inlet. A table detailing the deployment time, model, tidal stage, and wave height for all SWIFT deployments that resulted in surf-zone-trapped drifters is shown in the appendix (Table A1). A map of drifter tracks subset by tidal stage is available in Kastner et al. (2019) (Fig. 3). Two models of SWIFTs were used, as detailed in Table 1. Figures 2 and 3 show examples of raw, high-resolution observations of salinity (Figs. 2b and 3b), drift speed (Figs. 2c and 3c), and heave (Figs. 2d and 3d), as well as estimates of TKE dissipation rate, stratification, and the material derivative of salinity (Figs. 2e and 3e) from two SWIFTv3s that were trapped in the surf zone on 30 April and 1 May 2017. The estimates of mixing rate and $\partial \overline{S}/\partial z$ (hereinafter referred to as stratification or ∂S/∂z, dropping the overbar) shown in Figs. 2f, 2g, 3f, and 3g are calculated using the methods outlined in section 2b. We use the SWIFT measured raw heave to estimate significant wave height on 1-min intervals as H = 4σ_z, where σ_z is the standard deviation of the heave data over 1 min. This combination of measurements effectively resolves wave-driven processes, plume dynamics, and mixing on time scales varying from seconds to the drifter’s deployment length.

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The observational field campaign at the Quinault River mouth: (a) an example UAS perspective image from 28 Apr 2017, when the offshore wave height H_s = 1.62 m, the river volume flux Q_T = 181 m³ s⁻¹, and the tidal stage is −0.41 m, and (b) a map of deployed instruments and bathymetry. Color shading indicates depth, and isobaths are given as white lines with notations in meters. Orange squares indicate moorings using Nortek Aquadopp ADCPs, yellow circles indicate moorings using SWIFTs, and the red triangle indicates the offshore AWAC. The UAS field of view is given as a dashed–dotted red line, and SWIFT drift tracks on 30 Apr 2017 are thin solid lines (black for trapped drifters and gray for escaped drifters). The shoreline is indicated with a thick solid brown line.

Citation: Journal of Physical Oceanography 53, 3; 10.1175/JPO-D-21-0286.1

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Data from a SWIFTv3 deployed on 1 May 2017; this SWIFT was trapped in the surf zone: (a) the drift track overlaid on the bathymetry map from Fig. 1b transformed into a local coordinate system; (b) the 0.5-Hz salinity data at 0.1-, 0.5-, and 1.05-m depth; (c) the raw 4-Hz drift speed data from the GPS in black, with 1-min averages in red; (d) the raw 15-Hz inertial measurement unit heave data in black, with 1-min calculations of H_S = 4σ_z in red (gray lines show data screened as poor quality by SWIFT data processing routines); (e) the vertical average over the top 0.5 m of structure function estimates of TKE dissipation rate from 1 min of uplooking Aquadopp HR ADCP data (Thomson 2012; Thomson et al. 2019); (f) the 1-min mixing rate DS/Dt at 0.5-m depth determined by the fitting method in section 2b; and (g) stratification ∂S/∂z, also determined by the fitting method in section 2b. Alternating crosses and circles indicate the beginning of each SWIFT data collection burst.

Citation: Journal of Physical Oceanography 53, 3; 10.1175/JPO-D-21-0286.1

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Data from one SWIFTv3 and one SWIFTv4 deployed on 30 Apr 2017; both SWIFTs were trapped in the surf zone as the tidal stage rose from η = −0.4 m to η = 1.7 m as shown below the x-axis label (in all panels, data from only one SWIFT is shown for clarity; however, the two SWIFTS followed similar drift tracks and parameters matched well when data products overlapped): (a) the SWIFTv3 drift track overlaid on the bathymetry map from Fig. 1b transformed into a local coordinate system (the SWIFT track is colored by time, with black corresponding to times near deployment and light gray corresponding to times near recovery); (b) the 0.5-Hz salinity data at 0.1-, 0.5-, and 1.05-m depth from the SWIFTv3; (c) the raw 4-Hz drift speed data from the GPS on the SWIFTv3 in black, with 1-min averages in red; (d) the raw 15-Hz inertial measurement unit heave data in black, with 1-min calculations of H_S = 4σ_z in red (gray lines shows data screened as poor quality by SWIFT data processing routines); (e) the vertical average over the top 0.5 m of structure function estimates of SWIFTv4 TKE dissipation rate from 1 min of downlooking Signature1000 HR ADCP data (Thomson 2012; Thomson et al. 2019); (f) the 1-min mixing rate DS/Dt at 0.5-m depth on the SWIFT v3 determined by the fitting method in section 2b; and (g) stratification ∂S/∂z, also determined by the fitting method in section 2b.

Citation: Journal of Physical Oceanography 53, 3; 10.1175/JPO-D-21-0286.1

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Table 1.

The instrumentation and specifications of the two versions of SWIFT drifters used during the Quinault observational campaign.

