1. Introduction

The estuarine circulation transports buoyant material such as floating debris, marine organisms, surface foam, air bubbles, and pollutants, e.g., oil and microplastics (Kennish 2002). Previous observations and hydrodynamic simulations indicate that buoyant material concentrates in laterally narrow near-surface patches that may extend from several hundred meters to a few kilometers along the estuary (e.g., Nunes and Simpson 1985; Cohen et al. 2019). These material patches greatly facilitate interactions between particles related to mating behavior, predator–prey dynamics, access to nutrients, and the exposure to pollutants. A common mechanism for patch formation is attributed to converging near-surface currents as part of the Eulerian estuarine circulation (e.g., Nunes and Simpson 1985; MacCready and Geyer 2010). This study explores an alternative aggregation mechanism due to tidally averaged Lagrangian residual flow (Zimmerman 1979).

Eulerian surface convergence flows can be induced by lateral density variability, for example, resulting from differential advection during a tidal cycle (Nunes and Simpson 1985; MacCready and Geyer 2010). During flood, density in the thalweg becomes elevated relative to the flanks and the resulting baroclinic torques drive a two-cell flow structure with lateral surface currents toward the channel center (Lerczak and Geyer 2004; Burchard et al. 2011; Li et al. 2014). This mechanism suggests that along-estuary surface aggregation zones are tightly linked to lateral density variability. Here, we demonstrate that aggregation zones can form independently of the lateral density distribution.

The Coriolis force presents another critical ingredient in driving flows across the estuary (Lerczak and Geyer 2004; Valle-Levinson 2008; Li et al. 2014). The cell structure of this lateral circulation can be complex and generally depends on the estuary geometry, turbulent mixing, and density variations. For example, Lerczak and Geyer (2004) explore a single lateral circulation cell due to tidally varying Ekman dynamics. Such circulations are associated with vanishing tidally averaged Eulerian velocities. However, this study finds that in a Lagrangian framework, Eulerian currents with zero tidal average nevertheless contribute to particle transport due the lateral Lagrangian residual velocities. The Lagrangian residual velocity is defined as the velocity of a marked fluid parcel averaged over one tidal cycle (Longuet-Higgins 1969; Zimmerman 1979).

The importance of the Lagrangian framework to understand mass transport by time-varying ocean currents has long been recognized (Longuet-Higgins 1969; Zimmerman 1979); yet, except for a handful of exceptions (e.g., Feng et al. 1986; Jay 1991; Ridderinkhof and Zimmerman 1992; Lemagie and Lerczak 2015), few studies previously explored a Lagrangian approach to estuarine transport. A tidally averaged Lagrangian residual velocity (Stokes drift velocity for small-amplitude motions) in the direction of phase progression of the oscillatory currents is well known to occur for surface waves including tides (Longuet-Higgins 1969; Li and O’Donnell 1997; LeBlond 1978). This study shows that time-dependent secondary flows in estuaries result in Lagrangian residual flows that laterally organize buoyant material and control material transport, playing an important role in estuarine transport processes.

To understand the Stokes drift intuitively, it is insightful to cite Chris Garrett (Garrett 2004): “The Stokes drift is like surfing. The more you stay with a wave, the more you drift forward; that is, you stay longer with the forward flow than if were standing still (Eulerian) in which case you would see the forward and backward flow for exactly the same amount of time.” A time series of surface cross-channel velocities υ₀ further illustrates this concept (Fig. 1). Because the phase of υ₀ depends on cross-estuary location y, the particle of the shown path (black line) spends more time in positive υ₀ than in negative υ₀, so that the particle moves forward over a tidal cycle by “surfing” positive υ₀.

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Modeled cross-estuary surface velocity υ₀(t, υ) as function of time t and cross-estuary location y with associated particle trajectory (black line). This trajectory shows how the particle released at y = −1000 m follows the phase to converge in an oscillating tidal motion over the channel center. The model is discussed in detail in section 3 and is based on solution (9) with parabolic bathymetry (Fig. 2) with h_max = 15 m and h_min = 5 m and half-width B = 2 km. Other model parameters are A = 0.0022 m² s⁻¹, U₀ = 0.75 m s⁻¹, f = 10⁻⁴ s⁻¹.

