Mixing of the Connecticut River Plume During Ambient Flood Tides: Spatial Heterogeneity and Contributions of Bottom-Generated and Interfacial Mixing

Abstract

The Connecticut River plume is influenced by energetic ambient tides in the Long Island Sound receiving waters. The objectives of this modeling study are (a) characterizing the spatial heterogeneity of turbulent buoyancy fluxes, (b) partitioning turbulent buoyancy fluxes into bottom-generated and interfacial shear contributions, and (c) quantifying contributions to plume-integrated mixing within the tidal plume. The plume formed during ambient flood tides under low river discharge, spring tides, and no winds is analyzed. Turbulent buoyancy fluxes (B) and depth-integrated B through the plume (Bd) are characterized by pronounced spatial heterogeneity. Strong mixing (Bd ∼ 10⁻⁵-10⁻⁴ m³/s³) occurs near the mouth, in the nearfield plume turning region, over shoals, and nearshore shallow areas. Low to moderate mixing (Bd ∼ 10⁻⁸-10⁻⁶ m³/s³) occupies half the plume. Buoyancy fluxes are first partitioned based on the depth of the shear stress minimum between plume-generated and bottom-generated shear maxima. Four other tested partitioning methods are based on open channel flow and stratified shear flow parameterizations. Interfacial and bottom-generated shear contribute to different areas of intense and moderate mixing. All methods indicate a significant plume mixing role for bottom-generated mixing, but interfacial mixing is a bigger contributor. Plume-integrated total and interfacial mixing peak at max ambient flood and the timing of peak bottom-generated mixing varies among partitioning methods. Two-thirds of the mixing occurs in concentrated intense mixing areas. A parameter space with the ambient tidal Froude number and plume thickness to depth ratio as axes indicates many tidally modulated plumes are moderately to dominantly influenced by bottom-generated tidal mixing.

Key Points

• Turbulent buoyancy fluxes span several orders of magnitude over the river plume

• Plume-generated interfacial shear is the most important mixing contributor overall

• Bottom-generated mixing importance increases with tidal currents and plume thickness

Plain Language Summary

A plume of freshwater flows from the Connecticut River into Long Island Sound. Strong tidal currents influence the plume. How freshwater river plumes mix is an important area of research. This study explores where mixing occurs in the plume and which processes contribute to the mixing. A realistic computer simulation of the Connecticut River plume is analyzed. The strongest mixing areas are near the river mouth, in an area beyond the mouth where the plume turns, and over shallow waters. Mixing created by faster plume speeds relative to the surrounding waters is the most important contributor. Mixing driven by tidal currents interacting with the stationary bottom is a significant secondary contributor. The importance of this tidal contribution increases for faster tidal flow and plumes reaching closer to the bottom.

1 Introduction

River plumes carry freshwater, terrestrial materials, nutrients, and contaminants into the marine environment (Meybeck, 2003). The mixing of plume waters and the materials they transport occurs through many processes at fronts and in nearfield, midfield, and farfield plume regions (Horner-Devine et al., 2015). Leading-edge fronts are characterized by vigorous mixing associated with convergence, downwelling, and convection (e.g., Delatolas et al., 2023; Garvine & Monk, 1974; Kilcher & Nash, 2010; Luketina & Imberger, 1987; Marmorino & Trump, 2000). In the nearfield (relatively close to the mouth), interfacial mixing at the plume base driven by plume-generated shear is a major source of mixing (e.g., Kilcher et al., 2012; MacDonald et al., 2007). Stratified-shear flow instabilities including Kelvin-Helmholtz (K-H) and Holmboe instabilities drive interfacial mixing (e.g., Ayouche et al., 2022; MacDonald et al., 2013; Smyth & Moum, 2000; Thorpe, 1973). Tides are a powerful mixing agent in coastal waters that can influence plumes. Bottom-generated mixing, associated with tidal bottom stress, within macrotidal estuaries is important in setting plume outflow conditions at the mouth (Nash et al., 2009). Tides also can control source conditions by forming ebb-pulsed plumes (e.g., Cole et al., 2020; Cudaback & Jay, 1996; Horner-Devine et al., 2009; MacDonald et al., 2007; Pritchard & Huntley, 2006). Some ebb-pulsed plumes enter ambient waters with strong tidal currents (e.g., Basdurak et al., 2020; Bricker et al., 2006; Garvine, 1974; Hessner et al., 2001; O’Donnell et al., 1998; van Alphen et al., 1988; Warrick & Stevens, 2011; Whitney et al., 2021). In such plumes, the nearfield region also is referred to as the tidal plume and is strongly time varying (Horner-Devine et al., 2015).

Within idealized tidally influenced river plumes, Spicer et al. (2021) showed that bottom-generated shear associated with tides can contribute to vertical turbulent buoyancy fluxes (B) to a degree comparable to interfacial mixing. This is a consequential result since plume mixing studies have focused on frontal and interfacial mixing, while bottom-generated mixing in plumes remains understudied. The study also finds frontal mixing is less important to plume-integrated B relative to mixing throughout the interior of the simulated idealized plumes (Spicer et al., 2021). The present study is motivated by the Spicer et al. (2021) mixing analysis and builds upon it by applying it to a realistic plume. Particular attention is paid to the spatial heterogeneity of B and the role of bottom-generated shear relative to interfacial shear in mixing the plume. The study investigates mixing of the Connecticut River plume and builds on the modeling foundation of Whitney et al. (2021).

The Connecticut River plume is an example of a plume influenced by strong ambient tides. The Connecticut River flows into Long Island Sound (LIS, Figure 1a) with an average annual discharge (from 2009 to 2019) of 600 m³/s; the average monthly discharge ranges from 1,294 to 304 m³/s in April and September, respectively (USGS 01193050, Middle Haddam, CT). Summers are characterized by low discharge (e.g., Jia & Whitney, 2019; Whitney et al., 2016). The Connecticut River has a tidal salt-wedge estuary that typically extends 5–15 km from the mouth (Garvine, 1975; Meade, 1966; Ralston et al., 2017). Two jetties (known as the Saybrook Jetty) bound the navigational channel and extend beyond the mouth (Figure 1b); the 1 km wide mouth also has wide shoals adjacent to the channel. Plume waters enter into the ambient macrotidal environment of eastern LIS, with strong (∼1 m/s) along-shore tidal currents (Bennett et al., 2010; O’Donnell et al., 2014). The basin-wide subtidal exchange flow is fueled by buoyancy introduced by the Connecticut River and other smaller rivers; exchange flow magnitudes are an order of magnitude smaller than tidal currents (Whitney et al., 2016). Plume waters are tidally advected eastward during ambient ebbs and westward during flood tides (Garvine, 1974; Garvine & Monk, 1974). Depth-averaged tidal current amplitudes in the plume region vary from approximately 0.8 to 1.3 m/s from neap to strongest spring tides (Whitney et al., 2021). The plume flows over complex bathymetry including inshore shallow areas, Crane Reef and Hen and Chickens shoals, Long Sand Shoal farther offshore, and deeper areas in between. Winds can influence LIS circulation patterns; events with an eastward wind component are twice as frequent as westward events (Whitney & Codiga, 2011). Wind events can occur in any season, but strong storms are most frequent during winter while summers are characterized by low-wind conditions (e.g., Whitney & Codiga, 2011; Whitney et al., 2016; Jia & Whitney, 2019).

Figure 1

(a) Location map showing the Connecticut River, Long Island Sound (LIS), Block Island Sound (BIS), and the continental shelf. The outer (dashed red), intermediate (dashed red), and inner highest-resolution (solid red) grid domains. (b) Bathymetry in the inset region demarcated with the blue rectangle in panel (a) The 5, 10, and 15 m isobaths are contoured with dashed, solid, and dashed-dot brown lines. The defined river mouth, the Saybrook Jetties, Cornfield Point, Long Sand Shoals, Hen and Chicken shoals, and Crane Reef are labeled. The jetties show as thin white lines and are land points in the model.

As in many river plumes, observations of the Connecticut plume indicate turbulent dissipation rates (ε) are high at the narrow (∼10 m wide) frontal region and decrease to orders of magnitude lower farther into the plume interior. O’Donnell et al. (2008) observed ε ∼ 10⁻³ m²/s³ at the front and an exponential decrease (with 15 m decay scale) to ε ∼ 10⁻⁶ m²/s³ for 100 m into the plume interior. Delatolas et al. (2023) observed similar dissipation rates at the front, ε ∼ 10⁻⁴-10⁻⁵ m²/s³ in a transitional zone (10–100 m from the front), and ε ∼ 10⁻⁵-10⁻⁷ m²/s³ farther (100–400 m) into the plume. Applying the relationship B = Γε with a constant mixing efficiency (Γ = 0.2; Ivey & Imberger, 1991; Gregg et al., 2018) suggests B is of order 10⁻⁴, 10⁻⁵-10⁻⁶, and 10⁻⁶-10⁻⁸ m²/s³ at the front, in the transition zone, and in the plume interior, respectively. These observations, however, sample only limited locations along the plume front and measure conditions in an exceedingly small portion of the plume interior. Nevertheless, the Delatolas et al. (2023) observations point to at least two orders of magnitude variability in mixing rates within the plume interior. As pointed out in Spicer et al. (2021), the interior area of tidally pulsed plumes is many times larger than the frontal area so interior turbulent buoyancy fluxes can account for most of the mixing in plume-integrated budgets, even though interior mixing rates are lower than at the front. Whitney et al. (2021) simulated the Connecticut River plume and analyzed volume-averaged tracer budgets to assess the relative importance of mouth inputs, entrainment, and mixing. The model results indicate entrainment and mixing time scales are short enough to rapidly diminish tracer variance (and the buoyancy signature) within half a tidal period (Whitney et al., 2021). New mixing analysis is required to investigate spatial variability in mixing rates and assess the relative importance of mixing processes.