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We use nearshore moorings with YSI Sonde 600LS CTD sensors ∼1 and ∼3 m below the water surface deployed at the surf-zone edge in ∼4-m water depth (relative to mean lower low water at the USGS Point Grenville station) to measure the salinity field along the onshore edge of the inner shelf. At the boundaries of the surf-zone-plume system, we use measurements of spectral wave properties from a Nortek Acoustic Waves and Currents (AWAC) instrument deployed in ∼6-m water depth as an offshore wave condition as well as a downlooking Nortek Aquadopp and bottom-mounted Onset HOBO pressure sensor deployed at the river mouth to capture the tidally varying river momentum and volume flux. The results presented in section 3 use data from the entire 2-week study period, focusing on two drifter deployments that occurred on 30 April and 1 May 2017. Mean, maximum, and minimum offshore wave height H_s, river discharge Q_R, and tidal stage for the whole study period and for the specific deployments examined in section 3a are shown in Table 2 (see also Kastner et al. 2019, their Fig. 2).

Table 2.

Conditions over the course of the entire Quinault field campaign and on 30 Apr and 1 May 2017. Significant wave height H_s is calculated from the AWAC, River discharge Q_R is calculated as 150% of the Quinault Lake discharge per information from the Quinault Nation Division of Natural Resources, and tidal stage in meters above lower low water is taken from the USGS Point Grenville tide gauge.

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b. Estimating mixing and stratification in the surf zone

We use drifter measurements of salinity to quantify mixing as the material derivative of salinity, DS/Dt. This can be estimated from a single drifter and is therefore robust to the deployment constraints of the Quinault River mouth. Measurements from SWIFTs have been used in previous surf-zone studies (Zippel and Thomson 2015; Moghimi et al. 2016). Salinity measurements reported in this work are 1-min averages of the 0.5-Hz SWIFT CT data, except as noted.

The material derivative of salinity, DS/Dt, is calculated for each salinity sensor on a SWIFT drifter, either at three depths for the SWIFTv3 (0.1, 0.5, and 1.05 m) or at a single depth for the SWIFTv4 (0.2 m). We apply a third-order median filter to minimize the influence of bubbles, which cause spurious low salinity spikes in the raw data collected at 0.5 Hz, and obtain 1-min salinity averages. We calculate DS/Dt as the slope of a linear best fit to a 1-min time series of salinity data from each SWIFT CT sensor. We assume that an approximation of local linearity is reasonable over the 1-min period. The CT sensor resolution leads to uncertainty in DS/Dt of 1 × 10⁻⁵ psu s⁻¹. We exclude very fresh water, where S < 0.5 psu, from fits. This primarily occurs inside and near the river mouth at the beginning of drift tracks.

The SWIFT buoys are not perfectly Lagrangian tracers. This can be partially quantified by comparing the drift speeds of the two SWIFT versions, the larger 1.05-m-draft SWIFTv3 and the smaller 0.25-m-draft SWIFTv4. The v3 SWIFT is ∼O(10⁻²) m s⁻¹ slower than the smaller-draft v4 SWIFT; this slight discrepancy is similar to previous comparisons of the v3 SWIFT with high-frequency radar measurements of surface currents (Lund et al. 2018). Note that this comparison accounts for the difference in SWIFT draft but cannot account for non-Lagrangian bias in the SWIFTv4 measurements. Multiplying this velocity difference by the cross-shore salinity gradient, ∂S/∂x ∼ O(10⁻²) psu m⁻¹, as observed by the nearshore moorings and drifters, and integrating over the SWIFTv3 draft of 1.05 m yields an overestimate of DS/Dt due to the difference in SWIFT draft of O(10⁻⁴) psu s⁻¹. This analysis establishes a threshold of DS/Dt = 1 × 10⁻⁴ psu s⁻¹, below which DS/Dt estimates may be biased by the non-Lagrangian behavior of the drifters. We therefore exclude values below this threshold from our analysis, which eliminates approximately 1% of DS/Dt estimates.

We can similarly assess the role of vertical advection [w_adv(∂S/∂z)] by calculating the vertical velocity associated with the surfacing of the 4-psu isohaline shown between Figs. 5a and 5c. This isohaline transits 2 m vertically in approximately 2 h, corresponding to a vertical advective velocity of O(10⁻⁴) m s⁻¹. This velocity occurs when ∂S/∂z = 0.5 psu m⁻¹, yielding a possible vertical advective contribution to the material derivative of O(10⁻⁴) psu s⁻¹, the same as estimated from draft and thus not altering the cutoff value chosen for exclusion as outlined above. Moreover, the values of DS/Dt throughout this period from which this vertical advective contribution is estimated are higher than O(10⁻⁴) psu s⁻¹, showing that vertical mixing contributes more significantly than advection to the temporal evolution of salinity.

Previous drifter-based studies of river plumes have shown that drifters tend to be trapped in river plume fronts, as surface-following floats are unable to follow the strong frontal downwelling velocities (McCabe et al. 2008; Kakoulaki et al. 2014; Zippel and Thomson 2017; Kastner et al. 2018). It is impossible to exclude the possibility that fronts are present in the near-field Quinault plume without transect data, which are not feasible for practical and safety reasons. However, we see little evidence of fronts in the surf zone from uncrewed aerial system (UAS) imagery and no evidence that drifters are trapped in front based on analysis of the drifter tracks at the Quinault River mouth (Figs. 1b, 2a, and 3a), or in the drifter behavior described in Kastner et al. (2019). Previous studies where drifters are trapped in river plume fronts show that the drifters trace out plume streamlines as they move away from the river mouth, spreading slightly away from each other; at any given time in these studies, all drifters are roughly the same radial distance from the river mouth, spaced out laterally along the river plume front (McCabe et al. 2008; Kakoulaki et al. 2014; Kastner et al. 2018). We do not observe this drift track pattern in SWIFT deployments at the Quinault River mouth.