Citation: Journal of Physical Oceanography 53, 2; 10.1175/JPO-D-22-0067.1

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Laterally varying surface currents organize buoyant material and thus play an important role in estuarine transport processes. Burchard et al. (2011) provide examples where the Eulerian mean flows at the surface in the channel center can either be (i) landward or (ii) seaward depending on the details of the controlling dynamics. As such, a steady convergence of surface material to the thalweg via lateral Stokes drift would result in the (i) retention or (ii) export of material.

The goal of this paper is to provide a proof-of-concept demonstrating the importance of Lagrangian residual flows in tidally driven estuaries for aggregating buoyant material in laterally narrow zones along the estuary. The following section introduces basic theoretical concepts and discusses favorable conditions for converging Lagrangian flows in estuaries. Section 3 reviews an idealized analytic model for the tidally driven estuarine circulation following Lerczak and Geyer (2004), which is applied in section 4. We conclude in section 5 that Lagrangian convergence zones are efficient in forming persistent aggregation regions of buoyant material in estuaries.

2. Theory

a. Lagrangian residual velocity

We explore the horizontal Lagrangian residual velocity of an oscillating tidal flow with along-estuary velocity u(t, y, z) and cross-estuary velocity υ(t, y, z). The velocity shall only depend on vertical coordinate (increasing upward with z = 0 at that air–sea interface) and on cross-estuary (lateral) coordinate y (with y = 0 at the channel center), see definition sketch in Fig. 2. The velocity (u, υ) is assumed to be independent of the along-channel location x. The dependence of u and υ on time t shall be prescribed by a sinusoidal oscillation, so that the Eulerian average $\overline{u}$ and $\overline{\upsilon }$ over one tidal cycle is zero that is $\overline{u}=\overline{\upsilon }=0$. Here, the overbar denotes the temporal average over one tidal period T from t − T/2 to t + T/2.

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Definition sketch of idealized estuary with parabolic cross-estuary bathymetry. Along-channel tidal currents are characterized by velocity scale U₀, the Coriolis force −fu drives a cross-channel flow, while turbulent mixing induces a bottom boundary layer whose height scales as β. These processes result in a tidal cross-channel velocity whose phase depends on depth h.

Citation: Journal of Physical Oceanography 53, 2; 10.1175/JPO-D-22-0067.1

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The Lagrangian velocity is often decomposed in Eulerian and Stokes drift velocities (e.g., Bühler 2009). This decomposition is particularly meaningful for small-amplitude motions in which the Stokes drift is the first-order approximation of the difference between Lagrangian and Eulerian velocity. Following Longuet-Higgins (1969) and Zimmerman (1979), we distinguish in this study between Stokes drift for small-amplitude motions and a more general residual circulation, which is the Lagrangian velocity with zero Eulerian mean motion.

To define the Lagrangian residual velocity, we first consider the time-dependent horizontal trajectory [X(t), Y(t)] of a particle marked at time t₀ and horizontal position (X₀, Y₀). For this proof-of-concept study, we examine only Lagrangian surface motion with z = 0. The corresponding Lagrangian particle velocity is then obtained from the Eulerian velocity fields through the relation (U, V) = [u(t, Y, 0), υ(t, Y, 0)]. Finally, the Lagrangian residual velocity is defined by the tidal average of the Lagrangian velocity $(\overline{U},\overline{V})$, where the overbar indicates the tidal average (Zimmerman 1979). In this study, we focus on convergence regions of $\overline{V}$; the explicit expression for $\overline{V}$ is

$\overline{V}(t,t_0,Y_0)=\frac{1}{T}{\displaystyle {\int }_{t-T/2}^{t+T/2}V}(t^{\prime },t_0,Y_0)d{t}^{\prime }=\frac{1}{T}{\displaystyle {\int }_{t-T/2}^{t+T/2}\upsilon }[t^{\prime },Y(t^{\prime },t_0,Y_0),0]d{t}^{\prime }.$(1)

3. Idealized model for Ekman-forced lateral flow

The balance (7) highlights that the phase of υ critically depends on depth h, which is directly seen by scaling the acceleration term (first left-hand side term) relative to the stress terms (two last right-hand side terms) whose ratio scales as (h/β)², where β is the boundary layer thickness $\beta =\sqrt{2A/\sigma }$. For relatively small h, h/β ≪ 1, the acceleration term is relatively small, and υ is expected to be in phase with the Coriolis term. On the other hand, for relatively large h, h/β ≫ 1, acceleration is expected to be dominant so that υ is 90° out of phase with the Coriolis term. Therefore, φ depends on h which results in a lateral Stokes drift if h varies across the estuary, as discussed above.