This study focuses on the Connecticut River plume that forms during ambient flood tides, as in Whitney et al. (2021). The plume during low river discharge, spring tides, and no winds is analyzed since these conditions should yield the upper-bound for bottom-generated shear-driven mixing. These conditions are considered because the contribution of bottom-generated shear to plume mixing is understudied relative to frontal and interfacial mixing. The low-discharge, spring-tide plume has a smaller buoyancy signature and shorter mixing time scale than cases with high-discharge and/or neap tides (Whitney et al., 2021). Situations with low discharge and light or no winds occur often during summers. The main objectives are (a) characterizing the spatial heterogeneity of turbulent buoyancy fluxes among different areas of the plume, (b) partitioning turbulent buoyancy fluxes into bottom-generated and interfacial shear contributions and rating their relative importance throughout the plume, and (c) quantifying contributions to plume-integrated mixing as the tidally influenced plume evolves. The high-resolution numerical modeling approach in Whitney et al. (2021) is used. This study is a companion to similar analysis for idealized river plumes in Spicer et al. (2021). The buoyancy flux partitioning based on shear stress (Arevalo et al., 2022; Ralston et al., 2010; Spicer et al., 2021) is applied and compared to alternative partitioning methods based on parameterizations for open channel flow (e.g., Nezu, 2005; Voulgaris & Trowbridge, 1998) and stratified shear flow in plumes (e.g., Kunze et al., 1990; MacDonald & Chen, 2012; MacDonald et al., 2007). Robust results evident across partitioning approaches are emphasized. Results are discussed in context of other plumes. A parameter space is developed to identify situations where bottom-generated shear is most likely to be important for mixing plumes.

2 Materials and Methods

2.1 Model Setup

Modeling methods exactly follow Whitney et al. (2021); complete details of model setup are included in that publication. As a brief summary, the hydrostatic Regional Ocean Modeling System (ROMS, Haidvogel et al., 2000; Haidvogel et al., 2008) is applied to model the Connecticut River plume and surrounding waters with third-order upstream for 3D momentum, fourth-order centered for 2D (depth-averaged) and vertical momentum, and the Wu and Zhu (2010) third-order scheme for salinity, temperature, and passive tracers. Vertical turbulent viscosity and diffusivity are parameterized with the generic length scale method k-ε closure scheme. The outer (mother) grid domain includes the tidal Connecticut River, LIS, and adjacent coastal waters at 500 m maximum horizontal resolution with 30 sigma levels evenly distributed through the water column. The outer model is forced along the open ocean boundaries with eight tidal coefficients, subtidal velocities, temperatures, and salinities. Surface forcing includes winds, humidity, air temperature, surface pressure, net shortwave radiation, downward longwave radiation, and bulk flux formulae (Fairall et al., 2003) for surface fluxes of momentum and sensible, latent, and longwave heat. The nested intermediate grid has 100 m horizontal resolution. The inner grid covers the lower Connecticut River estuary and tidal plume region with 20 m resolution (Figure 1a). The nested runs are initialized with a 2.5-yearlong mother grid simulation and are evolved for 1.5 days to capture the intratidal evolution of the plume. Light wind periods are selected and then wind stress is set to zero to remove all immediate wind influence during the short nested runs. The low-discharge and spring-tide case from Whitney et al. (2021) is analyzed; discharge is 250 m³/s and ambient tidal currents in the plume region are 1.0 m/s. ROMS source code, compilation files, and input files necessary to run ROMS and generate the ROMS output are included in the Whitney (2023) data set as described in the Open Research statement.

Instantaneous fields (in ROMS ‘history’ files) output at 30-min intervals are analyzed. As in Whitney et al. (2021), results are presented in a rotated coordinate system with x directed approximately westward alongshore, y directed approximately southward offshore, and z positive upwards from the static surface. Analysis relies on model bottom depth (h), surface elevation (η, positive when above z = 0), salinity (Sal, measured on the Practical Salinity Scale), temperature (Temp), freshwater concentration (C), horizontal velocity components (u and v in the x and y directions, respectively), bottom stress components (τ_bx and τ_by, with magnitude τ_b) eddy viscosity for momentum (K_m), and eddy diffusivities (K_Sal and K_Temp for salinity and temperature, respectively). The freshwater concentration is tracked with a conservative passive tracer (evolved within ROMS) that is normalized such that C = 1 indicates pure Connecticut River water and C represents the freshwater fraction associated with recent Connecticut River inputs. The continuous tracer is released at the up-river boundary of the inner grid (4 km up-river from the mouth) with the same volume flux and freshwater fraction as the Connecticut River waters flowing through the boundary. There are no other passive tracer sources and no sinks. Note that the hourly tracers analyzed in Whitney et al. (2021) to assess plume connectivity sum to the continuous tracer analyzed here. All analysis is completed with MATLAB using standard functions and routines in addition to custom routines. The custom routines, MATLAB data files, and all data included in figures are in the Whitney (2023) data set as described in the Open Research statement. The main analysis methods are described below.

2.2 Buoyancy Flux Analysis

The analysis focuses on vertical turbulent buoyancy fluxes in the river plume. Three-dimensional (3D) time-varying B fields are calculated based on the eddy diffusivity model according to Equation 1 as in Spicer et al. (2021) and many other studies; g is gravitational acceleration, ρ is potential density, ρ_o = 1,025 kg/m³ is the reference density, and K_ρ is the eddy diffusivity for density.

$$$B=-\frac{g}{{\rho }_{o}}{K}_{\rho }\frac{\partial \rho }{\partial z}$$$ (1)

Note that Equation 1 also can be expressed as B = K_ρN², where N² is the buoyancy frequency squared. In the ROMS run, K_Temp and K_Sal are the same and K_ρ is set equal to them. Potential density is calculated from model temperature and salinity fields with the nonlinear equation of state for seawater (UNESCO, 1981). B values are calculated from 3D time-varying fields of K_ρ and $$\frac{\partial \rho }{\partial z}$$ at the same vertical levels as vertical velocity (w) in the model. B is set to zero at the surface and bottom, reflecting no buoyancy sources or sinks at these boundaries. B values are positive for the stable stratification associated with buoyant plumes (i.e., B > 0 reflects a downward flux of buoyancy due to turbulent mixing). The focus on shear-driven mixing requires analysis involving shear squared (S²) and the associated shear stress magnitude (τ) as defined in (2) and (3).

$$${S}^{2}={\left(\frac{\partial u}{\partial z}\right)}^{2}+{\left(\frac{\partial v}{\partial z}\right)}^{2}$$$ (2)

$$$\tau ={\rho }_{o}\sqrt{{\left({K}_{m}\frac{\partial u}{\partial z}\right)}^{2}+{\left({K}_{m}\frac{\partial v}{\partial z}\right)}^{2}}={\rho }_{o}{K}_{m}\sqrt{{S}^{2}}$$$ (3)

S² and values are calculated from 3D time-varying fields of K_m, $$\frac{\partial u}{\partial z}$$, and $$\frac{\partial v}{\partial z}$$ at the same vertical levels as w in the model. The shear stress magnitude is set to zero at the surface (since wind stress is zero) and to τ_b at the bottom. It is also worthwhile to refer to the turbulent shear production (Ps) as in Equation 4 (e.g., Reichl et al., 2022).

$$$Ps={K}_{m}{S}^{2}$$$ (4)

Ps is calculated from 3D time-varying fields of K_m and S² at the same locations. It is useful to consider the local turbulent kinetic energy (TKE) balance Ps-B = ε for approximation purposes (B > 0 indicates buoyancy destruction of turbulence, a TKE sink). The mixing efficiency can be applied to alternately express the local TKE balance as Ps=(1 + Γ)ε or Ps=(1 + 1/Γ)B.

Boundaries, both horizontal and vertical, for the active tidal river plume are identified based on freshwater tracer concentrations. In this study, the boundary threshold concentration is set at C_threshold = 0.01 (corresponding to a 1% freshwater fraction); this threshold concentration is similar to the threshold in Spicer et al. (2021). The selected threshold is four orders of magnitude higher than in Whitney et al. (2021), where the threshold was set extremely low to facilitate analysis of plume-averaged tracer variance budgets. For an example situation with a 30 salinity ambient endmember, a 1% freshwater fraction is equivalent to a 0.2 kg/m³ density anomaly. The depth of the plume base (d_p) is defined as the depth range (from the surface downward) where freshwater tracer concentrations meet or exceed the threshold (C ≥ C_threshold), as in Spicer et al. (2021). Note that d_p is calculated locally for the water column at each grid location, is spatially and temporally variable, and varies between 0 (where tidal plume waters are not present) and η+h (where the plume extends to the bottom). The horizontal surface area of the plume (A_p) is the region beyond the river mouth within the plume (bounded by C ≥ C_threshold) at the surface (marked on Figure 1b).

The bulk buoyancy frequency squared and bulk shear squared are averaged over the plume layer and the bulk Richardson number (Ri_b) is calculated as their ratio in Equation 5.

$$${N}_{b}^{2}=\frac{1}{{d}_{p}}\underset{-{d}_{p}{}}{\overset{\eta }{\int }}{N}^{2}dz{S}_{b}^{2}=\frac{1}{{d}_{p}}\underset{-{d}_{p}{}}{\overset{\eta }{\int }}{S}^{2}dz{Ri}_{b}=\frac{{N}_{b}^{2}}{{S}_{b}^{2}}$$$ (5)

The corresponding scales for the plume reduced gravity scale (g′) and the velocity difference between the plume and underlying ambient waters (Δu) are set with Equation 6.

$$${g}^{\prime }={N}_{b}^{2}{d}_{p}{\increment}u=\sqrt{{S}_{b}^{2}}{d}_{p}$$$ (6)

The vertically integrated turbulent buoyancy flux within the plume (Bd) is calculated with Equation 7.

$$$Bd=\underset{-{d}_{p}{}}{\overset{\eta }{\int }}Bdz$$$ (7)

Bd units are m³/s³ and multiplying Bd by ρ_o yields power per unit area in W/m². Bd fields indicate the horizontal heterogeneity of mixing within the plume. The total mixing power associated with the plume-integrated buoyancy flux (M) is calculated with Equation 8, as in Spicer et al. (2021). The M units are W.

$$$M={\rho }_{o}\underset{{A}_{p}}{\int }BddA$$$ (8)

2.3 Buoyancy Flux Partitioning

Partitioning in this study splits turbulent buoyancy fluxes into interfacial (IF) and bottom-generated (BG) parts such that B = B_IF + B_BG, Bd = Bd_IF + Bd_BG, and M = M_IF + M_BG. Since tides generate much of the bottom stress, Spicer et al. (2021) refer to the bottom-generated part as tidal and their “M_T” is synonymous with M_BG here. In this study, several partitioning methods are applied (Table 1) for comparison to the standard method in Spicer et al. (2021) and to identify the most robust patterns in the relative contributions of interfacial and bottom-generated shear to plume mixing.