To generate SWIFT TKE dissipation rate estimates we use a structure function method to estimate ε in the top 0.5 m of the water column (Thomson 2012; Thomson et al. 2019). Many of the raw Doppler velocity measurements are obscured by the high void fraction in the surf zone and so we apply a multiplicative correction factor of 3 to all dissipation rate estimates following Derakhti et al. (2020). Most SWIFT measurements do not resolve the wave energy flux gradient ∂F/∂x, because the drifters frequently transit as much or more alongshore than cross-shore once they enter the surf zone. Such SWIFT estimates of the cross-shore wave energy flux gradient are therefore contaminated by lateral variability driven by alongshore variability in bathymetry and noninfinite wave crest lengths due to directional spread (Dalrymple 1975). We therefore compare the dissipation rate estimates with a mean value for ∂F/∂x over the course of the SWIFT deployment, where F = Ec_g is determined using the AWAC measurements of incident wave energy spectra E and the group velocity c_g. After applying the void correction factor, depth-averaged values of SWIFT-estimated TKE dissipation rate in the top 0.5 m agree well with depth-averaged bulk estimates of surf-zone dissipation based on the wave energy flux gradient (1/d)∂F/∂x, where d is the water depth (not shown).

For drifts that recirculate (Fig. 3), we are able to assess the agreement between dissipation estimates from the SWIFT HR ADCPs and a calculation of (1/d)∂F/∂x from SWIFT measured wave properties and position. The SWIFT dissipation estimates in the top 0.5 m are roughly one-third as large as (1/d)∂F/∂x, showing reasonable agreement in the limited data available. The bottom CT sensor of the SWIFTv3 is at 1.05-m depth; only about 4% of our measurements occur in the bottom 20% of the water column, where the role of the bottom boundary layer is expected to be important (Feddersen and Trowbridge 2005). We therefore assume that our turbulence measurements are due to surface injection from breaking waves, and we do not examine the role of surf-zone bottom boundary turbulence in mixing the Quinault River plume.

The beach near the Quinault River mouth is dissipative during moderate to high wave forcing conditions, when river water to be trapped in the surf zone. Depth decreases monotonically onshore of the 4-m isobath; therefore, the beach is always dissipative at low water (Fig. 5). Wave breaking does not always occur in the deeper river channel; however, such breaking is required for river water to be trapped in the surf zone, especially at high water (Kastner et al. 2019). Therefore, the beach near the Quinault River mouth is dissipative when river water is trapped in the surf zone, the focus of this study.

We define $\overline{S_t^{\prime 2}}$ as the mean of the squared differences of the median filtered salinity value from the fitted salinity value such that

$\overline{S_t^{\prime 2}}=\frac{1}{f_s{T}_{\text{win}}}{\displaystyle \sum {[S_{\text{fit}}(t)-S(t)]}^2},$(4)

where f_s = 0.5 Hz is the CT sampling frequency, T_win = 60 s is the averaging window, the sigma operator indicates a summation over T_win, S(t) is the median filtered salinity at a time t within the window, and S_fit(t) is the linear salinity fit found in order to calculate DS/Dt, evaluated at the same time t. Propagating error from the SWIFT’s onboard Aanderaa 4319 CT sensor indicates that the error associated with this calculation is ∼1% of the calculated value. The mean square salinity error as defined above represents the scatter around the linear fit DS/Dt. In these calculations, we assume that our 1-min averaging window and salinity thresholds are sufficient to achieve local linearity. In this case, $\overline{S_t^{\prime 2}}$ only represents the scatter around the best fit line. If instrument noise and non-Lagrangian bias are small, then the local turbulent fluctuations dominate $\overline{S_t^{\prime 2}}$, and we expect $\overline{S_t^{\prime 2}}$ to be related to DS/Dt, as shown in Fig. 4b. The strong relationship between DS/Dt and $\overline{S_t^{\prime 2}}$ indicates that DS/Dt is a good estimate of Quinault River plume mixing and is not significantly influenced by changes in salinity driven by plume spreading and shoaling of the plume interface (McCabe et al. 2008; Kastner et al. 2018). We will therefore use DS/Dt as an estimate of mixing throughout this paper.

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Estimates of DS/Dt and $\overline{S_t^{\prime 2}}$ using salinity observations from a SWIFTv3 deployed on 30 Apr 2017: (a) median filtered salinity data and the linear fits used to calculate DS/Dt at each CT sensor depth, as indicated by symbols (all r² > 0.99), and (b) the comparison of $\overline{S_t^{\prime 2}}$ and DS/Dt for each CT sensor depth (indicated by symbols) for all SWIFTv3 1-min averages on 30 Apr 2017.