Solutions to (6) and (7) are derived by imposing the boundary conditions u = υ = 0 at z = −h and a zero surface stress condition ∂u/∂z = ∂υ/∂z = 0 at z = 0 (see details in Lerczak and Geyer 2004). Furthermore, solutions are determined so that the depth-integrated cross-channel transport is zero for all t. Analytic solutions are obtained by imposing a time dependence exp(−iσt) for u and υ resulting with (6) and (7) in the complex cross-channel flow structure functions for υ in (3)

$\left|\upsilon \right|\exp (i\phi )=\frac{i}{2}{U}_0\frac{f}{\sigma }\left[\alpha \frac{\cosh (\kappa z)}{\cosh (\kappa h)}-\alpha \frac{\tanh (\kappa h)}{\kappa h}-\kappa z\frac{\sinh (\kappa z)}{\cosh (\kappa h)}-\frac{\tanh (\kappa h)}{\kappa h}+1\right]$(8)

with κ² = −2iβ⁻² and $\alpha =[{(\kappa h)}^2\tanh (\kappa h)]/[\kappa h-\tanh (\kappa h)]-1.$ The lateral velocity depends on the nondimensional boundary layer thickness β/h (Fig. 3). If β is comparable to h, the lateral velocity is characterized by a single lateral cell structure, whereas for small β and greater h/β, the boundary layer is shallower and lateral currents intensify closer to the bottom (Lerczak and Geyer 2004).

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Nondimensional lateral surface (a) phase φ₀ and (b) velocity magnitude |υ₀| from (9) as a function of nondimensional depth h/β. For constant β, the phase φ₀ changes as the particle moves across the estuary, resulting in a Stokes drift according to (4).

Citation: Journal of Physical Oceanography 53, 2; 10.1175/JPO-D-22-0067.1

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Thus, the normalized surface velocity magnitudes and phases only depend on h/β because κ ∝ β⁻¹ (Fig. 3). Lerczak and Geyer (2004) considered the range of 0.1 < β/h < 0.4 or 2.5 < h/β < 10, which is a typical range found in estuarine systems. For this parameter range, Fig. 3 suggests a significant |υ| and a substantial dependence of φ on h for h/β > 8, indicating nonzero Stokes drift according to (4) and a significant Lagrangian residual circulation. Figure 1 illustrates how the lateral surface velocity υ₀ changes over a tidal cycle across the estuary enabling surface trapped particles to move with the phase.

4. Convergence zones due to the Lagrangian residual circulation

Applying the idealized model, we will first contrast differences between the Stokes drift and the full Lagrangian residual velocity, before considering illustrative example solutions for a parabolic cross-estuary bathymetry (Fig. 2).

a. Stokes drift

Substitution of (8) into (4) results in the nondimensional lateral Stokes drift

${\upsilon }_s\frac{\beta \sigma }{{(U_{0}f/\sigma )}^2}\frac{1}{h_y}=\frac{1}{2}\frac{{\left|\upsilon \right|}^2}{{(U_{0}f/\sigma )}^2}\frac{\partial \phi }{\partial (h/\beta )},$(10)

which only depends on the single nondimensional parameter (h/β) (Fig. 4). As expected, the normalized Stokes drift peaks near where |υ| and ∂φ/∂(h/β) peak, compare with Fig. 3. Note that the Stokes drift changes sign near h/β = 6, which is associated with a Lagrangian convergence zone as discussed above. Equation (10) highlights that the Stokes drift is proportional to h_y. Recall that h_y must not be too large for an accurate Stokes drift estimate because the small-amplitude motion requires that υ does only change approximately linearly with Y − Y₀. This requires that |σ⁻¹∂υ/∂y| ≪ 1 or $h_y{(\beta \sigma )}^{-1}\left|\partial \upsilon /\partial (h/\beta )\right|\ll 1$, so that h_y also critically controls higher-order contributions to the Lagrangian residual velocity (1).

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Cross-channel Lagrangian residual velocity $\overline{V}$ from (1) for (a) a dimensional example and (b) nondimensional $\overline{V}\beta \sigma /{(U_{0}f/\sigma )}^2/h_y$, which depends only on h/β and h_y. Gray line shows the Stokes drift estimate (10). The dimensional case uses parameters h_y = 0.01, A = 0.0022 m² s⁻¹, U₀ = 0.75 m s⁻¹, f = 10⁻⁴ s⁻¹.