Table 1. Interfacial (IF) and Bottom-Generated (BG) Partitioning Methods With Terminology and Contributions to Plume-Integrated Mixing Near Max Ambient Flood (M = 121 kW at t = 3 hr)

Partitioning method	BdIF names	BdBG names	MIF (10^3 W)	MBG (10^3 W)>	MIF/M (%)	MBG/M (%)
Standard shear-stress minimum depth^a^, ^b^, ^c	BdIFdτ	BdBGdτ	88	33	73	27
Standard open channel flow^d^, ^e	BdIFocf	BdBGocf	104	16	87	13
Universal open channel flow^f^, ^g^, ^h	BdIFuni	BdBGuni	99	22	82	18
Stratified shear flow with spreading^i^, ^j^, ^k	BdIFssf	BdBGssf	71	49	59	41
Alternate kwb stratified shear flow^l^, ^m	BdIFkwb	BdBGkwb	105	15	87	12

• ^a Ralston et al., 2010.
• ^b Spicer et al., 2021.
• ^c Arevalo et al., 2022.
• ^d Voulgaris and Trowbridge, 1998.
• ^e Orton et al., 2010.
• ^f Nezu, 1977.
• ^g Nezu, 2005.
• ^h Johnson and Cowen, 2017.
• ⁱ MacDonald and Geyer, 2004.
• ^j MacDonald et al., 2007.
• ^k MacDonald and Chen, 2012.
• ^l Kunze et al., 1990.
• ^m Jurisa et al., 2016.

A standard way to partition buoyancy fluxes relies on identifying a shear-stress depth (d_τ) marking the local minimum between the near-surface stress maximum associated with plume-generated interfacial shear and the deeper maximum associated with bottom-generated shear. This standard method relying on the shear-stress minimum (Table 1) was developed for estuaries (Ralston et al., 2010) and applied in plumes (Arevalo et al., 2022; Spicer et al., 2021). Spicer et al. (2021) includes a schematic and detailed description of this approach. Where there is only one τ maximum, d_τ = 0 and all the shear is linked to bottom stress. Shear stress profiles can be noisy, even in models. In this study, τ is rounded with 0.01 Pa precision before running the local minimum search routine to avoid insignificant wiggles; alternately, τ profiles can be smoothed first.

The present study searches upward from the bottom through the water column and sets z = -d_τ at the first τ minimum above the deepest τ maximum. The rationale is only the deepest τ maximum is associated with bottom-generated shear. In Spicer et al. (2021), d_τ is found by searching downward from the surface for the first τ minimum below the shallowest τ maximum. The downward and upward search methods produce the same results where there are only two τ maxima in the water column. Such is the case for a single buoyant layer likely to be encountered in idealized plumes. Realistic settings may also include water columns with more than two τ maxima and more than one τ minimum. Such situations can be created when a new plume overrides stratification from older plume waters so that there are maxima associated with the active plume, the older stratified layer, and bottom stress. In this case, the bottom upwards approach identifies more interfacial mixing and better isolates the bottom-generating mixing contribution.

The standard shear-stress minimum partitioning method (Table 1) assigns all B above d_τ to B_IF and all B at or below d_τ to B_BG. This partitioning method is indicated with a d_τ subscript and the B parts are determined with Equation 9.

$$${B}_{IF{d}_{\tau }}=\left\{\begin{array}{@{}l@{}}B,\,\eta -{d}_{\tau }< z\le \eta \\ 0,\,-h\le z\le \eta -{d}_{\tau }\end{array}\right.\hspace*{.5em}{B}_{BG{d}_{\tau }}=\left\{\begin{array}{@{}l@{}}0,\,\eta -{d}_{\tau }< z\le \eta \\ B,\,-h\le z\le \eta -{d}_{\tau }\end{array}\right.$$$ (9)

Note that B at each vertical level at any given point in time and space is considered either entirely interfacial or bottom-generated with this method. This binary split does not accommodate regions where both interfacial and bottom-generated shear contributions overlap. Thus, this partitioning method is best viewed as splitting the plume water column into parts dominated by interfacial and bottom-generated shear. Depth-integrating B_IFdτ and B_BGdτ from z = η-d_p to z = η as in Equation 7 yields Bd_IFdτ and Bd_BGdτ, respectively. Note that Bd_IFdτ = Bd and Bd_BGdτ = 0 where d_τ  ≥ d_p and bottom-generated shear is considered to not penetrate through the plume base. Bd_IFdτ = 0 and Bd_BGdτ = Bd where d_τ = 0 and bottom-generated shear reaches throughout the plume to the surface. Integrating the Bd parts over the plume area as in Equation 8 gives the mixing parts M_IFdτ and M_BGdτ.

Another partitioning approach involves parameterizing bottom-generated mixing with solutions for unstratified open channel flow (Table 1). Parameterizations for Ps in open channel flow, which equals ε for a local TKE balance in this unstratified reference regime, depend on the frictional velocity ($${u}_{\ast }=\sqrt{{\tau }_{b}/{\rho }_{o}}$$), the von Karman constant (κ = 0.4), the depth, and the height above the bottom (z_ab = z + h). Following Voulgaris and Trowbridge (1998) and applied in Orton et al. (2010), shear production for unstratified open channel flow (Ps_ocf) can be approximated as in Equation 10.

$$${Ps}_{ocf}=\frac{{{u}_{\ast }}^{3}}{\kappa {z}_{ab}}\left(1-\frac{{z}_{ab}}{\eta +h}\right)$$$ (10)

Model τ_b and η fields and z_ab values coinciding with w vertical levels are applied to produce time-varying 3D fields of Ps_ocf. The ratio of Ps_ocf to the full Ps in the model (calculated with 4) estimates the portion of shear production associated with open channel flow. Ps_ocf/Ps can exceed one if the open channel flow parameterization overestimates shear production. In such cases, all of Ps is attributed to Ps_ocf and a minimum function (min) is applied to cap the ratio at one: min (1,Ps_ocf/Ps). This ratio is used as a weighting to estimate the portion of the B profile consistent with bottom-generated shear for open channel flow (B_BGocf) according to Equation 11. The interfacial shear part (B_IFocf) is calculated as the residual Equation 11.

$$${B}_{IFocf}=B-{B}_{BGocf}{\,B}_{BGocf}=\min \left(1,\frac{{Ps}_{ocf}}{Ps}\right)B$$$ (11)

Note that this method allows for overlapping contributions of interfacial and bottom-generated shear within the water column. Thus, B at a particular vertical level and location can be assigned to contributions from both interfacial and bottom-generated mixing with this method. The B parts are integrated over the plume water column (η − d_p ≤ z ≤ η) as in Equation 7 to yield Bd_IFocf and Bd_BGocf and then integrated over the entire plume area as in Equation 8 to calculate M_IFocf and M_BGocf.

Another unstratified open channel flow parameterization is based on the “universal” empirical relationship for dissipation (Table 1; Nezu, 1977; Nezu, 2005; Johnson & Cowen, 2017). Equating Ps to ε in this unstratified reference situation, yields the shear production expression for open channel flow (Ps_uni) in Equation 12.

$$${Ps}_{uni}=\frac{9.8\,{{u}_{\ast }}^{3}}{(\eta +h)\sqrt{\frac{{z}_{ab}}{\eta +h}}}{e}^{-3\tfrac{{z}_{ab}}{\eta +h}}$$$ (12)

Model τ_b, η, and z_ab fields are applied to produce time-varying 3D fields of Ps_uni. The B partitioning is accomplished with Equation 13, in analogous fashion to the other open channel flow partitioning.

$$${B}_{IFuni}=B-{B}_{BGuni}{\,B}_{BGuni}=\min \left(1,\frac{{Ps}_{uni}}{Ps}\right)B$$$ (13)

The B parts are integrated over the plume water column Equation 7 to yield Bd_IFuni and Bd_BGuni and then integrated over the entire plume area Equation 8 to calculate M_IFuni and M_BGuni.

Another partitioning approach begins with a parameterization for stratified shear flows (Table 1) that characterizes interfacial mixing in river plumes. Following Ivey and Imberger (1991), MacDonald and Geyer (2004) and MacDonald et al. (2007) arrive at the parameterization ε/(g′Δu)=(1-Ri_f)C_Di/Ri_b = 2 × 10⁻³ where Ri_f is the flux Richardson number (Ri_f = B/Ps) and C_Di is an interfacial drag coefficient. The flux Richardson number can be written in terms of the mixing efficiency Ri_f = Γ/(1 + Γ); a constant Γ = 0.2 (Gregg et al., 2018; Ivey & Imberger, 1991) and corresponding Ri_f = ⅙ are used in the present study. C_Di varies with Richardson numbers in this parameterization. MacDonald and Geyer (2004) notes that C_Di is of order 5 × 10⁻⁴ in plume liftoff regions and MacDonald et al. (2007) notes that the dissipation estimate is an average for the plume layer. MacDonald and Chen (2012) points to the importance of plume spreading in stretching K-H instabilities and modifies the C_Di formulation to accommodate spreading-enhanced mixing. This parameterization is formed around a variable dimensionless mixing parameter ξ = B/(g′Δu). The ξ formulation includes a constant ξ_o that represents the base-level mixing without spreading. The formulation also uses a dimensionless parameter Ŧ, which is the time scale of K-H instability evolution multiplied by the shear scale Δu/d_p. The parameterization also involves the dimensionless spreading rate Φ, which is the spreading rate dv_⊥/dy_⊥ (with ⊥ indicating the direction perpendicular to the local flow) divided by Δu/d_p. With these definitions, the mixing parameter is expressed as ξ = ξ_o (1+ ŦΦ)². Manipulating the MacDonald and Chen (2012) ξ formulation with the ξ definition and the N²_b, S²_b, and Ri_b definitions (7) yields a parameterization of the shear production for stratified shear flow (Ps_ssf) with the corresponding spreading-enhanced interfacial drag coefficient (C_Dssf) (14).