Citation: Journal of Physical Oceanography 53, 3; 10.1175/JPO-D-21-0286.1

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The mean square salinity error [Eq. (4)] is similar to the spatial salinity variance introduced by MacCready et al. (2018) in that it is a squared salinity deviation, but differs in two key ways. First, the salinity variance is calculated using deviations from the spatial salinity mean, whereas $\overline{S_t^{\prime 2}}$ is calculated using residuals from a fit line. Second, intense mixing is indicated by high $\overline{S_t^{\prime 2}}$, whereas a sharp decrease in spatial salinity variance indicates intense mixing.

a. The structure of a surf-zone-trapped river plume

We show data from two example SWIFT deployments in Figs. 2 and 3, which took place during wave, river discharge, and tidal conditions that were close to average for the 2-week observational period (Table 2). On 1 May 2017, one SWIFTv3 traveled 1750 m through the surf zone over 48 min before beaching south of the river mouth. It reached a maximum speed of 2.35 m s⁻¹ and experienced a maximum wave height of 1 m. Average near-surface TKE dissipation rate estimates ε were of order ∼2 × 10⁻³ W kg⁻¹, and the onboard CT sensors measured a maximum salinity value of 23.3 psu at 1.05 m, concurrent with a salinity of 0.7 psu at 0.1 m (Fig. 2). As the drifter transited through the surf zone, the measured salinity increased continuously, the measured significant wave height and the estimated TKE dissipation rate increased before leveling off, and the drift speed peaked near the river mouth before decreasing to a near-constant value of ∼0.75 m s⁻¹.

On 30 April 2017, one SWIFTv3 and one SWIFTv4 were deployed concurrently in the river mouth. The drifters were deployed at low water and recirculated in a region within ∼500 m south of the river mouth for a period of 5 h before beaching (until almost high water; Fig. 3a). A similar event occurred on 5 May 2017, but the SWIFTs deployed on that day did not have functioning CT sensors after several beaching events. Both of these events occurred during normal wave conditions for our deployment period and at different river discharges (Table 2), suggesting that this recirculation may be a common feature of the surf-zone circulation at the Quinault River mouth. The 30-min-average cross-shore position (normalized with respect to the surf-zone width) of the SWIFTs deployed on 30 April 2017 did not change with tidal stage. SWIFTv3 measurements from 30 April 2017 show that surf-zone salinity increases sharply at first as the drifter enters the surf zone, slowly over the next ∼3 h as it recirculated within the surf zone, and then sharply again at the end of the deployment as it exited the recirculation zone, transited southward, and beached (Fig. 3b). Heave (from the SWIFTv3) and TKE dissipation measurements (from the SWIFTv4) showed a similar pattern to the deployment on 1 May 2017, increasing as the drifter exited the river mouth and entered the surf zone before leveling off in magnitude. TKE dissipation rate is approximately constant in the surf zone, with ε ≈ 10⁻², consistent with previous studies (Figs. 3d,e).

The deployments on 30 April and 1 May 2017 show similar patterns of evolution in 0.5 m depth mixing rate and stratification in their initial 40 min of deployment (Figs. 2f,g and 3f,g). As the drifters leave the river mouth, enter the surf zone, and salinity begins to increase, the mixing rate drops from DS/Dt > 10⁻² psu s⁻¹ near the river mouth to under 10⁻² psu s⁻¹. During the longer deployment on 30 April 2017, the mixing rate increases as the drifters recirculate in the surf zone over 5 h; by the end of the deployment, DS/Dt was larger than its local maximum near the river mouth. Stratification also falls for both deployments during the first 40 min, in both cases by roughly one order of magnitude. Stratification is higher during the 40-min deployment on 1 May than during the initial 40 min of the deployment on 30 April, but increases during the 30 April deployment and reaches values similar to those estimated on 1 May (∼10⁻¹ psu m⁻¹) by the end of the drift track. These increases in mixing rate and stratification on 30 April 2017 occur as tidal stage increased from low water at the beginning of the deployment to high water at the end of the deployment; higher mixing rates and stratification occur at higher tidal stage.

At the Quinault, total volume flux from the river mouth is maximum at low water, and minimum at high water due to volume storage within the estuary during flood tides. We define this volume flux as

$Q_T=u_T{d}_T{b}_T,$(5)

where u_T is the depth-averaged velocity at the river mouth (including both river and tidally driven components), d_T is the tidally varying river mouth depth, and the effective river mouth width b_T ≈ 87 m is defined as in Kastner et al. (2019). A time series of u_T can be found in Kastner et al. (2019), Fig. 2, and values of Q_T during the example drifts shown in Figs. 2 and 3 are shown in Table 2. The discharge at the river mouth is always completely fresh; thus, total freshwater discharge is also tidally modulated, with a minimum at high water and a maximum at low water. The surf zone is also widest and extends farthest offshore at low water (given a constant wave condition) due to the concave beach shape (Kastner et al. 2019). Salinity measurements from 30 April 2017 show that minimum salinity occurs at low water, when the drifters are deployed, and increases with the tidal stage (Figs. 3b and 5). Salinity at the inner shelf moorings also increases with tidal stage during this time period, particularly at the south mooring (Fig. 5). Simultaneously, the mooring salinity difference between 1 and 3 m decreases with tidal stage (Fig. 5). Water at the moorings is always saltier than in the surf zone. Thus, from low to high water on 30 April 2017, the surf zone makes a transition from fresh and unstratified to saltier and stratified. As mixing rate estimates increase with tidal stage coincident with increased stratification, we might expect that variability in vertical eddy diffusivity would be small, or even that vertical eddy diffusivity would be constant [Eqs. (1) and (2)]. This relationship will be examined in section 3c.