Citation: Journal of Physical Oceanography 53, 2; 10.1175/JPO-D-22-0067.1

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c. Illustrative examples

We apply the idealized model to a parabolic channel with depth $h(y)=h_{\max }-(h_{\max }-h_{\min }){(y/B)}^2$, where B is half the channel width, and the maximum and minimum depths are, respectively, h_max = 15 m and h_min = 5 m (Fig. 2). To explore the dependence on slope, we design three experiments with B = 5, 2, and 1 km with constant eddy viscosity of A = 2.2 × 10⁻³ m² s⁻¹. In these experiments, B controls the slope with steeper slopes for smaller B. To investigate model result for a greater h/β range, we also explore a case with smaller A of A = 3.3 × 10⁻⁴ m² s⁻¹, where B is set to B = 2 km. These A values are consistent with those from Lerczak and Geyer (2004), similar to other model parameters that are held constant at U₀ = 0.75 m s⁻¹, f = 10⁻⁴ s⁻¹, and T = 12 h. For all cases, particles are released at t = 0 and y = −B/2 and tracked over six full semidiurnal tidal cycles.

For the three experiments with greater A, h/β is in the range h/β = 0.5–3 so that the phase increases with depth (Fig. 3). Consistent with the analysis above, particles converge in the channel center, even for the mildest sloping case with B = 5 km for which particles aggregate more slowly (Figs. 5a–f). As expected, the Stokes drift υ_s is greater for steeper slopes (smaller B), so that the particles accumulate more quickly around channel center where ∂φ/∂y = 0. In agreement with the analysis presented above, υ_s accurately approximates $\overline{V}$ near the channel center, but deviations between υ_s and $\overline{V}$ are found away from the channel where slopes are steeper. These Lagrangian convergence zones quickly drive particles toward the channel center regardless of initial particle location (Fig. 6a). As observed in Fig. 1, particles surf the cross-channel velocity υ ₀, resulting in a net transport over a tidal cycle. Note that without tidal averaging particle aggregation zones oscillate around the channel center because of the tidal flow.

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Lagrangian cross-channel transport solutions with parabolic bathymetry with h_max = 15 m, h_min = 5 m and (a),(b) B = 5000 m, A = 2.2 × 10⁻³ m² s⁻¹; (c),(d) B = 2000 m, A = 2.2 × 10⁻³ m² s⁻¹; (e),(f) B = 1000 m, A = 2.2 × 10⁻³ m² s⁻¹; (g),(h) B = 2000 m, A = 3.3 × 10⁻⁴ m² s⁻¹. (left) Particle locations Y(t) and $\overline{Y}(t)$ and (right) $\overline{V}(y)$. Other model parameters are U₀ = 0.75 m s⁻¹, f = 10⁻⁴ s⁻¹, T = 12 h.

Citation: Journal of Physical Oceanography 53, 2; 10.1175/JPO-D-22-0067.1

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Surface velocity υ₀ and particle trajectories Y(t) (lines) for (a) A = 2.2 × 10⁻³ m² s⁻¹ with ∂h/∂y = 0 and a convergence zone in the channel center at y = 0 m, and (b) A = 3.3 × 10⁻⁴ m² s⁻¹ and ∂φ/∂h = 0 and convergence zones near y ≈ ±1000 m; other parameters as discussed in the main text.

Citation: Journal of Physical Oceanography 53, 2; 10.1175/JPO-D-22-0067.1

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For the case with smaller A, h/β is in the range h/β = 2–8, which includes h/β ≈ 6 at about y ≈ ±1000 m. Since the phase is nearly constant at this cross-channel location with ∂φ/∂h = 0 (Fig. 3), the phase is also nearly constant across the channel and ∂φ/∂y = 0. Consistently, Lagrangian convergence zones occur at those lateral locations where h/β ≈ 6. As a consequence, particles converge for this case off the channel center at y = ±1000 m (Figs. 5g,h), and particles that are evenly released over the channel aggregate in those two convergence zones (Fig. 6b). In the channel center between y ≈ −1000 and y ≈ 1000 m, the depth is deeper so that h/β > 6 here. In this region φ depends only weakly on h/β (Fig. 3), so that particles converge only slowly to the lateral location where h/β ≈ 6 and particles are trapped between y ≈ −1000 to y ≈ 1000 m for long times (Fig. 6b). These results suggest that the Lagrangian residual transport and Lagrangian convergence zones strongly depend on turbulent boundary layer mixing (represented by A) and depth. The Stokes drift estimate is accurate for small $\overline{V}$ and thus insightful for identifying convergence regions.