(14)

Note that shear-layer thickness is set to d_p in the present study. In MacDonald and Chen (2012) and in the present study, ξ_o = 5 × 10⁻⁵ and Ŧ = 360. Model-derived fields of Δu, d_p, Ri_b, and dv_⊥/dy_⊥ (from surface velocities) are applied to produce time-varying 2D fields of Ps_ssf. Since Ps_ssf reflects a depth-average within the plume, there is no ability to calculate vertical variations of the mixing contributions within the plume for this method. Consequently, this partitioning approach is expressed in terms of Bd, rather than in terms of B. Bd_IFssf is based on the Ps_ssf parameterization Equation 14 and Bd_BGssf is calculated as the residual Bd Equation 15.

$$${Bd}_{IFssf}=\min \left(1,\frac{{Ps}_{ssf}}{{Ps}_{b}}\right)Bd{\,Bd}_{BGssf}=Bd-{Bd}_{IFssf}$$$ (15)

Note that Ps_b is the bulk (depth-averaged) shear production in the plume (η − d_p ≤ z ≤ η). With this parameterization, the Bd parts are integrated over the entire plume area Equation 8 to calculate M_IFssf and M_BGssf.

An alternate stratified shear flow parameterization (Table 1) based on Kunze et al. (1990) is described and tested in Jurisa et al. (2016), where it is referred to with a kwb subscript signifying the Kunze et al. (1990) author names. The kwb method parameterizes the TKE dissipated by K-H instabilities to return the shear layer to a marginally stable state with a critical Richardson number (Ri_c) (Jurisa et al., 2016). Assuming Ps = ε/(1-Ri_f) as above, the kwb parameterization for depth-averaged shear production in the plume (Ps_kwb) and corresponding interfacial drag coefficient (C_Dkwb) are expressed in Equation 16.

$$${Ps}_{kwb}={C}_{Dkwb}\frac{{{\increment}u}^{3}}{{d}_{p}}\,\text{with}\,{C}_{Dkwb}=\left\{\begin{array}{@{}ll@{}}\frac{1}{96}\left(1-\frac{{Ri}_{b}}{{Ri}_{c}}\right)\left(1-\sqrt{\frac{{Ri}_{b}}{{Ri}_{c}}}\right),& {Ri}_{b}\le {Ri}_{c}\\ 0,& {Ri}_{b} > {Ri}_{c}\end{array}\right.$$$ (16)

In Jurisa et al. (2016) and the present study, Ri_c = 0.5. Note that shear-layer thickness is set to d_p in the present study, but can be set more generally as a fraction of the plume thickness as in Jurisa et al. (2016). Model-derived fields of Δu, d_p, and Ri_b are applied to produce time-varying 2D fields of Ps_kwb. Ps_kwb reflects a depth-average within the plume. Consequently, this approach does not provide vertically varying B partitioning and is expressed in terms of Bd. Bd_IFkwb is based on the Ps_kwb parameterization Equation 16 and Bd_BGkwb is the residual Bd Equation 17.

$$${Bd}_{IFkwb}=\min \left(1,\frac{{Ps}_{kwb}}{{Ps}_{b}}\right)Bd{\,Bd}_{BGkwb}=Bd-{Bd}_{IFkwb}$$$ (17)

The Bd parts are integrated over the entire plume area Equation 8 to calculate M_IFkwb and M_BGkwb.

Description of partitioning results focuses first on the standard d_τ method. This approach is closest to the data (model results in this case), since the partitioning is based on the actual shear stress profiles. The other approaches parameterize either the bottom-generated or interfacial shear and may reflect somewhat simplified versions of shear production profiles; nevertheless they are based on accepted and previously applied scaling estimates. The parameterized methods provide benchmark comparisons for the standard partitioning method and offer links to prior theoretical and empirical results. The main findings that are robust among methods are highlighted.

3 Results

3.1 Plume Features

The Connecticut River plume is described in detail in Whitney et al. (2021) for situations with low and high river discharge and neap and spring tides. The present study analyzes the low-discharge spring-tide case. The plume stretches westward along the coast as it evolves during the flood tidal phase in ambient LIS waters (Figure 2). Because of the net river flow and the phase relationship between ebb tides in the river and ambient LIS tides (described in Whitney et al., 2021), surface source waters flow out through the mouth from early ambient ebb to max ambient flood. Source waters exiting during late ambient ebb form the leading edge of the plume. Waters exiting during early ambient flood feed the rest of the plume and the plume is cutoff from source waters through late ambient flood. Near max ambient flood, the bulk salinity anomaly is 16 at the mouth and 4 near the downstream end of the plume (Figure 2c). At this time, the plume is 3.5 km wide and has extended 10 km along the coast due to the combined action of plume propagation and tidal advection. The plume internal wave speeds (c_p) for d_p = 2 m and 16 and 4 salinity anomalies (ΔSal) are 0.49 and 0.25 m/s, respectively. The scale for the ambient tidal current amplitude (U_Ta) is 1.00 m/s and the corresponding Froude number comparing tidal currents to plume propagation (Fr_Ta=U_Ta/c_p) lies between 2.0 and 4.1. Additional details on plume structure and results for other cases are included in Whitney et al. (2021). Ambient tidal currents continue to stretch the plume farther along the coast during late ambient flood (Figures 2d–2f). The enhanced velocities within the plume (relative to surrounding ambient waters) are associated with plume propagation and suggest plume-generated interfacial shear is important.

Figure 2

Surface salinity (shaded) and velocity (arrows) map for (a) 1, (b) 2, (c) 3, (d) 4, (e) 5, and (f) 6 hr after onset of ambient flood tide. Salinity in the river and in parts of the plume are lower than the contoured range. Velocity arrows are shown for every 40th point (800 m spacing) and every twentieth point (400 m spacing) in x and y, respectively. Plume boundaries (the C_threshold freshwater tracer contour) are shown in white. The white dashed line indicates the defined river mouth. Profile locations in Figure 3 are labeled with white points and letters in panel (c)

Profiles at selected locations along the plume near max ambient flood (Figure 2c) indicate the vertical structure of the density and velocity fields (Figure 3). The plume is deeper (d_p = 4 m) at the station closest downstream from the jetties (Profile B) than at the other locations, where d_p is around 2 m. Peak stratification and shear are relatively strong (N²>0.03 s⁻², S²>0.05 s⁻²) at the selected profile locations. The points close to the jetties (Profiles A and B) have two stratification/shear peaks, suggesting two shear layers with the plume. All of the profiles indicate increased velocities and shear associated with the plume that augment the tidal flow, which is sheared from the bottom up through the water column. The B and τ profiles are described in subsequent sections.

Figure 3

Profiles of (a) ρ, (b) speed, (c) N², (d) S², (e) B, and (f) τ magnitude at max ambient flood. Thick lines indicate data within the plume (depth ≤ d_p). The shear stress minima associated with d_τ are marked with circles on the |τ| profiles in panel (f) Profile D extends to 17 m deep (beyond the range shown). Profile locations are shown in Figure 2c.

The d_p field at max ambient flood (Figure 4) indicate the plume occupies the entire water column immediately outside the mouth and near the jetties prior to the plume liftoff point. In most other places, the plume is in the upper-half of the water column and the d_p range is 1–6 m. The plume is thicker near the western (downstream) leading edge, along the coast, the arrested eastern (tidally upstream) front, and the portion of the offshore boundary relatively close to the.mouth. The plume also deepens and occupies more of the water column over shoals and reefs (Figures 1b and 3). The shallower waters over these bathymetric features favors greater bottom interaction and the influence of bottom-generated shear. The plume-averaged d_p throughout ambient flood tides (not shown) ranges from 1.5 to 2.8 m, with a thicker plume during early flood and a thinning plume through late flood.

Figure 4

(a) Plume thickness (d_p) and (b) plume thickness to depth ratio (d_p/h) at max ambient flood. The 5, 10, and 15 m isobaths are contoured with dashed, solid, and dashed-dot brown lines. The black dashed line marks the defined river mouth.

3.2 Mixing Patterns

As the plume grows and evolves during ambient flood tides, mixing patterns vary and are characterized by a high degree of spatial heterogeneity spanning several orders of magnitude (Figure 5). One hour into ambient flood (Figure 5a), depth-integrated turbulent buoyancy fluxes within the plume are strongest (Bd ∼ 10⁻⁵-10⁻⁴ m³/s³) along the main estuary channel, reaching between and around the jetties, and in patches farther offshore. Bd varies from 10⁻⁸ to 10⁻⁴ m³/s³ in the plume, with large variability in all directions and the lowest values farther offshore. An hour later (Figure 5b), mixing has decreased within the estuary, but levels between and around the jetties are comparable to earlier levels. At this stage, the eastern plume boundary near the mouth is arrested by the strengthening westward tidal flow, the plume turns tightly beyond the jetties, and the downstream boundary is extending westward. Bd values are highest (10⁻⁵-10⁻⁴ m³/s³) at the arrested eastern front, the turning region, near the downstream boundary where it flows over Long Sand Shoal, and along the coast. There is strong spatial heterogeneity in Bd and B both along and across the plume.

Figure 5

Depth-integrated mixing within the plume (Bd) for (a) 1, (b) 2, (c) 3, (d) 4, (e) 5, and (f) 6 hr after onset of ambient flood tide. Isobaths and the river mouth are shown as in Figure 4.

Near max ambient flood (3 hr into the flood tidal stage) (Figure 5c), mixing is at peak levels. Close to the mouth in and around the jetties, Bd reaches 10⁻⁴ m³/s³. Elevated mixing levels (Bd > 10⁻⁶ m³/s³) extend beyond the turning region through the core of the plume in the eastern half. Near the western (downstream) end, Bd is highest (>10⁻⁵ m³/s³) offshore near the edge of Long Sand Shoal, centrally over the shoals encompassing Crane Reef and Hen and Chickens shoals, and over nearshore shallow areas. Mixing also is high along the inshore boundary, particular near Cornfield Point (where Bd ∼ 10⁻⁴ m³/s³). These patterns highlight bathymetric interaction and intensified mixing over shallow areas. About half the plume area has low to moderate mixing levels (Bd ∼ 10⁻⁸-10⁻⁶ m³/s³), particularly in the plume middle. These expansive areas of low to moderate mixing contribute to the overall mixing of the plume. Along- and across-plume variability continues to be pronounced. The profiles at selected locations near max ambient flood (Figure 2c) indicate the variety of B vertical structure (Figure 3e). At many locations (Profiles A, B, and E), B peaks both near the surface and close to the base of the plume; a pattern consistent with multiple mixing layers. Peak B values vary from 3 × 10⁻⁷ m²/s³ (in the plume middle) to 3 × 10⁻⁵ m²/s³ (near the jetties) among the selected profiles.