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(a),(c),(e) Color scatterplots showing the salinity structure across the surf zone to the inner shelf as measured by the SWIFTv3 and the moorings on 30 Apr 2017 alongside (b),(d),(f) maps showing the location of the drifter measurements shown in the corresponding cross section, colored by the salinity difference between the 1.05- and 0.1-m CT sensors. Time advances, and tidal stage increases, from (a) and (b) to (e) and (f). The solid black line and tan-filled region in (a), (c), and (e) indicate the seabed (note that depth changes as the tidal stage increases), and the vertical dashed black line indicates the surf-zone width, calculated using linear wave energy flux conservation as described in Kastner et al. (2019). The breaker depth increases as the offshore significant wave height increases during the deployment from close to the minimum to close to the maximum values for 30 Apr 2017 (Table 2); this surf-zone widening is also visible in the UAS imagery (not shown). The colored dots inside the surf zone in (a), (c), and (e) indicate all 1-min salinity averages from the SWIFTv3 in the 30-min window centered on the time indicated in the panel title, and the colored dots outside the surf zone indicate the average salinity measured by the moorings within the 30-min window. The mooring data points are shown larger to emphasize that they are 30-min-average salinity values, as opposed to the 1-min-averaged SWIFT data. In (b), (d), and (f), the solid black line indicates the coastline, as in Fig. 1b, and the gray contours indicate bathymetry, with values in meters relative to mean lower low water.

Citation: Journal of Physical Oceanography 53, 3; 10.1175/JPO-D-21-0286.1

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TKE dissipation rate estimates from the SWIFTv4 do not show any dependence of ε over the course of the deployment (Fig. 3e). Together with the changes in DS/Dt observed in Fig. 3f the lack of dependence on ε suggests that DS/Dt increases irrespective of ε; in other words, increases in mixing are not driven by increases in turbulence and instead correspond to an increase in ∂S/∂z. This differs from many river plume studies, which have shown that more intense mixing generally occurs in regions with higher TKE dissipation rate (McCabe et al. 2008; Jurisa et al. 2016; Geyer et al. 2017). We attribute the increase in ∂S/∂z to tidal modulation of the riverine freshwater flux in section 4a. Additionally, we do not see evidence of an increase in TKE dissipation rate with offshore wave height (not shown), indicating that the surf zone at the Quinault River mouth may be saturated by breaking waves. In a saturated surf zone, TKE dissipation rate at any cross-shore location is primarily a function of bathymetry, as depth-limited breaking occurs from the break point to the shoreline on a dissipative beach (Wright et al. 1982; Raubenheimer et al. 1996).

Figure 6 shows the ensemble average of S, u, H, ε, DS/Dt, and ∂S/∂z along a drifter track in 450-m bins over the first 2 km of along-track distance. Salinity is normalized by the mean measured salinity at the nearshore moorings at the time of drifter deployment (or the time-averaged nearshore mooring salinity for drifter deployments when the moorings were not in the water), drift speed is normalized by the near-mouth maximum drift speed, and significant wave height is normalized by the wave height at breaking. These dimensionless quantities follow similar patterns to the along-track examples shown in Figs. 2 and 3 as a drifter leaves the river mouth: salinity increases before leveling off, wave height increases and levels off, and drift speed peaks early in the drift track before decreasing to a near-constant value. TKE dissipation rate increases slightly at the beginning of the drift track before leveling off, while mixing rate decreases continuously along an average drift track and stratification decreases quickly after an initial increase. These initial increases in salinity, wave height, TKE dissipation rate, and stratification, and peak in drift speed are associated with the drifter exiting the river mouth and entering the surf zone. The initial increase in stratification (Fig. 6f) is likely associated with shoaling of the plume interface after liftoff (MacDonald and Geyer 2004).

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Along-track plots of observed and estimated quantities from SWIFT drifters that were trapped in the surf zone: (a) salinity, normalized by the mean measured salinity at the nearshore moorings at the time of drifter deployment (or the averaged nearshore mooring salinity over the entire study period for drifter deployments when the moorings were not in the water); (b) drift speed, normalized by the maximum drift speed reached near the river mouth; (c) significant wave height, normalized by the height at breaking; (d) vertical averages in the top 0.5 m of structure function estimates of TKE dissipation rate estimates using SWIFT ADCP HR measurements; (e) the material derivative of salinity, estimated as detailed in section 2b; and (f) the stratification in the top 1.05 m from SWIFTv3s. Measurements from all SWIFTs are shown in (a)–(e); only measurements from SWIFTv3s are shown in (f). Blue points show 1-min-average quantities, and black open circles show ensemble averages of all observations and estimates of trapped SWIFTs in 400-m bins, as described in section 3a. The gray-shaded region shows 1 standard deviation away from the mean in the same bins.

Citation: Journal of Physical Oceanography 53, 3; 10.1175/JPO-D-21-0286.1

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The along-track evolution of plume structure shown by the ensemble averages in Fig. 6 contrasts drifter observations of plume structure in the Columbia and Fraser River plumes outlined in McCabe et al. (2008) and Kastner et al. (2018), respectively. These plumes (and drift tracks) have a much larger spatial footprint than the Quinault River plume, with drift tracks extending for 10–15 km at the Fraser and ∼35 km at the Columbia, but also have a larger discharge (Q_R ∼ 950 m³ s⁻¹for Fraser, and Q_R ∼ 7000 m³ s⁻¹ for Columbia) and do not encounter surf-zone wave breaking. In both cases drifters were deployed seaward of plume liftoff; salinity tends to increase sharply at the beginning of the drift tracks before leveling off farther away from the river mouth, while velocity tends to behave inversely, decreasing sharply at the beginning of the drift tracks before leveling off.