Note that A may vary across the estuary due complex topography and related bottom roughness. For example, if roughness increases in shallower regions, one may assume an inverse relation between A and h. If this inverse relation is approximated by A ∝ h⁻¹, then β ∝ h^−1/2 and h/β ∝ h^3/2, which increases the phase change as the fluid moves across the channel relative to constant A (Fig. 3) and thus enhances the Stokes drift. Alternatively, for A ∝ h, h/β ∝ h^1/2 and Stokes drift would be reduced.

5. Conclusions

This proof-of-concept study demonstrates the importance of Lagrangian residual velocities in transporting and aggregating surface-trapped buoyant material in tidally driven estuaries. Any cross-estuary (lateral) Eulerian velocity with zero tidally averaged velocity yet with a lateral tidal phase variability will result in a net Lagrangian velocity that will follow this phase and converge at a location where phase reaches an extremum, often the center of the channel. For small-amplitude motions this velocity is characterized by Stokes velocity described by the low-order approximation in (2).

A straightforward analysis of the lateral momentum balance reveals that the tidal phase generally depends on tidal frequency, turbulent mixing, and estuary depth. For greater depth and tidal frequency but weaker turbulent mixing, the lateral acceleration increases relative to the stress divergence term, inducing a phase shift of the later velocity toward 90° out of phase with the Coriolis force term. Therefore, the tidal phase varies across the estuary with varying estuarine depth. Our results show that this lateral phase variability has critical implications for the lateral transport and aggregation of surface trapped material. For steeper bathymetry, the lateral change in tidal phase is greater and the corresponding lateral Lagrangian residual flow faster. At local depth extrema, e.g., in the thalweg, depth does not vary laterally, so that the associated tidal phase reaches an extremum. Therefore, the Stokes drift changes sign at this phase extremum at the depth extrema resulting in Lagrangian convergence zones where buoyant material concentrates.

We examined these ideas employing the idealized analytic model from Lerczak and Geyer (2004) in which the along-estuary tidal flow is driven by an imposed barotropic pressure gradient, whereas cross-estuary flow is induced by the Coriolis force. Analytic solutions of the lateral tidal phase and nondimensional velocity at the surface only depend on a nondimensional depth given by the estuarine depth normalized by a tidally driven turbulent boundary layer depth scale. The nondimensional Stokes drift also only depends on this nondimensional depth and the Stokes drift is proportional to the bathymetry slope. Therefore, the Stokes drift changes sign across depth maxima coinciding with a Lagrangian convergence zone. Furthermore, analytic phase solutions demonstrate a local peak of lateral phase at a particular normalized depth revealing Stokes drift convergence across the channel that is not related to bathymetry extrema but may occur away from the channel center. Although the Stokes drift is only an accurate estimate for the Lagrangian residual velocity for smaller-amplitude motions, and thus smaller slopes, it still provides expectations for Lagrangian convergence zones.

More comprehensive approaches need to incorporate competing influences due to density variations and lateral advection, which affects the lateral circulation and mixing in the estuary. Some mixing modifications due to stratifications may be captured by adjusting A, and thus β, though others may require that A depends on the tidal phase and vertical coordinate (Jay and Musiak 1994). Furthermore, differential advection of saltier or fresher water induces lateral density gradients and associated Eulerian secondary circulations (Nunes and Simpson 1985; MacCready and Geyer 2010) that could either reinforce or compete with Lagrangian convergence zones. Note that even in the presence of variable densities, the tidal phase of the lateral circulation still generally depends on cross-channel location which will introduce Lagrangian residual flows focus of our study.

Overall, this study highlights that convergence zones due to Lagrangian residual velocities are efficient in forming persistent aggregation regions of buoyant material along the estuary and should be taken into account for comprehensive estuarine transport investigations.

Data availability statement.

This research is based on theoretical work without numerical or observational datasets that can be shared beyond what is presented here.