Immediately after max ambient flood (Figure 5d), the plume has stretched farther westward and mixing levels are beginning to decrease. The strongest mixing areas still are near the mouth, in the turning region, and in areas near the downstream end. Bd values tend to be higher over shallow areas closer to shore. These high mixing areas are stretched out in streaks along the plume. Relatively low mixing levels (Bd∼10⁻⁸-10⁻⁶ m³/s³) occupy more than half the plume. During late ambient flood (5 hr after onset of ambient flood) (Figure 5e), the plume is cutoff from riverine source-water inflow. Mixing levels now are low between and around the jetties (Bd < 10⁻⁷ m³/s³). The eastern (upstream) part of the plume still has elevated levels, but maximum intensities have decreased (Bd < 10⁻⁵ m³/s³). There are fewer areas of high mixing near the downstream half, the strongest area (with B ∼ 10⁻⁵ m³/s³) is in nearshore shallow waters. Most of the plume has low mixing levels (Bd∼10⁻⁸-10⁻⁶ m³/s³). The plume is fragmenting as the plume buoyancy signature has been mixed away in many areas, particularly in nearshore waters. An hour farther into late ambient flood (Figure 5f), the plume is completely cutoff from the source region and the plume has not stretched much farther westward. Maximum mixing areas are less intense (Bd∼10⁻⁶ m³/s³) and almost all of the plume has relatively low mixing rates (Bd < 10⁻⁷ m³/s³). The plume has been mixed away in nearshore shallow waters (inshore of the 10 m isobath). The plume is narrower at this stage due to this shore-side loss of plume area.

The surface maps (Figure 5) and most of the results description are in terms of Bd, the vertically integrated B within the plume, to facilitate the overall plume-integration described later. Selected B profiles indicate the variety of vertical structure throughout the plume (Figure 3e). Converting Bd to more familiar B values (in m²/s³) involves locally dividing Bd by d_p values. Since Bd varies by orders of magnitude and d_p has a small range (1–6 m), an appropriate estimate for vertically averaged B within the plume is obtained by dividing the Bd field by the 2 m average d_p during ambient flood tides. The vertically averaged B range is 10⁻⁸ to 10⁻⁴ m²/s³ with the spatial structure following the Bd fields. Maximum B values are highest (∼10⁻⁴ m²/s³) during early and max ambient flood and drop down to ∼10⁻⁶ m²/s³ during late ambient flood. While the 100-fold temporal variation in maximum mixing values is considerable, it is dwarfed by the 1000-fold to 10,000-fold spatial variability. The spatial heterogeneity patterns are characterized by the most intense mixing near the mouth and jetties, at the arrested eastern front, in the nearfield plume turning region, over shoals, and in nearshore shallow waters.

3.3 Interfacial and Bottom-Generated B Partitioning

B profiles (Figure 3e) are first partitioned using the standard d_τ method (Table 1, Figure 6). Near max ambient flood, the shear stress minimum between two maxima is within the plume (d_τ <d_p) for the selected profiles that are near the jetties (Profiles A and B, Figure 3f). The profile a third of the way along the plume (Profile C) only has a surface shear stress minimum (and d_τ = 0). For these three profiles, the d_τ partitioning points to a significant role for bottom-generated shear in mixing the plume and B_BGdτ >B_IFdτ at two of three locations (Profiles A and C, Figures 6a and 6b). For the two other selected τ profiles located farther along the plume (Profiles D and E, Figure 3f), d_τ >d_p and B_BGdτ = 0; indicating interfacial shear dominates plume mixing at these locations (Figures 6a and 6b). Note that there are other locations where the shear stress profile has more than one candidate for d_τ (a τ minimum between maxima). Selecting d_τ as the deepest minimum (as in this study) yields a smaller role for bottom-generated shear in plume mixing that searching down from the surface and selecting the shallowest minimum (as in Spicer et al., 2021). Using the deepest minimum to set d_τ (and the corresponding B_BGdτ) is consistent with bottom-generated shear working up from the bottom.

Figure 6

Mixing partitions with the dτ method for profiles of (a) interfacial and (b) bottom-generated B and maps of (c) interfacial and (d) bottom-generated Bd. Max ambient flood conditions are shown. Profile segments within the plume are shown with thick lines as in Figure 3. Isobaths and the river mouth are shown as in Figures 4 and 5. Profile locations are labeled with white or black points and letters in panels (c) and (d)

Bd_IFdτ and Bd_BGdτ both contribute to the total depth-integrated turbulent buoyancy flux within the plume near max ambient flood (Figures 6c and 6d). Bd_IFdτ is strong (Figure 6c) and the same order of magnitude as Bd (Figure 5c) in most of the intense mixing areas (where Bd_IFdτ and Bd > 10⁻⁵ m³/s³) near the mouth and jetties and near the western (downstream) end, particularly over shoals and inshore shallow areas. The Bd_IFdτ field, however, does not contribute to the elevated mixing area starting beyond the turning region, around Profile C (Figure 6a), and extending to halfway along the plume. Bd_IFdτ also is at least an order of magnitude lower than Bd in many other areas of the plume interior. Bottom-generated shear influence is strong near the mouth where Bd_BGdτ (Figure 6d) is intense (Bd_BGdτ > 10⁻⁵ m³/s³). Bd_BGdτ also is intense at several locations where there are gaps in Bd_IFdτ over shoals and inshore shallow areas near the downstream end. Bd_BGdτ accounts for most of Bd (Figure 5c) in the area around Profile C (Figure 6b) and in the surrounding moderate mixing areas (Bd_BGdτ and Bd > 10⁻⁶ m³/s³).

Both Bd_IFdτ and Bd_BGdτ are strong over shoals and in nearshore shallow waters along the downstream half. The relative influence of Bd_IFdτ and Bd_BGdτ varies throughout the plume. Both interfacial and bottom-generated mixing contribute throughout other times during ambient flood (not shown).

Partitioning based on open channel flow parameterizations provides alternate estimates of interfacial and bottom-generated mixing contributions (Table 1). The Bd_IFocf field near max ambient flood (Figure 7a) has similar magnitudes and spatial structure to the Bd field (Figure 5c). Bd_IFocf does not exhibit the low areas in the middle of the plume that the Bd_IFdτ (Figure 6c) has. Bd_IFocf (Figure 7a) generally is larger than Bd_BGocf (Figure 7b) in most locations.

Figure 7

Mixing partitions: (a) Bd_IFocf, (b) Bd_BGocf, (c) Bd_IFuni, (d) Bd_BGuni, (e) Bd_IFssh, (f) Bd_BGssh, (g) Bd_IFkwb, and (h) Bd_BGhwb at max ambient flood. Isobaths and the river mouth are shown as in Figures 4-6.

Bd_BGocf shares some similar spatial patterns with Bd_BGdτ (Figure 6d). These two estimates of bottom-generated shear-driven plume mixing are generally large in the same areas, particularly near the mouth and jetties, the turning region, the extension into the plume center, and some areas downstream including over shoals and nearshore shallow areas. The partitioning method based on the universal empirical relationship for open channel flow (Figures 7c and 7d) has similar results to the first open channel flow method (Figures 7a and 7b). The chief differences are Bd_IFuni < Bd_IFocf in some areas and correspondingly Bd_BGuni > Bd_BGocf in these areas. Overall, the universal open channel flow partitioning indicates a somewhat larger role for bottom-generated mixing than the first open channel flow method. Both open channel flow methods estimate a smaller influence for bottom-generated mixing than the standard d_τ partitioning, but all share similar spatial patterns.

Applying stratified shear flow parameterizations for plumes yields another set of estimates for interfacial and bottom-generated mixing contributions (Table 1). The Bd_IFssf field near max ambient flood (Figure 7e) has similar patterns to the Bd field (Figure 5c), but with generally weaker intensity near the downstream end. Bd_BGssf (Figure 7f) is large in these downstream areas, particularly over shoals and inshore shallow areas. Bd_BGssf also is intense in focused areas near the jetties and eastern end. Bd_BGssf is moderate along a narrow band in the plume center and patches in other interior areas. A sensitivity test (not shown) with C_Di = 2 × 10⁻³Ri_b/(1-Ri_f) (consistent with C_Di ∼ 5 × 10⁻⁴ as in MacDonald & Geyer, 2004), rather than using the spreading-enhanced C_Dssf, has similar results near the mouth and turning region, stronger interfacial mixing in interior and downstream areas, and overall weaker bottom-generated mixing. The simpler C_Di parameterization performs well in the intense-mixing areas near the liftoff zone that it was designed for, but the spreading-dependent C_Dssf provides additional flexibility to accommodate other plume regions (MacDonald & Chen, 2012). Compared to Bd_IFssf (Figure 7e), Bd_IFkwb (Figure 7g) estimates stronger interfacial mixing near the mouth, turning region, and near the downstream end. Bd_IFkwb is shutoff in large areas where Ri_b > Ri_c. Bd_BGkwb (Figure 7h) is moderate to intense in these interior regions and over some areas near the mouth and over shoals. The kwb method estimates a smaller role for bottom-generated shear in the most intense mixing areas.

Across all partitioning methods, interfacial mixing is a bigger contributor overall than bottom-generated mixing. Other robust results include the importance of bottom-generated near the jetties and turning region, over shoals and inshore shallow areas near the downstream end, and in some other parts of the plume interior. Nevertheless, the distribution and intensity of estimated interfacial and bottom-generated mixing vary among methods. The combined analysis points to a significant role for bottom-generated shear generated by tidal currents that augments the interfacial mixing associated with plume-generated shear. The analysis also emphasizes the spatial heterogeneity of the partitioning between mixing processes as well as the total mixing levels within the plume.

3.4 Plume-Integrated Mixing

The time series of plume-integrated mixing, M calculated with (8), shows the increase through early ambient flood, the 1.21 × 10⁵ W (121 kW) peak at max flood, and the more rapid decline through late flood (Figure 8). It is noteworthy that early and late ebb account for approximately the same amount of plume-integrated mixing even though the plume area (and volume) increases through ambient flood. Thus, the mixing intensity (corresponding to plume-averaged mixing) is much higher during early flood. The higher mixing intensity is associated with the sustained large turbulent buoyancy fluxes close to the mouth (described in Section 3.2).

Figure 8

Time series of plume-integrated mixing (M) and interfacial and bottom-generated mixing partitions for (a) standard dτ approach, (b) open channel flow methods, and (c) stratified shear flow methods. Onset and end times of ambient flood tide are marked with black dotted vertical lines. Max ambient flood is indicated with a black dashed vertical line.