Estimates of salt flux and stratification also differ from the Quinault versus the Columbia and Fraser river plumes. Vertical salt fluxes, calculated using a control volume method in both McCabe et al. (2008) and Kastner et al. (2018), are higher near the river mouth and decrease farther afield. A similar pattern is seen in stratification; these behaviors are analogous to the behavior of the mixing rates shown in Figs. 2, 3e, and 6e. McCabe et al. (2008) use Eq. (2) to calculate eddy diffusivity, finding that the decrease in vertical salt flux along a drift track outpaces the decrease in stratification, resulting in a decline in eddy diffusivity. TKE dissipation rate estimates from the Columbia River plume also decline offshore due to stratification suppressing turbulence (Kilcher et al. 2012; Jurisa et al. 2016), unlike the estimates presented in Figs. 2, 3d, and 6d, which show no trend in along-track distance. This lack of trend is likely due to the fact that the breaking wave generated turbulence in the surf zone is not related to the river volume flux, which controls stratification. This is a significant difference from larger plumes such as the Columbia where the two are linked and the combined tidal and river discharge is the primary source of turbulence and buoyancy (Horner-Devine et al. 2015; Jurisa et al. 2016).

Care must be taken in interpreting the along-track TKE dissipation rate, as we would expect a cross-shore gradient of TKE dissipation rate in the surf zone, with stronger dissipation toward the break point. The drifters mostly transit alongshore during any given deployment (Figs. 1b and 2), and so the ensemble-averaged TKE dissipation rate shown in Fig. 6d does not explicitly include cross-shore variability. The recirculating drifts shown in Figs. 3 and 5 allow us an opportunity to assess the impact of this cross-shore variability in TKE dissipation rate on DS/Dt; we do not observe a significant dependence of mixing rate on TKE dissipation rate (time series of each quantity shown in Fig. 3, direct dependence not shown). While we cannot rule out a dependence of mixing rate on TKE dissipation rate more generally, we find it more likely that stratification is instead a limiting factor on mixing, as described in more detail in sections 3c and 4c.

b. Cross-shore structure

High wave forcing (offshore H_S up to 2.3 m; Table 2) at the Quinault River mouth results in river water being mostly trapped in the surf zone during our observational period (Kastner et al. 2019). Salinity measurements from all surf-zone-trapped SWIFT drifters over the course of the entire study period show that the surf zone often remain fresh (S < 10 psu; 83% of measurements), whereas salinity measurements from the moorings just outside of the surf zone are significantly higher (S > 25 psu; 91% of measurements) (Fig. 7a). At ∼1-m depth, the surf-zone SWIFTv3 salinity measurements are frequently more than 10 psu fresher than the inner shelf mooring measurements. Salinity does not vary significantly in the cross-shore direction within the surf zone, resulting in a large horizontal cross-shore salinity change near the surf-zone edge when the surf-zone salinity is low (Fig. 7). Although we observe a large salinity difference between the surf zone and inner shelf, there is no evidence of trapping in the drifter tracks as described in section 2b. We conclude that the intensity of mixing in this system prohibits the generation of a strong, convergent front.

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Cross-shore profiles of quantities measured from trapped SWIFT drifters and nearshore moorings at alongshore positions less than 1500 m from the river mouth: (a) Cross-shore bin average of salinity with a bin size of 100 m; the shaded area represents the standard deviation of salinity in each bin. The deployment-average salinity for the nearshore moorings is also shown, with error bars indicating standard deviation. Salinity measurements from all surf-zone-trapped SWIFT drifters within the alongshore spatial constraint detailed above are included (see Table 1 for measurement depths). There are two points for each mooring, offset in the horizontal for clarity; the fresher point for each mooring is the 1-m CTD, and the saltier point is the 3-m CTD. (b) Cross-shore bin average of stratification with a bin size of 100 m; the shaded area represents the range of stratification in each bin, as well as the deployment-average salinity for the nearshore moorings, along with the mean and range of mid-water-column stratification from the moorings. The mean surf-zone width is given by the black dashed vertical line, with the black dotted lines indicating 1 standard deviation from the mean break point.