The standard d_τ partitioning indicates that plume-integrated interfacial mixing (M_IFdτ) is the largest contribution and it has a similar tidal phasing to M (Table 1, Figure 8a). The plume-integrated interfacial mixing (M_IFdτ) represents almost three-quarters of the mixing near max ambient flood (at t = 3 hr). Bottom-generated mixing (M_BGdτ) accounts for 27% of the mixing at this stage. M_BGdτ generally has higher values in late ambient flood than early flood.

Plume-integrated results for both open channel flow partitionings are similar (Table 1, Figure 8b). Bottom-generated mixing (M_BGocf and M_BGuni) represents 13%–18% of the mixing near max ambient flood (at t = 3 hr). M_BGuni is slightly larger than M_BGocf, but both are weaker than the peak M_BGdτ estimate. The two stratified shear flow methods generate different results (Table 1, Figure 8c). M_BGssh is larger than M_BGdτ at max ambient flood, when it accounts for 41% of the mixing. M_BGkwb is much smaller than M_BGssh and is similar in magnitude to M_BGocf at peak ambient flood, when it represents on 15% of the mixing. Both M_BGssh and M_BGkwb have strong tidal asymmetry with late peaks and much higher mixing values during late ambient flood. M_BGssh grows larger than M_IFssh during the late tidal stage. All partitioning methods indicate that plume-integrated interfacial mixing is greater than bottom-generated mixing near peak ambient flood (Table 1). Though smaller, bottom-generated mixing makes a significant contribution to plume-integrated mixing.

Results presented in previous sections indicate mixing levels vary by orders of magnitude throughout the plume. Further analysis can determine how the expansive areas of weak and moderate mixing compare to concentrated zones of intense mixing in setting the overall plume-integrated mixing. The histogram in Figure 9 breaks M into Bd bins at max ambient flood (t = 3 hr). Intense (with Bd ≥ 10⁻⁵ m³/s³), moderate (with 10⁻⁶ ≤ Bd < 10⁻⁵ m³/s³), and weak (with Bd < 10⁻⁶ m³/s³) mixing areas represent 67%, 30% and 3% of M, respectively. Thus, two-thirds of plume mixing occurs in concentrated intense mixing areas and most of the rest occurs in expansive moderate mixing zones. The relative importance of interfacial and bottom-generated processes in these mixing categories is estimated with the standard d_τ partitioning method (Figure 9).

Figure 9

Histogram of M in bins of Bd. Partitions from interfacial and bottom-generated mixing are calculated with the d_τ method. Bars sum to 100% of M.

Interfacial mixing is strongly skewed toward the higher mixing regions; the intense, moderate, and weak mixing areas represent 52%, 19%, and 2% of M, respectively. Bottom-generated mixing is more evenly split between intense and moderate areas; the intense, moderate, and weak areas account for 15%, 11%, and 1% of M, respectively.

4 Discussion

Turbulent buoyancy fluxes vary over several orders of magnitude within the plume, indicating pronounced spatial variations in mixing. Near max ambient flood, B diminishes from maximum values above 3 × 10⁻⁵ m²/s³ outside the jetties, to 3 × 10⁻⁶ m²/s³ in the plume core 2 km farther downstream, to 3 × 10⁻⁷ m²/s³ following the flow into the plume middle 3–4 km from the jetties. This pattern of decreasing mixing along the plume away from the river mouth is consistent with the trend derived from observations with the control volume technique for the Merrimack River mouth (MacDonald et al., 2007). Furthermore, B has similar intensities and decreases by two orders of magnitude over similar distance from the mouth in both plumes (MacDonald et al., 2007). Model results for the Connecticut River plume also highlight the importance of lateral variability. B varies to the same degree across the plume as along the plume and over shorter length scales. Maximum B values in the modeled plume also fall within the 10⁻⁵-10⁻⁴ m²/s³ range of maximum B generally observed near plume fronts and within nearfield plume regions (Horner-Devine et al., 2015). Prior studies have observed ε in the Connecticut River plume inward from a bounding front. Corresponding B values can be estimated by applying the B = Γε relationship. O’Donnell et al. (2008) observed dissipation rates consistent with B ∼ 10⁻⁴ m²/s³ at the front that exponentially decrease to three orders of magnitude lower (10⁻⁷ m²/s³) within 100 m into the plume interior. Recent dissipation observations in Delatolas et al. (2023) are consistent with B of similar magnitude at the front, 10⁻⁴-10⁻⁶ m²/s³ in the transition zone (10–100 m from the front), and 10⁻⁶-10⁻⁸ m²/s³ farther (100–400 m) into the plume interior. The combined width of the frontal and transition zone is also of order 100 m wide for plumes such as the Merrimack and Columbia River plumes (e.g., Cole et al., 2020; Orton & Jay, 2005). The 20-m horizontal resolution and hydrostatic nature of the model do not resolve the full texture of frontal processes, but it does resolve the transition zone. Increased mixing levels associated with interfacial mixing (B ∼ 10⁻⁴-10⁻⁵ m²/s³) are evident along many parts of the modeled bounding plume front. In many areas, the frontal mixing band is around 100 m wide. Thus, the decrease in mixing levels away from the bounding front is seen in observations and model results including this study and Spicer et al. (2021).

Bathymetric variations play a large role in setting mixing levels within the plume. There are large areas of intense mixing in the western interior, particularly over shoals and along a shallow coastal band. Plume velocities are generally faster in these areas and interfacial mixing is intense. Bottom-generated mixing also is intense in these shallow waters. These areas point to the role of bathymetric interactions in promoting mixing within the plume. The findings emphasize that bathymetric variations are a key contributor to the pronounced spatial heterogeneity of mixing (and mixing contributions) throughout the plume.

The analysis of this study, across all partitioning methods, indicates that interfacial mixing is a bigger contributor overall than bottom-generated mixing. Near the mouth and jetties, in the nearfield plume turning region, over shoals and inshore shallow areas near the downstream end, and in other parts of the plume interior bottom-generated shear is an important contributor. All tested partitioning methods yield reasonable results. The d_τ method directly uses information from model shear stress profiles. The other methods rely on parameterizations either for open channel flow or stratified shear flow. The d_τ method is preferred in this study and likely is preferable in other studies where shear stress profiles are available from observations or models. The methods based on parameterizations can be relied on in situations where such detailed information is not available and in idealized applications.

The companion study (Whitney et al., 2021) provides context and perspective on plume evolution. The plume expands to a voluminous and diluted state by the end of flood (evident in Figure 2). The expansion is fueled both by buoyant inputs through the mouth and entrainment. The dilution occurs through entrainment and mixing within the plume, which outpace buoyancy injection at the mouth. Using a tracer variance approach, the companion study finds that entrainment and mixing contribute about equally to reducing plume tracer variance over the entire ambient flood tidal stage. Entrainment tends to be larger than mixing during early flood and smaller during max and late flood. At max ambient flood for the low-discharge spring-tide case (analyzed in the present study), the corresponding entrainment and mixing time scales are 2 and 1 hr, respectively. These short time scales are consistent with the rapid evolution of the plume due to these processes. The companion study indicates that plume-averaged thickness increases linearly over time, which points to the increased susceptibility to bottom-generated mixing.

Future observational and modeling studies of the Connecticut River plume can focus additional attention on localized mixing hotspots (e.g., over shoals) within the plume interior. Furthermore, the mechanisms and types of vertical shear instabilities can be identified. K-H instabilities almost certainly are involved and it is possible that shorter-wavelength secondary Holmboe instabilities and/or braided secondary instabilities form as they do in non-hydrostatic models of idealized river plumes and estuary changes (Ayouche et al., 2022; Zhou et al., 2017). K-H and braided secondary instabilities have been observed within the Connecticut River estuary (Geyer et al., 2010) and it is plausible that both instability types occur within the plume, particularly closer to the mouth. Future research can apply a non-hydrostatic model to the plume during ambient flood tides and vertical shear instabilities can be analyzed over the plume interior. An apt starting point is the non-hydrostatic model of the plume during ambient ebb tide and comparison to observations in Simpson et al. (2022). The ambient ebb plume also is important for delivering and mixing buoyant waters. The ambient ebb plume expands farther offshore than the ambient flood plume (Garvine, 1975). The spatial patterns of mixing and relative importance of bottom-generated mixing are likely very different for the plumes formed during the two tidal stages.

The idealized modeling results of Spicer et al. (2021) point to a stronger role for bottom-generated mixing than in the realistic simulations of the Connecticut River plume in the present study. This difference is partially due to different approaches for calculating d_τ. In locations where there are multiple eligible shear stress minima, this study selects the deepest and Spicer et al. (2021) selects the shallowest. Applying the Spicer et al. (2021) technical detail increases M_BG/M from 27% (Table 1) to 45%. The lower d_τ is used in this study because searching upwards from the bottom is consistent with isolating shear generated from bottom stress. Another main contributing factor for the differences is this study's realistic setting has more complicated bathymetry with shoals and deeper areas. On average, the plume flows over deeper bathymetry in the realistic model than the idealized runs in Spicer et al. (2021).

Bulk scalings for bottom-generated tidal shear and plume-generated interfacial shear at various plume locations can illustrate the role of local bottom depth. Bottom-generated tidal shear can be crudely approximated as U_Ta/h, where U_Ta is a velocity scale for ambient tidal currents (as defined in Section 3.1). Plume-generated interfacial shear can be approximated as c_p/d_p, assuming Δu within the plume scales with the internal wave speed. With these approximations, the ratio of bottom-generated shear (S_BG) to interfacial shear (S_IF) is:

$$$\frac{{S}_{BG}}{{S}_{IF}}\approx {Fr}_{Ta}\frac{{d}_{p}}{h}$$$ (18)

where Fr_Ta=U_Ta/c_p is the ambient tidal Froude number (as defined in Section 3.1). The dependence on Fr_Ta favors the relative importance of bottom-generated shear since Fr_Ta > 1 for the Connecticut River plume and other tidally pulsed plumes. The dependence on the local plume-depth to bottom depth ratio (d_p/h) favors interfacial shear since d_p/h ≤ 1 in all situations and d_p/h<<1 over deep bathymetry. The on-average deeper bathymetry in the realistic situation has smaller d_p/h and correspondingly smaller S_BG/S_IF than in the idealized situations in Spicer et al. (2021).