Citation: Journal of Physical Oceanography 53, 3; 10.1175/JPO-D-21-0286.1

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Calculations of stratification (section 2b) indicate that the surf zone can be stratified (Fig. 7b). Specifically, the stratification in the upper 1.05 m of the surf zone is slightly higher on average (mean ∼ 1.8 psu m⁻¹) than the offshore stratification between 1 and 3 m (mean ∼ 0.72 psu m⁻¹). This differs from surf-zone studies conducted far from river mouths, which have shown the surf zone to be unstratified (e.g., Hally-Rosendahl and Feddersen 2016). Our results show that this is not necessarily true; in the vicinity of a moderate river discharge the surf zone can sustain levels of stratification similar to those observed in larger plumes. For example, Fig. 6 from Kastner et al. (2018) provides values of near-surface stratification from the Fraser: ∼10 psu m⁻¹ at the liftoff peak and ∼3 psu m⁻¹ in the near-field plume. Thus, the Quinault stratification values of ∼2 psu m⁻¹ are similar in magnitude to the values observed in the near-field Fraser plume. We do not observe any significant cross-shore trend in surf-zone stratification (Fig. 7b). Note that the center of mass of the drifter locations remains roughly in the middle of the surf zone ($\overline{x}/L_{\text{sz}}=0.67,0.48,\text{and}0.5$ for the times shown in Fig. 5), confirming that the observed changes in stratification are not the result of drifters sampling the edges of the surf zone where other processes may influence stratification.

c. Turbulence and mixing in the surf zone

We consider two surf-zone-mixing models based on Eq. (3); the simpler version assumes a constant value of vertical eddy diffusivity, K_z. To assess the validity of this simple model, we calculate an average value of the vertical eddy diffusivity based on all the available mixing rate and stratification measurements. We first bin average the vertically integrated mixing rate in logarithmic space stratification bins of width 0.2 (i.e., bin edges of 10⁻¹, 10^−0.8 psu m⁻¹, etc.). The 1-min mixing rate and stratification products shown in Fig. 8a are significantly scattered from the bin averages. Allowing for a nonzero y intercept in the fit of vertically integrated mixing rate to stratification, Eq. (3) becomes

${\displaystyle \int \frac{DS}{Dt}}dz=K_z\frac{\partial S}{\partial z}+\mathcal{C},$(6)

where $\mathcal{C}$ represents background mixing (perhaps due to lateral processes) as a constant value. We fit the relationship between stratification and vertically integrated mixing rate in four main ways: first, using all 1-min SWIFT estimates shown in Fig. 8a and allowing a nonzero y intercept as show in Eq. (6) (red line in Fig. 8, method I in Table 3), second, using the bin-averaged vertically integrated mixing rate and stratification and allowing a nonzero y intercept as show in Eq. (6) (gold line on Fig. 8a; method II in Table 3), third, using all 1-min SWIFT estimates shown in Fig. 8a and forcing a y intercept of zero as show in Eq. (3) (not shown in Fig. 8a; method III in Table 3), and finally using the bin-averaged vertically integrated mixing rate and stratification and forcing a y intercept of zero as show in Eq. (6) (not shown in Fig. 8a; method IV in Table 3). For fits to bin averages, we do not fit bins that contain less than 5% of all data. We tested to see whether the scatter in the data about these fits is dependent on wave, river, or tidal forcing but found no strong relationship for any of these forcing parameters (not shown).

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We calculate a single, representative eddy diffusivity for the entire study period and show that eddy diffusivity can vary with stratification: (a) A demonstration of how eddy diffusivity can be estimated by the relationship between vertically integrated Lagrangian mixing rate on the y axis and the stratification on the x axis. SWIFTv3 data are shown as blue data points, and bin averages of vertical salt flux in stratification bins are shown as black squares. Ensemble-averaged values from Fig. 6 (i.e., binned in distance offshore) are shown as open black circles. Linear fits with a 68% confidence interval are shown with lines and a shaded area to represent the confidence interval. The solid red line indicates a fit to all estimates shown on this figure (blue dots); the dashed gold line indicates a fit to the bin averages (black squares). The slopes of these linear fits are estimates of the eddy diffusivity; the y intercept of the fits is representative of background mixing processes (Table 3). (b) The eddy diffusivity obtained by simply dividing the vertical salt flux and stratification values of the bin averages (black squares) in (a); the decreasing trend of this eddy diffusivity estimate with increasing vertical salt flux is well described by a log-linear power-law relationship of the form K_z = a(∂S^b/∂z), where a = 0.005 and b = −0.4 ± 0.18 m³ psu⁻¹ s⁻¹ (fitting is done in log space; 95% confidence interval of a: 0.004 < a < 0.0065).

Citation: Journal of Physical Oceanography 53, 3; 10.1175/JPO-D-21-0286.1

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Table 3.

Eddy diffusivity and background mixing linear fitting parameters from different fitting methods, as specified in section 3c. Method I fits all 1-min SWIFT estimates of vertically integrated mixing rate and stratification using Eq. (6); method II fits bin averages of these same quantities using Eq. (6); method III fits all estimates of these quantities using Eq. (3), which has a y intercept of zero; method IV fits the bin averages using Eq. (3), which has a y intercept of zero. The p values shown in this table in parentheses are calculated using the robust standard error method, which does not require data to be homoscedastic (White 1980).

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Fitting all data shown in Fig. 8a yields a vertical eddy diffusivity K_z = (2.1 ± 0.55) × 10⁻³ m² s⁻¹ (p < 1 × 10⁻⁴), with $\mathcal{C}$ = (5.8 ± 2.8) × 10⁻³ psu m s⁻¹ (p < 1 × 10⁻⁴) (Fig. 8a; method I on Table 3). The analysis results in a modest correlation (R² = 0.27; Fig. 8a) but yields a statistically significant linear fit and supports the conclusion that near-surface vertical mixing in the surf zone can be reasonably parameterized by a constant eddy diffusivity (statistics calculated in linear space). The data presented in Fig. 8a are heteroscedastic in linear space because the variance of salt flux increases as stratification increases. Therefore, the p values reported above and in Table 3 are calculated using the robust standard error method, which does not require data to be homoscedastic (White 1980). Note that in Fig. 8 error bars indicate 1 standard deviation from the mean to facilitate log-scale plotting; the uncertainty intervals in the text above, Table 3, and the figure caption are 95% confidence intervals obtained from fitting. Fitting the bin averages (method II on Table 3) yields similar values of K_Z and $\mathcal{C}$ as described above from method I. While this method is less statistically significant than method I because of the reduced sample size, we will use the bin-average calculations of vertically integrated mixing rate and stratification in section 4a and so include these fitting parameters for completeness. Using Eq. (3) to fit all the data shown in Fig. 8a (i.e., forcing the y intercept through zero; methods III and IV in Table 3), yields a slightly higher eddy diffusivity than found using Eq. (6) by methods I and II. This could be due to mixing caused by lateral processes being erroneously included as vertical mixing (e.g., Hally-Rosendahl et al. 2014; see section 4a). We do not observe strong trends in the residual of the fit calculated using Eq. (6) with either cross-shore or alongshore position (not shown).

The range of stratification for the study spans from O(10⁻¹) to O(10¹) psu m⁻¹, while the binned near-surface vertical salt flux spans a smaller range from O(10⁻³) to O(10⁻²) psu m s⁻¹. We therefore investigate whether the large range in stratification results in different values of eddy diffusivity. We take the vertically integrated mixing rate to be equal to the vertical salt flux (as described in section 2b) and calculate a variable eddy diffusivity by dividing bin averages of this vertically integrated mixing rate by stratification, yielding a range of 1.4 × 10⁻³ < K_z <1 × 10⁻² m² s⁻¹. The bin-averaged eddy diffusivity displays a strong dependence on stratification; eddy diffusivity is highest where stratification is low (Fig. 8b). A power law fit (linear in logarithmic space) represents the decrease in eddy diffusivity with increasing stratification well (r² = 0.78; p = 1 × 10⁻³). This fit takes the form K_z = a(∂S^b/∂z), where a = 0.005 and b = −0.4 ± 0.18 m³ psu⁻¹ s⁻¹ (fitting done in log space; 95% confidence interval of a: 0.004 < a < 0.0065). We note that there is autocorrelation in this fit, as K_z ∼ a(∂S⁻¹/∂z); however, the 95% confidence interval of the exponent b does not include −1, so the result presented above is robust to the autocorrelation.

The higher correlation and significance of the relationship between binned stratification and eddy diffusivity than the relationship between binned stratification and salt flux from Eq. (6) (gold line in Fig. 8a; method II in Table 3), which has the same number of fitting parameters, suggests that there may be limits on the effectiveness of a constant eddy diffusivity to model mixing in this system. Vertical mixing of a river plume in the surf zone may therefore be described by both a constant eddy diffusivity K_Z ≈ 2 × 10⁻³ m² s⁻¹ and a stratification-dependent eddy diffusivity, 1.4 × 10⁻³ < K_z < 1 × 10⁻² m² s⁻¹. While incorporating variation with stratification improves the overall fit, both methods are statistically significant, suggesting that using a constant K_Z (which may be easier to apply in some situations) would be acceptable. This is further explored in section 4. The values of K_z reported here fall within the range of values of eddy diffusivity observed previously in river plumes [typically from O(10⁻⁴) to O(10⁻²) m² s⁻¹] (MacDonald and Geyer 2004; McCabe et al. 2008). These are the first estimates of the vertical eddy diffusivity of salt in the surf zone, where prior studies in the absence of a river plume have shown lateral dispersion to be dominant (Clark et al. 2010; Spydell and Feddersen 2012; Hally-Rosendahl et al. 2014).

a. Tidal dependence of stratification and mixing

While a constant value of eddy diffusivity K_z statistically significantly models the mixing of the Quinault plume in the surf zone, calculations of eddy diffusivity using bin-averaged values of vertical salt flux and stratification have a range of one order of magnitude (section 3c; Fig. 8b). To determine the source of this variability, we examine tidal influence on river volume flux [Q_T in Eq. (5)] and surf-zone stratification (Figs. 9a,b). Of relevance here is that Q_T is a maximum at low water and a minimum at high water because of estuarine storage.

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River volume flux, stratification, and salt flux depend on tidal stage η, whereas TKE dissipation rate does not: (a) volume flux at the mouth Q_T normalized by upstream river discharge Q_R, decreasing with tidal stage η, calculated as described in Eq. (5) and Kastner et al. (2019); (b) stratification ∂S/∂z, increasing with tidal stage (averages of SWIFTv3 1-min estimates); (c) salt flux $\int (DS/Dt)dz$, increasing with tidal stage (averages of SWIFTv3 1-min estimates); and (d) TKE dissipation rate ε, with no clear trend with respect to tidal stage (averages of SWIFTv3 and SWIFTv4 1-min estimates). Black squares are bin averages in 0.5-m tidal stage bins of the relevant quantity, with an error bar showing the standard deviation. Statistics are calculated in linear space for volume flux [in (a)] and log space for all other quantities [(b)–(d)].

Citation: Journal of Physical Oceanography 53, 3; 10.1175/JPO-D-21-0286.1

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