The shear ratio scaling above points to the importance of Fr_Ta and d_p/h, but a more detailed approach involving the ratio of depth-integrated shear production contributions at various locations within the plume is necessary to see the power dependence on Fr_Ta and d_p/h. Integrating the open channel flow representation for bottom-generated shear production (Ps_ocf, Equation 10) from z = -d_p to z = 0 yields:

$$$\underset{-{d}_{p}{}}{\overset{0}{\int }}{Ps}_{ocf}dz=\frac{{{u}_{\ast }}^{3}}{\kappa }\left(-\mathrm{ln}\left(1-\frac{{d}_{p}}{h}\right)-\frac{{d}_{p}}{h}\right)$$$ (19)

Note that a Taylor series can be applied for ln(1-d_p/h) if d_p/h is small, as for thin plume layers. Retaining the two largest terms in the Taylor series, depth-integrated Ps_ocf is approximately as:

$$$\underset{-{d}_{p}{}}{\overset{0}{\int }}{Ps}_{ocf}dz\approx \frac{{{u}_{\ast }}^{3}}{2\kappa }{\left(\frac{{d}_{p}}{h}\right)}^{2}\quad \text{for}\quad \frac{{d}_{p}}{h}\ll 1$$$ (20)

The Taylor series approximation Equation 20 is an underestimate, but is ≥ 86% of Equation 19 for d_p/h ≤ 0.2 and is ≥ 65% for d_p/h ≤ 0.5. The stratified shear flow representation for interfacial shear production (Ps_ssf, Equation 14) is already depth-averaged through the plume layer, so the corresponding depth integral is:

$$$\underset{-{d}_{p}{}}{\overset{0}{\int }}{Ps}_{ssf}dz={C}_{Dssf}{{\increment}u}^{3}$$$ (21)

The frictional velocity in Equations 19 and 20 can be scaled as (C_DU_Ta²)^½ (following a quadratic bottom drag formulation), Δu in Equation 21 can be scaled as c_p, and C_Dssf in Equation 21 cam be approximated with a constant C_Di. With these scalings the bottom-generated to interfacial depth-integrated shear-production ratio (BGI) at a given plume location can be approximated as:

$$$BGI=\frac{\underset{-{d}_{p}{}}{\overset{0}{\int }}{Ps}_{ocf}dz}{\underset{-{d}_{p}{}}{\overset{0}{\int }}{Ps}_{ssf}dz}\approx \frac{{{C}_{D}}^{\tfrac{3}{2}}}{\kappa {C}_{Di}}{{Fr}_{Ta}}^{3}\left(-\mathrm{ln}\left(1-\frac{{d}_{p}}{h}\right)-\frac{{d}_{p}}{h}\right)$$$ (22)

For thin plume locations with small d_p/h, the ratio approximation (BGI_thin) of Equation 22 is:

$$${BGI}_{thin}=\frac{\underset{-{d}_{p}{}}{\overset{0}{\int }}{Ps}_{ocf}dz}{\underset{-{d}_{p}{}}{\overset{0}{\int }}{Ps}_{ssf}dz}\approx \frac{{{C}_{D}}^{\tfrac{3}{2}}}{2\kappa {C}_{Di}}{{Fr}_{Ta}}^{3}{\left(\frac{{d}_{p}}{h}\right)}^{2}\quad \text{for}\quad \frac{{d}_{p}}{h}\ll 1$$$ (23)

The ratio is strongly sensitive to Fr_Ta and d_p/h, with faster tidal flows and increased plume thickness increasing the relative contribution of bottom-generated mixing. Note that the approximate ratio BGI_thin also can be expressed in terms of U_Ta³/h from the Simpson-Hunter parameter (Simpson & Hunter, 1974), an analogous expression for interfacial shear production (c_p³/d_p), and d_p/h:

$$${BGI}_{thin}=\frac{\underset{-{d}_{p}{}}{\overset{0}{\int }}{Ps}_{ocf}dz}{\underset{-{d}_{p}{}}{\overset{0}{\int }}{Ps}_{ssf}dz}\approx \frac{{{C}_{D}}^{\tfrac{3}{2}}}{2\kappa {C}_{Di}}\left(\frac{{{U}_{Ta}}^{3}}{h}\right){\left(\frac{{{c}_{p}}^{3}}{{d}_{p}}\right)}^{-1}\frac{{d}_{p}}{h}\quad \text{for}\quad \frac{{d}_{p}}{h}\ll 1$$$ (24)

With frictional parameters set at κ = 0.4, C_D = 2.5 × 10⁻³ and C_Di = 5 × 10⁻⁴ (the order of magnitude value in MacDonald & Geyer, 2004), the combined coefficients C_D^3/2/(κC_Di) and C_D^3/2/(2κC_Di) in Equations 22-23 and 24 are 0.63 and 0.32, respectively.

The BGI can calculated with local conditions at various locations within a plume or with plume-averaged parameters. The plume-averaged approach is valuable for broadly characterizing an plume and for intercomparison of different plumes, but it must be remembered that mixing conditions and BGI values can vary considerably within a plume. For the Connecticut River plume during low discharge and spring tides, plume-averaged conditions are represented with ΔSal = 10, d_p = 2 m, U_Ta = 1.0 m/s, and h = 10 m. With these settings, c_p = 0.4 m/s, Fr_Ta = 2.6, and d_p/h = 0.2. The BGI value is 0.25; indicating significant, but not dominant, bottom-generated shear-driven mixing of the plume. The scaling results are consistent with the detailed findings based on the model. It is important to emphasize that the salinity anomaly (and therefore c_p) and d_p/h vary considerably over the plume. BGI also varies throughout the plume, with larger values near the mouth prior to liftoff and over shoals and along inshore shallow areas (where d_p/h approaches 1) near the downstream end where Fr_Ta is larger due to weaker ΔSal and c_p.

A parameter space for the relative importance of bottom-generated mixing of plumes is constructed with axes of Fr_Ta and d_p/h and contours of BGI (Figure 10). The parameter axes and BGI values can be calculated either from water-column observations or model results for a given plume. The Connecticut River plume scaling described above is included within the context of other tidally modulated plumes around the world. Values for the other plumes are based on published research (Table 2). Locations are representative conditions within nearfield plumes, rather than at the river mouth. Characteristics at the mouth have been linked to processes including tidal mixing within the estuary (e.g., Nash et al., 2009). In contrast, this parameter space is designed to rate the importance of additional tidal mixing within the plume after exiting the mouth. Spring-tide conditions are selected where available to emphasize situations that most likely favor bottom-generated mixing. For each plume ΔSal, d_p, U_Ta, and h are specified to calculate c_p, Fr_Ta, d_p/h, and BGI (Table 2). In cases where the upper layer is continuously stratified, ΔSal is estimated as the average of the salinity at the surface and at d_p. In cases where the stratification is near constant throughout the water column but there is a flow reversal, d_p is estimated as ½h. Comparisons to other plumes are meant to be illustrative of different regimes with varying importance of tidal mixing. It is important to note that each plume can exhibit a range of BGI values due to spatial heterogeneity in mixing and temporal variability in forcing conditions. The Connecticut plume is within the moderate category for the relative importance of bottom-generated mixing (0.1 ≤ BGI < 1) along with several other plumes. The Connecticut contrasts with plumes in the weak tidal mixing category (BGI < 0.1) such as the Merrimack which has both weaker ambient tides (smaller Fr_Ta) and a relatively thinner plume (smaller d_p/h). Several plumes such as the Rhine (which has similar Fr_Ta and larger d_p/h) have stronger tidal mixing influence (1 ≤ BGI < 10). A few plumes are dominated by tidal mixing (10 ≤ BGI); these range from the small Fukuda to the large Changjiang. The Daly plume falls in this dominant category during the low-flow dry season and is in the moderate category during the wet season. Data points include surface-advected (e.g., Connecticut, small d_p/h) and bottom-advected (e.g., Delaware, large d_p/h) plumes (Yankovsky & Chapman, 1997). The large d_p/h associated with bottom-advected plumes favors tidal mixing (higher BGI) provided that tidal currents are strong. This parameter space analysis is broadly applicable for characterizing the role of bottom-generated mixing in nearfield tidally modulated river plumes throughout the world.

Figure 10

Parameter space for bottom-generated to interfacial mixing ratio (BGI) in terms of Fr_Ta and d_p/h. BGI is calculated using Equation 22 with the coefficient C_D^3/2/(2κC_Di) set to 0.63. Plume data points are from Table 2 estimates. BGI = 0.1, 1, and 10 thresholds associated with moderate, strong, and extreme bottom-generated mixing importance are shown with black dotted, solid, and dashed lines, respectively.

Table 2. Estimated Bottom-Generated to Interfacial Mixing Ratios (BGI, Calculated With Equation 22) for Tidally Modulated River Plumes Around the World Based on Plume Parameters (ΔSal, d_p, c_p), Ambient Conditions (U_Ta and h), and Corresponding Nondimensional Parameters (Fr_Ta and d_p/h)

Plume (receiving waters)	ΔSal	dp (m)	cp (m/s)	UTa (m/s)	h (m)	FrTa	dp/h	BGI
St John (Bay of Fundy, Canada)^1^, ^2	5	8	0.55	0.7	15	1.28	0.53	0.301
Merrimack (Gulf of Maine, USA)^3^, ^4	8	1	0.24	0.08	10	0.33	0.10	0.000
Connecticut (Long Island Sound, USA) ^5	10	2	0.39	1	10	2.59	0.20	0.251
Delaware (Atlantic, USA)^6^, ^7	4	10	0.55	0.45	15	0.82	0.67	0.151
Rappahannock (Chesapeake Bay, USA)^8^, ^9	5	6.5	0.49	0.25	13	0.51	0.50	0.016
York (Chesapeake Bay, USA)^8^, ^9	6	6	0.52	0.35	20	0.68	0.30	0.011
James (Chesapeake Bay, USA)^8^, ^9	7	9	0.69	0.35	26	0.51	0.35	0.007
Choctawhatchee (Gulf of Mexico, USA)^10	5	2.5	0.31	0.11	10	0.36	0.25	0.001
Fraser (Strait of Georgia, Canada)^11^, ^12	8	4	0.49	0.5	50	1.02	0.08	0.002
Skagit (Skagit Bay, Puget Sound, USA)^13	12	2	0.42	0.5	3.5	1.18	0.57	0.285
Elwha (Strait of Juan de Fuca, USA)^14	10	3	0.47	0.7	15	1.48	0.20	0.047
Columbia (Pacific, USA)^15	7.5	11	0.78	0.8	25	1.02	0.44	0.093
Lena (Laptev Sea, Russia)^16	5	5	0.43	0.1	10	0.23	0.50	0.001
Teign (English Channel, England)^17	2	3	0.21	0.18	12	0.85	0.25	0.015
Rhine (North Sea, Netherlands)^18	4	5	0.39	1	12	2.59	0.42	1.327
Gironde (Bay of Biscay, France)^19^, ^20	1	16	0.35	0.25	28	0.72	0.57	0.065
Yellow (Bohai Sea, China)^21	2	3	0.21	0.5	5	2.36	0.60	2.611
Fukuda (Osaka Bay, Japan)^22	0.6	3	0.12	1	7	8.63	0.43	52.662
Changjiang (East China Sea, China)^23^, ^24	0.5	7	0.16	1.5	14	9.28	0.50	96.615
Lien-Tong (Yin-Yang Bay, East China Sea, Taiwan)^25	0.2	2	0.05	0.2	4	3.66	0.50	5.928
Hongqili/Hengmen (Pearl River (Zhujiang) Estuary, China)^26^, ^27	3	2	0.21	1	5	4.73	0.40	7.318
Red (Gulf of Tonkin, Vietnam)^28^, ^29	13	2	0.44	0.6	7	1.36	0.29	0.080
Negro (Atlantic, Argentina)^30^, ^31	8	4	0.49	1	6	2.05	0.67	2.316
Santa Cruz (Atlantic, Argentina)^30^, ^31^, ^32^, ^33	9	2.5	0.41	1	5	2.44	0.50	1.756
Fly (Gulf of Papua, Papua New Guinea)^34	1	10	0.27	0.6	27	2.20	0.37	0.612
Daly (dry) (Anson Bay, Australia)^35	0.5	2	0.09	0.6	4	6.95	0.50	40.488
Daly (wet) (Anson Bay, Australia)^35	10	2.5	0.43	0.6	5	1.39	0.50	0.324

• Note. The coefficient C_D^3/2/(2κC_Di) in Equation 22 is set to 0.63. Parameter values are based on the references cited in the footnotes.
• ¹ Xue et al., 2000.
• ² Wu et al., 2014.
• ³ NOAA Buoy 44030 data.
• ⁴ Cole et al., 2020.
• ⁵ This study.
• ⁶ Whitney and Garvine, 2006; 2008.
• ⁷ Muscarella et al., 2011.
• ⁸ Kuo and Neilson, 1987.
• ⁹ Zhong and Li, 2006.
• ¹⁰ Huguenard et al., 2016.
• ¹¹ Halverson and Pawlowicz, 2008.
• ¹² Kastner et al., 2018.
• ¹³ Yang and Khangaonkar, 2009.
• ¹⁴ Warrick and Stevens, 2011.
• ¹⁵ Nash et al., 2009.
• ¹⁶ Fofonova et al., 2015.
• ¹⁷ Pritchard and Huntley, 2006.
• ¹⁸ Flores et al., 2017.
• ¹⁹ Toublanc et al., 2023.
• ²⁰ Le Cann, 1990.
• ²¹ Yu et al., 2021.
• ²² Bricker et al., 2006.
• ²³ Gao et al., 2009.
• ²⁴ Wu et al., 2011.
• ²⁵ Lin et al., 1994.
• ²⁶ Mao et al., 2004.
• ²⁷ Gong et al., 2020.
• ²⁸ van Maren and Hoekstra, 2004.
• ²⁹ van Maren and Hoekstra, 2005.
• ³⁰ Piccolo and Perillo, 1999.
• ³¹ Moreira et al., 2011.
• ³² Espinosa and Isla, 2015.
• ³³ Brun et al., 2020.
• ³⁴ Wolanski et al., 1995.
• ³⁵ Wolanski et al., 2006.

The parameter space analysis for BGI is based on buoyant, tidal, and bathymetric conditions within the plume. The plume g′ and d_p are partially related to conditions at the mouth with the expectation that g′ reduces and d_p thickens as mixing occurs over the plume region. Mouth conditions can be linked to river discharge (Q) and estuarine tidal mixing (e.g., Nash et al., 2009). The ΔSal and g′ at the mouth tend to increase with increased Q and/or reduced estuarine tidal mixing, though the dependence is nonlinear (e.g., MacCready & Geyer, 2010; Nash et al., 2009; Sun et al., 2017). The d_p at the mouth tends to decrease with higher Q and/or reduced estuarine mixing (Nash et al., 2009). BGI is smaller for larger g′ due to its g′^−3/2 dependence arising from the denominator of Fr_Ta³ (Equations 22 and 23). BGI is smaller for thinner d_p. The dependence is weaker because the inclusion of d_p in the BGI numerator is partially countered by its presence in the denominator via Fr_Ta³ in Equations 22 and 23; the power dependence is d_p^1/2 for the BGI_thin approximation Equation 23. With these power dependencies on g′ and d_p, BGI should be larger for lower Q and strong estuarine mixing conditions; as in the low-discharge spring-tide case analyzed in this study. Mouth conditions have been shown to depend on the estuarine Richardson number (Ri_E), a nondimensional ratio of Q and the cube of estuarine tidal currents (Nash et al., 2009). Spicer et al. (2021) finds that bottom-generated tidal mixing is most important to plume mixing (consistent with large BGI) when Ri_E is smaller than the mouth Rossby radius. The role of estuarine conditions in setting BGI is important, particularly near the mouth. The evolving g′ and d_p, ambient tidal currents, and changing bathymetry within the plume region are crucial for determining plume-averaged BGI values and potentially large local variations in BGI as in the Connecticut River plume.

5 Conclusions

The Connecticut River plume is influenced by energetic ambient tides in the receiving waters of Long Island Sound. This modeling study focuses on plume mixing with an emphasis on spatial heterogeneity and the relative importance of bottom-generated to interfacial shear-driven mixing. The plume formed during ambient flood tides under low river discharge, spring tides, and no winds is analyzed to highlight conditions that favor increased importance of bottom-generated mixing.

At max ambient flood, the tidal plume is elongated by tidal flow and plume propagation. The ambient tidal Froude number is large (Fr_Ta = 2.6) and the representative plume thickness to bottom depth ratio is small (d_p/h = 0.2). Near the mouth, over shoals, and in inshore shallow waters, there is a higher degree of bottom interaction since the plume occupies most of the water column (d_p/h ∼ 1).

Mixing rates, quantified with depth-integrated integrated turbulent buoyancy fluxes through the plume (Bd), are characterized by a high degree of spatial heterogeneity (both along and across the plume) spanning several orders of magnitude. At max ambient flood, strong mixing (Bd ∼ 10⁻⁵-10⁻⁴ m³/s³) occurs near the mouth, in the plume turning region, parts of the downstream end (particularly over shoals), and along a shallow coastal band. Low to moderate mixing levels (Bd ∼ 10⁻⁸-10⁻⁶ m³/s³) occupy about half of the plume, particularly in the middle.

The standard buoyancy flux partitioning method is based on the depth of the shear stress minimum between plume-generated and bottom-generated shear maxima (the d_τ method). Results indicate interfacial and bottom-generated shear contribute to different areas of intense and moderate mixing. The relative influence of these contributions varies throughout the plume. Other tested partitioning methods are based on parameterizations for open channel flow and stratified shear flow. All partitioning methods indicate a significant plume mixing role for bottom-generated shear, but interfacial mixing is a bigger contributor overall than bottom-generated mixing. All methods point to the importance of bottom-generated mixing near the mouth and jetties, in the nearfield plume turning region, over shoals and inshore shallow areas near the downstream end, and in some other parts of the plume interior.

Plume-integrated mixing (M) peaks at max ambient flood (M = 121 kW). Plume-integrated interfacial mixing also peaks near max ambient flood and bottom-generated mixing peak near max or late flood depending on the partitioning method. Two-thirds of plume mixing occurs in concentrated intense mixing areas (with Bd ≥ 10⁻⁵ m³/s³) and most of the rest occurs in expansive moderate mixing zones. The d_τ partitioning method indicates interfacial mixing accounts for almost three-quarters of M near max ambient flood. All partitioning methods point to the significant but mostly secondary role of bottom-generated shear to overall plume mixing.

The bottom-generated to interfacial depth-integrated shear-production ratio (BGI) is calculated with open channel flow and stratified shear flow parameterizations. BGI is strongly sensitive to the tidal Froude number and the plume depth ratio (BGI ∼ Fr_Ta³(d_p/h)² for thin plumes). The Connecticut plume is within the moderate BGI mixing (0.1 ≤ BGI < 1); indicating significant, but not dominant, bottom-generated shear-driven mixing. The BGI parameter space analysis suggests that many tidally modulated plumes around the world are moderately to dominantly mixed by bottom-generated tidal mixing.

Acknowledgments

This research was supported by the NSF Ocean Sciences Physical Oceanography grants 1756578, 1756599, 1756690, and 2242070. The manuscript was revised following the helpful suggestions of anonymous reviewers.

Open Research

Data Availability Statement

The hydrodynamic model used in this study is the Regional Ocean Modeling System (ROMS); the current code version is publicly available at https://github.com/myroms. The ROMS code branch used in this study was built from the ROMS trunk with last changed revision 783 downloaded from https://www.myroms.org/svn/src/trunk/ROMS/Version with user credentials (obtainable at no cost by creating a user account through https://www.myroms.org). Model data were processed with MATLAB by Mathworks (https://mathworks.com). The MATLAB versions used are R2017b (9.3.0.713,579) and R2019b (9.7.0.1,190,202); study analysis works interchangeably among these versions and likely many others. ROMS and MATLAB files used in the study are available in the data set hosted by Mendeley Data and publicly accessible at https://doi.org/10.17632/674yyd3drw.1 (Whitney, 2023). This data set includes (a) ROMS source code for the branch used, (b) files for compiling this source code and the run-specific Fortran 90 code built after C pre-processing, (c) ROMS ASCII and NetCDF input files for the model run, (d) MATLAB custom analysis routines and generated data files used in this study, and (e) MATLAB files associated with figures in this paper including a custom routine and data files with all data included in figures. Note that the ROMS NetCDF output files are not archived in this data set due to the prohibitively large size. The output files can be reproduced by compiling and executing ROMS with the source code, compilation files, and input files included in the data set. All final analysis fields are included in the MATLAB data files associated with the figures. The files associated with the Whitney (2023) data set are licensed under a Creative Commons Attribution 4.0 International license that allows sharing, copying, and modification with appropriate credit and without endorsement by the data set rights holder.